scholarly journals A numerical scheme to solve variable order diffusion-wave equations

2019 ◽  
Vol 23 (Suppl. 6) ◽  
pp. 2063-2071 ◽  
Author(s):  
Ganji Moallem ◽  
Hossein Jafari ◽  
Abdullahi Adem
Keyword(s):  
Collocation Method ◽  
Algebraic System ◽  
Numerical Scheme ◽  
Wave Equations ◽  
Bernstein Polynomials ◽  
Operational Matrices ◽  
Variable Order ◽  
Bernstein Basis ◽  
Diffusion Wave ◽  
Variable Order Derivative

In this work, we consider variable order diffusion-wave equations. We choose variable order derivative in the Caputo sense. First, we approximate the unknown functions and its derivatives using Bernstein basis. Then, we obtain operational matrices based on Bernstein polynomials. Finally, with the help of these operational matrices and collocation method, we can convert variable order diffusion-wave equations to an algebraic system. Few examples are given to demonstrate the accuracy and the competence of the presented technique.

scholarly journals Using Fractional Bernoulli Wavelets for Solving Fractional Diffusion Wave Equations with Initial and Boundary Conditions

2021 ◽  
Vol 5 (4) ◽  
pp. 212
Author(s):  
Monireh Nosrati Sahlan ◽  
Hojjat Afshari ◽  
Jehad Alzabut ◽  
Ghada Alobaidi
Keyword(s):  
Fractional Order ◽  
Fractional Derivatives ◽  
Wave Equations ◽  
Bernoulli Polynomials ◽  
Fractional Diffusion ◽  
Computational Time ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Diffusion Wave ◽  
Klein Gordon

In this paper, fractional-order Bernoulli wavelets based on the Bernoulli polynomials are constructed and applied to evaluate the numerical solution of the general form of Caputo fractional order diffusion wave equations. The operational matrices of ordinary and fractional derivatives for Bernoulli wavelets are set via fractional Riemann–Liouville integral operator. Then, these wavelets and their operational matrices are utilized to reduce the nonlinear fractional problem to a set of algebraic equations. For solving the obtained system of equations, Galerkin and collocation spectral methods are employed. To demonstrate the validity and applicability of the presented method, we offer five significant examples, including generalized Cattaneo diffusion wave and Klein–Gordon equations. The implementation of algorithms exposes high accuracy of the presented numerical method. The advantage of having compact support and orthogonality of these family of wavelets trigger having sparse operational matrices, which reduces the computational time and CPU requirements.

A new operational matrix method based on the Bernstein polynomials for solving the backward inverse heat conduction problems

2014 ◽  
Vol 24 (3) ◽  
pp. 669-678 ◽  
Author(s):  
Davood Rostamy ◽  
Kobra Karimi
Keyword(s):  
Heat Conduction ◽  
Collocation Method ◽  
Bernstein Polynomials ◽  
High Order ◽  
System Of Equations ◽  
Bernstein Basis ◽  
Content Type ◽  
Order Matrix ◽  
Ill Posed ◽  
Matrix Derivative

Purpose – The purpose of this paper is to introduce a novel approach based on the high-order matrix derivative of the Bernstein basis and collocation method and its employment to solve an interesting and ill-posed model in the heat conduction problems, homogeneous backward heat conduction problem (BHCP). Design/methodology/approach – By using the properties of the Bernstein polynomials the problems are reduced to an ill-conditioned linear system of equations. To overcome the unstability of the standard methods for solving the system of equations an efficient technique based on the Tikhonov regularization technique with GCV function method is used for solving the ill-condition system. Findings – The presented numerical results through table and figures demonstrate the validity and applicability and accuracy of the technique. Originality/value – A novel method based on the high-order matrix derivative of the Bernstein basis and collocation method is developed and well-used to obtain the numerical solutions of an interesting and ill-posed model in heat conduction problems, homogeneous BHCP with high accuracy.

scholarly journals Numerical Solution of Time-Fractional Diffusion-Wave Equations via Chebyshev Wavelets Collocation Method

