An inverse problem of finding a source parameter in a semilinear parabolic equation

Abstract An inverse problem concerning diffusion equation with source control parameter is considered. Several finite-difference schemes are presented for identifying the control parameter. These schemes are based on the classical forward time centred space (FTCS) explicit formula, and the 5-point FTCS explicit method and the classical backward time centred space (BTCS) implicit scheme, and the Crank–Nicolson implicit method. The classical FTCS explicit formula and the 5-point FTCS explicit technique are economical to use, are second-order accurate, but have bounded range of stability. The classical BTCS implicit scheme and the Crank–Nicolson implicit method are unconditionally stable, but these schemes use more central processor (CPU) times than the explicit finite difference mehods. The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. The results of a numerical experiment are presented, and the accuracy and CPU time needed for this inverse problem are discussed.

[1]  John A. MacBain,et al.  Existence and uniqueness properties for the one‐dimensional magnetotellurics inversion problem , 1986 .

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

[3]  Curtis F. Gerald.  Applied numerical analysis , 1970 .

[4]  Mehdi Dehghan.  Alternating direction implicit methods for two-dimensional diffusion with a non-local boundary condition , 1999, Int. J. Comput. Math..

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

[6]  John A. Macbain,et al.  Inversion theory for parameterized diffusion problem , 1987 .

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

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

[9]  J. Crank,et al.  A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type , 1947 .

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

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

[12]  R. F. Warming,et al.  The modified equation approach to the stability and accuracy analysis of finite-difference methods , 1974 .

[13]  Graeme Fairweather,et al.  The Reformulation and Numerical Solution of Certain Nonclassical Initial-Boundary Value Problems , 1991, SIAM J. Sci. Comput..

[14]  G. Pinder,et al.  Numerical solution of partial differential equations in science and engineering , 1982 .

[15]  Mehdi Dehghan.  Finite difference schemes for two-dimensional parabolic inverse problem with temperature overspecification , 2000, Int. J. Comput. Math..

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

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

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