Contribution to the Numerical Solution of Nonlinear Heat Transfer Equation Subject to a Boundary Integral Specification

The work presents one dimensional heat transfer in a media with temperature-dependent thermal conductivity. We solve numerically the one-dimensional unsteady heat conduction equation subject to initial condition and integral boundary conditions. We first discertize the equation in time, using the implicite Euler time method. A sequence of nonlinear two-point boundary value problems is obtained. This discretisation reduce the problem to the second spatial derivative of temperature wich is a nonlinear function of the temperature and the temperature gradient. For the implementation of Newton method, we derive expressions for the partial derivative of the nonlinear function. Using higher order parallel splitting finite difference method and the Simpson's composite quadrature method, we solve the the resulting nonlinear systems by the multivariate Newton method. The MATLAB 2013a provides the approximate solution.