A Novel Numerical Method for Solving Fractional Diffusion-Wave and Nonlinear Fredholm and Volterra Integral Equations with Zero Absolute Error

Mohammad, Mutaz; Trounev, Alexandre; Alshbool, Mohammed

doi:10.3390/axioms10030165

Open AccessArticle

A Novel Numerical Method for Solving Fractional Diffusion-Wave and Nonlinear Fredholm and Volterra Integral Equations with Zero Absolute Error

by

Mutaz Mohammad

^1,*

,

Alexandre Trounev

² and

Mohammed Alshbool

^1,3,*

¹

Department of Mathematics & Statistics, Zayed University, Abu Dhabi 144543, United Arab Emirates

²

Department of Computer Technology and Systems, Kuban State Agrarian University, 350044 Krasnodar, Russia

³

Department of Applied Mathematics, Abu Dhabi University, Abu Dhabi P.O. Box 59911, United Arab Emirates

^*

Authors to whom correspondence should be addressed.

Axioms 2021, 10(3), 165; https://doi.org/10.3390/axioms10030165

Submission received: 19 April 2021 / Revised: 6 June 2021 / Accepted: 8 June 2021 / Published: 28 July 2021

(This article belongs to the Special Issue Fractional Calculus - Theory and Applications)

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Abstract

In this work, a new numerical method for the fractional diffusion-wave equation and nonlinear Fredholm and Volterra integro-differential equations is proposed. The method is based on Euler wavelet approximation and matrix inversion of an

M \times M

collocation points. The proposed equations are presented based on Caputo fractional derivative where we reduce the resulting system to a system of algebraic equations by implementing the Gaussian quadrature discretization. The reduced system is generated via the truncated Euler wavelet expansion. Several examples with known exact solutions have been solved with zero absolute error. This method is also applied to the Fredholm and Volterra nonlinear integral equations and achieves the desired absolute error of

0 \times 10^{- 31}

for all tested examples. The new numerical scheme is exceptional in terms of its novelty, efficiency and accuracy in the field of numerical approximation.

Keywords: time-fractional diffusion-wave equations; Euler wavelets; integral equations; numerical approximation

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MDPI and ACS Style

Mohammad, M.; Trounev, A.; Alshbool, M. A Novel Numerical Method for Solving Fractional Diffusion-Wave and Nonlinear Fredholm and Volterra Integral Equations with Zero Absolute Error. Axioms 2021, 10, 165. https://doi.org/10.3390/axioms10030165

AMA Style

Mohammad M, Trounev A, Alshbool M. A Novel Numerical Method for Solving Fractional Diffusion-Wave and Nonlinear Fredholm and Volterra Integral Equations with Zero Absolute Error. Axioms. 2021; 10(3):165. https://doi.org/10.3390/axioms10030165

Chicago/Turabian Style

Mohammad, Mutaz, Alexandre Trounev, and Mohammed Alshbool. 2021. "A Novel Numerical Method for Solving Fractional Diffusion-Wave and Nonlinear Fredholm and Volterra Integral Equations with Zero Absolute Error" Axioms 10, no. 3: 165. https://doi.org/10.3390/axioms10030165

APA Style

Mohammad, M., Trounev, A., & Alshbool, M. (2021). A Novel Numerical Method for Solving Fractional Diffusion-Wave and Nonlinear Fredholm and Volterra Integral Equations with Zero Absolute Error. Axioms, 10(3), 165. https://doi.org/10.3390/axioms10030165

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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A Novel Numerical Method for Solving Fractional Diffusion-Wave and Nonlinear Fredholm and Volterra Integral Equations with Zero Absolute Error

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