Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time

Abstract We consider the inverse problem of finding the temperature distribution and the heat source whenever the temperatures at the initial time and the final time are given. The problem considered is one dimensional and the unknown heat source is supposed to be space dependent only. The existence and uniqueness results are proved.

[1]  A. F. Tsang.  The solution of a nonlocal boundary value problem , 2006 .

[2]  A. Castro,et al.  A TWO POINT BOUNDARY VALUE PROBLEM , 2010 .

[3]  N. Kosmatov.  Integral equations and initial value problems for nonlinear differential equations of fractional order , 2009 .

[4]  I. A. Kaliev,et al.  Problems of determining the temperature and density of heat sources from the initial and final temperatures , 2010 .

[5]  O. Agrawal,et al.  Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering , 2007 .

[6]  S. Wearne,et al.  Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions , 2009, Journal of mathematical biology.

[7]  J. A. Tenreiro Machado,et al.  New Trends in Nanotechnology and Fractional Calculus Applications , 2010 .

[8]  F. Mainardi,et al.  Fractals and fractional calculus in continuum mechanics , 1997 .

[9]  V. Lakshmikantham,et al.  Theory of Fractional Dynamic Systems , 2009 .

[10]  S. G. Pyatkov,et al.  Certain inverse problems for parabolic equations , 2008 .

[11]  V. Isakov.  Inverse parabolic problems with the final overdetermination , 1991 .

[12]  Salman A. Malik,et al.  The profile of blowing-up solutions to a nonlinear system of fractional differential equations , 2010 .

[13]  R. Gorenflo,et al.  Fractional Calculus: Integral and Differential Equations of Fractional Order , 2008, 0805.3823.

[14]  T. R. Prabhakar.  A SINGULAR INTEGRAL EQUATION WITH A GENERALIZED MITTAG LEFFLER FUNCTION IN THE KERNEL , 1971 .

[15]  Afet Golayoglu Fatullayev,et al.  Determination of an unknown source parameter in two-dimensional heat equation , 2004, Appl. Math. Comput..

[16]  W. Rundell.  The determination of a parabolic equation from initial and final data , 1987 .

[17]  Ravi P. Agarwal,et al.  On the concept of solution for fractional differential equations with uncertainty , 2010 .

[18]  V. Lakshmikantham,et al.  A Krasnoselskii–Krein-type uniqueness result for fractional differential equations , 2009 .

[19]  Hao Cheng,et al.  Determining surface heat flux in the steady state for the Cauchy problem for the Laplace equation , 2009, Appl. Math. Comput..

[20]  D. O’Regan,et al.  Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations , 2010 .

[21]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[22]  Ali Sever.  A stability estimate of an inverse problem in financial prospection , 2004, Appl. Math. Comput..

[23]  W. Olmstead,et al.  Source identification for the heat equation , 1988 .

[24]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .