meter system described by linear partial differential equations, its two point boundary value pro blem is reduced to the problem of solving the Fredholm type integral equation of the second kind. In this paper, metal piece heating process with the furnace of first-order lag is taken up as an example, and two different algorithms are proposed in order to solve the above problem. First of all, numerical solutions are derived applying Step-wise Steepest Descent Method in which the optimal

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... is derived analytically by minimizing the performance value of each iteration, and the convergence of the performance value is compared with that of Continous Steepest Descent Method. Next, an application of the eigen function expansion method to the above problem is described. As the kernel function is separable in this heating process problem, its kernel takes approximately the from of Pincherle-Goursat type and the eigen value is obtained easily to give an analytical solution, and its numerical result is compared with that of Steepest Descent Method.