Iterative refinement for a system of linear integro-differential equations of fractional type

Abstract

A novel technique based on iterative refinement is developed to approximate the analytical solution of a system of linear fractional integro-differential equations. While the study focuses mainly on Fredholm-type equations, adaptation to the Volterra-type is also presented. A comparison is made with the method of successive approximations on the basis of convergence speed and accuracy. Several numerical examples are given to demonstrate the efficacy of our algorithm. The authors also present formulations of some error bounds.

An Approximate Approach for Systems of Fractional IntegroDifferential Equations Based on Taylor Expansion

“…Using the present method, the first-order and the second-order approximate solutions at equidistant points are computed. The obtained results and the results given in [5,29] are listed in Tables 1 and 2. From Tables 1 and 2, we observe that the second-order approximate solution yields the exact solution as expected, since the exact solution is a polynomial function of degree 2.…”

“…Example 5.1. Consider the following system of fractional integro-differential equations (see [5,29]):…”

A New Efficient Method for Solving System of Weakly Singular Fractional Integro‐Differential Equations by Shifted Sixth‐Kind Chebyshev Polynomials

“…In the next year, for the first time, the hybrid functions composed of the Block-pulse functions and Bernoulli polynomials were applied for problems with fractional-order differential equations in [14]. Also, a novel technique based on iterative refinement was presented to analytically approximate a system of linear fractional integro-differential equations [15]. In 2018, Hesameddini and Shahbazi developed the concept of [14] and used it to solve the FDIE system in [16].…”

Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method

“…Yüzbasi [12] solved the system of linear Fredholm-Volterra integro-differential equations, which includes the derivatives of unknown functions in integral parts using the Bessel collocation method. In 2016, Deif and Grace [13] developed a new iterative method to approximate the solution of a system of linear fractional differential integral equations. In 2019, Xie and Yi [14] developed a numerical method for solving a nonlinear system of fractional-order Volterra-Fredholm integro-differential equations based on block-pulse functions.…”