2017 ◽  
Vol 2017 ◽  
pp. 1-17 ◽  
Author(s):  
Fengying Zhou ◽  
Xiaoyong Xu
Keyword(s):  
Collocation Method ◽  
Wave Equations ◽  
Integral Formula ◽  
Fractional Diffusion ◽  
Two Dimensions ◽  
Algebraic Equations ◽  
Diffusion Wave ◽  
Chebyshev Wavelets ◽  
Linear Algebraic Equations ◽  
Chebyshev Wavelet

The second-kind Chebyshev wavelets collocation method is applied for solving a class of time-fractional diffusion-wave equation. Fractional integral formula of a single Chebyshev wavelet in the Riemann-Liouville sense is derived by means of shifted Chebyshev polynomials of the second kind. Moreover, convergence and accuracy estimation of the second-kind Chebyshev wavelets expansion of two dimensions are given. During the process of establishing the expression of the solution, all the initial and boundary conditions are taken into account automatically, which is very convenient for solving the problem under consideration. Based on the collocation technique, the second-kind Chebyshev wavelets are used to reduce the problem to the solution of a system of linear algebraic equations. Several examples are provided to confirm the reliability and effectiveness of the proposed method.

scholarly journals NUMERICAL SOLUTION OF VARIABLE-ORDER TIME FRACTIONAL WEAKLY SINGULAR PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS WITH ERROR ESTIMATION

2020 ◽  
Vol 25 (4) ◽  
pp. 680-701
Author(s):  
Haniye Dehestani ◽  
Yadollah Ordokhani ◽  
Mohsen Razzaghi
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Collocation Method ◽  
Error Estimation ◽  
Operational Matrices ◽  
Variable Order ◽  
Algebraic Equations ◽  
Laguerre Functions ◽  
Weakly Singular Kernel ◽  
Weakly Singular

In this paper, we apply Legendre-Laguerre functions (LLFs) and collocation method to obtain the approximate solution of variable-order time-fractional partial integro-differential equations (VO-TF-PIDEs) with the weakly singular kernel. For this purpose, we derive the pseudo-operational matrices with the use of the transformation matrix. The collocation method and pseudo-operational matrices transfer the problem to a system of algebraic equations. Also, the error analysis of the proposed method is given. We consider several examples to illustrate the proposed method is accurate.

scholarly journals An Operational Matrix of Fractional Differentiation of the Second Kind of Chebyshev Polynomial for Solving Multiterm Variable Order Fractional Differential Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Jianping Liu ◽  
Xia Li ◽  
Limeng Wu
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Chebyshev Polynomial ◽  
Algebraic System ◽  
Operational Matrix ◽  
Main Idea ◽  
Operational Matrices ◽  
Variable Order ◽  
Fractional Differential ◽  
Variable Order Fractional Derivative

The multiterm fractional differential equation has a wide application in engineering problems. Therefore, we propose a method to solve multiterm variable order fractional differential equation based on the second kind of Chebyshev Polynomial. The main idea of this method is that we derive a kind of operational matrix of variable order fractional derivative for the second kind of Chebyshev Polynomial. With the operational matrices, the equation is transformed into the products of several dependent matrices, which can also be viewed as an algebraic system by making use of the collocation points. By solving the algebraic system, the numerical solution of original equation is acquired. Numerical examples show that only a small number of the second kinds of Chebyshev Polynomials are needed to obtain a satisfactory result, which demonstrates the validity of this method.

scholarly journals Chebyshev wavelets operational matrices for solving nonlinear variable-order fractional integral equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Y. Yang ◽  
M. H. Heydari ◽  
Z. Avazzadeh ◽  
A. Atangana
Keyword(s):  
Integral Equations ◽  
Algebraic System ◽  
Fractional Integral ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Variable Order ◽  
Wavelet Method ◽  
Chebyshev Wavelets ◽  
The Galerkin Method

Abstract In this study, a wavelet method is developed to solve a system of nonlinear variable-order (V-O) fractional integral equations using the Chebyshev wavelets (CWs) and the Galerkin method. For this purpose, we derive a V-O fractional integration operational matrix (OM) for CWs and use it in our method. In the established scheme, we approximate the unknown functions by CWs with unknown coefficients and reduce the problem to an algebraic system. In this way, we simplify the computation of nonlinear terms by obtaining some new results for CWs. Finally, we demonstrate the applicability of the presented algorithm by solving a few numerical examples.

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