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# Numerical Algorithm for the Solutions of Fractional Order Systems of Dirichlet Function Types with Comparative Analysis

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... In this section, four different problems of the designed model have been and numerical solutions have been presented by applying the artificial neural networks (ANNs), optimized with genetic algorithm (GA), sequential quadratic programming (SQP) scheme and the hybrid of GA-SQP scheme. The details of this technique are extensively practice in the literature; see, for example, [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55]. The feed-forward ANN models for proposed outcomes and their respective derivatives are mathematically given as: ...
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The aim of the present study is to present a new model based on the nonlinear singular second order delay differential equation of Lane-Emden type and numerically solved by using the heuristic technique. Four different examples are presented based on the designed model and numerically solved by using artificial neural networks optimized by the global search, local search methods and their hybrid combinations, respectively, named as genetic algorithm (GA), sequential quadratic programming (SQP) and GA-SQP. The numerical results of the designed model are compared for the proposed heuristic technique with the exact/explicit results that demonstrate the performance and correctness. Moreover, statistical investigations/assessments are presented for the accuracy and performance of the designed model implemented with heuristic methodology.
... Similarly, many fluidic problems are investigated with numerical and analytical solver such as magnetic field effect on nanofluid flow using AGM and ADM [24][25][26] , heat transfer rate in nuclear waste 27 , fins arrangement in cubic enclosure 28 , heat transfer simulation in a channel with rectangular cylinder 29,30 and 3D optimization of baffle arrangement 31 . Besides these, the research community has exploited different numerical schemes in diversified applications [32][33][34][35][36][37][38][39] . ...
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The objective of the current investigation is to examine the influence of variable viscosity and transverse magnetic field on mixed convection fluid model through stretching sheet based on copper and silver nanoparticles by exploiting the strength of numerical computing via Lobatto IIIA solver. The nonlinear partial differential equations are changed into ordinary differential equations by means of similarity transformations procedure. A renewed finite difference based Lobatto IIIA method is incorporated to solve the fluidic system numerically. Vogel's model is considered to observe the influence of variable viscosity and applied oblique magnetic field with mixed convection along with temperature dependent viscosity. Graphical and numerical illustrations are presented to visualize the behavior of different sundry parameters of interest on velocity and temperature. Outcomes reflect that volumetric fraction of nanoparticles causes to increase the thermal conductivity of the fluid and the temperature enhances due to blade type copper nanoparticles. The convergence analysis on the accuracy to solve the problem is investigated viably though the residual errors with different tolerances to prove the worth of the solver. The temperature of the fluid accelerates due the blade type nanoparticles of copper and skin friction coefficient is reduced due to enhancement of Grashof Number.
... Bowling and Velson [10,11] proposed a classic method to evaluate MARL algorithm. The MARL has recently been extensively used in wireless sensor networks [12], event-triggered consensus system [13], traffic signal controllers [14], numerical algorithm [15], comparative analysis [16], integrodifferential algebraic [17], computational * Corresponding author. Email: llpmath@163.com ...
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In this paper, we study the value function with regret minimization algorithm for solving the Nash equilibrium of multi-agent stochastic game (MASG). To begin with, the idea of regret minimization is introduced to the value function, and the value function with regret minimization algorithm is designed. Furthermore, we analyze the effect of discount factor to the expected payoff. Finally, the single-agent stochastic game and spatial prisoner's dilemma (SDP) are investigated in order to support the theoretical results. The simulation results show that when the temptation parameter is small, the cooperation strategy is dominant; when the temptation parameter is large, the defection strategy is dominant. Therefore, we improve the level of cooperation between agents by setting appropriate temptation parameters.
... There are many published works related to the fractional differential equations in the literature. Arqub et al. [21][22][23] have studied the numerical solutions of singular time-fractional partial differential equations, numerical simulation of time-fractional partial differential equations arising in fluid flows via reproducing kernel method and numerical algorithm for the solutions of fractional order systems of Dirichlet function types with comparative analysis. Baleanu et al. [24] have investigated the fractional operator combining proportional and classical differintegrals. ...
Article
We investigate a couple of different financial/economic models based on market equilibrium and option pricing with three different fractional derivatives in this paper. We obtain the fundamental solutions of the models by Sumudu transform and Laplace transform. We demonstrate our results by illustrative figures to point out the difference between the fractional operators that have power kernel, exponential kernel and Mittag-Leffler kernel. We prove the efficiency and accuracy of the Sumudu transform and decomposition series method constructed by the Laplace transform in providing the solutions of several different linear/nonlinear financial models by considering the theoretical results and illustrative applications. It seems that the proposed method is an efficient way to solve such problems that contain different types of fractional operators and one is able to point out the differences between these mentioned operators. One of the valuable features of the method is the possibility of using it in solving other similar equations including fractional derivatives having a singular or nonsigular kernel. This paper also suggests a good initiative and profitable tool for those who want to invest in these types of options either individually or institutionally.
... Various valuable definitions have been presented by Caputo, Hadamard, Riemann and Liouville etc. [23,24] . Different approaches, such as iterative and numerical methods, have been used to investigate fractional mathematical models under the usual Caputo derivative sense, see, [25][26][27] . ...
Article
In this article we study a fractional-order mathematical model describing the spread of the new coronavirus (COVID-19) under the Caputo-Fabrizio sense. Exploiting the approach of fixed point theory, we compute existence as well as uniqueness of the related solution. To investigate the exact solution of our model, we use the Laplace Adomian decomposition method (LADM) and obtain results in terms of infinite series. We then present numerical results to illuminate the efficacy of the new derivative. Compared to the classical order derivatives, our obtained results under the new notion show better results concerning the novel coronavirus model.
... Recently, machine learning (ML) models have become more popular than traditional data mining techniques. ML is one of the artificial intelligence (AI) techniques that use soft computer algorithms to explore and predict information by learning from training datasets [18][19][20]. ML algorithms that have been used for GESM techniques include boosted regression tree (BRT) [21], random forest (RF) [14,22], classification and regression trees (CART) [14], support vector machine (SVM) [23], artificial neural networks (ANN) [24], maximum entropy (ME) [17,25], linear discriminant analysis (LDA) [26], stochastic gradient tree-boost (SGT), and multivariate adaptive regression splines (MARS) [27], and logistic model tree (LMT) [9]. ...
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Gully erosion is one of the advanced forms of water erosion. Identifying the effective factors and gully erosion predicting is one of the important tools to control and manage such phenomenon. The main purpose of this study is to evaluate the effect of four different resampling algorithms including cross-validation (5-fold and 10-fold) and bootstrapping (Bootstrap and Optimism bootstrap) on boosted regression tree (BRT), support vector machine (SVM), and random forest (RF) models in spatial modeling and evaluation of head-cut gully erosion in Konduran watershed. For this purpose, based on an extensive field survey, the points of the head-cut of the gully erosion were identified first, and a map of the distribution of head-cut gully erosion in the study area was prepared. Then 18 variable identify and prepare as factors affecting the occurrence of head-cut gully erosion. To assess the efficiency of the models, receiver operating characteristics (ROC) and area under the curve (AUC) were used.
