We describe a new Maple package for treating boundary problems for linear ordinary differential equations, allowing two-/multipoint as well as Stieltjes boundary conditions. For expressing differential operators, boundary conditions, and Green’s operators, we employ the algebra of integro-differential operators. The operations implemented for regular boundary problems include computing Green’s ...

In the first part of this paper we study the regularity properties of solutions to initial or boundary-value problems of Fredholm integro-differential equationswithweakly singular or other nonsmooth kernels.We then use these results in the analysis of a piecewise polynomial collocation method for solving such problems numerically. Presented numerical examples display that theoretical results ar...

We consider a fractional order integro-differential equation with a weakly singular convolution kernel. The equation with homogeneuos Dirichlet boundary conditions is reformulated as an abstract Cauchy problem, and well-posedness is verified in the context of linear semigroup theory. Then we formulate a continuous Galerkin method for the problem, and we prove stability estimates. These are then...

We consider traveling front and pulse solutions to a system of integro-differential equations used to describe the activity of synaptically coupled neuronal networks in a single spatial dimension. Our first goal is to establish a series of direct links between the abstract nature of the equations and their interpretation in terms of experimental findings in the cortex and other brain regions. T...

whose kernel k(t, s) is either discontinuous or not smooth along the main diagonal, is presented. This scheme is of spectral accuracy when k(t, s) is infinitely differentiable away from the diagonal t = s, and is also applicable when k(t, s) is singular along the boundary, and at isolated points on the main diagonal. The corresponding composite rule is described. Application to integro-differen...

We present an hp-error analysis of the discontinuous Galerkin time-stepping method for Volterra integro-differential equations with weakly singular kernels. We derive new error bounds that are explicit in the time-steps, the degrees of the approximating polynomials, and the regularity properties of the exact solution. It is then shown that start-up singularities can be resolved at exponential r...

A spectral collocation method is proposed to solve Volterra or Fredholm integral equations with weakly singular kernels and corresponding integro-differential equations by virtue of some identities. For a class of functions that satisfy certain regularity conditions on a bounded domain, we obtain geometric or supergeometric convergence rate for both types of equations. Numerical results confirm...