Mehrdad Lakestani

University of Tabriz · Department of Applied Mathematics

An effective method based upon cardinal Hermite interpolant multiscaling functions is proposed for the solution of the one-dimensional parabolic partial differential equation with given initial condition and known boundary conditions and subject to overspecification at a point in the spatial domain. The properties of multiscaling functions are firs...

In this paper, a numerical method for solving nonlinear quadratic optimal control problems with inequality constraints is presented. The method is based upon cardinal Hermite interpolant multiscaling function approximation. The properties of these multiscaling functions are presented first. These properties are then utilized to reduce the solution...

In this paper, we propose a novel image deblurring approach that utilizes a new mask based on the Grünwald-Letnikov fractional derivative. We employ the first five terms of the Grünwald-Letnikov fractional derivative to construct three masks corresponding to the horizontal, vertical, and diagonal directions. Using these matrices, we generate eight...

In this article, we solve fractional differential equations in the Caputo fractional derivative sense using cubic Hermite spline functions. We first construct the operational matrix to the fractional derivative of the cubic Hermite spline functions. Then using this matrix and some properties of these functions, we convert a fractional differential...

It is a critical issue to observe a sample in the laboratory during the experiment stages, as well as the possibility of recording such images as documents in research works. In addition, the chance of observing the changes in the thickness of the samples relative to the height provides the possibility of obtaining more realistic results compared t...

Vascular-related diseases pose significant public health challenges and are a leading cause of mortality and disability. Understanding the complex structure of the vascular system and its processes is crucial for addressing these issues. Recent advancements in medical imaging technology have enabled the generation of high-resolution 3D images of va...

In this paper, the new optical wave solutions to the truncated M-fractional (2 + 1)-dimensional non-linear Schrödinger’s complex hyperbolic model by utilizing the generalized Kudryashov method are obtained. The obtained solutions are in the form of trigonometric, hyperbolic and mixed form. These solutions have many applications in nonlinear optics,...

Images are frequently corrupted by various sorts of mixed or unrecognized noise, including mixed Poisson-Gaussian noise, rather than just a single kind of noise. In this work, we propose a time-space fractional differential equation to remove mixed Poisson-Gaussian noise. Combining fixed- and variable-order fractional derivatives allows us to maint...

The equation of the shallow water wave in oceanography and atmospheric science is extended to (3+1) dimensions, which is a known equation. To achieve this, an illustrative example of the VC generalized shallow water wave equation is provided to demonstrate the feasibility and reliability of the used procedure in this study. It is shown that Hirota...

In the current research, we develop a collocation method based on the biorthogonal Hermite cubic spline functions to solve a class of fractional optimal control problems using Caputo–Fabrizio derivative operator. We design dual bases for Hermite cubic spline functions for the first time in this work. So, we present two direct and efficient algorith...

Simultaneous image denoising and deblurring is a challenging issue because noise and edges are both high-frequency signals, and eliminating noise while enhancing edges counteracts each other. In this paper, we deal with this difficulty by designing a novel coupled time-fractional diffusion along with a fractional shock filter and developing a power...

An efficient algorithm based on the wavelet collocation method is introduced in order to solve nonlinear fractional optimal control problems (FOCPs) with inequality constraints. By using the interpolation properties of Hermite cubic spline functions, we construct an operational matrix of the Caputo fractional derivative for the first time. Using th...

In this paper, the time-varying system control is resolved with the aid of forward Riccati formulation and hybrid functions. The applied Riccati equations need neither advanced knowledge of the system dynamics nor the assumption of the future information of system matrices. Herein, the state space equation of LTV system is rewritten in the Hybrid f...

Here, we present a numerical scheme to solve optimal control problems with time-varying delay system. This method is based on Lucas wavelets and Galerkin method. Operational matrices of integration and delay for Lucas wavelets are proposed. Then, Galerkin method is used to solve the mentioned problems. Numerical results are included to demonstrate...

