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.

scholarly journals Sinc-Chebyshev Collocation Method for a Class of Fractional Diffusion-Wave Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Zhi Mao ◽  
Aiguo Xiao ◽  
Zuguo Yu ◽  
Long Shi
Keyword(s):  
Variable Coefficient ◽  
Fractional Derivatives ◽  
Wave Equations ◽  
Fractional Diffusion ◽  
Algebraic Equations ◽  
Chebyshev Collocation Method ◽  
Diffusion Wave ◽  
Linear Algebraic Equations ◽  
Collocation Technique ◽  
Sinc Functions

This paper is devoted to investigating the numerical solution for a class of fractional diffusion-wave equations with a variable coefficient where the fractional derivatives are described in the Caputo sense. The approach is based on the collocation technique where the shifted Chebyshev polynomials in time and the sinc functions in space are utilized, respectively. The problem is reduced to the solution of a system of linear algebraic equations. Through the numerical example, the procedure is tested and the efficiency of the proposed method is confirmed.

scholarly journals Approximate Solutions of Time Fractional Diffusion Wave Models

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 923 ◽  
Author(s):  
Abdul Ghafoor ◽  
Sirajul Haq ◽  
Manzoor Hussain ◽  
Poom Kumam ◽  
Muhammad Asif Jan
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Quadrature Rule ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Haar Wavelets ◽  
Test Problems ◽  
Algebraic Equations ◽  
Diffusion Wave ◽  
Wave Models

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

scholarly journals Finite Difference and Sinc-Collocation Approximations to a Class of Fractional Diffusion-Wave Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Zhi Mao ◽  
Aiguo Xiao ◽  
Zuguo Yu ◽  
Long Shi
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Fractional Diffusion ◽  
Stability And Convergence ◽  
Algebraic Equations ◽  
Time Step ◽  
Diffusion Wave ◽  
Linear Algebraic Equations ◽  
Sinc Functions ◽  
Exponential Rate Of Convergence

We propose an efficient numerical method for a class of fractional diffusion-wave equations with the Caputo fractional derivative of orderα. This approach is based on the finite difference in time and the global sinc collocation in space. By utilizing the collocation technique and some properties of the sinc functions, the problem is reduced to the solution of a system of linear algebraic equations at each time step. Stability and convergence of the proposed method are rigorously analyzed. The numerical solution is of3-αorder accuracy in time and exponential rate of convergence in space. Numerical experiments demonstrate the validity of the obtained method and support the obtained theoretical results.

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.

Subordination principle for fractional diffusion-wave equations of Sobolev type

2020 ◽  
Vol 23 (2) ◽  
pp. 427-449
Author(s):  
Rodrigo Ponce
Keyword(s):  
Banach Space ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Wave Equations ◽  
Mild Solutions ◽  
Fractional Diffusion ◽  
Sobolev Type ◽  
Diffusion Wave ◽  
Fractional Differential ◽  
Resolvent Families

AbstractIn this paper we study subordination principles for fractional differential equations of Sobolev type in Banach space. With the help of the theory of Sobolev type resolvent families (known also as propagation family) as well as these subordination principles, we obtain the existence of mild solutions for this kind of equations. We study simultaneously the case 0 < α < 1 and 1 < α < 2 for the Caputo and Riemann-Liouville fractional derivatives.

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