Nonlinear Variable-order Time Research Articles

Orthonormal shifted discrete Legendre polynomials for solving a coupled system of nonlinear variable-order time fractional reaction-advection-diffusion equations

In this paper, we generalize a coupled system of nonlinear reaction-advection-diffusion equations to a variable-order fractional one by using the Caputo-Fabrizio fractional derivative, which is a non-singular fractional derivative operator. In order to establish an appropriate method for this system, we introduce a new formulation of the discrete Legendre polynomials namely the orthonormal shifted discrete Legendre polynomials. The operational matrices of classical and fractional derivatives of these basis functions are extracted. The devised method uses these polynomials and their operational matrices together with the collocation technique to transform the system under consideration into a system of algebraic equations which is uncomplicated for solving. Two numerical examples are analyzed to examine the accuracy of the method.

A meshless approach for solving nonlinear variable-order time fractional 2D Ginzburg-Landau equation

This paper introduces the nonlinear variable-order (VO) time fractional 2D Ginzburg-Landau equation by replacing the conventional derivative with the Atangana–Baleanu–Caputo fractional derivative. An efficient moving least squares (MLS) meshfree approximation method is considered to devise an algorithm for solving this enhanced equation. To be precise, first, the finite difference scheme is used to evaluate the fractional differentiation. Then, a recurrence formula is derived by applying the θ-weighted technique. Next, the real and imaginary components of the solution are expanded in terms of meshless functions. Last, these expansions which include unknown coefficients are input in the original equation. Therefore, the equation is converted into a system of linear algebraic equations which is uncomplicated for solving by mathematical software. To verify the validity of the devised method and demonstrate its precision, several problems are put to the test.

A wavelet method for nonlinear variable-order time fractional 2D Schrödinger equation

<p style='text-indent:20px;'>In this study, an efficient semi-discrete method based on the two-dimensional Legendre wavelets (2D LWs) is developed to provide approximate solutions for nonlinear variable-order time fractional two-dimensional (2D) Schrödinger equation. First, the variable-order time fractional derivative involved in the considered problem is approximated via the finite difference technique. Then, by help of the finite difference scheme and the theta-weighted method, a recursive algorithm is derived for the problem under examination. After that, the real functions available in the real and imaginary parts of the unknown solution of the problem are expanded via the 2D LWs. Finally, by applying the operational matrices of derivative, the solution of the problem is transformed to the solution of a linear system of algebraic equations in each time step which can simply be solved. In the proposed method, acceptable approximate solutions are achieved by employing only a small number of the basis functions. To illustrate the applicability, validity and accuracy of the wavelet method, some numerical test examples are solved using the suggested method. The achieved numerical results reveal that the method established based on the 2D LWs is very easy to implement, appropriate and accurate in solving the proposed model.

An operational matrix method for nonlinear variable-order time fractional reaction–diffusion equation involving Mittag-Leffler kernel

This paper is concerned with an operational matrix (OM) scheme based on the shifted Chebyshev cardinal functions (SCCFs) of the second kind for numerical solution of the variable-order time fractional nonlinear reaction–diffusion equation. The fractional derivative operator is defined in the sense of Atangana–Baleanu–Caputo. Through the way, a new OM of variable-order fractional derivative is derived for the mentioned cardinal functions. More precisely, the unknown solution is expanded by the SCCFs with undetermined coefficients. Then the expansion substituted in the equation and the generated OM is utilized to extract some algebraic equations. The precision of the established approach is examined through various types of test examples. Numerical simulations confirm the suggested approach is highly accurate to provide satisfactory results.

A cardinal method to solve coupled nonlinear variable-order time fractional sine-Gordon equations

In this study, a computational approach based on the shifted second-kind Chebyshev cardinal functions (CCFs) is proposed for obtaining an approximate solution of coupled variable-order time-fractional sine-Gordon equations where the variable-order fractional operators are defined in the Caputo sense. The main ideas of this approach are to expand the unknown functions in tems of the shifted second-kind CCFs and apply the collocation method such that it reduces the problem into a system of algebraic equations. To algorithmize the method, the operational matrix of variable-order fractional derivative for the shifted second-kind CCFs is derived. Meanwhile, an effective technique for simplification of nonlinear terms is offered which exploits the cardinal property of the shifted second-kind CCFs. Several numerical examples are examined to verify the practical efficiency of the proposed method. The method is privileged with the exponential rate of convergence and provides continuous solutions with respect to time and space. Moreover, it can be adapted for other types of variable-order fractional problems straightforwardly.

A meshless method to solve nonlinear variable-order time fractional 2D reaction–diffusion equation involving Mittag-Leffler kernel

In this paper, an efficient and accurate meshless method based on the moving least squares (MLS) shape functions is developed to solve the generalized variable-order (V-O) time fractional nonlinear 2D reaction–diffusion equation. The V-O fractional derivative is considered in the Atangana–Baleanu–Caputo sense with Mittag-Leffler non-singular kernel. The numerical method is based on the following steps: First, the V-O fractional derivative is approximated by finite differences, and the $$\theta $$ -weighted method has been used to derive a recursive algorithm. Then, the solution of the problem is expanded by the MLS shape functions. Finally, by a substitution of this series expansion and corresponding its partial derivatives into the main equation, the problem is reduced to a linear system of algebraic equations to be solved at each time step. Several numerical examples are also given to illustrate the applicability, validity and accuracy of the presented method. The achieved numerical results reveal that the proposed method is highly accurate in solving the introduced V-O fractional model.

