A NONLINEAR HEAT-TRANSFER PROBLEM - SOLUTION OF NONLINEAR, WEAKLY SINGULAR VOLTERRA INTEGRAL-EQUATIONS OF THE 2ND KIND

This paper revisits a classical nonlinear heat transfer problem using modern computational methods based on an equivalent integral formulati on. Heat conduction in a semi-infinite medium having a nonlinear bound ary condition is considered. This paper investigates the use of two nu merical methods for solving the resulting nonlinear, singular Volterra integral equation of the second kind. The Volterra integral equation considered in this paper has a weakly singular kernel of the form k(t, t0) = (t - t0)-x where 0 < x < 1. Integral equations having this type of kernel appear in many facets of engineering and physics. The prese nt paper investigates the global and local use of singularity subtract ion and product integration. These methods effectively overcome the nu merical difficulties associated with weakly singular kernels of this t ype. Next, a numerical procedure based on trapezoidal integration and a Newton-Raphson method is shown to produce accurate results rapidly.