Determination of space–time-dependent heat source in a parabolic inverse problem via the Ritz–Galerkin technique

Three inverse problems of reconstructing the time-dependent, spacewise- dependent and both initial condition and spacewise-dependent heat source in the one-dimensional heat equation are considered. These problems are reformulated by eliminating the unknown functions using some special assumptions concerning the points in space or time as additional measurements. Then direct techniques are proposed to solve the non-classical boundary value problems. For obtaining the robust and stable approximations, Bernstein multi-scaling and B-spline basis functions in the context of the Ritz–Galerkin method are utilized to immediate passage from differential equations to algebraic equations and afterwards, a Newton-type method is used to produce the admissible solution. The numerical convergence and stability are discussed in the test examples to show that the presented schemes provide accurate and acceptable approximations.

[1]  Damian Slota,et al.  Direct and inverse one-phase Stefan problem solved by the variational iteration method , 2007, Comput. Math. Appl..

[2]  A. A. Samarskii,et al.  Numerical Methods for Solving Inverse Problems of Mathematical Physics , 2007 .

[3]  Mehdi Dehghan,et al.  High-order scheme for determination of a control parameter in an inverse problem from the over-specified data , 2010, Comput. Phys. Commun..

[4]  William Rundell,et al.  Recovering a time dependent coefficient in a parabolic differential equation , 1991 .

[5]  Peter Jochum,et al.  To the Numerical Solution of an Inverse Stefan Problem in Two Space Variables , 1982 .

[6]  Daniel Lesnic,et al.  The boundary-element method for the determination of a heat source dependent on one variable , 2006 .

[7]  Per Christian Hansen,et al.  Analysis of Discrete Ill-Posed Problems by Means of the L-Curve , 1992, SIAM Rev..

[8]  Mehdi Dehghan,et al.  Parameter determination in a partial differential equation from the overspecified data , 2005, Math. Comput. Model..

[9]  Satya N. Atluri,et al.  Computational heat transfer , 1986 .

[10]  Mehdi Dehghan,et al.  Ritz–Galerkin method for solving an inverse heat conduction problem with a nonlinear source term via Bernstein multi-scaling functions and cubic B-spline functions , 2013 .

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

[12]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[13]  Mehdi Dehghan,et al.  Recovering a time‐dependent coefficient in a parabolic equation from overspecified boundary data using the pseudospectral Legendre method , 2007 .

[14]  E. G. SAVATEEV,et al.  On problems of determining the source function in a parabolic equation , 1995 .

[15]  Mehdi Dehghan,et al.  Numerical solutions of the generalized Kuramoto–Sivashinsky equation using B-spline functions , 2012 .

[16]  Jacques-Louis Lions,et al.  The method of quasi-reversibility : applications to partial differential equations , 1969 .

[17]  Mehdi Dehghan,et al.  Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions , 2006, Math. Comput. Model..

[18]  Afet Golayoglu Fatullayev,et al.  Numerical solution of the inverse problem of determining an unknown source term in a two-dimensional heat equation , 2004, Appl. Math. Comput..

[19]  Mehdi Dehghan,et al.  Direct numerical method for an inverse problem of a parabolic partial differential equation , 2009, J. Comput. Appl. Math..

[20]  P. N. Vabishchevich,et al.  Numerical solution of the inverse problem of reconstructing a distributed right-hand side of a parabolic equation , 2000 .

[21]  Derek B. Ingham,et al.  An iterative boundary element method for solving the one-dimensional backward heat conduction problem , 2001 .

[22]  M. Dehghan,et al.  A method based on the tau approach for the identification of a time-dependent coefficient in the heat equation subject to an extra measurement , 2012 .

[23]  On the Unique Solvability of an Inverse Problem for Parabolic Equations under a Final Overdetermination Condition , 2003 .

[24]  Mehdi Dehghan,et al.  Computation of two time-dependent coefficients in a parabolic partial differential equation subject to additional specifications , 2010, Int. J. Comput. Math..

