<p style='text-indent:20px;'>We are devoted with singular integro-differential abstract Cauchyproblems. Required conditions on spaces and operators givenguaranteeing existence uniqueness of solutions. Applications from partial differential equations given to illustrate the problem.</p>

For one class of the singular integro-differential equations with Cauchy kernel on an interval, a Galerkin method is justified. The convergence is proved and the error estimation is given.

* Correspondence: [email protected] Department of Mathematical Sciences, Princess Nora Bint Abdulrahman University, Riyadh 84428, Saudi Arabia Full list of author information is available at the end of the article Abstract In this article, fractional integro-differential inequalities with singular coefficients have been considered. The bounds obtained for investigating the behavior of the ...

We discuss a possibility to construct high order methods on uniform or mildly graded grids for the numerical solution of linear Volterra integro-differential equations with weakly singular or other nonsmooth kernels. Using an integral equation reformulation of the initial value problem, we apply to it a smoothing transformation so that the exact solution of the resulting equation does not conta...

A Legendre multiwavelet based method is developed in this paper to solve second kind hypersingular integral equation by converting it into a Cauchy singular integro-differential equation. Multiscale representation of the singular and differential operators is obtained by employing Legendre multiwavelet basis. An estimate of the error of the approximate solution of the integral equation is obtai...

To solve the weakly-singular Volterra integro-differential equations, the combined method of the Laplace Transform Method and the Adomian Decomposition Method is used. As a result, series solutions of the equations are constructed. In order to explore the rapid decay of the equations, the pade approximation is used. The results present validity and great potential of the method as a powerful al...

We study the singular perturbation problem (E2) 2 2u′′ 2 (t) + u ′ 2(t) = Au2(t) + (K ∗Au2)(t) + f2(t), t ≥ 0, 2 > 0, for the integrodifferential equation (E) w′(t) = Aw(t) + (K ∗Aw)(t) + f(t), t ≥ 0, in a Banach space, when 2 → 0. Under the assumption that A is the generator of a strongly continuous cosine family and under some regularity conditions on the scalar-valued kernel K we show that p...