Superconvergence in the Maximum Norm of a Class of Piecewise Polynomial Collocation Methods for Solving Linear Weakly Singular Volterra Integro-Differential Equations

A note on collocation methods for Volterra integro-differential equations with weakly singular kernels

will be employed in the analysis of the principle properties of the collocation approximations; the extension to nonlinear equations is straightforward (cf. [1, p. 225]). High-order numerical methods for VIDEs with weakly singular kernels may be found in [1,2,6,7,8]. In this note we shall consider collocation methods for VIDE (1.1), based on Brunner's approach [1]. The following method and nota...

A Method to Estimate the Solution of a Weakly Singular Non-linear Integro-differential Equations by Applying the Homotopy Methods

Application of Tau Approach for Solving Integro-Differential Equations with a Weakly Singular Kernel

In this work, the convection-diffusion integro-differential equation with a weakly singular kernel is discussed. The  Legendre spectral tau method is introduced for finding the unknown function. The proposed method is based on expanding the approximate solution as the elements of a shifted Legendre polynomials. We reduce the problem to a set of algebraic equations by using operational matrices....

The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations

Second-kind Volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration. Using an adaptation of the analysis originally developed for nonlinear weakly singular Fredholm integral equations, we present a complete discussion of the optimal (global and local) order of convergence of piecewise polynomial ...

fractional differential transform method for solving a class of weakly singular volterra integral equations

a method for solving a class of weakly singular volterra integral equations is given by using the fractional differential transform method. the approximate solution of these equa­tions is calculated in the form of a finite series with easily computable terms. while in some examples this series solution increased up to the exact closed solution, in some other examples, we can see the accuracy an...