Document Type : Research Paper

**Author**

Salman Farsi University of Kazerun

**Abstract**

It is well known that the parabolic partial differential equations in two or more space dimensions with overspecified boundary data, feature in the mathematical modeling of many phenomena. In this article, an inverse problem of determining an unknown time-dependent source term of a parabolic equation in general dimensions is considered. Employing some transformations, we change the inverse problem to a Volterra integral equation of convolution-type. By using an explicit procedure based on Sinc function properties, the resulting integral equation is replaced by a system of linear algebraic equations. The convergence analysis is included, and it is shown that the error in the approximate solution is bounded in the infinity norm by the condition number and the norm of the inverse of the coefficient matrix multiplied by a factor that decays exponentially with the size of the system. Some numerical examples are given to demonstrate the computational efficiency of the method.

[1] P. Amore, A variational Sinc collocation method for Strong-Coupling problems, Journal

of Physics A, 39 (22) (2006) 349-355.

[2] J.R. Cannon, Y. Lin, Determination of parameter p(t) in Holder classes for some semi-

linear parabolic equations, Inverse Problems 4, (1988) 595-606.

of Physics A, 39 (22) (2006) 349-355.

[2] J.R. Cannon, Y. Lin, Determination of parameter p(t) in Holder classes for some semi-

linear parabolic equations, Inverse Problems 4, (1988) 595-606.

[3] J.R. Cannon, Y. Lin, An inverse problem of nding a parameter in a semi-linear heat

equation, Journal of Mathematical Analysis and Applications, 145(2) (1990) 470-484.

[4] J.R. Cannon, Y. Lin, S. Wang, Determination of source parameter in parabolic equa-

tions, Meccanica 27, (1992) 85-94.

[5] J.R. Cannon, Y. Lin, Determination of a parameter p(t) in some quasilinear parabolic

dierential equations, Inverse Problems 4, (1988) 35-45.

[6] J.R. Cannon, The one dimensional heat equation, 1984 (Reading, MA: Addison-Wesley).

[7] M. Dehghan, M. Tatari, Determination of a control parameter in a one-dimensional par-

abolic equation using the method of radial basis functions, Mathematical and Computer

Modelling 44, (2006) 1160-1168.

[8] M. Dehghan, An inverse problem of nding a source parameter in a semilinear parabolic

equation, Applied Mathematical Modelling 25, (2001) 743-754.

[9] M. Dehghan, A. Saadatmandi, A tau method for the one-dimensional parabolic inverse

problem subject to temperature overspecication, Computational Mathematics with Ap-

plications, 52 (2006) 933-940.

[10] M. Dehghan, Finding a control parameter in one-dimensional parabolic equations, Ap-

plied Mathematics and Computation, 135 (2003) 491-503.

[11] M. Dehghan, Numerical solution of one-dimensional parabolic inverse problem, Applied

Mathematics and Computation, 136 (2003) 333-344.

[12] M. Dehghan, Determination of a control function in three-dimensional parabolic equa-

tions, Mathematics and Computers in Simulation, 61 (2003) 89-100.

[13] M. Dehghan, Determination of a control parameter in the two-dimensional diusion

equation,Applied Numerical Mathematics, 37 (2001) 489-502.

[14] M. Dehghan, Fourth order techniques for identing a control parameter in the parabolic

equations, International Journal of Engineering Science, 40 (2002) 433-447.

[15] M. Dehghan, Method of lines solutions of the parabolic inverse problem with an over-

specication at apoint, Numerical Algorithms, 50 (2009) 417-437.

[16] M. Dehghan, Finite dierence schemes for two-dimensional parabolic inverse problem

with temperature overspecication, International Journal of Computer Mathematics,

75 (3) (2000) 339-349.

[17] F. Li, Z. Wu, Ch. Ye, A nite dierence solution to a two-dimensional parabolic inverse

problem,Applied Mathematical Modelling, 36 (2012) 2303-2313.

[18] Y. Lin, An inverse problem for a cleass of quasilinear parabolic equations, SIAM Journal

on Mathematical Analysis, 22(1) (1991) 146-156.

[19] J. Lund, K. Bowers, Sinc methods for quadrature and dierential equations,SIAM,

Philadelphia, 1992.

[20] J. Lund, C. Vogel, A Fully-Galerkin method for the solution of an inverse problem in a

parabolic partial dierential equation, Inverse Problems, 6 (1990) 205-217.

[21] A.I. Prilepko, D.G. Orlovskii, Determination of the evolution parameter of an equation

and inverse problems of mathematical physics, Part I. Journal of Dierential Equations,

21 (1985) 119-129 [and part II, 21 (1985) 694-701].

[22] A.I. Prilepko, V.V. Soloev, Solvability of the inverse boundary value problem of nd-

ing a coecient of a lower order term in a parabolic equation. Journal of Dierential

Equations, 23(1) (1987) 136-143.

[23] W. Rundell, Determination of an unknownnon-homogeneous term in a linear partial

dierential equation from overspecied boundary data, Applicable Analysis, 10 (1980)

231-242.

equation, Journal of Mathematical Analysis and Applications, 145(2) (1990) 470-484.

[4] J.R. Cannon, Y. Lin, S. Wang, Determination of source parameter in parabolic equa-

tions, Meccanica 27, (1992) 85-94.

[5] J.R. Cannon, Y. Lin, Determination of a parameter p(t) in some quasilinear parabolic

dierential equations, Inverse Problems 4, (1988) 35-45.

