In this paper we prove existence theorems for integro – differential equations x(t) = f(t, x(t), ∫ t 0 k(t, s, x(s))∆s), x(0) = x0 t ∈ Ia = [0, a] ∩ T, a ∈ R+, where T denotes a time scale (nonempty closed subset of real numbers R), Ia is a time scale interval. Functions f, k are Carathéodory functions with values in a Banach space E and the integral is taken in the sense of Henstock-Kurzweil d...

The main intention of the present research study is focused on analysis a Caputo fractional integro-differential boundary problem (CFBVP) in which right-hand side supposed differential equation represented as sum two nonlinear terms. Under integro-derivative conditions, we extract an equivalent integral and then define new operators based it. With help three distinct fixed-point theorems attrib...

In this paper, a hybrid technique based on the combining of differential transform approach, Laplace transformation, and Pade approximant is successfully implemented for solving Volterra Integro (VID) equations subject to appropriate conditions. The proposed method known as modified (MDTM) enlarges domain convergence with less computational time. One interesting features algorithm ability produ...

In this article the problem of an interface fracture between two isotropic linear elastic materials is studied within a continuum modeling framework, which incorporates important nanoscale effects. The proposed model of bi-material crack ascribes curvature-dependent surface tension to both the fracture surfaces and the solid-solid interface. Further, it uses as boundary conditions the jump mome...

The main purpose of this paper is to approximate several non-local evolution equations by zero-sum repeated games in the spirit of the previous works of Kohn and the second author (2006 and 2009): general fully non-linear parabolic integro-differential equations on the one hand, and the integral curvature flow of an interface (Imbert, 2008) on the other hand. In order to do so, we start by cons...

This study aims to solve the most general form of linear mixed Volterra-Fredholm integro-differential equations multi-fractional order in Caputo sense (MV-FIFDEs), which is solved by using orthogonal generalized Bernstein’s polynomial expansion with collocation and moment discrete weighted residual method under suitable conditions; then, Clenshaw-Curtis formula applied approximate integral term...

the spline collocation method  is employed to solve a system of linear and nonlinear fredholm and volterra integro-differential equations. the solutions are collocated by cubic b-spline and the integrand is approximated by the newton-cotes formula. we obtain the unique solution for linear and nonlinear system $(nn+3n)times(nn+3n)$ of integro-differential equations. this approximation reduces th...

This article presents a numerical method for solving nonlinear two-dimensional fractional Volterra integral equation. We derive the Hat basis functions operational matrix of order integration and use it to solve integro-di?erential equations. The is described illustrated with examples. Also, we give error analysis.