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 qualitative and quantitative analyses of numerical methods delay differential equations (DDEs) are now quite well understood, as reflected in the recent monograph by Bellen and Zennaro (2003). This is in remarkable contrast to the situation in the numerical analysis of more general Volterra functional equations in which delays occur in connection with memory terms described by Volterra inte...

Shell structures are widely used in various fields of technology and construction. Often, they play the role a bearing surface with assemblies, overlays, aggregates installed on them. At same time, solving problems, such attached elements considered as concentrated at points rigidly connected. Vibrations an orthotropic viscoelastic shallow shell masses geometrically nonlinear setting considered...

In this paper we give details of a new numerical method for the solution of a singular integral equation of Volterra type that has an infinite class of solutions. The split-interval method we have adopted utilises a simple robust numerical method over an initial time interval (which includes the singularity) combined with extrapolation. We describe the method and give details of its order of co...

An efficient numerical method to integrate the constitutive response of fractional order viscoelasticity is developed. The method can handle variable time steps. To overcome the problem of the growing amount of data that has to be stored and used in each time step we introduce sparse quadrature. We use an internal variable formulation of the viscoelastic equations where the internal variable is...

We prove local unique solvability of the wave equation for a large class of weakly singular, locally bounded space-time metrics in a suitable space of generalised functions.

In this article, we consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. In the first part, we deal with two-dimensional integralalgebraic equations. Next, we analyze Volterra integral equations of the first kind with a degenerate matrix-kernel on the diagonal. Finally, the third part of the work is devoted to the analysis of degenerate integro...

We prove local unique solvability of the wave equation for a large class of weakly singular, locally bounded space-time metrics in a suitable space of generalised functions.

We show that various (discrete) methods for the approximate solution of Volterra (and Abel) integral equations of the first kind correspond to some discrete version of the method of (recursive) collocation in the space of (continuous) piecewise polynomials. In a collocation method no distinction has to be made between equations with regular or weakly singular kernels; the regularity or nonregul...