In this paper, we present a numerical method for pricing European options. This approximation method is based on the characteristic function and family of B-Spline function (including: Linear, Quadratic and Cubic B-Spline).

The main idea of this study is to introduce a novel set of wavelet functions named Touchard wavelets for solving time-fractional Black-Scholes equations. The numerical method is discussed based on the pseudo-operational matrix of Riemann-Liouville fractional integration and the least square approximation method. The methodology of deriving the pseu...

This paper aims to compute solitary wave solutions and soliton wave solutions based on the ansatz methods to the perturbed nonlinear Schrödinger equation (NLSE) arising in nano-fibers. The improved tan(θ/2)-expansion method and the rational extended sinh–Gordon equation expansion method are used for the first time to obtain the new optical solitons...

This work deals with the pseudospectral method to solve the Sturm–Liouville eigenvalue problems with Caputo fractional derivative using Chebyshev cardinal functions. The method is based on reducing the problem to a weakly singular Volterra integro-differential equation. Then, using the matrices obtained from the representation of the fractional inte...

The object of this article is to present our most spectral methods that have been developed over the years for solving a class of fractional optimal control problems. Recently, operational matrices of fractional derivatives and integrals for various bases were adapted for solving these types of problems. By using operational matrices in conjunction...

In this article, we develop a new set of functions called fractional-order Alpert multiwavelet functions to obtain the numerical solution of fractional pantograph differential equations (FPDEs). The fractional derivative of Caputo type is considered. Here we construct the Riemann–Liouville fractional operational matrix of integration (Riemann–Liouv...

This paper presents a new fast iterative shrinkage-thresholding algorithm, termed AFISTA. The essential idea is to improve the convergence rate of FISTA using a new continuation strategy leading to a less number of iterations compared to FISTA. The convergence theorem of the AFISTA is proposed. In order to further accelerate the AFISTA method, it i...

In this work, we design, analyze, and test the multiwavelets Galerkin method to solve the two‐dimensional Burgers equation. Using Crank–Nicolson scheme, time is discretized and a PDE is obtained for each time step. We use the multiwavelets Galerkin method for solving these PDEs. Multiwavelets Galerkin method reduces these PDEs to sparse systems of...

The current study utilizes the generalized \(\tan (K(\rho )/2)\)-expansion method, the generalized \(\tanh \)-\(\coth \) method and He’s semi-inverse variational method in constructing various soliton and other solutions to the (2+1)-dimensional coupled variant Boussinesq equations which describes the elevation of water wave surface for slowly modu...

In this article, a class of fractional optimal control problems (FOCPs) are solved using a direct method. We present a new operational matrix of the fractional derivative in the sense of Caputo based on the B-spline functions. Then we reduce the solution of fractional optimal control problem to a nonlinear programming (NLP) one, where some existing...

In this paper a novel image enhancement model based on shock filter for image deblurring is proposed in three cases. For the weight of shock filter, the fractional order derivative of initial blurry image is used. This fractional order weight can be adjusted adaptively according to the gradient of blurred image. Compared with the traditional intege...

In this paper, we have presented an accurate and impressive spectral algorithm for solving fractional telegraph equation with Riesz-space derivative and Dirichlet boundary conditions. The proposed method is based on Jacobi tau spectral procedure together with the Jacobi operational matrices of Riemann–Liouville fractional integral and left- and rig...

The Hirota bilinear method is prepared for searching the diverse soliton solutions for the fractional generalized Calogero‐Bogoyavlenskii‐Schiff‐Bogoyavlensky‐Konopelchenko (CBS‐BK) equation. Also, the Hirota bilinear method is used to finding the lump and interaction with two stripe soliton solutions. Interaction among lumps, periodic waves, and m...

An efficient algorithm based on wavelet Galerkin method is proposed for solving the time-varying delay systems. According to the useful properties of Alpert’s multiwavelets such as interpolation, orthonormality, flexible vanishing moments and sparsity, a time-varying system is reduced to the sparse linear system of algebraic equations. The results...