... These matrices are obtained from different orthogonal functions such as Legendre polynomials Bhrawy et al. [18], Lotfi et al. [41], Jacobi polynomials Doha et al. [23], and Bernstein polynomials Alipour et al. [9]. Heydari et al. [30] used Legendre wavelet orthogonal functions for solving one-dimensional FLQTVOCPs (see other works Arqub [11], Arqub [12], Arqub [13], Djennadi [22], Baleanu et. al. [16], Baleanu and Shiri [17], Gu et al. [26], Kashfi Sadabad et al. [37], Shiri et al. [55], Shiri and Baleanu [56]). ...
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This paper presents a numerical indirect method based on shifted Legendre polynomials for solving fractional linear quadratic time-variant optimal control problems (FLQTVOCPs) in the Caputo sense. At first, using the well-known optimality conditions for FLQTVOCPs, a fractional two-point boundary value problem (FBVP) is obtained. For solving the FBVP, we introduce a new operational matrix of Riemann–Liouville fractional integrals based on the shifted Legendre polynomial, and then we show that an approximate solution can be computed by solving only an algebraic system of equations. Numerical experiments confirm the efficiency and accuracy of our proposed method.
... The authors in Refs. [8,9] proposed a computational algorithm based on reproducing kernel Hilbert space for solving time-fractional partial differential equations in porous media and nonlinear homogeneous and nonhomogeneous time-fractional equations. In Ref. [10], a numerical method based on multiple fractional power series solution was introduced to deal with the Schrödinger equation. ...
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This article proposes new strategies for solving two-point Fractional order Nonlinear Boundary Value Problems (FNBVPs) with Robin Boundary Conditions (RBCs). In the new numerical schemes, a two-point FNBVP is transformed into a system of Fractional order Initial Value Problems (FIVPs) with unknown Initial Conditions (ICs). To approximate ICs in the system of FIVPs, we develop nonlinear shooting methods based on Newton’s method and Halley’s method using the RBC at the right end point. To deal with FIVPs in a system, we mainly employ High-order Predictor–Corrector Methods (HPCMs) with linear interpolation and quadratic interpolation (Nguyen and Jang in Fract. Calc. Appl. Anal. 20(2):447–476, 2017) into Volterra integral equations which are equivalent to FIVPs. The advantage of the proposed schemes with HPCMs is that even though they are designed for solving two-point FNBVPs, they can handle both linear and nonlinear two-point Fractional order Boundary Value Problems (FBVPs) with RBCs and have uniform convergence rates of HPCMs, $\mathcal{O}(h^{2})$ O ( h 2 ) and $\mathcal{O}(h^{3})$ O ( h 3 ) for shooting techniques with Newton’s method and Halley’s method, respectively. A variety of numerical examples are demonstrated to confirm the effectiveness and performance of the proposed schemes. Also we compare the accuracy and performance of our schemes with another method.
... Recently, meshless methods such as the smoothed particle hydrodynamics (SPH) approach emerge as useful alternative in terms of idealizing nonlinear free surface fluid dynamics [14][15][16]. Modern matrix decomposition and reduction methods that aim to reduce computational time and improve the fidelity of numerical methods used for fluid structure interaction problems are also under development [25][26][27][28]. Notwithstanding this, conventional grid based numerical methods such as the finite difference method (FDM) [17,18], finite element method (FEM) [19][20] and especially the boundary element method (BEM) [21][22][23][24], despite their limitation to idealize discontinuity of liquid motion or breaking waves, remain attractive as they are academically robust, computationally economic and hence useful for design development and rapid assessment. ...
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The analysis of liquid sloshing remains a challenging computational mechanics topic due to its complex underlying physics. The rapid simulation of sloshing problems requires accurate modelling of two-phase fluid dynamics and sloshing impacts on solid tank boundaries by suitable Flexible Fluid Structure Interaction (FFSI) models. This paper presents a hydroelastic model for the prediction of sway induced sloshing loads on flexible trapezoidal and rectangular tanks. Tank walls and a vertical baffle in way of the mid span of the tank bottom are idealized by Timoshenko beam structural dynamics. Hydroelastic analysis is enabled by a Boundary Element Method (BEM) that couples tank wall and baffle structural dynamics with free surface hydrodynamics to evaluate excitation forces and peak hydrodynamic pressures in way of the tank perimeter. Results show that for the case study presented accounting for the influence of hydroelasticity in a rectangular tank may lead to decrease of free surface oscillations and peak pressure by 20%. This is because the dynamics of tank flexibility are coupled with the angular frequency of the sway motion. These benefits amplify further for the case of trapezoidal tank designs for which the free surface and pressure of the trapezoidal tank with lateral angle θ=80° are decreased relative to the rectangular one by about 80% and 65%, respectively.
... Several algorithms, such as the finite difference and finite element, as well as spectral techniques, have been used in the approximation of mathematical models [1][2][3][4]. Nowadays, considerable attention is being paid to distinct types of wavelet methods to improve the formulation of mathematical models. Wavelets have relevant features such as orthogonality, capability of representing functions with different levels of resolution, and the exact representation of polynomials, just to mention a few. ...
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In the past decade, various types of wavelet-based algorithms were proposed, leading to a key tool in the solution of a number of numerical problems. This work adopts the Chebyshev wavelets for the numerical solution of several models. A Chebyshev operational matrix is developed, for selected collocation points, using the fundamental properties. Moreover, the convergence of the expansion coefficients and an upper estimate for the truncation error are included. The obtained results for several case studies illustrate the accuracy and reliability of the proposed approach.
... The literature reveals that different dynamical systems are analyzed by different fractional derivative e.g., Riemann and Liouville, Hadamard and Caputo, etc., [17][18][19][20]. Many numerical and iterative approaches have been developed to solve the Caputo fractional order epidemic models [21][22][23][24], while the complications arise due to a singular kernel. Therefore, a novel idea has been reported by Caputo and Fabrizio to the fractional-order derivative based upon non-singular kernel subject to various important results for the Caputo-Fabrizio fractional integral [25]. ...
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The pandemic of SARS-CoV-2 virus remains a pressing issue with unpredictable characteristics which spread worldwide through human interactions. The current study is focusing on the investigation and analysis of a fractional-order epidemic model that discusses the temporal dynamics of the SARS-CoV-2 virus in a community. It is well known that symptomatic and asymptomatic individuals have a major effect on the dynamics of the SARS-CoV-2 virus therefore, we divide the total population into susceptible, asymptomatic, symptomatic, and recovered groups of the population. Further, we assume that the vaccine confers permanent immunity because multiple vaccinations have commenced across the globe. The new fractional-order model for the transmission dynamics of SARS-CoV-2 virus is formulated via the Caputo-Fabrizio fractional-order approach with the maintenance of dimension during the process of fractionalization. The theory of fixed point will be used to show that the proposed model possesses a unique solution whereas the well-posedness (bounded-ness and positivity) of the fractional-order model solutions are discussed. The steady states of the model are analyzed and the sensitivity analysis of the basic reproductive number is explored. Moreover to parameterize the model a real data of SARS-CoV-2 virus reported in the Sultanate of Oman from January 1st, 2021 to May 23rd, 2021 are used. We then perform the large scale numerical findings to show the validity of the analytical work.
