A method of fundamental solutions for radially symmetric and axisymmetric backward heat conduction problems

We propose and investigate an application of the method of fundamental solutions (MFS) to the radially symmetric and axisymmetric backward heat conduction problem (BHCP) in a solid or hollow cylinder. In the BHCP, the initial temperature is to be determined from the temperature measurements at a later time. This is an inverse and ill-posed problem, and we employ and generalize the MFS regularization approach [B.T. Johansson and D. Lesnic, A method of fundamental solutions for transient heat conduction, Eng. Anal. Boundary Elements 32 (2008), pp. 697–703] for the time-dependent heat equation to obtain a stable and accurate numerical approximation with small computational cost.

[1]  Daniel Lesnic,et al.  A method of fundamental solutions for the one-dimensional inverse Stefan problem , 2011 .

[2]  V. D. Kupradze,et al.  A method for the approximate solution of limiting problems in mathematical physics , 1964 .

[3]  Ean Hin Ooi,et al.  A dual-reciprocity boundary element approach for solving axisymmetric heat equation subject to specification of energy. , 2008 .

[4]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[5]  B. Tomas Johansson,et al.  A method of fundamental solutions for theaxisymmetric backward heat equation , 2011 .

[6]  Salim A. Messaoudi,et al.  Initial inverse problem in heat equation with Bessel operator , 2002 .

[7]  A. S. Muleshkov,et al.  The Method Of Fundamental Solutions For Time-Dependent Problems , 1970 .

[8]  J. A. Kołodziej,et al.  Transient Heat Conduction by Boundary Collocation Methods and FEM — A Comparison Study , 1992 .

[9]  M. Li,et al.  A DISCREPANCY PRINCIPLE FOR THE SOURCE POINTS LOCATION IN USING THE MFS FOR SOLVING THE BHCP , 2009 .

[10]  Somchart Chantasiriwan.  Methods of fundamental solutions for time‐dependent heat conduction problems , 2006 .

[11]  B. Tomas Johansson,et al.  A method of fundamental solutions for transient heat conduction in layered materials , 2009 .

[12]  D. L. Young,et al.  Time-dependent fundamental solutions for homogeneous diffusion problems , 2004 .

[13]  F. B..  Introduction to Bessel Functions , 1939, Nature.

[14]  B. Tomas Johansson,et al.  A comparative study on applying the method of fundamental solutions to the backward heat conduction problem , 2011, Math. Comput. Model..

[15]  A. Haji-sheikh,et al.  Heat Conduction Using Green's Function , 1992 .

[16]  Graeme Fairweather,et al.  The method of fundamental solutions for elliptic boundary value problems , 1998, Adv. Comput. Math..

[17]  Paulo R. M. Lyra,et al.  An axisymmetric finite volume formulation for the solution of heat conduction problems using unstructured meshes , 2005 .

[18]  M. Biot.  General Theory of Three‐Dimensional Consolidation , 1941 .

[19]  B. Tomas Johansson,et al.  A method of fundamental solutions for transient heat conduction , 2008 .

[20]  B. Tomas Johansson,et al.  Numerical approximation of the one-dimensional inverse Cauchy–Stefan problem using a method of fundamental solutions , 2011 .

[21]  Nguyen Van Duc,et al.  Regularization of parabolic equations backward in time by a non-local boundary value problem method , 2010 .

[22]  B. D. Brewster Geelen.  Accurate solution for the modified Bessel function of the first kind , 1995 .

[23]  Carlos Alberto Brebbia,et al.  A formulation of the boundary element method for axisymmetric transient heat conduction , 1981 .

[24]  Jean-Claude Saut,et al.  Unique continuation for some evolution equations , 1987 .

[25]  R. Johnson,et al.  A priori estimates and unique continuation theorems for second order parabolic equations , 1971 .

[26]  Chih-Wen Chang,et al.  A new algorithm for direct and backward problems of heat conduction equation , 2010 .

[27]  Damon Honnery,et al.  Radiant heating of a Bio-oil droplet: A quest for a suitable model and scaling of pre-explosion conditions , 2004 .

[28]  N. S. Mera.  The method of fundamental solutions for the backward heat conduction problem , 2005 .

[29]  B. Johansson,et al.  A procedure for determining a spacewise dependent heat source and the initial temperature , 2008 .

[30]  Sergey E. Mikhailov.  Axisymmetric fundamental solutions for the equations of heat conduction in the case of cylindrical anisotropy of a medium , 1990 .

[31]  Wei Cheng,et al.  Source term identification for an axisymmetric inverse heat conduction problem , 2010, Comput. Math. Appl..

[32]  B. Tomas Johansson,et al.  A method of fundamental solutions for two-dimensional heat conduction , 2011, Int. J. Comput. Math..

[33]  Svilen S. Valtchev,et al.  A time-marching MFS scheme for heat conduction problems , 2008 .