This article proposes a new numerical approach for solving fractional optimal control problems including state and control inequality constraints using new biorthogonal multiwavelets. The properties of biorthogonal multiwavelets are first given. The Riemann‐Liouville fractional integral operator for biorthogonal multiwavelets is utilized to reduce...

The present article deals with M-lump solution and N-soliton solution of the (2+1)-dimensional variable-coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada equation by virtue of Hirota bilinear operator method. The obtained solutions for solving the current equation represent some localized waves including soliton, breather, lump and their interactions i...

In this article, a pair of wavelets for Hermite cubic spline bases are presented. These wavelets are in C1 and supported on [−1,1]. These spline wavelets are then adapted to the interval [0,1] and we prove that they form a Riesz wavelet in L2([0,1]). The wavelet bases are used to solve the linear optimal control problems. The operational matrices o...

In this paper, we use the Hirota bilinear method. With the help of symbolic calculation and applying this method, we solve the \((2+1)\)-dimensional bidirectional Sawada–Kotera (bSK) equation to obtain some new lump-kink, lump-solitons, periodic kink-wave, periodic soliton and periodic wave solutions.

Medical Ultrasound is a diagnostic imaging technique based on the application of ultrasound in various branches of medical sciences. It can facilitate the observation of structures of internal body, such as tendons, muscles, vessels and internal organs such as male and female reproductive system. However, these images usually degrade by a special k...

Under investigation in this work is a (\(2+1\))-dimensional the space–time fractional coupled nonlinear Schrödinger equations, which describes the amplitudes of circularly-polarized waves in a nonlinear optical fiber. With the aid of conformable fractional derivative and the fractional wave transformation, we derive the analytical soliton solutions...

Numerical method based on the Crank–Nicolson scheme and the Tau method is proposed for solving nonlinear Klein–Gordon equation. Nonlinear Klein–Gordon equation is reduced by Crank–Nicolson scheme to the system of ordinary differential equations then Tau method is used to solve this system by using interpolating scaling functions and operational mat...

In this paper, we have proposed a hybrid denoising algorithm based on combining of the shearlet transform method, as a pre-processing step, with the Yaroslavsky’s filter, as a kernel smoother, on a wide class of images with various properties such as thin features and textures. In the other word, proposed algorithm is a two-step algorithm, where in...

This paper presents a numerical technique for solving the nonlinear Benjamin-Bona-Mahony equation. As a first step, we discretize the time by approximating the first-order time derivative via θ- weighted scheme. A system of ordinary differential equations is obtained and we solve this system using the wavelet Galerkin method by use of the Alpert mu...

A numerical technique based on the Hermite interpolant multiscaling functions is presented for the solution of Convection-diffusion equations. The operational matrices of derivative, integration and product are presented for multiscaling functions and are utilized to reduce the solution of linear Convection-diffusion equation to the solution of alg...

This paper retrieves soliton solutions to an equation in nonlinear electrical transmission lines using the semi-inverse variational principle method (SIVPM), the exp(−�(�))-expansion method (EEM) and the improved tan(�/2)-expansion method (ITEM), with the aid of the symbolic computation package Maple. As a result, the SIVPM, EEM and ITEM methods ar...

In this paper, a TV-based shearlet shrinkage is proposed for discontinuity-preserving denoising using a combination of shearlet with a total variation scheme. For TV denoising numerical procedure, we use two approaches. In the first approach, we apply semi-implicit method for total variation. To solve Euler–Lagrange equation associated with total v...

The aim of present paper is to obtain the analytical solution of modified Fornberg–Whitham equation by using \(\tan (\phi /2)\)-expansion and \(\tanh (\phi /2)\)-expansion methods. These methods are used to construct solitary and soliton solutions of nonlinear evolution equation. These methods are straightforward and concise, and their applications...

Under investigation in this paper is a nonlinear conformable time-fractional Boussinesq equations as an important class of fractional differential equations in mathematical physics. The extended trial equation method, the exp (- Ω (η)) -expansion method and the tan (/ 2) -expansion method are used in examining the analytical solution of the nonline...