... The proper treatments of BCs are one of the basic difficulties in evaluating any numerical scheme, study based on this concept is carried out in [36,37,38,39,40,41,42]. Among the computational methods, one of the important class of methods were spectral methods to solve linear and non-linear partial differential equations, somehow spectral methods have few drawbacks because these methods do not represent the physical process in spectral space. ...
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In this study, numerical treatment of liquid crystal model described through Hunter-Saxton equation (HSE) has been presented by sinc collocation technique through theta weighted scheme due to its enormous applications including, defects, phase diagrams, self-assembly, rheology, phase transitions, interfaces, and integrated biological applications in mesophase materials and processes. Sinc functions provide the procedure for function approximation over all types of domains containing singularities, semi-infinite or infinite domains. Sinc functions have been used to reduce HSE into an algebraic system of equations that makes the solution quite superficial. These algebraic equations have been interpreted as matrices. This projected that sinc collocation technique is considerably efficacious on computational ground for higher accuracy and convergence of numerical solutions. Stability analysis of the proposed technique has ensured the accuracy and reliability of the method, moreover, as the stability parameter satisfied the condition the proposed solution of the problem converges. The solution of the HSE is presented through graphical figures and tables for different cases that are constructed on various values of 𝜃 and collocation points. The accuracy and efficiency of the proposed technique is analyzed on the basis of absolute errors.
... This study reveal the elastic connections among different physical models, and the simplicity to recover the dynamics of one application by exploring other related applications and models. For future work, we plan to use analytical schemes such as reproducing kernel and residual power series algorithms [33,34,35,36,37,38] to solve similar proposed models as well as other fractional-classical nonlinear models. ...
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In this paper we are interested in investigating the physical shape-changed propagations to the generalized Equal-Width equation through studying the explicit solutions of Wazwaz-Benjamin-Bona-Mahony model. Both models are of considerable importance in many disciplines of research, including ocean engineering and science, and describe the propagation of equally-width waves. We highlight the effect of the coefficients of both nonlinearity and dispersion terms on changing the physical shape of both models by implementing the new exponential-expansion scheme. 2D and 3D graphical plots are provided to validate the findings of the paper.
... In addition to these, fractional calculus has clearer physical importance and a simpler representation with a nonlinear model. Due to these advantages, fractional calculus became a very popular topic nowadays [16][17][18][19][20][21][22][23][24]. There are many different definitions of fractional calculus. ...
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In this study, the authors obtained the soliton and periodic wave solutions for time fractional symmetric regularized long wave equation (SRLW) and Ostrovsky equation (OE) both arising as a model in ocean engineering. For this aim modified extended tanh-function (mETF) is used. While using this method, chain rule is employed to turn fractional nonlinear partial differential equation into the nonlinear ordinary differential equation in integer order. Owing to the chain rule, there is no further requirement for any normalization or discretization. Beta derivative which involves fractional term is used in considered mathematical models. Obtaining the exact solutions of these equations is very important for knowing the wave behavior in ocean engineering models.
... The ANN is a black-box model (Akrami et al., 2013) mimicking the biological neurons of the human brain (Anctil et al., 2009;Lek et al., 1999;Sharma et al., 2003). Similar to fractional calculus theory (Arqub, 2017(Arqub, , 2019, numerical methods such as ANNs have also gained popularity because of their applicability in a variety of problems in physics and engineering (Arqub & Abo-Hammour, 2014). ANNs have been used continuously, and without limitation to the prediction of various water quality variables. ...
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Advanced human activities, including modern agricultural practices, are responsible for alteration of natural concentration of nitrogen compounds in rivers. Future prediction of nitrogen compound concentrations (especially nitrate-nitrogen and ammonia-nitrogen) are important for countries where household water is obtained from rivers after treatment. Increased concentrations of nitrogen compounds result in the suspension of household water supplies. Artificial Neural Networks (ANNs) have already been deployed for the prediction of nitrogen compounds in various countries. But standalone ANN have several limitations. However, the limitations of ANNs can be resolved using hybrid models. This study proposes a new ACO-ENN hybrid model developed by integrating Ant Colony Optimization (ACO) with an Elman Neural Network (ENN). The developed ACO-ENN hybrid model was used to improve the prediction results of nitrate-nitrogen and ammonia-nitrogen prediction models. The results of new hybrid models were compared with multilayer ANN models and standalone ENN models. There was a significant improvement in the mean square errors (MSE) (0.196→0.049→0.012, i.e. ANN→ENN→Hybrid), mean absolute errors (MAE) (0.271→0.094→0.069) and Nash–Sutcliffe efficiencies (NSE) (0.7255→0.9321→0.984). The hybrid model had outstanding performance compared with the ANN and ENN models. Hence, the prediction accuracy of nitrate-nitrogen and ammonia-nitrogen has been improved using new ACO-ENN hybrid model.
... Several new numerical [11][12][13][14] and analytical methods [15][16][17][18][19][20][21] are proposed for fractional order problems by getting the ideas from integer order calculus, but, it's still challenge for the researchers. Several researchers are studying the fractional order problems in different way [43][44][45][46][47]. ...
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The higher dimensional Fokas equation is the integrable expansion of the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations. The Fokas model has an important role in wave theory, to describe the physical phenomena of waves on the surface and inside the water. This article deals with the (4+1)-dimensional conformable space-time fractional-order Fokas partial differential equation. Two efficient methods, namely the generalized exp(-ϕ(ξ))-expansion and improved F-expansion methods, are formulated for conformable fractional-order partial differential equation and new wave structures of fractional order Fokas model are constructed. The different kinds of new solitons are achieved such as bright soliton, dark soliton, Kink and anti-kink solitons, periodic solitary waves, and traveling waves. These new soliton waves are constructed at some values of fractional order α and using different parametric values of the methods by using the software package Mathematica. Newly obtained soliton solutions are compared with the available soliton solutions with different fractional derivatives in the literature. Some of the achieved results are explained 2D and 3D graphically. The new results interpreting that these obtained solutions can be a part, to complete the family of solutions and considered methods are effective, simple, and easy to use. Furthermore, this paper gives an idea, how can reduce the conformable fractional order higher dimensional partial differential equation into an ODE of one variable to obtain the exact solutions. These results and methods can be help to investigate the other higher-dimensional conformable fractional-order models which appear in nonlinear wave theory such as optics, quantum gases, hydrodynamics, photonics, plasmas, and solid-state physics.
... With the global economic development entering a new stage, the aviation industry has ushered in a larger space for development [1,4]. Civil aviation industry, as an important component of aviation industry, plays a significant role in promoting economic development and technological innovation [2,15]. In the development process of civil aircraft, safety and comfort are important principles in the early stage of design [7,9,13]. ...
