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Inverse heat conduction problem consists of finding an initial temperature distribution from the knowledge of a distribution of the temperature at the present time. Here, we assume that the associated boundary conditions are known. The heat conduction problem backward in time is a typical example of ill-posed problems in the sense that the solution exists only for regular functions of some kind describing the present temperature distribution and also the solution is unstable for the present temperature distribution function. Conventional numerical methods often suffer from instability of the problem itself when high accuracy is intended in the approximation. Our aim is to create a meshless method which is applicable to the ill-posed inverse heat conduction problem. We construct a high order finite difference method in which quadrature points do not need to have a lattice structure. In order to develop our new method we show a tool in using exponential functions in Taylor's expansion. From numerical experiments we confirmed that our method is effective for solving two-dimensional inverse heat conduction problem numerically subject to mixed boundary conditions.

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Abstract. Real numbers are usually represented in the computer as a finite number of digits hexa-decimal floating point numbers. Accordingly the numerical analysis is often suffered from rounding errors. The rounding errors particularly deteriorate the precision of numerical solution in inverse and ill-posed problems. We attempt to use a multi-precision arithmetic for reducing the rounding error evil. The use of the multi-precision arithmetic system is by the courtesy of Dr. Fujiwara of Kyoto University. In this paper we try to show effectiveness of the multi-precision arithmetic by taking two typical examples; the Cauchy problem of the Laplace equation in two dimensions and the shape identification problem by inverse scattering in three dimensions. It is concluded from a few numerical examples that the multi-precision arithmetic works well on the resolution of those numerical solutions, as it is combined with the high order finite difference method for the Cauchy problem and with the eigenfunction expansion method for the inverse scattering problem.

Applications of a high order finite difference method with free quadrature points to Cauchy problems of two-dimensional Laplace equation Kentaro IijimaWe construct a high order finite difference method in which the quadrature points can be set at arbitrary locations.It is known that the Cauchy problem of the Laplace equation is unstable with respect to L2-norm.We apply our method to Cauchy problems of the Laplace equation.The finite difference approximation in this research is formulated as follows: the derivative of a function is replaced with a linear combination of values of the function for each quadrature points which are located arbitrarily.We propose an algorithm in order to determine weights in this linear combination.In numerical experiments we obtain a highly precise numerical solution.These results imply effectiveness on solving the Cauchy problem of the Laplace equation by our method.