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

Monireh Nosrati Sahlan,Ghada Alobaidi,Jehad Alzabut,Hojjat Afshari

doi:10.3390/fractalfract5040212

Abstract

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.

Highlights

Fractional partial differential equations have appeared in variety of applications [1,2,3,4,5,6]
We aim to solve the general form of Time Fractional Diffusion Wave Equation (TFDWE), which is defined as:
By specifying the unknown matrix U, we find the approximate solution of TFDWE as (40)

Summary

Introduction

As an important branch of these equations, have attracted considerable attention amongst researchers due to their extensive applications in science and engineering. These equations have been widely utilized in some significant physical materials, such as percolation clusters, amorphous, colloid, glassy and porous materials through fractals, dielectrics, semiconductors, polymers and biological systems [7]. We aim to solve the general form of Time Fractional Diffusion Wave Equation (TFDWE), which is defined as: ∂ξ u( x, t) ∂u( x, t) ∂2 u( x, t) +θ + μu( x, t) = θ.

Objectives

Results

Conclusion