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With the rapid development of civil aviation industry and the continuous improvement in passenger demand, higher requirements have been put forward for safety and comfort in the design process of civil aircraft. In order to develop a special ergonomic simulation experimental system for civil aircraft cabin seats, from the perspective of “human”, this paper comprehensively considers the impact on civil aircraft cabin seats environment on human physiology and psychology, and puts forward an evaluation method of civil aircraft cabin ergonomic simulation experimental system based on passengers’ perception of key design features. Based on the principles of ergonomics, the connection between key design features of seat comfort and the requirements of user preference is comprehensively considered, a simulation experimental system based on comfort evaluation is constructed using subjective and objective evaluation methods. Moreover, according to the experimental requirements of the simulation experimental system and the characteristics of seat structure, an ergonomic simulation experimental system for civil aircraft cabin is developed. The final experimental results prove the practicability and validity of the simulation experiment system designed.
... For example, one can point out some of these instances in which we observe new generalized models and techniques in relation to existence theory and mathematical models such as the investigation of a system of pantograph equations via Caputo conformable operators [4], the role of fuzzy Caputo-Fabrizio operators in studying analytical solutions of a 2D heat equation [5], the role of approximate endpoint criterion for a quantum integro-difference equation [6], modeling of different diseases with respect to fractional operators [7][8][9][10], analysis of non-periodic p-Laplacian FDEs [11], application of the wighted ðk; sÞ-RL fractional operators in Kinetic equation [12], studying some fractional sum inequalities in the context of discrete operators [13], analytical and numerical treatment of some BVPs [14][15][16][17][18][19][20][21][22][23] etc. ...
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In the present paper, we analyze a coupled system of nonlinear three point boundary value problems (BVPs) consisting of a coupled system of higher order hybrid sequential differential equations formulated by fractional operators. We study several new conditions in the direction of existence theory for solutions of the given hybrid system under the weaker hypotheses based on the method of measure of non-compactness in degree theory. The required conditions confirming the uniqueness and stability are derived in relation to the coupled system of hybrid sequential BVPs. Lastly, we extend our hybrid sequential system to a real mathematical model of typhoid treatment with four compartments and by deriving numerical schemes for this generalized hybrid sequential system, we analyze the variations of different fractional orders and their impact on dynamical behaviors of the mentioned hybrid system of typhoid model.
... the Thomas model [13], the Schnakenberg model [14], the Gierer-Meinhardt model [15], the Glycolysis model [16], the cubic autocatalytic reaction system [17] and others [18,19]. In recent years, fractional partial differential equations (FPDEs) have pulled the notice of investigators in many fields, especially the biological and biochemical phenomena because of their extensive applications [20][21][22][23][24][25][26]. The general reaction-diffusion model of two-component is defined as follows: ...
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In this paper, three examples of fractional reaction–diffusion systems, including cubic autocatalytic reaction system, Glycolysis model and Gray–Scott model have been studied. Q-homotopy analysis transform method (q-HATM) is the main objective method to solve these problems. Comparisons are made between q-HATM, homotopy analysis method (HAM) and exact solutions. The uniqueness theorem, convergence analysis and maximum truncation error have been proved. It is found that the results obtained indicate greater accuracy for any fractional partial differential equations.
... Subsequently, reproducing kernel theory was used by the mathematician, scientist [2,3,4,5,6,7,14,15] like to solve the theoretical problems of many special fields. In 1986, Cui [13] construct the reproducing kernel space and corresponding kernel in the Sobolev space. ...
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In this paper, we introduced a reproducing kernel space which is a particular class of Hilbert space. We discuss various properties of the reproducing kernel. In particular, our aim to construct kernel in reproducing kernel Hilbert space of the specific function space (Sobolev space) with the inner product and norm. Also, we derive the reproducing kernel for Robin boundary conditions.
... Those interested should refer to [26,27] to obtain sufficient information about the RKHSM, including its applications, properties, theories, and limitations. Indeed, those interested should refer to [28][29][30][31][32][33][34][35][36][37] to obtain sufficient information about the CFD and other kinds of fractional derivatives, including its applications, properties, theories, and limitations. ...
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In this study, we will discuss numerical solutions of conformable fractional systems of second-order integrodifferential equations concerning couple types Volterra and Fredholm. These conformable fractional systems will be considered in the sense of periodic boundary conditions in two points. The reproducing kernel Hilbert space method is used with the help of Mathematica 11 to utilize different tabulated data and graphical results. Series representation and required theorems are utilized and proved in the constructed Hilbert spaces. Several applications of linear and nonlinear function types have been studied and solved to ensure the applicability and power of the method. From the gained results, one can note that the approach algorithm is accurate and provides numerical results in a faster time and with less effort. Several notes and highlights with the latest references are discussed and presented at the end of the article.
... In engineering, science, and physics, many issues have been refined by many techniques to solve fractional partial differential equations (FPDEs) especially in nonlinear optics, fluid dynamics, biology, plasma physics, and so forth. [1][2][3][4][5][6][7][8][9][10][11][12][13] Recently, a lot of approximate techniques for solving such equations have been shown, for example, homotopy analysis transform method (HATM), [14][15][16][17] multiple-scale Lindstedt-Poincare method, 18 modified Lindstedt-Poincare method, 19 linearized perturbation method, 20 generalized two-dimensional differential transform method (GDTM), 21 and so forth. [22][23][24] The Laplace substitution method (LSM) is an approximate method that has been widely used to obtain the solution of FPDEs including mixed partial derivatives. ...
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In this paper, we present new approximate solutions to some linear and nonlinear fractional partial differential equations (FPDEs), which include mixed partial derivatives using Laplace substitution method (LSM). This method is entirely based on the well‐known Laplace transform and Adomian polynomial. Looking at several studies, we find that the results obtained from this technique converge well with the exact solutions.
... Arqub and Shawagfeh [26] proposed a problem in porous media with Dirichlet conditions and developed an algorithm of reproducing kernel of time-fractional PDEs. Arqub [27,28] gave numerical solution for the time-fractional differential equations in porous media. Arqub [29] studied an iterative approach, also called RPS method to solve the time-fractional Schrodinger equation. ...
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We consider a device which consists of a floating structure over a cylindrical plate placed at a finite height from the impermeable ocean floor. This paper developes the interaction of linear water waves with such a device. The whole fluid domain is divided into a number of sub-domains and boundary value problems are formulated for each identified sub-domain. The channel multipoles, separation of variables and matched eigenfunction expansion methods allow us to solve boundary value problems for the diffracted velocity potentials in each sub-domain. We investigate the wave forces exerted on the proposed device. Consequently, the effects of the various parameters, e.g., drafts, radii, the gap between the cylinders and mainly channel width of the device on the wave forces exerted by the cylinders are presented graphically. We observe a small oscillation nature near the peak value of the exciting force for the particular value of channel width w=2.4m. The peak value of the exciting forces occurs near the wavenumber kr1=1.0 for different width of the channel walls. The obtained results are compared with some available results, and it shows a good agreement between the obtained and available results.
