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Numerical Solution of Nonlinear Reaction-Advection-Diffusion Equation
Abstract
In the present article, the advection-diffusion equation (ADE) having a nonlinear type source/sink term with initial and boundary conditions is solved using finite difference method (FDM). The solution of solute concentration is calculated numerically and also presented graphically for conservative and nonconservative cases. The emphasis is given for the stability analysis, which is an important aspect of the proposed mathematical model. The accuracy and efficiency of the proposed method are validated by comparing the results obtained with existing analytical solutions for a conservative system. The novelty of the article is to show the damping nature of the solution profile due to the presence of the nonlinear reaction term for different particular cases in less computational time by using the reliable and efficient finite difference method.
... The transport equations are one type of differential equation that contains parameters. The parameters associated with these equations are used to govern the diffusion process, convection process, and transportation of particles from the high to low absorption portion of the particle (see, [11]). For instance, energy can be converted within a physical system through convection and diffusion processes. ...
... For instance, energy can be converted within a physical system through convection and diffusion processes. So, this process is governed by the parameter associated with the problems used to study the conservation of energy [11,12]. These equations are also used to describe natural and anthropogenic processes. ...
... Nonlinear PDEs arise in various fields, including fluid dynamics, finance, and biology, and solving them numerically is often more complex due to their inherent nonlinearity [11,40]. The Crank-Nicolson method, originally designed for linear PDEs, faces challenges when applied to nonlinear problems, particularly the requirement to solve a nonlinear system of equations at each time step. ...
The study examines the convergence analysis of a higher order approximation for a transport equation with both nonhomogeneous and homogeneous boundary conditions by using the Crank–Nicolson and their modified schemes. Using the Taylor series expressions, the suggested numerical schemes are created. Von Neumann stability analysis combined with the error-boundedness criterion provides a complete examination of stability and convergence. The analysis’s findings show that both Crank–Nicolson systems exhibit unconditional stability and are convergent with second-order accuracy in both directions. Using the current two approaches, extensive numerical experiments are carried out. The outcomes of carried-out trials are compared to the accuracy of the previous schemes in the literature. The comparative analysis demonstrates that the current approaches yield higher accuracy than the earlier techniques. In addition, we contrast the results of the two approaches and compare them. Based on these findings, the accuracy of the modified Crank–Nicolson scheme is better than that of the original Crank–Nicolson scheme. Overall, this research demonstrates that current methods effectively solve partial differential equations.
... Time, space, and time-space FDEs are extensively used to describe physical and engineering problems like anomalous diffusion, solute transport, signal processing, control, etc. For the description and understanding of dispersion phenomena, the FDEs have fundamental importance and have received a lot of attention during last few decades [8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Thus scientists and engineers are involved to find the solutions of FDEs for their non-local behaviour, greater flexibility in models and its convergence to the integer order systems. ...
... Now on using above equations (16) in the equations (14) and (15), we have ...
... In this problem a comparison of errors with other existing methods are given in Table 2. It is very clear from the table that our method is performing well as compared to an existing method [16] even for less temporal discretization. Bhatt et al. [29] have also solved above problem with fourth-order scheme and obtained the maximum error of order 10 −5 for n = 16 and n = 20. ...
In this article a non-standard finite difference collocation method is developed with the help of Fibonacci polynomial to solve the coupled type fractional order Burgers’ equation. To show the efficiency of the method, it is used to solve the coupled Burgers’ equation having exact solutions and compared the obtained numerical results with the existing results through error analysis. Through the tabular presentation of the results, it is shown that the proposed method is performing much better as compared to the existing methods even for less degree of approximation and less order of temporal discretization. After validation, the method is used for solving an unsolved nonlinear fractional order coupled Burgers’ equation and simulate the results for different fractional order spatial derivative for different values of the parameters.
... Also, Johari et al., [18] employed the ADI methods for 2D ADEs, while Hutomo et al. [19] applied the Du-Fort Frankel method for solving a 2D ADE with variable coefficients. Singh et al., [20] solved a non-linear reactive ADE by explicit FDM and calculated the stability of their model using the von Neumann method. The solution was validated with the existing analytical solution and found a substantial similarity between them. ...
... where, j = 1, 2, . . . , N − 1 and i = M. Applying Equations (20) and (22) to Equation (18) yields: ...
In this study, a two-dimensional contaminant transport model with time-varying axial input sources subject to non-linear sorption, decay, and production is numerically solved to find the concentration distribution profile in a heterogeneous, finite soil medium. The axial input sources are assigned on the coordinate axes of the soil medium, with background sources varying sinusoidally with space. The groundwater velocities are considered space-dependent in the longitudinal and transversal directions. Various forms of axial input sources are considered to study their transport patterns in the medium. The alternating direction implicit (ADI) and Crank-Nicolson (CN) methods are applied to approximate the two-dimensional governing equation, and the obtained algebraic system of equations in each case is further solved by MATLAB scripts. Both approximate solutions are illustrated graphically for various hydrological input data. The influence of various hydrogeological input parameters, such as the medium’s porosity, density, sorption conditions, dispersion coefficients, etc., on the contaminant distribution is analyzed. Further, the influence of constant and varying velocity parameters on groundwater contaminant transport is studied. The stability of the proposed model is tested using the Peclet and Courant numbers. Substantial similarity is observed when the approximate solution obtained using the CN method is compared with the finite element method in a special case. The proposed approximate solution is compared with the existing numerical solutions, and an overall agreement of 98–99% is observed between them. Finally, the stability analysis reveals that the model is stable and robust.
