Two-dimensional Bernoulli wavelets with satisfier function in the Ritz–Galerkin method for the time fractional diffusion-wave equation with damping

Abstract

In this paper, the two-dimensional Bernoulli wavelets (BWs) with Ritz-Galerkin method are applied for the numerical solution of the time fractional diffusionwave equation. In this way, a satisfier function which satisfies all the initial and boundary conditions is derived. The two-dimensional BWs and Ritz-Galerkin method with satisfier function are used to transform the problem under consideration into a linear system of algebraic equations. The proposed scheme is applied for numerical solution of some examples. It has high accuracy in computation that leads to obtaining the exact solutions in some cases.

Numerical Solution of Fractional Diffusion Wave Equation and Fractional Klein–Gordon Equation via Two-Dimensional Genocchi Polynomials with a Ritz–Galerkin Method

“…The comparison of the numerical examples shows that our scheme is more accurate and less computational compared to other published methods. More specifically, we show that our new scheme is able to solve nonlinear cases, whereas existing similar methods as in [14,15] are only limited to linear cases.…”

“…Recently, Ref. [14,15] had successfully incorporated the Ritz-Galerkin method with Bernoulli polynomials and Bernoulli wavelets to solve a number of fractional calculus problems. Since the results were encouraging, we hope to apply this method to Genocchi polynomials.…”

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

“…In [15], the authors applied fractional order Legendre functions method depending on the choices of two parameters to solve the fractional diffusion-wave equations. Two-dimensional Bernoulli wavelets with satisfier function in the Ritz-Galerkin method were proposed for the time-fractional diffusion-wave equation in [16]. The author of [17] proposed a numerical method based on the Legendre wavelets with their operational matrix of fractional integral to solve the time-fractional diffusionwave equations.…”