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In this paper, a modified Sinusoidal Pulse width Modulation (MSPWM) technique and a modified single-phase H-bridge seven-level inverter is proposed. The switching pulses for the proposed seven-level inverter are generated using a single triangular carrier waveform, a fully rectified sinusoidal signal, and three stepped reference signals (Uref1, Uref2 and Uref3). Using optimization technique, the magnitude of the stepped reference signal is determined so that the total harmonic distortion (THD) of the output voltage waveform is minimum and the fundamental component, RMS value of the voltage is improved for a given modulation index Ma as compared to the Sinusoidal Pulse width Modulation (SPWM). By the implementation of the new scheme, the seven-level of the inverter output voltage level (+Vdc, +2Vdc/3, +Vdc/3, 0, −Vdc, −2Vdc/3, −Vdc) is obtained for any given modulation index. Similarly, if only two stepped reference signals are used then the inverter will act as a five-level inverter for any modulating index ma. The proposed MSPWM and seven-level inverter are simulated on MATLAB/SIMULINK for R, R-L load and on a single-phase capacitor-start and capacitor-start-run Induction Motor.
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Mathematical modeling of uncertain fractional integrodifferentials (FIDEs) is an extremely significant topic in electric circuits, signal processing, electromagnetics, and anomalous diffusion systems. Based on the reproducing kernel algorithm (RKA), a touching numerical approach is considering in this study to solve groups of fuzzy fractional integrodifferentials (FFIDEs) with Atangana–Baleanu–Caputo (ABC) fractional distributed order derivatives. The solution‐based approach lies in generating infinite orthogonal basis from kernel functions, where an uncertain condition is fulfilled. Thereafter, an orthonormal basis is erected to figurate fuzzy ABC solutions with series shape in idioms of η‐cut extrapolation in Hilbert space A D and B D . In this orientation, fuzzy ABC fractional integral, fuzzy ABC fractional derivative, and fuzzy ABC FIDE are utilized and comprised. The competency and accuracy of the suggested approach are indicated by employing several experiments. From theoretical viewpoints, the acquired results signalize that the utilization approach has several merits in feasibility and opportunity for treating with many fractional ABC distributed order models. In the end, highlights and future suggested research work are eluded.
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In this research study, fuzzy fractional differential equations in presence of the Atangana–Baleanu–Caputo differential operators are analytically and numerically treated using extended reproducing kernel Hilbert space technique. With the utilization of a fuzzy strongly generalized differentiability form, a new fuzzy characterization theorem beside two fuzzy fractional solutions is constructed and computed. To besetment the attitude of fuzzy fractional numerical solutions, analysis of convergence and conduct of error beyond the reproducing kernel theory are explored and debated. In this tendency, three computational algorithms and modern trends in terms of analytic and numerical solutions are propagated. Meanwhile, the dynamical characteristics and mechanical features of these fuzzy fractional solutions are demonstrated and studied during two applications via three‐dimensional graphs and tabulated numerical values. In the end, highlights and future suggested research work are eluded.
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For solving nonlinear elliptic equations given in arbitrary plane domains, the meshless methods of radial-polynomial and Pascal-polynomial are easy to programming, which are employed as the bases to expand the solution. After a simple collocation technique, we can derive nonlinear equations to determine the expansion coefficients. We adopt a splitting parameter to split the nonlinear term into two nonlinear parts, which are separately placed on both sides of the nonlinear elliptic equation. Then, a new linearization technique is used to treat the nonlinear part on the left-hand side. In each iteration, the linear system of equations is regularized by the multiple-scale technique. The proposed methods converge very fast to obtain very accurate numerical solutions, which confirm the validity of the presented novel splitting and linearizing technique (NSLT) to solve nonlinear elliptic equations in arbitrary plane domains.
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This research considers an inverse source problem for fractional diffusion equation, containing fractional derivative with non-singular and non-local kernel, namely, Atangana-Baleanu fractional derivative. In our study, an explicit solution set is acquired via the expansion method and the overdetermination condition at a final time. The problem is ill-posed in the meaning of Hadamard and thus the solution does not continuously depend on the input data. We have applied the Tikhonov regularization method to regularize the unstable solution. For the estimation of convergence between the exact and the regularized solutions, we focus on two parameter choice rules, a-prioi and a-posteriori parameter.
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Many problems arising in different fields of sciences and engineering can be reduced, by applying some appropriate discretization, either to a system of integrodifferential algebraic equations or to a sequence of such systems. The aim of the present analysis is to implement a relatively recent computational method, reproducing kernel Hilbert space, for obtaining the solutions of integrodifferential algebraic systems of temporal two-point boundary value problems. Two extended inner product spaces W[0, 1] and H[0, 1] are constructed in which the boundary conditions of the systems are satisfied, while two smooth kernel functions Rt(s) and rt(s) are used throughout the evolution of the algorithm in order to obtain the required grid points. An efficient construction is given to obtain the numerical solutions for the systems together with an existence proof of the exact solutions based upon the reproducing kernel theory. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed.
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In this paper, we investigate the analytic and approximate solutions of second-order, two-point fuzzy boundary value problems based on the reproducing kernel theory under the assumption of strongly generalized differentiability. The solution methodology is based on generating the orthogonal basis from the obtained kernel functions, while the orthonormal basis is constructing in order to formulate and utilize the solutions with series form in terms of their r-cut representation in the space $$\oplus _{j=1}^2 W_2^3 \left[ {a,b}\right]$$. An efficient computational algorithm is provided to guarantee the procedure and to confirm the performance of the proposed method. Results of numerical experiments are provided to illustrate the theoretical statements in order to show potentiality, generality, and superiority of our algorithm for solving such fuzzy equations. Graphical results, tabulated data, and numerical comparisons are presented and discussed quantitatively to illustrate the possible fuzzy solutions.
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In this article, we propose the reproducing kernel Hilbert space method to obtain the exact and the numerical solutions of fuzzy Fredholm–Volterra integrodifferential equations. The solution methodology is based on generating the orthogonal basis from the obtained kernel functions in which the constraint initial condition is satisfied, while the orthonormal basis is constructing in order to formulate and utilize the solutions with series form in terms of their r-cut representation form in the Hilbert space (Formula presented.). Several computational experiments are given to show the good performance and potentiality of the proposed procedure. Finally, the utilized results show that the present method and simulated annealing provide a good scheduling methodology to solve such fuzzy equations.
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For a differentiable function fI→ℝ k , where I is a real interval and k∈ℕ, a counterpart of the Lagrange mean-value theorem is presented. Necessary and sufficient conditions for the existence of a mean MI 2 →I such that f(x)-f(y)=(x-y)f ' (M(x,y)),x,y∈I, are given. Similar considerations for a theorem accompanying the Lagrange mean-value theorem are presented.
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A numerical method, modified simple-shooting method (MSSM), to solve two-point boundary-value problem (TPBVP) was presented. The method include favourable aspects of the short-shooting method (SSM) and multiple-shooting method (MSM). A comparison of computation time of MSSM, MSM, collocation methods (CM), and finite difference methods (FDM) was illustrated. It was observed that the convergence of the MSSM was proved under mild conditions on the TPBVP. The solutions of MSSM were resulted in a trajectory which satisfied the system differential equations.