... Even if there are several analytical methods for accomplishing this, they are restricted to particular unique types to reduce errors [16][17][18][19]. Additionally, several numerical techniques such as the boundary element method (BEM), the finite volume method (FVM), the finite difference method (FDM), and the finite element method (FEM) are designed to address constraints associated with the model structure [20][21][22]. These methods often need the domain to be discretized into finite numbers, which is an issue that has an implementation on complex domains and considerably affects the efficiency. ...
This study aims to provide insights into new areas of artificial intelligence approaches by examining how these techniques can be applied to predict behaviours for difficult physical processes represented by partial differential equations, particularly equations involving nonlinear dispersive behaviours. The current advection-dispersion-reaction equation is one of the key formulas used to depict natural processes with distinct characteristics. It is composed of a first-order advection component, a third-order dispersion term, and a nonlinear response term. Using the deep neural network approach and accounting for physics-informed neural network awareness, the problem has been elaborately discussed. Initial and boundary conditions are added as constraints when the neural networks are trained by minimizing the loss function. In comparison to the existing results, the approach has produced qualitatively correct kink and anti-kink solutions, with losses often remaining around 0.01%. It has also outperformed several traditional discretization-based methods.
... As analytic solutions of the FRADEs are available only for few simple cases, therefore developing very accurate and efficient numerical methods is of great interest. Many numerical methods have been developed to solve differential equations of fractional order, for example, finite difference method (FDM) [15], finite element method (FEM) [16], and homotopy perturbation method (HPM) [17] to solve the fractional order diffusion equation. Rose [18] has solved the diffusion equation with the compact finite volume method, Zhang and Chen [19] solved the diffusion equation with the weak Galerkin finite element method, Zhang et al. [20] developed an approximate scheme to solve diffusion equation, Polyanin [21] has solved the diffusion equation with variable coefficient and Fairweather et al. [22] have used an orthogonal spline collocation method to solve the two-dimensional fractional diffusion-wave equation. ...
In this article, the approximate solution of the fractional-order reaction advection-diffusion equation with the prescribed initial and boundary conditions is found with the help of a cubic B-spline collocation method, which is unconditionally stable and convergent. The accuracy of the scheme is validated by applying the method on four existing problems having analytical solutions and through the evaluation of the absolute errors between numerical results and the exact solutions for different particular cases. Applying the proposed method on the last two numerical problems, it is shown that the method performs better than the existing methods even for very less number of spatial and temporal discretizations. The main contribution of the article is to develop an efficient method to solve the proposed fractional order nonlinear problem and to find the effect on solute concentration graphically due to increase in the non-linearity in the diffusion term for different particular values of parameters.
... The computational mesh that represents the physical domain in the tests was modeled, followed by the numerical model described by the Navier-Stokes equations [16] [17] [18] [19] [20] and the pollutant transport equation [21] [22] [23]. ...
This work proposes a way of modelling two-dimensional complex meshes using elliptic equations, in which the computational grid coincides with the problem geometry, making computational processing more suitable. A multiblock technique was used in order to achieve a better representation of the problem domain. In this way, numerical simulations of the movement and dispersion of pollutant emissions in the atmosphere are presented in the generated domains, using the Navier-Stokes pollutant transport equations. The curvilinear coordinates and the finite difference method are used for the discretization. The model was verified in two tests. In the first test, three cases were proposed, with geometries containing a chimney followed by an obstacle, using different chimney heights, and the obstacle height was fixed. The test aims to verify the vortices appearance, in the blocks, to obtain agreement as presented in the literature. In the second test, the geometry is described by a chimney and an obstacle that represents one of the mountains in the valley. The performed tests made it possible to verify that the height of the chimney can be considered a determining factor to describe the dispersion of pollutants, as well as their concentration in the proximity of industrial areas.
... A Adomain decomposition method for solving fractional partial differential equations (FFPDEs) is given in article [29]. An implicit finite difference method dealing with fuzzy fractional diffusion equation is given in References [30,31,26]. ...
In this article, we will study the fuzzy fractional advection diffusion model in which both space and time are fractional. This model has fuzzy unknown function, fuzzy coefficient and fuzzy numbers. First, we define all necessary fuzzy preliminaries and Caputo definition of fractional derivative in fuzzy environment. Then, we derive the fuzzy operational matrix of fractional differentiation with the help of Chebyshev polynomial. With the help of this matrix, we solve our advection–diffusion model with initial and boundary conditions in fuzzy environment by Chebyshev collocation method. This method reduces the fuzzy fractional differential equation into a system of fuzzy algebraic equation which can be solved easily and we get the desired solution. We show the accuracy and validity of method by solving a particular case of model. We show the dynamics of unknown function for different fractional order in both space and time direction. The effect of advection term is presented through graphs.
... Hence, several analytical and numerical methods are available in the literature such as domain decomposition method [1,2], Wavelet operational method [3], homotopy analysis method [4], variational iterative method [5], etc. Moreover, there are several numerical methods to solve different types of fractional diffusion equations [5][6][7][8][9][10][11][12][13][14], etc. ...
In this article, a new algorithm is proposed to solve the nonlinear fractional-order one-dimensional solute transport system. The spectral collocation technique is considered with the Fibonacci polynomial as a basis function for the approximation. The Fibonacci polynomial is used to obtain derivative in terms of an operational matrix. The proposed algorithm is actually based on the fact that the terms of the considered problem are approximated through a series expansion of double Fibonacci polynomials and then collocated those on specific points, which provide a system of nonlinear algebraic equations which are solved by using Newton's method. To validate the precision of the proposed method, it is applied to solve three different problems having analytical solutions. The comparison of the results through error analysis is depicted through tables which clearly show the higher accuracy of order of convergence of the proposed method in less central processing unit (CPU) time. The salient feature of the article is the graphical exhibition of the movement of solute concentration for different particular cases due to the presence and absence of reaction term when the proposed scheme is applied to the considered nonlinear fractional-order space–time advection–reaction–diffusion model.
... However, these methods also have some drawbacks, and we cannot use it for any type of problems. To fulfill these need, researchers introduced many semi analytical techniques such as HPM [3], HPTM [4], HAM [5], FDM [6], RPSM [7], etc. ...
This article presents the Fractional Laplace transform with the help of new iterative method for finding the approximate solutions of coupled system of fractional order partial differential equations. The time fractional Whitham-Broer-Kaup system is taken as test example The fractional derivatives are described in the Caputo sense. Numerical results obtained by the proposed method are compared with that of Adomian Decomposition Method (ADM), Variational Iteration Method (VIM) and Optimal Homotopy Asymptotic Method (OHAM).Numerical results show that the proposed method is reliable and efficient for solution of fractional order coupled system partial differential equations. The accuracy of the method increases by taking higher order approximations.
... Thus, many researchers have introduced and developed numerical methods in order to get numerical solutions for these two classes of equations (see for example [7,8,9,10,11,12,13]). Moreover, partial differential equations of fractional order have also been investigated and various numerical methods for their approximation have been introduced recently [14,15]. ...
In this work, we present a collocation method based on the Legendre wavelet combined with the Gauss--Jacobi quadrature formula for solving a class of fractional delay-type integro-differential equations. The problem is considered with either initial or boundary conditions and the fractional derivative is described in the Caputo sense. First, an approximation of the unknown solution is considered in terms of the Legendre wavelet basis functions. Then, we substitute this approximation and its derivatives into the considered equation. The Caputo derivative of the unknown function is approximated using the Gauss--Jacobi quadrature formula. By collocating the obtained residual at the well-known shifted Chebyshev points, we get a system of nonlinear algebraic equations. In order to obtain a continuous solution, some conditions are added to the resulting system. Some error bounds are given for the Legendre wavelet approximation of an arbitrary function. Finally, some examples are included to show the efficiency and accuracy of this new technique.
... Thus, many researchers have introduced and developed numerical methods in order to get numerical solutions for these two classes of equations (see for example [7,8,9,10,11,12,13]). Moreover, partial differential equations of fractional order have also been investigated and various numerical methods for their approximation have been introduced recently [14,15]. ...
In this work, we present a collocation method based on the Legendre wavelet combined with the Gauss–Jacobi quadrature formula for solving a class of fractional delay-type integro-differential equations. The problem is considered with either initial or boundary conditions and the fractional derivative is described in the Caputo sense. First, an approximation of the unknown solution is considered in terms of the Legendre wavelet basis functions. Then, we substitute this approximation and its derivatives into the considered equation. The Caputo derivative of the unknown function is approximated using the Gauss–Jacobi quadrature formula. By collocating the obtained residual at the well-known shifted Chebyshev points, we get a system of nonlinear algebraic equations. In order to obtain a continuous solution, some conditions are added to the resulting system. Some error bounds are given for the Legendre wavelet approximation of an arbitrary function. Finally, some examples are included to show the efficiency and accuracy of this new technique.
Fractional reaction advection–diffusion problems have been of interest in the area of fractional calculus. In this paper, we consider two-dimension time and space nonlocal nonlinear reaction convection diffusion problem on a finite domain. We propose interpolation approximation to approximate the Caputo fractional derivative combined the weighted and shifted-Grünwald Letnikov difference scheme to discretize the Riesz space fractional derivative. In addition, we use fully implicit Crank Nicolson method combined with weighted and shifted Grünwald–Letnikov difference operator and get a numerical approximation method is unconditionally stable and convergent with the accuracy of . We take the alternative direction implicit approach for two-dimension problem, which is used to reduce a two-dimension problem to a one-dimension equation by applying x and y-direction alternately over the groundwater problem. Finally, the numerical simulations with examples of nonlinear source term are given, demonstrating the theoretical study and confirming the effectiveness of the proposed method.
Studying the interactions between advection and dispersion in natural systems, especially in cases where advection predominates, are important because it is necessary to accurately model physical phenomena in domains like hydrology, atmospheric science, environmental engineering, etc. Conventional analytical and numerical methods often have drawbacks, such as high computational costs and challenges in accurately modeling complex behaviors. Improved simulation and a deeper understanding of these interactions are made possible by neural networks’ capacity to learn from big datasets and simulate complex relationships. To demonstrate that deep neural networks effectively capture scenarios within physical processes where advection plays a dominant role, the effectiveness of applying deep neural networks to a third-order dispersive partial differential equation incorporating advection and dispersion terms has been investigated in this study.
The advection-diffusion-reaction (ADR) equation is a fundamental mathematical model used to describe various processes in many different areas of science and engineering. Due to wide applicability of the ADR equation, finding accurate solution is very important to better understand a physical phenomenon represented by the equation. In this study, a numerical scheme for solving two-dimensional unsteady ADR equations with spatially varying velocity and diffusion coefficients is presented. The equations include nonlinear reaction terms. To discretize the ADR equations, the Crank–Nicolson finite difference method is employed with a uniform grid. The resulting nonlinear system of equations is solved using Newton’s method. At each iteration of Newton’s method, the Gauss–Seidel iterative method with sparse matrix computation is utilized to solve the block tridiagonal system and obtain the error correction vector. The consistency and stability of the numerical scheme are investigated. MATLAB codes are developed to implement this combined numerical approach. The validation of the scheme is verified by solving a two-dimensional advection-diffusion equation without reaction term. Numerical tests are provided to show the good performances of the proposed numerical scheme in simulation of ADR problems. The numerical scheme gives accurate results. The obtained numerical solutions are presented graphically. The result of this study may provide insights to apply numerical methods in solving comprehensive models of physical phenomena that capture the underlying situations.
The solution of a time-fractional differential equation usually exhibits a weak singularity near the initial time t = 0. It causes traditional numerical methods with uniform mesh to typically lose their accuracy. The technique of nonuniform mesh based on reasonable regularity of the solution has been found to be a very effective way to recover the accuracy. The current work aims to devise a compact finite difference scheme with temporal graded and harmonic meshes for solving time-fractional diffusion-advection-reaction equations with non-smooth solutions. The time-fractional operator involved in this is taken in the Caputo sense. A theoretical analysis of the stability and convergence of the proposed numerical technique is presented by von Neumann's method. The scheme's proficiency, robustness, and effectiveness are examined through three numerical experiments. A comparison of the numerical results on the uniform, graded, and harmonic meshes is presented to demonstrate the advantage of the suggested meshes over the uniform mesh. The tabular and graphical representations of numerical results confirm the high accuracy and versatility of the scheme.
The time-fractional advection–diffusion reaction equation (TFADRE) is a fundamental mathematical model because of its key role in describing various processes such as oil reservoir
simulations, COVID-19 transmission, mass and energy transport, and global weather production. One of the prominent issues with time fractional differential equations is the design of
efficient and stable computational schemes for fast and accurate numerical simulations. We
construct in this paper, a simple and yet efficient modified fractional explicit group method
(MFEGM) for solving the two-dimensional TFADRE with suitable initial and boundary
conditions. The proposed method is established using a difference scheme based on L1 discretization in temporal direction and central difference approximations with double spacing
in spatial direction. For comparison purposes, the Crank–Nicolson finite difference method
(CNFDM) is proposed. The stability and convergence of the presented methods are theoretically proved and numerically affirmed. We illustrate the computational efficiency of the
In the present article, a finite domain is considered to find the numerical solution of a two‐dimensional nonlinear fractional‐order partial differential equation (FPDE) with Riesz space fractional derivative (RSFD). Here two types of FPDE–RSFD are considered, the first one is a two‐dimensional nonlinear Riesz space‐fractional reaction–diffusion equation (RSFRDE) and the second one is a two‐dimensional nonlinear Riesz space‐fractional reaction‐advection‐diffusion equation (RSFRADE). SFRDE is obtained by simply replacing second‐order derivative term of the standard nonlinear diffusion equation by the Riesz fractional derivative of order (β+1)∈(1,2) whereas the SFRADE is obtained by replacing the first‐order and second‐order space derivatives from the standard order advection–dispersion equation with the Riesz fractional derivatives of order β∈(0,1). A numerical method is provided to deal with the RSFD with the weighted and shifted Grünwald–Letnikov (WSGD) approximations, for the spatial discretization. The SFRDE and SFRADE are transformed into a system of ordinary differential equations (ODEs), which have been solved using a fast compact implicit integration factor (FcIIF) with nonuniform time meshes. Finally, the demonstration of the validation and effectiveness of the numerical method is given by considering some existing models.
The nonlinear wave phenomena are described by Partial Differential Equations (PDEs) and the solution of these PDEs plays a significant role in understanding the existence and behaviour of wave propagation under different material media associated with the physical system. This paper presents approximate solution of nonlinear Burgers’ equation with linear source term by using Laplace Decomposition Method (LDM).The analysis of the combined effect of source term and initial condition on the wave profiles is presented in this paper. As the solutions of nonlinear PDEs are greatly influenced by the presence of source term under different initial conditions, the results obtained can be readily extended to study the wave profiles of various types of initial value problems describing nonlinear reaction-dissipative phenomena. Moreover, the solutions can be used to study the internal mechanisms of the physical phenomena of more general system with diffusive effects.
This article is concerned with developing a method to find a numerical solution of a one‐dimensional variable‐order non‐linear partial integro‐differential equation (PIDE) viz., reaction–advection–diffusion equation with initial and boundary conditions. The proposed numerical scheme shifted Legendre collocation method is based on operational matrices. The operational matrices using one‐dimensional wavelets are derived to solve the said variable‐order model. First operational matrices have been introduced for integration and variable‐order derivatives using one‐dimensional Legendre wavelets (LWs). After that, using the shifted Legendre collocation points, the model is reduced to a system of algebraic equations, which are solved using Newton–Cotes method. The error is then calculated by comparing the numerical solution obtained from the system of algebraic equations and the known exact solution of an existing problem to validate the efficiency of the proposed numerical scheme. The main contribution of the article is the graphical exhibitions of the solution profile for different variable order derivatives in presence of different values of the parameters.
In this work, we introduce a novel reaction-diffusion equation, based on an anisotropic behavior and a time-fractional order derivative in the sense of Caputo. Thanks to some recent compactness results we could provide an existence and uniqueness results using the Schauder fixed-point theorem. A finite difference scheme is introduce for discretization and simulations. In order to show the efficiency of the proposed approach, we use textured images and the peak signal-to-noise ratio (PSNR) with the structural similarity index (SSIM) to measure the quality of the result images in the numerical tests. Thanks to the time-fractional derivative and Weickert term, the proposed approach shows some remarkable results with respect to the PSNR and SSIM visually and quantitatively, compared to the competitive models.
Many problems in mathematical physics are modelled by second order ordinary differential equations (ODES). Consequently, numerically solving second order ODES has attracted much attention of many mathematicians and physicists. Most of the existing methods reduce second order ODES to a system of first order ODES. In the present study, we are motivated to develop methods which are suitable to solve second order initial value problems and second order boundary value problems of ODES directly. The methods are based on the universal approximation capability of a three layer feedforward sigmoid-weighted neural networks optimized with error back propagation algorithm. In addition, the proposed methods are tested for some examples and compared to classical methods using computer simulations. The obtained results show the efficiency and accuracy of the proposed methods.
A reliable and efficient numerical technique for the approximation of Riesz fractional partial differential equations with chaotic and spatiotemporal chaos properties is developed in this work. In such a system, the standard anomalous diffusion terms are modeled using the Riesz fractional derivative which can be extended from one to high dimensional cases. The methodology adopts finite difference schemes, as well as the novel Fourier spectral methods for the approximation of the Riesz fractional derivatives in space. The resulting system of ordinary differential equations is advanced in time with the exponential time-differencing Runge–Kutta method. The performance and applicability of these methods were tested with some practical and real-life problems which are of current and recurring interests arising from computational physics, engineering and other areas of applied sciences. Simulation results are given for different instances of fractional parameter α in the intervals (0,1) and (1,2) which correspond to subdiffusion and superdiffusion scenarios, respectively.
In this study, we give the numerical scheme to the stochastic nonlinear advection diffusion equation. This models is considered with white noise (or random process) having same intensity by changing frequencies. Furthermore, the stability and consistency of proposed scheme are also discussed. Moreover, it is concerned about the analytical solutions, the Riccati equation mapping method is adopted. The different families of single (shock and singular) and mixed (complex solitary-shock, shock-singular, and double-singular) form solutions are obtained with the different choices of free parameters. The graphical behavior of solutions is also depicted in 3D and corresponding contours.
In this article, a new high-order explicit group iterative scheme is developed for the solution of the two-dimensional time fractional cable equation. The proposed scheme is derived from the Crank-Nicolson (C-N) high-order compact finite difference method, where the Caputo discretization and C-N high-order approximations are used for the time fractional and space derivative respectively. The stability and convergence of the proposed scheme are also established. Finally, two numerical examples are presented to show the accuracy and efficiency of the scheme.
In this paper, we present numerical solution for the fractional Bratu-type equation via fractional residual power series method (FRPSM). The fractional derivatives are described in Caputo sense. The main advantage of the FRPSM in comparison with the existing methods is that the method solves the nonlinear problems without using linearization, discretization, perturbation or any other restriction. Three numerical examples are given and the results are numerically and graphically compared with the exact solutions. The solutions obtained by the proposed method are in complete agreement with the solutions available in the literature. The results reveal that the FRPSM is a very effective, simple and efficient technique to handle a wide range of fractional differential equations.
This paper is concerned with numerical solution of time fractional stochastic advection-diffusion type equation where the first order derivative is substituted by a Caputo fractional derivative of order (). This type of equations due to randomness can rarely be solved, exactly. In this paper, a new approach based on finite difference method and spline approximation is employed to solve time fractional stochastic advection-diffusion type equation, numerically. After implementation of proposed method, the under consideration equation is transformed to a system of second order differential equations with appropriate boundary conditions. Then, using a suitable numerical method such as the backward differentiation formula, the resulting system can be solved. In addition, the error analysis is shown in some mild conditions by ignoring the error terms in the system. In order to show the pertinent features of the suggested algorithm such as accuracy, efficiency and reliability, some test problems are included. Comparison achieved results via proposed scheme in the case of classical stochastic advection-diffusion equation () with obtained results via wavelets Galerkin method and obtained results for other values of with the values of exact solution confirm the validity, efficiency and applicability of the proposed method.
