A problem of coefficient determination in parabolic equations solved as moment problem


  • Maria Beatriz Pintarelli Universidad Nacional de La Plata






Generalized Moment Problem, Integral Equations, Inverse Problem, Parabolic Pdes, Truncated Expansion Method.


The problem is to find a(t) y w(x; t) such that wt = a(t) (wx)x+r(x; t) under the initial condition w(x; 0) =fi(x) and the boundary conditions w(0; t) = 0 ; wx(0; t) = wx(1; t)+alfa w(1; t) about a region D ={(x; t); 0 <x < 1; t >0}. In addition it must be fulfilled the integral of w (x, t) with respect to x is equal to E(t) where fi(x) , r(x; t) and E(t) are known functions and alfa is an arbitrary real number other than zero.
The objective is to solve the problem as an application of the inverse moment problem. We will find an approximated solution and bounds for the error of the estimated solution using the techniques on moments problem. In addition, the method is illustrated with several examples.


[1] Oussaeif Taki-Eddine and Bouzziani Abdelfatah, "An Inverse Coefficient Problem for a Parabolic Equation under Nonlocal Boundary and Integral Overdetermination Conditions",International Journal of Partial Differential Equations and Applications, Vol. 2, No 3, (2014), pp.38-43 .

[2] W. Liao, M. Dehghan, and A. Mohebbi, "Direct numerical method for an inverse problem of a parabolic partial differential equation", J. Comput. Appl. Math, 232, (2009), pp.351-360.

[3] J.R. Cannon, Y. Lin, and S. Wang, " Determination of a control parameter in a parabolic partial differential equation", J. Austral. Math. Soc. Ser. B., 33 ,(1991), pp.149-163.

[4] Nadiya Huzyk,"Inverse problem of determinig the coefficients in a degenerate parabolic equation", Electronic Journal of Differential Equations, Vol. 2014 , No. 172, (2014), pp.1–11.

[5] Baiyu Wang, Anping Liao and Wei Liu, "Simultaneous determination of unknown two parameters in parabolic equation", International Journalof Applied Mathematics and Computation,Vol. 4, (2012), pp.332-336.

[6] J. Biazar, T. Houlari, "Implementation of Sinc-Galerkin on Parabolic Inverse problem with unknown boundary condition",Int. J. Industrial Mathematics, Vol. 7, No. 4, (2015), pp.313–319 .

[7] Mehdi Dehghan and Mehdi Tatari, "Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions",Mathematical and Computer Modelling, 44, (2006), pp.1160–1168.

[8] Akheizer N I, The classical moment problem, Olivier and Boyd, Edinburgh, 1965

[9] Akheizer N I and Krein M G, Some questions in the theory of moment,Am. Math. Soc. Providence, RI, 1962.

[10] Shohat J A and Tamarkin J D, The problem of Moments, Math. Surveys,Am. Math. Soc., Providence, RI, 1943.

[11] G. Talenti, "Recovering a function from a finite number of moments", Inverse Problems, 3 ,(1987), pp.501-517.

[12] 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.

[13] M.B. Pintarelli and F. Vericat, "Stability theorem and inversion algorithm for a generalized moment problem", Far East Journal of Mathematical Sciences, 30,(2008), pp.253-274.

[14] M.B. Pintarelli and F. Vericat, "Bi-dimensional inverse moment problems",Far East Journal of Mathematical Sciences, 54, (2011), pp.1-23.

[15] M.B.Pintarelli, "Linear partial differential equations of first order as bi-dimensional inverse moment problem" ,Applied Mathematics, Vol. 6, No 6, (2015), pp.979-989.

[16] M.B.Pintarelli, "Parabolic partial differential equations as inverse moments problem",Applied Mathematics, Vol. 7, No 1, (2016),pp.77-99.

View Full Article: