Partial Differential Equations as Three-Dimensional Inverse Problem of Moments
Maria B. Pintarelli and Fernando Vericat
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DOI:10.17265/2159-5291/2014.10.001
1. Grupo de AplicacionesMatematicas y Estadisticas de la Facultad de Ingenieria (GAMEFI), UNLP, and Departamento de Matematica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata. 2. Grupo de AplicacionesMatematicas y Estadisticas de la Facultad de Ingenieria (GAMEFI), UNLP and Instituto de Fisica de Liquidos y SistemasBiologicos (IFLYSIB) CONICET La Plata-UNLP, Argentina.
We considerer partial differential equations of second order, for
example the Klein-Gordon equation, the Poisson equation, on a region 
. We will see that with a common procedure in all
cases, we can write the equation in partial derivatives as an Fredholm integral
equation of first kind and will solve this latter with the techniques of
inverse problem moments. We will find an approximated solution and bounds for
the error of the estimated solution using the techniques on problem of moments.
Partial differential equations (PDEs), Freholm integral equations, generalized moment problem
[2] Akheizer N I and Krein M G, Some questions in the theory of moment,Am. Math. Soc. Providence, RI, 1962.
[3] W.F. Ames, Numerical methods for partial differential equations, Academic Press, New York, 1992.
[4] D.D. Ang, R. Goreno, V.K. Le and D.D. Trong,Moment theory and some inverse problems in potential theory and heat conduction, Lectures Notes in Mathematics, Springer-Verlag, Berlin, 2002.
[5] H.W. Engl, C.W. Groetsch (Eds.) Inverse and ill-posed problems, Academic Press, Boston, 1987.
[6] P. Knabner and L. Angermann, Numerical methods for elliptic and parabolic partial differential equations (Springer-Verlag, New York,2003).
[7] L. Lapidus and G.F. Pinder, Numerical solution of partial differential equations in Science and Engineering, John Wiley and Sons, New York,1982.
[8] M.B. Pintarelli and F. Vericat, Stability theorem and inversion algorithm for a generalized moment problem, Far East Journal of Mathematical Sciences, 30 (2008), 253-274.
[9] M.B. Pintarelli and F. Vericat, Bi-dimensional inverse moment problems,Far East Journal of Mathematical Sciences, 54 (2011), 1-23.
[10] M.B. Pintarelli and F. Vericat, Klein-Gordon equation as a bi-dimensional moment problem, Far East Journal of Mathematical Sciences, 70 (2012),201-225.
[11] Shohat J A and Tamarkin J D, The problem of Moments, Math. Surveys,Am. Math. Soc., Providence, RI, 1943.
[12] G.D. Smith, Numerical solution of partial differential equations: Finite difference methods, Oxford University Press, New York, 1985.
[13] G. Talenti, Recovering a function from a finite number of moments, Inverse Problems 3 (1987), 501-517.
[14] J.W. Thomas, Numerical partial differential equations: Finite difference methods, Springer-Verlag, New Yorl, 1995.
[15] A. Tikhonov, V. Arsenine, Mthods de rsolution de problmes mal poss, MIR, Moscow, 1976.