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Second-order, two-point boundary-value problems are encountered in many engineering applications including the study of beam deflections, heat flow, and various dynamic systems. Two classical numerical techniques are widely used in the engineering community for the solution of such problems; the shooting method and finite difference method. These methods are suited for linear problems. However, when solving the non-linear problems, these methods require some major modifications that include the use of some root-finding technique. Furthermore, they require the use of other basic numerical techniques in order to obtain the solution. In this paper, the author introduces a novel method based on continuous genetic algorithms for numerically approximating a solution to this problem. The new method has the following characteristics; first, it does not require any modification while switching from the linear to the non-linear case; as a result, it is of versatile nature. Second, this approach does not resort to more advanced mathematical tools and is thus easily accepted in the engineering application field. Third, the proposed methodology has an implicit parallel nature which points to its implementation on parallel machines. However, being a variant of the finite difference scheme with truncation error of the order O(h2), the method provides solutions with moderate accuracy. Numerical examples presented in the paper illustrate the applicability and generality of the proposed method. Copyright © 2004 John Wiley & Sons, Ltd.
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Numerical modeling of partial integrodifferential equations of fractional order shows interesting properties in various aspects of science, which means increased attention to fractional calculus. This paper is concerned with a feasible and accurate technique for obtaining numerical solutions for a class of partial integrodifferential equations of fractional order in Hilbert space within appropriate kernel functions. The algorithm relies on the reproducing kernel Hilbert space method that provides the solutions in rapidly convergent series representations for the reproducing kernel based upon the Fourier coefficients of orthogonalization process. The Caputo fractional derivatives are introduced to address these issues. Moreover, the error estimate of the generated solutions is established as well as the convergence of the iterative method is investigated under some theoretical assumptions. The superiority and applicability of the present technique is illustrated by handling linear and nonlinear numerical examples. The outcomes obtained are compared with exact solutions and existing methods to confirm the effectiveness of the reproducing kernel method.
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The aim of the present analysis is to implement a relatively recent computational algorithm, reproducing kernel Hilbert space, for obtaining the solutions of systems of first-order, two-point boundary value problems for ordinary differential equations. The reproducing kernel Hilbert space is constructed in which the initial–final conditions of the systems are satisfied. Whilst, three smooth kernel functions are used throughout the evolution of the algorithm in order to obtain the required grid points. An efficient construction is given to obtain the numerical solutions for the systems together with an existence proof of the exact solutions based upon the reproducing kernel theory. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to such systems compared with other numerical methods. © 2018, Istituto di Informatica e Telematica del Consiglio Nazionale delle Ricerche.
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The subject of the fractional calculus theory has gained considerable popularity and importance due to their attractive applications in widespread fields of physics and engineering. The purpose of this research article is to present results on the numerical simulation for time-fractional Tricomi and Keldysh equations of Dirichlet functions types in Hilbert space that were found in the transonic flows. Those resulting mathematical models are solved using the reproducing kernel algorithm which provide appropriate solutions in term of infinite series formula. Convergence analysis, error estimations, and error bounds under some hypotheses which provide the theoretical basis of the proposed algorithm are also discussed. The dynamical properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. Finally, the prospects of the gained results and the algorithm are discussed through academic validations.
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Recently, many new applications in engineering and science are governed by a series of time-fractional partial integrodifferential equations, which will lead to new challenges for numerical simulation. In this article, we propose and analyze an efficient iterative algorithm for the numerical solutions of such equations subject to initial and Dirichlet boundary conditions. The algorithm provide appropriate representation of the solutions in infinite series formula with accurately computable structures. By interrupting the n -term of exact solutions, numerical solutions of linear and nonlinear time-fractional equations of nonhomogeneous function type are studied from mathematical viewpoint. Convergence analysis, error estimations, and error bounds under some hypotheses which provide the theoretical basis of the proposed algorithm are also discussed. The dynamical properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to such integrodifferential equations.
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In this article, improved residual power series method (RPSM) is effectively implemented to find the approximate analytical solution of a time fractional diffusion equations. The proposed method is an analytic technique based on the generalized Taylor's series formula which construct an analytical solution in the form of a convergent series. In order to illustrate the advantages and the accuracy of the RPSM, we have applied the method to two different examples. In case of first example, different cases of initial conditions are considered. Finally, the solutions of the time fractional diffusion equations are investigate through graphical representation, which interpret simplicity, accuracy and practical usefulness of the present method.
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We present monotone convergence results for general iterative methods in order to approximate a solution of a nonlinear equation defined on a partially ordered linear topological space. The main novelty of the paper is that the operators appearing in the iterative method are not necessarily linear. This way we expand of the applicability of iterative methods. Some applications are also provided from fractional calculus using Caputo and Canavati type fractional derivatives and other areas.
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Latterly, many problems arising in different fields of science and engineering can be reduced, by applying some appropriate discretization, to a series of time-fractional partial differential equations. Unlike the normal case derivative, the differential order in such equations is with a fractional order, which will lead to new challenges for numerical simulation. The purpose of this analysis is to introduce the reproducing kernel Hilbert space method for treating classes of time-fractional partial differential equations subject to Neumann boundary conditions with parameters derivative arising in fluid-mechanics, chemical reactions, elasticity, anomalous diffusion, and population growth models. The method provides appropriate representation of the solutions in convergent series formula with accurately computable components. Numerical experiments with different order derivatives degree are performed to support the theoretical analyses which are acquired by interrupting the -term of the exact solutions. Finally, the obtained outcomes showed that the proposed method is competitive in terms of the quality of the solutions found and is very valid for solving such time-fractional Neumann problems.
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Purpose The purpose of this study is to introduce the reproducing kernel algorithm for treating classes of time-fractional partial differential equations subject to Robin boundary conditions with parameters derivative arising in fluid flows, fluid dynamics, groundwater hydrology, conservation of energy, heat conduction and electric circuit. Design/methodology/approach The method provides appropriate representation of the solutions in convergent series formula with accurately computable components. This representation is given in the W(Ω) and H(Ω) inner product spaces, while the computation of the required grid points relies on the R(y,s) (x, t) and r(y,s) (x, t) reproducing kernel functions. Findings Numerical simulation with different order derivatives degree is done including linear and nonlinear terms that are acquired by interrupting the n-term of the exact solutions. Computational results showed that the proposed algorithm is competitive in terms of the quality of the solutions found and is very valid for solving such time-fractional models. Research limitations/implications Future work includes the application of the reproducing kernel algorithm to highly nonlinear time-fractional partial differential equations such as those arising in single and multiphase flows. The results will be published in forthcoming papers. Practical implications The study included a description of fundamental reproducing kernel algorithm and the concepts of convergence, and error behavior for the reproducing kernel algorithm solvers. Results obtained by the proposed algorithm are found to outperform in terms of accuracy, generality and applicability. Social implications Developing analytical and numerical methods for the solutions of time-fractional partial differential equations is a very important task owing to their practical interest. Originality/value This study, for the first time, presents reproducing kernel algorithm for obtaining the numerical solutions of some certain classes of Robin time-fractional partial differential equations. An efficient construction is provided to obtain the numerical solutions for the equations, along with an existence proof of the exact solutions based upon the reproducing kernel theory.