This paper proposes a new numerical approach for finding the solution of linear time-delay control systems with a quadratic performance index using new hybrid functions. This method is based on a hybrid of block-pulse functions and biorthogonal multiwavelets that consist of cubic Hermite splines on the primal side. The excellent properties of the h...

A numerical technique based on the finite difference and collocation methods is presented for the solution of Korteweg–de Vries (KdV) equation. The integration relations between any two families of B-spline functions are presented and are utilized to reduce the solution of KdV equation to the solution of linear algebraic equations. Numerical simula...

Background: Medical ultrasonic images are usually degraded by a special kind of noise called ’speckle’. The speckle noises usually have an effect more on edges and fine details of an ultrasound images which lead to reduction in their contrast resolution consequently create difficulties in the diagnosis of illnesses. Methods: In this paper, to reduc...

The aim of this paper is to introduce a novel study of obtaining an analytical solutions to the modified dispersive water-wave system. An analytical technique based on the improved \(\tan (\phi /2)\)-expansion method (ITEM) is extended to handle such a system. Description of the method is given and the obtained results reveal that the ITEM is a new...

To numerically solve the Burgers’ equation, in this paper we propose a general method for constructing wavelet bases on the interval derived from symmetric biorthogonal multiwavelets on the real line. In particular, we obtain wavelet bases with simple structures on the interval from the Hermite cubic splines. In comparison with all other known cons...

In this paper, we find exact solutions of some nonlinear evolution equations by using generalized tanh–coth method. Three nonlinear models of physical significance, i.e. the Cahn–Hilliard equation, the Allen– Cahn equation and the steady-state equation with a cubic nonlinearity are considered and their exact solutions are obtained. From the general...

This paper is concerned with the traveling wave solutions and analytical treatment of the infiltration equation. Based on the new technique, namely, the tan (phi/2)-expansion method, the solution procedure of the nonlinear infiltration equation is investigated. We obtained the exact solutions for the aforementioned nonlinear infiltration equation....

Theaimofthepresentpaper istopresentananalyticalmethodforthetimefractionalbiologicalpopulationmodel, time fractional Burgers, time fractional Cahn–Hilliard, space–time fractional Whitham–Broer–Kaup, space–time fractional Fokasequationsbyusingthegeneralizedtanh–cothmethod.Thefractionalderivativeisdescribedinthesenseofthemodified Riemann–Liouville deri...

In this paper, by introducing new approach, the improved tan(ϕ(ξ)/2)-expansion method (ITEM) is further extended into Gerdjikov–Ivanov (GI) model. As a result, the hyperbolic function solution, the trigonometric function solution, the exponential solution and the rational solution with free parameters are obtained. When the parameters are taken as...

In this paper, the improved tan (Phi(xi)/2)-expansion method is proposed to seek more general exact solutions of the Kundu-Eckhaus equation. Being concise and straightforward, this method is applied to the nonlinear Kundu-Eckhaus equation. The exact particular solutions containing five types hyperbolic function solution (exact soliton wave solution...

A improvement of the expansion methods namely the improved \(\tan \left( \varPhi (\xi )/2\right)\)-expansion method for solving the Tzitzéica type nonlinear evolution equations is proposed. In this work, the dispersive optical solitons that are governed by the Tzitzéica type nonlinear evolution equations. As a result, many new and more general exac...

An improvement of the expansion methods, namely, the improved tan ⁡ Φ ξ / 2 -expansion method, for solving nonlinear second-order partial differential equation, is proposed. The implementation of the new approach is demonstrated by solving the generalized Fitzhugh-Nagumo equation with time-dependent coefficients. As a result, many new and mor...

In this paper, a new numerical method for solving the optimal control problems with payoff term or fixed state endpiont by quadratic performance index is presented. The method is based on Bezier polynomial. The properties of Bezier polynomials in any intervel as [a; b] are presented. The operational matrices of integration and derivative are utiliz...

