A Numerical Study of Nonlinear Fractional Order Partial Integro-Differential Equation with a Weakly Singular Kernel

Abstract

Fractional differential equations can present the physical pathways with the storage and inherited properties due to the memory factor of fractional order. The purpose of this work is to interpret the collocation approach for tackling the fractional partial integro-differential equation (FPIDE) by employing the extended cubic B-spline (ECBS). To determine the time approximation, we utilize the Caputo approach. The stability and convergence analysis have also been analyzed. The efficiency and reliability of the suggested technique are demonstrated by two numerical applications, which support the theoretical results and the effectiveness of the implemented algorithm.

An efficient spline technique for solving time-fractional integro-differential equations

“…To show the reliability of ExCuBS estimation for solving non-linear FPIDEs, two examples are demonstrated in this section. The computed outcomes are compared with [30] . The maximum error and relative error are defined by: and where and are exact and approximate solutions respectively.…”

“…To discretize the time and space derivatives, the Caputo fractional derivative (CFD) approximation and new ExCuBS approximation for second derivative are applied, respectively. The non-linear FPIDE [30] with initial condition (IC): and the boundary conditions (BCs): where are fractional parameters, are any constants and , are given functions. The represents the CFD with Euler's Gamma function Γ, which is defined as: Non-integer order derivatives and operators in terms of integral are more appropriate than ordinary derivatives and integration at various phases of real systems because they give a more detailed explanation of the structural and genetic characteristics of a number of dynamical and physical processes.…”

“…Akram et al [30] solved the FPIDE by employing the ExCBS with the help of Caputo approach. They analyzed the stability and convergence to support the solutions.…”

Advanced shifted first-kind Chebyshev collocation approach for solving the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel

“…The exact solution of this problem is u(x, t) = t 5 2 + 1 x 2 (1 − x) 2 . Table 4 show the comparison of the absolute errors between our method at N = 12 with the method developed in Akram et al (2021) at β = 0.15 and different values of α. Also, Fig.…”

An Efficient Approach to Approximate the Solutions of Fractional Partial Integro-Differential Equations

“…Partial integro-differential equations of fractional order are a very extensive field which has a numerous application in different fields of science and engineering such as mathematical biology, population dynamics, chemical processes, hydraulic system and communication networks [4]. Partial integro-differential equations of fractional order have been attacked by many researchers such as [2,3,6,9,11].…”