This paper presents a dynamical and generalized synchronization (GS) of two dependent chaotic nonlinear advection‐diffusion‐reaction (ADR) processes with forcing term, which is unidirectionally coupled in the master‐slave configuration. By combining backward differentiation formula‐Spline (BDFS) scheme with the Lyapunov direct method, the GS is studied for designing controller function of the coupled nonlinear ADR equations without any linearization. The GS behaviors of the nonlinear coupled ADR problems are obtained to demonstrate the effectiveness and feasibility of the proposed technique without losing natural properties and reduce the computational difficulties on capturing numerical solutions at the low value of the viscosity coefficient. This technique utilizes the master configuration to monitor the synchronized motions. The nonlinear coupled model is described by the incompressible fluid flow coupled to thermal dynamics and motivated by the Boussinesq equations. Finally, simulation examples are presented to demonstrate the feasibility of the synchronization of the proposed model.
In this manuscript, we consider in detail numerical approach of solving Laplace’s equation in 2-dimensional region with given boundary values which is based on the Alternating Direction Implicit Method (ADI). This method was constructed using Taylor’s series expansion on the second order Laplace equation leading to a linear algebraic system. Solving the algebraic system, leads to the unknown coefficients of the basis function. The techniques of handling practical problems are considered in detail. The results obtained compared favorably with the results obtained from the Finite difference method constructed by Dhumal and Kiwne and the exact solution. Thus the ADI method can as well be used for the numerical solution of steady-state Laplace equations.
This paper presents an approximate solution of 3D convection diffusion equation (CDE) using DQM based on modified cubic trigonometric B-spline (CTB) basis functions. The DQM based on CTB basis functions are used to integrate the derivatives of space variables which transformed the CDE into the system of first order ODEs. The resultant system of ODEs is solved using SSPRK (5,4) method. The solutions are approximated numerically and also presented graphically. The accuracy and efficiency of the method is validated by comparing the solutions with existing numerical solutions. The stability analysis of the method is also carried out.
Mathematical modelling of Salmonella disease played a significant role in the understanding of the transmission of this epidemic. In this article, we study numerically a fractional dynamical, with time-delay, Salmonella disease model. The introduced model is written as a system of fractional-order delay differential equations. The fractional operator is defined in the Atangana–Baleanu–Caputo sense, where the parameters are varied regarding the fractional-order derivative. We study, for any time delay, the stability of the disease-free equilibrium point and the endemic equilibrium point. Implicit nonstandard finite difference method is introduced to simulate numerically the proposed model. The obtained schema was unconditionally stable. Some comparison with our numerical simulations is introduced to show the effectiveness and applicability of the introduced approach for approximating the solutions of such stiff problems of fractional delay differential equations. Our simulations confirm the theoretical studies in this paper.
Abstract This article mainly explores and applies a modified form of the analytical method, namely the homotopy analysis transform method (HATM) for solving time-fractional Cauchy reaction–diffusion equations (TFCRDEs). Then mainly we address the error norms L2 and L∞ for a convergence study of the proposed method. We also find existence, uniqueness and convergence in the analysis for TFCRDEs. The projected method is illustrated by solving some numerical examples. The obtained numerical solutions by the HATM method show that it is simple to employ. An excellent conformity obtained between the solution got by the HATM method and the various well-known results available in the current literature. Also the existence and uniqueness of the solution have been demonstrated.
In the current paper, a new approach is applied to solve time fractional advection-diffusion equation. The utilized fractional derivative operator is the Atangana–Baleanu (AB) derivative in Caputo sense. The mentioned fractional derivative involves the Mittag–Leffler function as the kernel that is both non-singular and non-local. A new operational matrix of AB fractional integration is obtained for the Bernstein polynomials (Bps). By applying the aforesaid matrix, the considered problems are reduced to a system of equations. The approximate solution is derived by solving the yielded system. Also, the error bound is studied. The obtained results show that the applied scheme is simple and powerful tool in finding numerical solutions of fractional equations.
The shifted Legendre collocation method is used to solve the one‐dimensional nonlinear reaction‐advection‐diffusion equation having spatial and temporal fractional‐order derivatives with initial and boundary conditions. The solution profiles of the normalized solute concentration of space‐time fractional‐order Burgers‐Fisher and Burgers‐Huxley equations are presented through graphs for different particular cases. The main purpose of the article is the graphical exhibition of the effect of the temporal, spatial fractional‐order derivatives and the reaction term on the solution profile of the space‐time fractional‐order Burgers‐Fisher and Burgers‐Huxley equations. The other purpose of the article is the error estimation of the proposed method. A drive has been taken to validate the effectiveness of the proposed method through tabular presentation of comparison of numerical results with analytical results for the existing problems through convergence analysis.
In this article dispersive optical soliton solutions that are governed by the time-fractional Schrödinger–Hirota equation have been presented. Some new soliton solutions of the time-fractional SH equations are obtained in this work. Using extended auxiliary equation method, optical soliton solutions are sought for the fractional SH equations with power law nonlinearity as well as Kerr law nonlinearity. Using fractional complex transform the time-fractional SH equation is converted into the nonlinear ordinary differential equation, and then, the resulting equation is solved using a novel analytical method viz. extended auxiliary equation method. As a result of this, some new optical soliton solutions have been successfully obtained. The obtained results show that the proposed method is a straightforward technique to find out new optical solutions of time-fractional SH equation.