In this paper, the improved tan(Φ(ξ)/2)-expansion method is proposed to seek more general exact solutions of the new partial differential equation. Being concise and straightforward, this method is applied to the Biswas-Milovic equation (BME). The exact particular solutions containing four types hyperbolic function solution, trigonometric function...

A numerical technique is presented for the solution of system of Fredholm integro-differential equations. The method consists of expanding the required approximate solution as the elements of Alpert multiwavelet functions (see Alpert B. et al. J. Comput. Phys. 2002, vol. 182, pp. 149–190). Using the operational matrix of integration and wavelet tra...

In this article, we establish the exact solutions for nonlinear Drinfeld-Sokolov (DS) and generalized Drinfeld-Sokolov (gDS) equations. Generalized (G G)-expansion method is proposed to seek exact solutions of nonlinear evolution equations. This method is used to construct solitary and soliton solutions of nonlinear evolution equations. Also, for f...

In this paper, we applied the improved \(\tan \left( \Phi (\xi )/2\right) \) -expansion scheme for the (2+1)-dimensional Zoomeron, the Duffing and the symmetric regularized long wave equations and exact particular solutions have been found. The exact particular solutions containing four types hyperbolic function solution, trigonometric function sol...

It is commonly accepted that fractional differential equations play an important role in the explanation of many physical phenomena. For this reason we need a reliable and efficient technique for the solution of fractional differential equations. This paper deals with the numerical solu- tion of a class of fractional differential equation. The frac...

This paper proposes new effective methods for analyze of time-varying delay systems using new hybrid functions. The excellent properties of the hybrid functions which consist of block- pulse functions and biorthogonal multiscaling functions are presented. we utilize operational matrices of product, delay and integration to reduce the solution of de...

This paper presents the one-soliton solution to the Biswas-Milovic equation with Kerr law nonlinearity. This paper studies the perturbed Biswas-Milovic equation by the aid of the Exp-function method. By means of the Exp-function method, we report further exact travelling wave solutions, in a concise form, to the Biswas-Milovic equation which admits...

In this paper, a new numerical method for solving the nonlinear constrained optimal control with quadratic performance index is presented. The method is based upon B-spline functions. The properties of B-spline functions are presented. The operational matrix of derivative (Dφ) and integration matrix (P) are introduced. These matrices are utilized t...

An application of the (G ′/G)-expansion method to search for exact solutions of nonlinear partial differential equations is analysed. This method is used for Burgers, Fisher, Huxley equations and combined forms of these equations. The (G ′/G)-expansion method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equ...

We establish exact solutions including periodic wave and solitary wave solutions for the integrable sixth-order Drinfeld-Sokolov-Satsuma-Hirota system. We employ this system by using a generalized (íµí°º í®í° /íµí°º)-expansion and the generalized tanh-coth methods. These methods are developed for searching exact travelling wave solutions of nonline...

The current paper proposes a technique for the numerical solution of Burgers equations. The method is based on finite difference formula combined with the Galerkin method, which uses the interpolating scaling functions. Several test problems are given, and the numerical results are reported to show the accuracy and efficiency of the new algorithm....

A numerical technique is presented for the solution of Riccati differential equation. This method uses the Legendre scaling functions. The method consists of expanding the required approximate solution as the elements of Legendre scaling functions. Using the operational matrix of integral, we reduce the problem to a set of algebraic equations. Some...

Four numerical techniques based on the linear B‐spline functions are presented for the numerical solution of the Lane–Emden equation. Some properties of the B‐spline functions are presented and are utilized to reduce the solution of the Lane–Emden equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the...

Three numerical techniques based on cubic Hermite spline functions are presented for the solution of Lane–Emden equation. Some properties of Hermite splines are presented and are utilized to reduce the solution of Lane–Emden equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicabi...

An application of the generalized tanh–coth method to search for exact solutions of nonlinear partial differential equations is analyzed. This method is used for variants of the KdV–Burger and the K(n, n)–Burger equations. The generalized tanh–coth method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equatio...

