A reliable mixed method for singular integro-differential equations of non-integer order

Abstract

It is our goal in this article to apply a method which is based on the assumption that combines two methods of conjugating collocation and multiple shooting method. The proposed method can be used to find the numerical solution of singular fractional integro-differential boundary value problems (SFIBVPs) Dϑ y(t) + η ∫0t (t−s)ς−1 y(s) ds = g(t), 1 < ϑ ≤ 2, 0 < ς< 1, η ∈ ℝ, where Dϑ denotes the Caputo derivative of order ϑ. Meanwhile, in a separate section the existence and uniqueness of this method is also discussed. Two examples are presented to illustrate the application and further understanding of the methods.

An Optimal computational method for Singular Perturbed second order boundary value problem

“…In 2013 Aronna et al, [2], proposed a shooting algorithm for a class of optimal control problems for which all control variables appear linearly. In 2015, Manjun Ma and Chunhua Ou [16] studied the existence, uniqueness, and asymptotic expansions to perturbed Poisson-Boltzmann equations on an unbounded domain in R 2 or R 3 . In 2016, Aditya Kaushik and Manju Sharma [13], presented higher order accurate numerical methods for convection-dominated diffusion problems.…”

“…In 2016, Aditya Kaushik and Manju Sharma [13], presented higher order accurate numerical methods for convection-dominated diffusion problems. In 2018, Baleanu et al, [3], combined two methods namely the collocation method and multiple shooting method to numerical solution to a fractional integrodifferential boundary value problem. In 2018, Matkowsky [17] applied singular perturbations in noisy dynamical systems.…”

An Uniform and Optimal Numerical method to solve Singular Perturbed second order initial value problem

“…In 2016, Aditya Kaushik and Manju Sharma [13], presented higher order accurate numerical methods for convection-dominated diffusion problems. In 2018, Baleanu et al, [3], combined two methods namely the collocation method and multiple shooting method to numerical solution to a fractional integrodifferential boundary value problem. In 2018, Matkowsky [19] applied singular perturbations in noisy dynamical systems.…”

Fractional-order Euler functions for solving fractional integro-differential equations with weakly singular kernel

“…Due to a wide range of applications of fractional calculus, the theoretical and numerical analysis of fractional differential equations has been receiving increasing attention by many researchers [16,17]. Consequently, many methods have been developed to give numerical solutions for fractional equations, including fractional differential transform method [18], Adomian's decomposition method [19,20], homotopy analysis method [21,22], wavelet method [23][24][25][26], mixed method [27], and other methods [28][29][30].…”