Numerical solution of one-dimensional parabolic inverse problem. (English) Zbl 1026.65078
Summary: This paper is concerned with numerical techniques for the solution of a one-dimensional parabolic inverse problem with a control parameter. The discrete approximation of the problem is based on the finite difference schemes. These techniques are presented for identifying the control parameter which produces at each time a desired temperature at a given point in a spatial domain. The main idea behind the finite difference methods for obtaining the solution of a given partial differential equation is to approximate the derivatives appearing in the equation by a set of values of the function at a selected number of points. The most usual way to generate these approximations is through the use of Taylor series.
The methods developed here are based on the modified equivalent partial differential equation as described by R. F. Warming and B. J. Hyett [J. Comput. Phys. 14, 159-179 (1974; Zbl 0291.65023)]. This approach allows the simple determination of the theoretical order of accuracy, thus allowing methods to be compared with one another. Also from the truncation error of the modified equivalent equation, it is possible to eliminate the dominant error terms associated with the finite-difference equations that contain free parameters (weights), thus leading to more accurate methods. The results of a numerical experiment are presented, and accuracy and the central processor (CPU) time needed for the parabolic inverse problem are discussed.
The methods developed here are based on the modified equivalent partial differential equation as described by R. F. Warming and B. J. Hyett [J. Comput. Phys. 14, 159-179 (1974; Zbl 0291.65023)]. This approach allows the simple determination of the theoretical order of accuracy, thus allowing methods to be compared with one another. Also from the truncation error of the modified equivalent equation, it is possible to eliminate the dominant error terms associated with the finite-difference equations that contain free parameters (weights), thus leading to more accurate methods. The results of a numerical experiment are presented, and accuracy and the central processor (CPU) time needed for the parabolic inverse problem are discussed.
MSC:
| 65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |
| 35K15 | Initial value problems for second-order parabolic equations |
| 35R30 | Inverse problems for PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
Keywords:
error bounds; weighted finite difference methods; modified equivalent equation; inverse problem; source control parameter; CPU time; numerical experimentCitations:
Zbl 0291.65023References:
| [1] | Cannon, J. R.; Lin, Y.; Xu, S., Numerical procedures for the determination of an unknown coefficient in semi-linear parabolic differential equations, Inverse Probl., 10, 227-243 (1994) · Zbl 0805.65133 |
| [2] | Cannon, J. R.; Yin, H. M., Numerical solutions of some parabolic inverse problems, Numer. Meth. Partial Differential Equations, 2, 177-191 (1990) · Zbl 0709.65105 |
| [3] | Cannon, J. R.; Lin, Y., An inverse problem of finding a parameter in a semi-linear heat equation, J. Math. Anal. Appl., 145, 2, 470-484 (1990) · Zbl 0727.35137 |
| [4] | 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 |
| [5] | Cannon, J. R.; Yin, H. M., On a class of nonlinear parabolic equations with nonlinear trace type functionals inverse problems, Inverse Probl., 7, 149-161 (1991) · Zbl 0735.35078 |
| [6] | Cannon, J. R.; Lin, Y.; Wang, S., Determination of source parameter in parabolic equations, Meccanica, 27, 85-94 (1992) · Zbl 0767.35105 |
| [7] | Crandall, S. H., An optimum implicit recurrence formulae for the heat conduction equation, Quart. Appl. Math., 13, 318-320 (1955) · Zbl 0066.10501 |
| [8] | Crank, J.; Nicolson, P., A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Proc. Camb. Philos. Soc., 43, 1, 50-67 (1947) · Zbl 0029.05901 |
| [9] | Gerald, C. F., Applied Numerical Analysis (1989), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0197.42901 |
| [10] | Lin, Y.; Tait, R. J., On a class of non-local parabolic boundary value problems, Int. J. Eng. Sci., 32, 3, 395-407 (1994) · Zbl 0792.73018 |
| [11] | Lapidus, L.; Pinder, G. F., Numerical Solution of Partial Differential Equations in Science and Engineering (1982), Wiley: Wiley New York · Zbl 0584.65056 |
| [12] | Macbain, J. A.; Bendar, 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] | Mitchell, A. R.; Griffiths, D. F., The Finite Difference Method in Partial Differential Equations (1980), Wiley: Wiley New York · Zbl 0417.65048 |
| [15] | Prilepko, A. I.; Orlovskii, D. G., Determination of the evolution parameter of an equation and inverse problems of mathematical physics, I and II, J. Differential Equations, 21, 1, 119-129 (1985) · Zbl 0571.35052 |
| [16] | 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) · Zbl 0661.35082 |
| [17] | Wang, S., Numerical solutions of two inverse problems for identifying control parameters in 2-dimensional parabolic partial differential equations, Mod. Dev. Numer. Simulation Flow Heat Transfer, 194, 11-16 (1992) |
| [18] | Wang, S.; Lin, Y., A finite difference solution to an inverse problem determining a control function in a parabolic partial differential equations, Inverse Probl., 5, 631-640 (1989) · Zbl 0683.65106 |
| [19] | 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 |
| [20] | M. Dehghan, Alternating direction implicit methods for two-dimensional diffusion with a non-local boundary condition, Int. J. Comput. Math. 72 (1999) 349-366; M. Dehghan, Alternating direction implicit methods for two-dimensional diffusion with a non-local boundary condition, Int. J. Comput. Math. 72 (1999) 349-366 · Zbl 0949.65085 |
| [21] | M. Dehghan, Implicit solution of a two-dimensional parabolic inverse problem with temperature overspecification, J. Computat. Anal. Appl. 3(4) (2001) 383-398; M. Dehghan, Implicit solution of a two-dimensional parabolic inverse problem with temperature overspecification, J. Computat. Anal. Appl. 3(4) (2001) 383-398 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
