A Tau Method for the One-Dimensional Parabolic Inverse Problem Subject to Temperature Overspecification

In this work, an approximational technique based on shifted Legendre-tau ideas is presented for the one-dimensional parabolic inverse problem with a control parameter. The method consists of expanding the required approximate solution as the elements of a shifted Legendre polynomial. Using the operational matrices we reduce the problem to a set of algebraic equations. A numerical example is included to demonstrate the validity and applicability of the technique and a comparison is made with existing results. The method is easy to implement and produces very accurate results.

[1]  M. Dehghan.  Efficient techniques for the second-order parabolic equation subject to nonlocal specifications , 2005 .

[2]  Yanping Lin,et al.  An inverse problem of finding a parameter in a semi-linear heat equation , 1990 .

[3]  Mehdi Dehghan,et al.  An inverse problem of finding a source parameter in a semilinear parabolic equation , 2001 .

[4]  Yanping Lin,et al.  On a class of non-local parabolic boundary value problems , 1994 .

[5]  Yanping Lin,et al.  A finite-difference solution to an inverse problem for determining a control function in a parabolic partial differential equation , 1989 .

[6]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[7]  Mohsen Razzaghi,et al.  A tau method approach for the diffusion equation with nonlocal boundary conditions , 2004, Int. J. Comput. Math..

[8]  Yanping Lin,et al.  Numerical procedures for the determination of an unknown coefficient in semi-linear parabolic differential equations , 1994 .

[9]  Mehdi Dehghan,et al.  Numerical solution of one-dimensional parabolic inverse problem , 2003, Appl. Math. Comput..

[10]  Mehdi Dehghan,et al.  Fourth-order techniques for identifying a control parameter in the parabolic equations , 2002 .

[11]  Shyam L. Kalla,et al.  Tau method approximation of the Hubbell rectangular source integral , 2000 .

[12]  John R. Cannon,et al.  A class of non-linear non-classical parabolic equations , 1989 .

[13]  Mohsen Razzaghi,et al.  Tau method approximation for radiative transfer problems in a slab medium , 2002 .

[14]  John R. Cannon,et al.  On a class of nonlinear parabolic equations with nonlinear trace type functionals , 1991 .

[15]  C. Lanczos.  Applied Analysis , 1961 .

[16]  Yanping Lin,et al.  Determination of source parameter in parabolic equations , 1992 .

[17]  T. A. Zang,et al.  Spectral Methods for Partial Differential Equations , 1984 .

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

[19]  John R. Cannon,et al.  Numerical solutions of some parabolic inverse problems , 1990 .