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An inverse problem of finding a source parameter in a semilinear parabolic equation. (English) Zbl 0995.65098

Summary: An inverse problem concerning diffusion equation with source control parameter is considered. Several finite-differences schemes are presented for identifying the control parameter. These schemes are based on the classical forward time centered 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 TFTCS 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 methods.
The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed by R. F. Warming and B. J. Hyett [J. Comput. Phys. 14, 159-179 (1974; Zbl 0291.65023)]. 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.

MSC:

65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
35R30 Inverse problems for PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Citations:

Zbl 0291.65023
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References:

[1] Cannon, J. R.; Yin, H. M., On a class of non-classical parabolic problems, J. Differential Equations, 79, 2, 266-288 (1989) · Zbl 0702.35120
[2] Cannon, J. R.; Yin, H. M., On a class of nonlinear parabolic equations with nonlinear trace type functionals inverse problems, Inverse Problems, 7, 149-161 (1991) · Zbl 0735.35078
[3] Cannon, J. R.; Lin, Y., An inverse problem of finding a parameter in a semilinear heat equation, J. Math. Anal. Appl., 145, 2, 470-484 (1990) · Zbl 0727.35137
[4] Cannon, J. R.; Lin, Y.; Xu, S., Numerical procedures for the determination of an unknown coefficient in semilinear parabolic differential equations, Inverse Problems, 10, 227-243 (1994) · Zbl 0805.65133
[5] Cannon, J. R.; Yin, H. M., Numerical solutions of some parabolic inverse problems, Numerical Methods for Partial Differential Equations, 2, 177-191 (1990) · Zbl 0709.65105
[6] Cannon, J. R.; Lin, Y.; Wang, S., Determination of source parameter in parabolic equations, Meccanica, 27, 85-94 (1992) · Zbl 0767.35105
[7] Fairweather, G.; Saylor, R. D., The reformulation and numerical solution of certain nonclassical initial-boundary value problems, SIAM J. Sci. Stat. Comput., 12, 127-144 (1991) · Zbl 0722.65062
[8] A.I. Prilepko, D.G. Orlovskii, Determination of the evolution parameter of an equation and inverse problems of mathematical physics, Part I and II, J. Differential Equations 21(1) (1985) 119-129; 649-701; A.I. Prilepko, D.G. Orlovskii, Determination of the evolution parameter of an equation and inverse problems of mathematical physics, Part I and II, J. Differential Equations 21(1) (1985) 119-129; 649-701
[9] Prilepko, A. I.; Soloev, V. V., Solvability of the inverse boundary value problem of finding a coefficient of a lower order term in a parabolic equation, J. Differential Equations, 23, 1, 136-143 (1987)
[10] Wang, S., Numerical solutions of two inverse problems for identifying control parameters in 2-dimensional parabolic partial differential equations, Numerical simulation of flow and heat transfer, 194, 11-16 (1992)
[11] Wang, S.; Lin, Y., A finite difference solution to an inverse problem determining a control function in a parabolic partial differential equations, Inverse Problems, 5, 631-640 (1989) · Zbl 0683.65106
[12] Macbain, J. A.; Bednar, J. B., Existence and uniqueness properties for one-dimensional magnetotelluric inversion problem, J. Math. Phys., 27, 645-649 (1986)
[13] Macbain, J. A., Inversion theory for a parametrized diffusion problem, SIAM J. Appl. Math., 18, 1386-1391 (1987) · Zbl 0664.35075
[14] Lin, Y.; Tait, R. J., On a class of non-local parabolic boundary value problems, Int. J. Engng. Sci., 32, 3, 395-407 (1994) · Zbl 0792.73018
[15] J. Crank, P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, In: Proceedings of the Cambridge Philosophical Society 43 (1) (1947) 50-67; J. Crank, P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, In: Proceedings of the Cambridge Philosophical Society 43 (1) (1947) 50-67 · Zbl 0029.05901
[16] A.R. Mitchell, D.F. Griffiths, The finite difference methods in partial differential equations, Wiley, 1980; A.R. Mitchell, D.F. Griffiths, The finite difference methods in partial differential equations, Wiley, 1980 · Zbl 0417.65048
[17] Warming, R. F.; Hyett, B. J., The modified equation approach to the stability and accuracy analysis of finite-difference methods, J. Comput. Phys., 14, 2, 159-179 (1974) · Zbl 0291.65023
[18] Lapidus, L.; Pinder, G. F., Numerical solution of partial differential equations in science and engineering (1982), Wiley: Wiley New York · Zbl 0584.65056
[19] Gerald, C. F., Applied Numerical Analysis (1989), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0197.42901
[20] Dehghan, M., Altering direction implicit methods for two-dimensional diffusion with a non-local boundary condition, Intern. J. Comp. Math., 72, 349-366 (1999) · Zbl 0949.65085
[21] Dehghan, M., Finite difference schemes for two-dimensional parabolic inverse problem with temperature overspecification, Intern. J. Comp. Math., 75, 339-349 (2000) · Zbl 0966.65068
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