In the present investigation, the q-homotopy analysis transform method (q-HATM) is applied to find approximated analytical solution for the system of fractional differential equations describing the unsteady flow of a polytropic gas. Numerical simulation has been conducted to prove that the proposed technique is reliable and accurate, and the outcomes are revealed using plots and tables. The comparison between the obtained solutions and the exact solutions shows that the proposed method is efficient and effective in solving nonlinear complex problems. Moreover, the proposed algorithm controls and manipulates the obtained series solution in a huge acceptable region in an extreme manner and it provides us a simple procedure to control and adjust the convergence region of the series solution.
In the present article, an effective Laguerre collocation method is used to obtain the approximate solution of a system of coupled fractional order non-linear reaction-advection-diffusion equations with prescribed initial and boundary conditions. In the proposed scheme, Laguerre polynomials are used together with operational matrix and collocation method to obtain approximate solutions of the coupled systems, so that our proposed model is converted into a system of algebraic equations which can be solved employing the Newton method. The solution profiles of the coupled systems are presented graphically for different particular cases. The salient features of the present article are finding the stability analysis of the proposed method and also the demonstration of the lower variations of solute concentrations with respect to the column length in fractional order system compared to integer order system. To show the higher efficiency, reliability and accuracy of the proposed scheme, a comparison between the numerical results of Burgers’ coupled systems and its existing analytical result is reported. There are high compatibility and consistency between the approximate solution and its exact solution to a higher order of accuracy. The exhibition of error analysis for each case through tables and graphs confirms the super-linearly convergence rate of the proposed method.
In this study, the numerical solution of the two-dimensional solute transport system in a homogeneous porous medium of finite length is obtained. The considered transport system has the terms accounting for advection, dispersion and first-order decay with first-type source boundary conditions. Initially, the aquifer is considered solute free, and a constant input concentration is considered at inlet boundary. The solution is describing the solute concentration in rectangular inflow region of the homogeneous porous media. The numerical solution is derived using a powerful method, viz. spectral collocation method. The numerical computation and graphical presentations exhibit that the method is effective and reliable during the solution of the physical model with complicated boundary conditions even in the presence of reaction term.
In this paper, we propose an efficient numerical scheme for the approximate solution of the time fractional diffusion-wave equation with reaction term based on cubic trigonometric basis functions. The time fractional derivative is approximated by the usual finite difference formulation and the derivative in space is discretized using cubic trigonometric B-spline functions. A stability analysis of the scheme is conducted to confirm that the scheme does not amplify errors. Computational experiments are also performed to further establish the accuracy and validity of the proposed scheme. The results obtained are compared with a finite difference schemes based on the Hermite formula and radial basis functions. It is found that our numerical approach performs superior to the existing methods due to its simple implementation, straight forward interpolation and very less computational cost.
In this article, Chebyshev collocation method is used to reduce the one-dimensional advection–dispersion equation having source/sink term with given boundary conditions and initial condition into a system of ordinary differential equations which are solved using finite difference method. The solution profile of normalized solute concentration factor for both conservative and non-conservative systems are calculated numerically which are presented through graphs for different particular cases. The salient feature of this article is the comparison of the trend of numerical solution with the existing analytical solution thereby validating our considered numerical technique.
In this paper we describe a spectrally accurate, unconditionally stable, efficient method using operational matrices to solve numerically two-dimensional advection-diffusion equations on a rectangular domain. The novelty of this paper is to relate for the first time evolution partial differential equations and Sylvester-type equations, avoiding Kronecker tensor products. Furthermore, to reach large times, the calculation of just two matrix exponentials is required, for which we compare different techniques based on Pade's approximations, matrix decompositions and Krylov spaces, as well as a new technique which avoids the computation of matrix exponentials. We also illustrate how to take advantage of multiple precision arithmetic. Finally, possible generalizations to non-linear problems and higher-dimensional problems, as well as to unbounded domains, are considered.
A numerical method is developed to solve the nonlinear Boussinesq equation using the quintic B-spline collocation method. Applying the Von Neumann stability analysis, the proposed method is shown to be unconditionally stable. An example has been considered to illustrate the efficiency of the method developed.
Most of the existing numerical schemes developed to solve Burgers’ equation cannot exhibit its correct physical behavior for very small values of viscosity. This difficulty can be overcome by using splitting methods derived for near-integrable system. This class of methods has positive real coefficients and can be used for non-reversible systems such as Burgers’ equation. It also has the advantage of being able to account small viscosity in the accuracy. The algorithm is based on the combination of implicit–explicit finite difference schemes to solve each simplified problem and filtering technique to treat nonlinear instability. The resulting algorithm is accurate, efficient and easy to implement. The new numerical results are compared with numerical and exact solutions reported in the literature and found that they are very accurate for small values of the viscosity.
Groundwater constitutes an important component of many water resource systems, supplying water for domestic use, for industry, and for agriculture. Management of a groundwater system, an aquifer, or a system of aquifers, means making such decisions as to the total quantity of water to be withdrawn annually, the location of wells for pumping and for artificial recharge and their rates, and control conditions at aquifer boundaries. Not less important are decisions related to groundwater qUality. In fact, the quantity and quality problems cannot be separated. In many parts of the world, with the increased withdrawal of ground water, often beyond permissible limits, the quality of groundwater has been continuously deteriorating, causing much concern to both suppliers and users. In recent years, in addition to general groundwater quality aspects, public attention has been focused on groundwater contamination by hazardous industrial wastes, by leachate from landfills, by oil spills, and by agricultural activities such as the use of fertilizers, pesticides, and herbicides, and by radioactive waste in repositories located in deep geological formations, to mention some of the most acute contamination sources. In all these cases, management means making decisions to achieve goals without violating specified constraints. In order to enable the planner, or the decision maker, to compare alternative modes of action and to ensure that the constraints are not violated, a tool is needed that will provide information about the response of the system (the aquifer) to various alternatives.
One can infer the past behaviour of an aquifer system from the available earlier field records, but there is no way of computing the future behaviour of the regional aquifer system unless it is actually subject to the stresses. A full-scale experiment would be prohibitively costly and time consuming. The only feasible recourse, therefore, is to appropriately simulate the aquifer and study its response to several input/output schemes and thereby evolve the optimal management schemes. A groundwater model is thus a simplified version of the real system that approximately simulates the input-output stresses and response relations of the system. One has to understand here that normally the real system is simplified to model the system as such there is no unique model for a given groundwater system. Normally, models are classified as predictive, interpretive and generic models. Predictive models are used to predict the future response of the aquifer, which needs a calibrated and validated model. Interpretive models are used for studying system dynamics and it is generally used for optimal data network design. Generic models are used to understand the flow dynamics in hypothetical situations.
In this paper, the space-time fractional diffusion equation related to the electromagnetic transient phenomena in transmission lines is studied, three cases are presented; the diffusion equation with fractional spatial derivative, with fractional temporal derivative and the case with fractional space-time derivatives. For the study cases, the order of the spatial and temporal fractional derivatives are 0 < β, γ ≤ 2 respectively. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional diffusion equation. The general solutions of the proposed equations are expressed in terms of the multivariate Mittag-Leffler functions; these functions depend only on the parameters β and γ and preserve the appropriated physical units for any value of the fractional derivative exponent. Furthermore, an analysis of the fractional time constant was made in order to indicate the change of the medium properties and the presence of dissipation mechanisms. The proposed mathematical representation can be useful to understand electrochemical phenomena, propagation of energy in dissipative systems, irreversible thermodynamics, quantum optics or turbulent diffusion, thermal stresses, models of porous electrodes, the description of gel solvents and anomalous complex processes.
The conceptual and mathematical models for seawater intrusion are presented in Chapter 7. In this chapter, two numerical models are presented for the analysis of groundwater flow in an aquifer saturated with two fluids (e.g., fresh and salt water), separated by a sharp interface. The first model applies to the flow in a vertical plane, in which the location of the interface is time-dependent, because of boundary conditions, pumping wells, etc. In this model, the flow domain (in the vertical plane) is subdivided into small elements, such that the interface is represented by a series of elements, connecting certain nodes in the mesh. As the interface moves, the location of the nodes upon it changes, so that the entire mesh is changing as a function of time.
The study of advection–diffusion equation continues to be an active field of research. The subject has important applications to fluid dynamics as well as many other branches of science and engineering. In this paper several different numerical techniques will be developed and compared for solving the three-dimensional advection–diffusion equation with constant coefficient. These techniques are based on the two-level fully explicit and fully implicit finite difference approximations. The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. Another nice feature of the modified equivalent partial differential equation approach is that a high order of accuracy can be combined with excellent stability properties. The new second-order accurate methods are free of numerical diffusion. The results of a numerical experiment are presented, and the accuracy and central processor (CPU) time needed are discussed and compared.
The modeling of groundwater pollution consists in describing the convection of a contaminant, the dispersion of the contaminant, the chemical and physicochemical reactions of the contaminant with the solid matrix of the porous medium, the biochemical reactions of the contaminant with its environment, the reactions within the contaminant, by means of mathematical tools, such as systems of partial differential equations or probabilistic processes, used in well-chosen functional or probabilistic spaces representing the porous medium. While describing the phenomena is certainly interesting as it allows a better understanding of their fundamentals especially when the mathematical model is coupled with a physical model in the laboratory, the modeling of groundwater pollution is mainly aimed at providing a descriptive and predictive management tool, practical for field problems, whose type and resulting accuracy depends on the considered management objectives and, of course, the socio-economical constraints of the study. Three types of models are discussed in this chapter: black box models, grey box models, and complete structural models.
Solute transport in soils is commonly simulated with the advective–dispersive equation, or ADE. It has been reported that this model cannot take into account several important features of solute movement through soil. Recently, a new model has been suggested that results in a solute transport equation with fractional spatial derivatives, or FADE. We have assembled a database on published solute transport experiments in soil columns to test the new model. The FADE appears to be a useful generalization of the ADE. The order of the fractional differentiation reflects differences in physical conditions of the solute transport in soil.
Contents I Basic Text 1 1 Finite difference approximations 3 1.1 Truncation errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Deriving finite difference approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Polynomial interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Second order derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Higher order derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Boundary Value Problems 9 2.1 The heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 The steady-state problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 A simple finite difference method . . .
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