In this work, we introduce the notion of a (weak) proportional Caputo fractional derivative order α∈(0,1) for continuous (locally integrable) function u:[0,∞)→E, where E is complex Banach space. our definition, do not require that u(·) continuously differentiable, which enables us to consider wellposedness corresponding relaxation problems in much better theoretical way. More precisely, systema...

The numerical stability of the polynomial spline collocation method for general Volterra integro-differential equation is being considered. The convergence and stability of the newmethod are given and the efficiency of the newmethod is illustrated by examples. We also proved the conjecture suggested by Danciu in 1997 on the stability of the polynomial spline collocation method for the higher-or...

In this paper, we prove the existence of mild solutions for the semilinear fractional order functional of Volterra-Fredholm type differential equations with impulses in a Banach space. The results are obtained by using the theory of fractional calculus, the analytic semigroup theory of linear operators and the fixed point techniques. ifx

Integral and integro-differential equations are one of the most useful mathematical tools in both pure and applied mathematics. In this article, we present a variational iteration method for solving Fredholm integro-differential equations. This study provides an analytical approximation to determine the behavior of the solution. To show the efficiency of the present method for our proble...

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of FredholmVolterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L∞ norm and weighted L2-norm. The numerical examp...

Ahmed A. Hamoud, Nedal M. Mohammed, Homan Emadifar;, Foroud Parvaneh, Faraidum K. Hamasalh, Soubhagya Kumar Sahoo, and Masoumeh Khademi. International Journal of Fuzzy Logic Intelligent Systems 2022;22:391-400. https://doi.org/10.5391/IJFIS.2022.22.4.391

In this research, the finite difference method is used to solve initial value problem of linear first order Volterra-Fredholm integro-differential equations with singularity. By using implicit rules and composite numerical quadrature rules, scheme established on a Shishkin mesh. The stability convergence proposed are analyzed two examples solved display advantages presented technique.