[25]  Mehdi Dehghan,et al.  Inverse problem of time-dependent heat sources numerical reconstruction , 2011, Math. Comput. Simul..

[26]  Mehdi Dehghan,et al.  Numerical solution of Riccati equation using the cubic B-spline scaling functions and Chebyshev cardinal functions , 2010, Comput. Phys. Commun..

[27]  Afet Golayoglu Fatullayev,et al.  An iterative procedure for determining an unknown spacewise-dependent coefficient in a parabolic equation , 2009, Appl. Math. Lett..

[28]  Mehdi Dehghan,et al.  Method of lines solutions of the parabolic inverse problem with an overspecification at a point , 2009, Numerical Algorithms.

[29]  D. Lesnic,et al.  Determination of a spacewise dependent heat source , 2007 .

[30]  Yanfei Wang,et al.  Optimization and regularization for computational inverse problems and applications , 2011 .

[31]  Victor Isakov,et al.  Inverse Source Problems , 1990 .

[32]  Mehdi Dehghan,et al.  Numerical solution of Fokker‐Planck equation using the cubic B‐spline scaling functions , 2009 .

[33]  A El Badia,et al.  A one-phase inverse Stefan problem , 1999 .

[34]  Wenyuan Liao.  A computational method to estimate the unknown coefficient in a wave equation using boundary measurements , 2011 .

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

[36]  Jessika Daecher,et al.  Computational Heat Transfer , 2016 .

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

[38]  Mehdi Dehghan,et al.  Application of the Ritz-Galerkin method for recovering the spacewise-coefficients in the wave equation , 2013, Comput. Math. Appl..

[39]  S. Yousefi,et al.  Satisfier function in Ritz–Galerkin method for the identification of a time-dependent diffusivity , 2012 .

[40]  J. Cannon,et al.  Remarks on the one-phase Stefan problem for the heat equation with the flux prescribed on the fixed boundary , 1971 .

[41]  B. Tomas Johansson,et al.  A variational method for identifying a spacewise-dependent heat source , 2007 .

[42]  D. Słota.  Using genetic algorithms for the determination of an heat transfer coefficient in three-phase inverse Stefan problem , 2008 .

[43]  A. Shidfar,et al.  Solving the inverse problem of identifying an unknown source term in a parabolic equation , 2010, Comput. Math. Appl..

[44]  Mehdi Dehghan,et al.  Determination of a control function in three‐dimensional parabolic equations by Legendre pseudospectral method , 2012 .

[45]  Mehdi Dehghan,et al.  The use of Chebyshev cardinal functions for the solution of a partial differential equation with an unknown time-dependent coefficient subject to an extra measurement , 2010, J. Comput. Appl. Math..

[46]  O. Alifanov.  Inverse heat transfer problems , 1994 .

[47]  Xiuming Li,et al.  Numerical Solution of the Inverse Problem of Determining an Unknown Source Term in a Heat Equation , 2012, J. Appl. Math..

[48]  Jaideva C. Goswami,et al.  Semi-orthogonal versus orthogonal wavelet basis sets for solving integral equations , 1997 .

[49]  Mehdi Dehghan,et al.  Ritz‐Galerkin method with Bernstein polynomial basis for finding the product solution form of heat equation with non‐classic boundary conditions , 2012 .

[50]  Adolf Ebel,et al.  Air, water and soil quality modelling for risk and impact assessment , 2007 .

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

[52]  Daniel Lesnic,et al.  Determination of a time-dependent diffusivity from nonlocal conditions , 2013 .

[53]  J. Cannon,et al.  Structural identification of an unknown source term in a heat equation , 1998 .

[54]  Barbara Kaltenbacher,et al.  Iterative Regularization Methods for Nonlinear Ill-Posed Problems , 2008, Radon Series on Computational and Applied Mathematics.

[55]  Mehdi Dehghan,et al.  The spectral methods for parabolic Volterra integro-differential equations , 2011, J. Comput. Appl. Math..