Book
This monograph provides a comprehensive overview of the author's work on the fields of fractional calculus and waves in linear viscoelastic media, which includes his pioneering contributions on the applications of special functions of the Mittag-Leffler and Wright types. It is intended to serve as a general introduction to the above-mentioned areas of mathematical modeling. The explanations in the book are detailed enough to capture the interest of the curious reader, and complete enough to provide the necessary background material needed to delve further into the subject and explore the research literature given in the huge general bibliography. This book is likely to be of interest to applied scientists and engineers.
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This paper presents novel reproducing kernel algorithm for obtaining the numerical solutions of differential algebraic systems with nonclassical boundary conditions for ordinary differential equations. The representation of the exact and the numerical solutions is given in the W [0, 1] and H [0, 1] inner product spaces. The computation of the required grid points is relying on the R t (s) and r t (s) reproducing kernel functions. An efficient construction is given to obtain the numerical solutions for the systems together with an existence proof of the exact solutions based upon the reproducing kernel theory. Numerical solutions of such nonclassical systems are acquired by interrupting the η-term of the exact solutions. In this approach, numerical examples were analyzed to illustrate the design procedure and confirm the performance of the proposed algorithm in the form of tabulate data and numerical comparisons. Finally, the utilized results show the significant improvement of the algorithm while saving the convergence accuracy and time.
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In this paper, new exact solutions of fractional nonlinear acoustic wave equations have been devised. The travelling periodic wave solutions of fractional Burgers–Hopf equation and Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation have obtained by first integral method. Nonlinear ultrasound modelling is found to have an increasing number of applications in both medical and industrial areas where due to high pressure amplitudes the effects of nonlinear propagation are no longer negligible. Taking nonlinear effects into account, the ultrasound beam analysis makes more accurate in these applications. The Burgers–Hopf equation is one of the extensively studied models in mathematical physics. In addition, the KZK parabolic nonlinear wave equation is one of the most widely employed nonlinear models for propagation of 3D diffraction sound beams in dissipative media. In the present analysis, these nonlinear equations have solved by first integral method. As a result, new exact analytical solutions have been obtained first time ever for these fractional order acoustic wave equations. The obtained results are presented graphically to demonstrate the efficiency of this proposed method.
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The aim of the present analysis is to implement a relatively recent computational method, reproducing kernel Hilbert space, for obtaining the solutions of differential algebraic systems for ordinary differential equations. The reproducing kernel Hilbert space ⊕j=1mW22a,b⊕⊕j=m+1nW_21a,b is constructed in which the initial conditions of the systems are satisfied. While, two smooth kernel functions are used throughout the evolution of the algorithm in order to obtain the required grid points. An efficient construction is given to obtain the numerical solutions for the systems together with an existence proof of the exact solutions based upon the reproducing kernel theory. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to such systems.
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Modeling of uncertainty differential equations is very important issue in applied sciences and engineering, while the natural way to model such dynamical systems is to use fuzzy differential equations. In this paper, we present a new method for solving fuzzy differential equations based on the reproducing kernel theory under strongly generalized differentiability. The analytic and approximate solutions are given with series form in terms of their parametric form in the space $$W_2^2 [a,b]\oplus W_2^2 [a,b].$$ The method used in this paper has several advantages; first, it is of global nature in terms of the solutions obtained as well as its ability to solve other mathematical, physical, and engineering problems; second, it is accurate, needs less effort to achieve the results, and is developed especially for the nonlinear cases; third, in the proposed method, it is possible to pick any point in the interval of integration and as well the approximate solutions and their derivatives will be applicable; fourth, the method does not require discretization of the variables, and it is not effected by computation round off errors and one is not faced with necessity of large computer memory and time. Results presented in this paper show potentiality, generality, and superiority of our method as compared with other well-known methods.
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This paper is devoted to the numerical treatment of a class of singularly perturbed delay boundary value problems with a left layer. The method is proposed based on the reproducing kernel theory and the error estimate of the present method is established. A numerical example is provided to show the effectiveness of the present method. Numerical results show that the present method is accurate and efficient.
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In this manuscript, we implement a relatively new analytic iterative technique for solving time–space-fractional linear partial differential equations subject to given constraints conditions based on the generalized Taylor series formula. The solution methodology is based on generating the multiple fractional power series expansion solution in the form of a rapidly convergent series with minimum size of calculations. This method can be used as an alternative to obtain analytic solutions of different types of fractional linear partial differential equations applied in mathematics, physics, and engineering. Some numerical test applications were analyzed to illustrate the procedure and to confirm the performance of the proposed method in order to show its potentiality, generality, and accuracy for solving such equations with different constraints conditions. Numerical results coupled with graphical representations explicitly reveal the complete reliability and efficiency of the suggested algorithm.
Book
1 Theory.- 2 RKHS AND STOCHASTIC PROCESSES.- 3 Nonparametric Curve Estimation.- 4 Measures And Random Measures.- 5 Miscellaneous Applications.- 6 Computational Aspects.- 7 A Collection of Examples.- to Sobolev spaces.- A.l Schwartz-distributions or generalized functions.- A.1.1 Spaces and their topology.- A.1.2 Weak-derivative or derivative in the sense of distributions.- A.1.3 Facts about Fourier transforms.- A.2 Sobolev spaces.- A.2.1 Absolute continuity of functions of one variable.- A.2.2 Sobolev space with non negative integer exponent.- A.2.3 Sobolev space with real exponent.- A.2.4 Periodic Sobolev space.- A.3 Beppo-Levi spaces.
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We study optimal control problems governed by semilinear parabolic equations with pointwise constraints on the state variable. We are mainly interested in optimality conditions for such problems. Since we consider control problems with pointwise state constraints, the adjoint state satisfies a parabolic equation with measures as data. We obtain some regularity results for solutions of such equations.
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Building fractional mathematical models for specific phenomena and developing numerical or analytical solutions for these fractional mathematical models are crucial issues in mathematics, physics, and engineering. In this work, a new analytical technique for constructing and predicting solitary pattern solutions of time-fractional dispersive partial differential equations is proposed based on the generalized Taylor series formula and residual error function. The new approach provides solutions in the form of a rapidly convergent series with easily computable components using symbolic computation software. For method evaluation and validation, the proposed technique was applied to three different models and compared with some of the well-known methods. The resultant simulations clearly demonstrate the superiority and potentiality of the proposed technique in terms of the quality performance and accuracy of substructure preservation in the construct, as well as the prediction of solitary pattern solutions for time-fractional dispersive partial differential equations.
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List of Examples Preface 1. Introduction. Boundary Value Problems for Ordinary Differential Equations Boundary Value Problems in Applications 2. Review of Numerical Analysis and Mathematical Background. Errors in Computation Numerical Linear Algebra Nonlinear Equations Polynomial Interpolation Piecewise Polynomials, or Splines Numerical Quadrature Initial Value Ordinary Differential Equations Differential Operators and Their Discretizations 3. Theory of Ordinary Differential Equations. Existence and Uniqueness Results Green's Functions Stability of Initial Value Problems Conditioning of Boundary Value Problems 4. Initial Value Methods. Introduction. Shooting Superposition and Reduced Superposition Multiple Shooting for Linear Problems Marching Techniques for Multiple Shooting The Riccati Method Nonlinear Problems 5. Finite Difference Methods. Introduction Consistency, Stability, and Convergence Higher-Order One-Step Schemes Collocation Theory Acceleration Techniques Higher-Order ODEs Finite Element Methods 6. Decoupling. Decomposition of Vectors Decoupling of the ODE Decoupling of One-Step Recursions Practical Aspects of Consistency Closure and Its Implications 7. Solving Linear Equations. General Staircase Matrices and Condensation Algorithms for the Separated BC Case Stability for Block Methods Decomposition in the Nonseparated BC Case Solution in More General Cases 8. Solving Nonlinear Equations. Improving the Local Convergence of Newton's Method Reducing the Cost of the Newton Iteration Finding a Good Initial Guess Further Remarks on Discrete Nonlinear BVPS 9. Mesh Selection. Introduction Direct Methods A Mesh Strategy for Collocation Transformation Methods General Considerations 10. Singular Perturbations. Analytical Approaches Numerical Approaches Difference Methods Initial Value Methods 11. Special Topics. Reformulation of Problems in 'Standard' Form Generalized ODEs and Differential Algebraic Equations Eigenvalue Problems BVPs with Singularities Infinite Intervals Path Following, Singular Points and Bifurcation Highly Oscillatory Solutions Functional Differential Equations Method of Lines for PDEs Multipoint Problems On Code Design and Comparison Appendix A. A Multiple Shooting Code Appendix B. A Collocation Code References Bibliography Index.
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In this paper, the tenth-order linear boundary value problems are solved using reproducing kernel method. The algorithm developed approximates the solutions, and their higher-order derivatives, of differential equations and it avoids the complexity provided by other numerical approaches. First a new reproducing kernel space is constructed to solve this class of tenth-order linear boundary value problems; then the approximate solutions of such problems are given in the form of series using the present method. Three examples compared with those considered by Siddiqi, Twizell and Akram [S.S. Siddiqi, E.H. Twizell, Spline solutions of linear tenth order boundary value problems, Int. J. Comput. Math. 68 (1998) 345-362; S.S.Siddiqi, G.Akram, Solutions of tenth-order boundary value problems using eleventh degree spline, Applied Mathematics and Computation 185 (1)(2007) 115-127] show that the method developed in this paper is more efficient.
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In this paper, the numerical solution of periodic Fredholm–Volterra integro–differential equations of first-order is discussed in a reproducing kernel Hilbert space. A reproducing kernel Hilbert space is constructed, in which the periodic condition of the problem is satisfied. The exact solution u x ð Þ is represented in the form of series in the space W 2 2. In the mean time, the n-term approximate solution u n x ð Þ is obtained and is proved to converge to the exact solution u x ð Þ. Furthermore, we present an iterative method for obtaining the solution in the space W 2 2. Some examples are displayed to demonstrate the validity and applicability of the proposed method. The numerical result indicates that the proposed method is straightforward to implement, efficient, and accurate for solving linear and non-linear equations.
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A deferred correction method for the numerical solution of nonlinear two-point boundary value problems has been derived and analyzed in two recent papers by the first author. The method is based on mono-implicit Runge–Kutta formulas and is specially designed to deal efficiently with problems whose solutions contain nonsmooth parts—in particular, singular perturbation problems of boundary layer or turning point type. This paper briefly describes an implementation of the method and gives the results of extensive numerical testing on a set of nonlinear problems that includes both smooth and increasingly stiff (and difficult) problems. Results on the test set are also given using the available codes COLSYS and COLNEW. Although the intent is not to make a formal comparison, the code described appears to be competitive in speed and storage requirements on these problems.
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The objective of this paper is to present a numerical method for solving singularly perturbed turning point problems exhibiting an interior layer. The method is based on the asymptotic expansion technique and the reproducing kernel method (RKM). The original problem is reduced to interior layer and regular domain problems. The regular domain problems are solved by using the asymptotic expansion method. The interior layer problem is treated by the method of stretching variable and the RKM. Four numerical examples are provided to illustrate the effectiveness of the present method. The results of numerical examples show that the present method can provide very accurate approximate solutions.
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In this article, we proposed a collocation method based on reproducing kernels to solve a modified anomalous subdiffusion equation problem. We give constructively the -approximate of the equation whose coefficients are determined optimally by solving a system of linear equations. The final numerical experiments demonstrate that the proposed method is simple, effective, and easy to implement. Copyright © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 289–300, 2014
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Based on the superconvergent approximation at some point (depending on the fractional order $\alpha$, but not belonging to the mesh points) for Gr\"{u}nwald discretization to fractional derivative, we develop a series of high order pseudo-compact schemes for space fractional diffusion equations. Because of the pseudo-compactness of the derived schemes, no points beyond the domain are used for all the high order schemes including second order, third order, fourth order, and even higher order schemes; moreover, the algebraic equations for all the high order schemes have the completely same matrix structure. The stability analyses for some typical schemes are made; and extensive numerical experiments are performed to confirm the theoretical analysis or verify the convergent orders.
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In this paper, we present a new numerical technique to obtain the approximation solution for linear Volterra integral equations of the second kind based on reproducing kernel theory. The approximation solution is expressed by n-term summation of reproducing kernel functions. The merit of the new method includes (1) it is easy to implement this method; (2) high accuracy. The numerical examples compared with other methods show that the new method is more efficient.
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In this study, the numerical solution of Fredholm integro–differential equation is discussed in a reproducing kernel Hilbert space. A reproducing kernel Hilbert space is constructed, in which the initial condition of the problem is satisfied. The exact solution u(x)ux is represented in the form of series in the space W22[a,b]. In the mean time, the n-term approximate solution un(x)un(x) is obtained and is proved to converge to the exact solution u(x)u(x). Furthermore, we present an iterative method for obtaining the solution in the space W22[a,b]. Some examples are displayed to demonstrate the validity and applicability of the proposed method. The numerical result indicates that the proposed method is straightforward to implement, efficient, and accurate for solving linear and nonlinear Fredholm integro–differential equations.
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The time-fractional diffusion-wave equation is considered. The time-fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α (0,2]. The fractional derivative is described in the Caputo sense. This paper presents the analytical solutions of the fractional diffusion equations by an Adomian decomposition method. By using initial conditions, the explicit solutions of the equations have been presented in the closed form and then their numerical solutions have been represented graphically. Four examples are presented to show the application of the present technique. The present method performs extremely well in terms of efficiency and simplicity.
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In this paper, the Continuous Genetic Algorithm (CGA), previously developed by the principal author, is applied for the solution of optimal control problems. The optimal control problem is formulated as an optimization problem by the direct minimization of the performance index subject to constraints, and is then solved using CGA. In general, CGA uses smooth operators and avoids sharp jumps in the parameter values. This novel approach possesses two main advantages when compared to other existing direct and indirect methods that either suffer from low accuracy or lack of robustness. First, our method can be applied to optimal control problems without any limitation on the nature of the problem, the number of control signals, and the number of mesh points. Second, high accuracy can be achieved where the performance index is globally minimized while satisfying the constraints. The applicability and efficiency of the proposed novel algorithm for the solution of different optimal control problems is investigated. Copyright © 2010 John Wiley & Sons, Ltd.