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

• Main Authors: Mutaz Mohammad, Alexandre Trounev, Mohammed Alshbool
• Format: Article
• Language: English
• Published: MDPI AG 2021-07-01
• Series: Axioms
• Subjects: time-fractional diffusion-wave equations; Euler wavelets; integral equations; numerical approximation
• Online Access: https://www.mdpi.com/2075-1680/10/3/165

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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>×</mo><mi>M</mi></mrow></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>31</mn></mrow></msup></mrow></semantics></math></inline-formula> for all tested examples. The new numerical scheme is exceptional in terms of its novelty, efficiency and accuracy in the field of numerical approximation.