A cardinal approach for nonlinear variable-order time fractional Schrödinger equation defined by Atangana–Baleanu–Caputo derivative

Abstract This paper is concerned with an operational matrix method based on the shifted Legendre cardinal functions for solving the nonlinear variable-order time fractional Schrodinger equation. The variable-order fractional derivative operator is defined in the Atangana–Baleanu–Caputo sense. Through the way, a new operational matrix of variable-order fractional derivative is derived for the shifted Legendre cardinal functions and used in the established method. More precisely, the unknown solution is separated into the real and imaginary parts, and then these parts are expanded in terms of the shifted Legendre cardinal functions with undetermined coefficients. These expansions are substituted into the main equation and the generated operational matrix is utilized to extract a system of nonlinear algebraic equations. Thereafter, the yielded system is solved to obtain an approximate solution for the problem. The precision of the established approach is examined through various types of test examples. Numerical simulations confirm that the suggested approach is high accurate in providing satisfactory results.

Legendre wavelets for the numerical solution of nonlinear variable-order time fractional 2D reaction-diffusion equation involving Mittag–Leffler non-singular kernel

Abstract In this study, an efficient semi-discrete method based on the two-dimensional Legendre wavelets (2D LWs) is developed to provide approximate solutions of nonlinear variable-order (V-O) time fractional 2D reaction-diffusion equations. The V-O time fractional derivative is defined in the Caputo sense with Mittag-Leffler non-singular kernel of order α ( x , t ) ∈ ( 0 , 1 ) (known as the Atangana–Baleanu–Caputo fractional derivative). First, the V-O fractional derivative is approximated via the finite difference scheme and the theta-weighted method is utilized to derive a recursive algorithm. Then, the unknown solution of the intended problem is expanded via the 2D LWs. Finally, by applying the differentiation operational matrices in each time step, the solution of the problem is reduced to solution of a linear system of algebraic equations. In the proposed method, acceptable approximate solutions are achieved by employing only a few number of the basis functions. To illustrate the applicability, validity and accuracy of the presented wavelet method, some numerical test examples are solved. The achieved numerical results reveal that the established method is highly accurate in solving the introduced new V-O fractional model.

Meshfree moving least squares method for nonlinear variable-order time fractional 2D telegraph equation involving Mittag–Leffler non-singular kernel

Abstract This paper investigates a novel version for the nonlinear 2D telegraph equation involving variable-order (V-O) time fractional derivatives in the Atangana–Baleanu–Caputo sense with Mittag–Leffler non-singular kernel. A meshfree method based on the moving least squares (MLS) shape functions is proposed for the numerical solution of this class of problems. More precisely, the V-O fractional derivatives in this model are approximated by the finite difference scheme at first. Then, the θ-weighted method is utilized to derive a recursive algorithm. Next, the solution of the problem is expanded in terms of the MLS shape functions with undetermined coefficients. Eventually, by substituting this expansion and its partial derivatives into the original equation, solution of the problem in each time step is reduced to the solution of a linear system of algebraic equations. Several numerical examples are investigated to show the applicability, validity and accuracy of the presented method. The achieved numerical results reveal that the established method is high accurate in solving such V-O fractional models.

A wavelet method to solve nonlinear variable-order time fractional 2D Klein–Gordon equation

In this study, we introduce the nonlinear variable-order time fractional two-dimensional (2D) Klein–Gordon equation by using the concept of variable-order fractional derivatives. The variable-order fractional derivative operator is defined in the Caputo type. Due to the useful properties of wavelets, we propose an efficient semi-discrete method based on the 2D Legendre wavelets (LWs) to numerically solve this equation. In fact, according to the proposed method, the variable-order time fractional derivative should be discretized in the first stage, and then the solution of the problem expanded in terms of the 2D LWs. However, the main objective of this study is to illustrate that the 2D LWs can be a useful tool for solving the nonlinear variable-order time fractional 2D Klein–Gordon equation. Stability analysis of the presented method is investigated theoretically and numerically. Moreover, the applicability and accuracy of the method are investigated by solving some numerical examples. Numerical results confirm the spectral accuracy of the established method.

A meshfree approach for solving 2D variable-order fractional nonlinear diffusion-wave equation

This paper is concerned with the moving least squares (MLS) meshless approach for the numerical solution of two-dimensional (2D) variable-order time fractional nonlinear diffusion-wave equation (V-OTFND-WE) on arbitrary domains. The variable-order fractional derivative of this equation is expressed in the Caputo type. The proposed method is based on the MLS approximation in conjunction with the finite difference technique. For spatial derivatives, the MLS approximation and for temporal derivative, the finite difference technique are utilized to discretize the equation. The aim of the present paper is to show that the meshfree method based on the MLS approximation and collocation scheme is suitable for solving the variable-order fractional partial differential equations (PDEs). The proposed method is validated, using some different nonlinear numerical examples with complex geometries. The results obtained demonstrate the accuracy and easy implementation of the method for nonlinear variable-order time fractional PDEs.

An optimization method based on the generalized polynomials for nonlinear variable-order time fractional diffusion-wave equation

In this paper, an optimization method based on a new class of basis functions, namely generalized polynomials (GPs), is proposed for nonlinear variable-order time fractional diffusion-wave equation. Variable-order time fractional derivative is expressed in the Caputo sense. In the proposed method, solution of the problem under consideration is expanded in terms of GPs with unknown free coefficients and control parameters. In this way, some new operational matrices of the ordinary and fractional derivatives are derived for these basis functions. The residual function and its 2-norm are employed for converting the problem under study to an optimization one and then choosing the unknown free coefficients and control parameters optimally. As a useful result, the necessary conditions of optimality are derived as a system of nonlinear algebraic equations with unknown free coefficients and control parameters. The validity and effectiveness of the method are demonstrated by solving some numerical examples. The results demonstrate that the proposed method is a powerful algorithm with good accuracy for solving such kind of problems.