The Numerical Solution of a Non-Characteristic Cauchy Probelm for a Parabolic Equation
We study the problem of solving numerically a parabolic partial differential equation in one space dimension, where boundary values are given on one boundary only. For the analysis and numerical solution the problem is reformulated as a Volterra integral equation of the first kind. The problem is ill-posed and we study the nature of the ill-posedness by computing approximately the singular values and functions of the Volterra operator. The integral equation is discretized giving a system of linear equations. Two methods for solving the discrete problem are briefly discussed. The first is the regularization method, and the other is derived using certain properties of the Volterra equation. It is shown that in the case of time-independent coefficients an approximate solution can be obtained in O(n2) operations, where n is the dimension of the linear system. Numerical examples are given.
KeywordsCauchy Problem Singular Value Decomposition Regularization Method Singular Vector Volterra Integral Equation
Unable to display preview. Download preview PDF.
- L.E. Andersson, A note on the regularity of the solution of a parabolic equation with discontinuous coefficient, Department of Mathematics, Linköping University, 1983, to appear.Google Scholar
- H.S. Carslaw, J.C. Jaeger, Conduction of heat in solids, Oxford University Press (Clarendon), London, 1948.Google Scholar
- J. Douglas, Jr., Approximate continuation of harmonic and parabolic functions, Numerical Solution of Partial Differential Equations, Ed. J.H. Bramble, Academic Press, New York, 1966.Google Scholar
- L. Eidén, An efficient algorithm for the regularization of ill-conditioned least squares problems with triangular Toeplitz matrix, SIAM J. Sci. Stat. Comp., to appear.Google Scholar
- L. Garifo, V.E. Schrock, E. Spedicato, On the solution of the inverse heat conduction problem by finite differences, Energia Nucleare 22(1975), 448–460.Google Scholar
- V.M. Isakov, On the uniqueness of the solution of the Cauchy problem, Soviet Math. Dokl. 22(1980), 539–642.Google Scholar
- Lavrentiev, Romanov, Vasiliev, Multidimensional inverse problems for differential equations, Lecture Notes in Mathematics 167, Springer, Berlin 1970.Google Scholar
- M.M. Lavrentiev, V.G. Romanov, S.P. Shishatskiy, Ill-posed problems in mathematical physics and analysis, Izdatelstvo Nauka, Moscow, 1980.Google Scholar
- P. Linz, A survey of methods for the solution of Vol terra integral equations of the first kind, in The application and numerical solution of integral equations, ed. R.S. Anderssen et al., Sijthoff & Noordhoff, Alphen an den Rijn, 1980.Google Scholar
- T.I. Seidman, Ill-posed problems arising in boundary control and observation for diffusion equations, in Inverse and improperly posed problems in differential equations, ed. Anger, Akademie-Verlag, Berlin, 1979.Google Scholar
- T. Seidman, Recovery of a diffused signal, Seminaires IRIA, 1979, (Analyse et controle de systemes), INRIA, Rocquencourt, 1980, pp. 71–82.Google Scholar
- F. Smithies, Integral equations, Cambridge University Press, Cambridge, 1965.Google Scholar