This paper is concerned with the construction of biorthogonal multiwavelet basis in the unit interval to form a biorthogonal flatlet multiwavelet system. Next a method to calculate integer and fractional derivatives of the dual flatlet multiwavelets by multiplying some matrices is suggested. The system is then used to solve a fractional convection–...

The main aim of this paper is to introduce the operational matrices of integral and fractional integral using the flatlet oblique multiwavelets. The operational matrices of integral and fractional integrals for flatlet scaling functions and wavelets are presented and are utilized to reduce the solution of the Abel integral equations of the first an...

Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. For that reason we need a reliable and efficient technique for the solution of fractional differential equations. Here we construct the operational matrix of fractional derivative of order α in the...

Two numerical techniques based on the interpolating scaling functions are presented for the solution of the generalized Burgers–Huxley equation. Some properties of the interpolating scaling functions are presented and are utilized to reduce the solution of the generalized Burgers–Huxley equation to the solution of a system of algebraic equations. A...

A numerical technique based on the finite difference and collocation methods is presented for the solution of generalized Kuramoto–Sivashinsky (GKS) equation. The derivative matrices between any two families of B-spline functions are presented and are utilized to reduce the solution of GKS equation to the solution of linear algebraic equations. Num...

A numerical technique is presented for the solution of Emden-Fowler equation. This method uses the Legendre scaling functions. The method consists of expanding the required approximate solution as the elements of Legendre scaling functions. Using the operational matrix of integration, the problem will be reduced to a set of algebraic equations. Som...

A numerical technique is presented for the solution of second-order one-dimensional linear hyperbolic equation. This method uses the trigonometric wavelets. The method consists of expanding the required approximate solution as the elements of trigonometric wavelets. Using the operational matrix of derivative, we reduce the problem to a set of algeb...

The main aim of this paper is to apply the trigonometric wavelets for the computation of eigenval-ues and eigenfunctions of Sturm-Liouville problem. The operational matrices of derivative for Trigonometric scaling functions and wavelets are presented and utilized to reduce the solution of the problem to find the eigenvalues of a matrix.

The main aim of this paper is to apply the trigonometric wavelets for the solution of the Fredholm integro-differential equations of nth-order. The operational matrices of derivative for trigonometric scaling functions and wavelets are presented and are utilized to reduce the solution of the Fredholm integro-differential equations to the solution o...

An effective method based upon Legendre multiwavelets is proposed for the solution of Fredholm weakly singular integro-differential equations. The properties of Legendre multiwavelets are first given and their operational matrices of integral are constructed. These wavelets are utilized to reduce the solution of the given integro-differential equat...

This paper is concerned with the construction of a biorthogonal multiwavelet basis in the unit interval to form a biorthogonal flatlet multiwavelet system. The system is then used to solve nonlinear ordinary differential equations. The biorthogonality and high vanishing moment properties of this system result in efficient and accurate solutions. Fi...

A numerical technique for solving a second-order nonlinear Neumann problem is presented. The authors’ approach is based on trigonometric scaling function on [0;π] which is constructed for Hermite interpolation. Two test problems are presented and error plots show the efficiency of the proposed technique for the studied problem.

A numerical technique is presented for the solution of Falkner-Skan equation. The nonlinear ordinary differential equation is solved using Chebyshev car-dinal functions. The method have been derived by first truncating the semi-infinite physical domain of the problem to a finite domain and expanding the required ap-proximate solution as the element...

A numerical technique is presented for the solution of a parabolic partial differential equation with a time-dependent coefficient subject to an extra measurement. The method is derived by expanding the required approximate solution as the elements of Chebyshev cardinal functions. Using the operational matrix of derivative, the problem can be reduc...

A numerical technique is presented for the solution of second order one dimensional linear hyperbolic equation. This method uses interpolating scaling functions. The method consists of expanding the required approximate solution as the elements of interpolating scaling functions. Using the operational matrix of derivatives, we reduce the problem to...