Frank de Hoog

Difference Methods for Boundary Value Problems with a Singularity of the First Kind

The application of certain difference schemes (box, trapezoidal, Euler and backward Euler) to the numerical solution of boundary value problems for nonlinear first order systems of ordinary differential equations with a singularity of the first kind is examined. The solution of the linear eigenvalue problem is also considered.

Collocation Methods for Singular Boundary Value Problems

The application of collocation methods based on piecewise polynomials to the numerical solution of boundary value problems for systems of ordinary differential equations with a singularity of the first kind is examined. The schemes are shown to be stable and convergent. Enhanced accuracy at the nodes for suitably chosen collocation points (superconvergence) is established for a class of important problems.

Asymptotic Expansions for Product Integration

A generalized Euler-Maclaurin sum formula is established for product integration based on piecewise Lagrangian interpolation. The integrands considered may have algebraic or logarithmic singularities. The results are used to obtain accurate con- vergence rates of numerical methods for Fredholm and Volterra integral equations with singular kernels.

The Numerical Solution of Boundary Value Problems with an Essential Singularity

The numerical solution by two difference schemes (box, trapezoidal) of boundary value problems for first order systems of ordinary differential equations with a singularity of the second kind is investigated. The techniques are applied to two concrete problems arising from differential equations on infinite intervals.

High Order Methods for a Class of Volterra Integral Equations with Weakly Singular Kernels

The solution of the Volterra integral equation, \[ ( * )\qquad x(t) = g_1 (t) + \sqrt {t}g_2 (t) + \int _0^t \frac {K(t,s,x(s))} {\sqrt {t - s} } ds, \quad 0 \leqq t \leqq T,\] where

The application of Runge-Kutta schemes to singular initial value problems

g_1 (t)

On the solution of volterra integral equations of the first kind

,

High Order Methods for Volterra Integral Equations of the First Kind

g_2 (t)

The Application of Linear Multistep Methods to Singular Initial Value Problems.

and

On the Solution of a Volterra Integral Equation with a Weakly Singular Kernel

K(t,s,x)