[6] J.R. Cannon, The one dimensional heat equation, 1984 (Reading, MA: Addison-Wesley).

[7] M. Dehghan, M. Tatari, Determination of a control parameter in a one-dimensional par-

abolic equation using the method of radial basis functions, Mathematical and Computer

Modelling 44, (2006) 1160-1168.

[8] M. Dehghan, An inverse problem of nding a source parameter in a semilinear parabolic

equation, Applied Mathematical Modelling 25, (2001) 743-754.

[9] M. Dehghan, A. Saadatmandi, A tau method for the one-dimensional parabolic inverse

problem subject to temperature overspecication, Computational Mathematics with Ap-

plications, 52 (2006) 933-940.

[10] M. Dehghan, Finding a control parameter in one-dimensional parabolic equations, Ap-

plied Mathematics and Computation, 135 (2003) 491-503.

[11] M. Dehghan, Numerical solution of one-dimensional parabolic inverse problem, Applied

Mathematics and Computation, 136 (2003) 333-344.

[12] M. Dehghan, Determination of a control function in three-dimensional parabolic equa-

tions, Mathematics and Computers in Simulation, 61 (2003) 89-100.

[13] M. Dehghan, Determination of a control parameter in the two-dimensional diusion

equation,Applied Numerical Mathematics, 37 (2001) 489-502.

[14] M. Dehghan, Fourth order techniques for identing a control parameter in the parabolic

equations, International Journal of Engineering Science, 40 (2002) 433-447.

[15] M. Dehghan, Method of lines solutions of the parabolic inverse problem with an over-

specication at apoint, Numerical Algorithms, 50 (2009) 417-437.

[16] M. Dehghan, Finite dierence schemes for two-dimensional parabolic inverse problem

with temperature overspecication, International Journal of Computer Mathematics,

75 (3) (2000) 339-349.

[17] F. Li, Z. Wu, Ch. Ye, A nite dierence solution to a two-dimensional parabolic inverse

problem,Applied Mathematical Modelling, 36 (2012) 2303-2313.

[18] Y. Lin, An inverse problem for a cleass of quasilinear parabolic equations, SIAM Journal

on Mathematical Analysis, 22(1) (1991) 146-156.

[19] J. Lund, K. Bowers, Sinc methods for quadrature and dierential equations,SIAM,

Philadelphia, 1992.

[20] J. Lund, C. Vogel, A Fully-Galerkin method for the solution of an inverse problem in a

parabolic partial dierential equation, Inverse Problems, 6 (1990) 205-217.

[21] A.I. Prilepko, D.G. Orlovskii, Determination of the evolution parameter of an equation

and inverse problems of mathematical physics, Part I. Journal of Dierential Equations,

21 (1985) 119-129 [and part II, 21 (1985) 694-701].

[22] A.I. Prilepko, V.V. Soloev, Solvability of the inverse boundary value problem of nd-

ing a coecient of a lower order term in a parabolic equation. Journal of Dierential

Equations, 23(1) (1987) 136-143.

[23] W. Rundell, Determination of an unknownnon-homogeneous term in a linear partial

dierential equation from overspecied boundary data, Applicable Analysis, 10 (1980)

231-242.

[24] A. Shidfar, R. Zolfaghari, J. Damirchi, Application of Sinc-collocation method for solv-

ing an inverse problem, Journal of Computational and Applied Mathematics, 233 (2009)

545-554.

[25] A. Shidfar, R. Zolfaghari, Determination of an unknown function in a parabolic in-

verse problem by Sinc-collocation method,Numerical Methods for Partial Dierential

Equations, 27 (6) (2011) 1584-1598.

[26] R. Smith, K. Bowers, A Sinc-Galerkin estimation of diusivity in parabolic problems,

Inverse Problems, 9 (1993).

[27] F. Stenger, Numerical methods based on Sinc and analytic functions, Springer, New

York, 1993.

[28] S. Wang, Y. Lin, A nite dierence solution to an invese problem determining a control

function in a parabolic partial dierential equation, Inverse Problems, 5 (1989) 631-640.

[29] S. A. Youse, M. Dehghan, Legendre multiscaling functions for solving the one-

dimensional parabolic inverse problem, Numerical Methods for Partial Dierential

Equations, 25 (2009) 1502-1510.

ing an inverse problem, Journal of Computational and Applied Mathematics, 233 (2009)

545-554.

[25] A. Shidfar, R. Zolfaghari, Determination of an unknown function in a parabolic in-

verse problem by Sinc-collocation method,Numerical Methods for Partial Dierential

Equations, 27 (6) (2011) 1584-1598.

[26] R. Smith, K. Bowers, A Sinc-Galerkin estimation of diusivity in parabolic problems,

Inverse Problems, 9 (1993).

[27] F. Stenger, Numerical methods based on Sinc and analytic functions, Springer, New

York, 1993.

[28] S. Wang, Y. Lin, A nite dierence solution to an invese problem determining a control

function in a parabolic partial dierential equation, Inverse Problems, 5 (1989) 631-640.

[29] S. A. Youse, M. Dehghan, Legendre multiscaling functions for solving the one-

dimensional parabolic inverse problem, Numerical Methods for Partial Dierential

Equations, 25 (2009) 1502-1510.

July 2013

Pages 55-70

**Receive Date:**16 December 2013**Accept Date:**16 December 2013