Ritz-least squares method for finding a control parameter in a one-dimensional parabolic inverse problem

• Autorzy: Khorshidi M. N. , Yousefi S. A.

Identyfikatory

DOI

10.1515/jaa-2016-0018

• Języki publikacji: EN

Abstrakty

EN

An inverse problem concerning a diffusion equation with source control parameter is considered. The approximation of the problem is based on the Ritz method with satisfier function. The Ritz method together with the least squares approximation (Ritz-least squares method) are utilized to reduce the inverse problem to the solution of algebraic equations. We extensively discuss the convergence of the method and finally present illustrative examples to demonstrate validity and applicability of the new technique.

Słowa kluczowe

EN

Ritz method; least-squares approximation; satisfier function; inverse problem; source control parameter; parabolic partial differential equations

PL

metoda Ritza; aproksymacja metoda najmniejszych kwadratów; zagadnienie odwrotne; paraboliczne cząstkowe równanie różniczkowe

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2016
• Tom: Vol. 22, nr 2
• Strony: 169--179
• Opis fizyczny: Bibliogr. 22 poz.

Twórcy

autor

Khorshidi M. N.

• Department of Mathematics, Shahid Beheshti University, G.C., Tehran, Iran (Islamic Republic of)

autor

Yousefi S. A.

• Department of Mathematics, Shahid Beheshti University, G.C., Tehran, Iran (Islamic Republic of)

Bibliografia

• [1] J. R. Cannon and Y. Lin, An inverse problem of finding a parameter in a semi-linear heat equation, J. Moth. Anal. Appl. 145 (1990), no. 2, 470-484. *
• [2] J. R. Cannon, Y. Lin and S. Wang, Determination of source parameter in parabolic equations, Meccanica 27 (1992), 85-94.
• [3] J. R. Cannon, Y. Lin and S. Xu, Numerical procedures for the determination of an unknown coefficient in semi-linear parabolic differential equations, inverse Problems 10 (1994), 227-243.
• [4] J. R. Cannon and H. M. Yin, On a class of non-classical parabolic problems, J. Differential Equations 79 (1989), no. 2, 266-288.
• [5] J. R. Cannon and H. M. Yin, Numerical solutions of some parabolic inverse problems, Numer. Methods Partial Differential Equations 2 (1990), 177-191.
• [6] J. R. Cannon and H. M. Yin, On a class of non-linear parabolic equations with non-linear trace type functionals inverse problems, Inverse Problems 7 (1991), 149-161.
• [7] A. Constantinides, Applied Numerical Methods with Personal Computers, McGraw-Hill, New York, 1987.
• [8] M. Dehghan, An inverse problem of finding a source parameter in a semilinear parabolic equation, Appl. Math. Model. 25 (2001), 743-754.
• [9] M. Dehghan, Fourth-order techniques for identifying a control parameter in the parabolic equations, Int. J. Eng. Sci. 40 (2002), 433-447.
• [10] M. Dehghan, Numerical solution of one-dimensional parabolic inverse problem, Appl. Math. Comput. 136 (2003), 333-344.
• [11] H. W. Engl, M. Hank and A. Neubauer, Regularization of Inverse Problems, Math. Appl. 357, Kluwer, Dordrecht, 1996.
• [12] P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems, SIAM, Philadelphia, 1998.
• [13] Y. Lin and R. J, Tait, On a class of non-local parabolic boundary value problems, Int. J, Eng. Sci. 32 (1994), no. 3, 395-407.
• [14] J. A. Macbain, Inversion theory for a parameterized diffusion problem, SIAM J. Appl. Math. 18 (1987), 1386-1391.
• [15] J. A. Macbain and J. B. Bendar, Existence and uniqueness properties for one-dimensional magnetotelluric inversion problem, A Math. Phys. 27 (1986), 645-649.
• [16] A. I. Prilepko and D. G. Orlovskii, Determination of the evolution parameter of an equation and inverse problems of mathematical physics. 1,7. Differential Equations 21 (1985), no. 1, 119-129.
• [17] A. I. Prilepko and V. V. Soloev, Solvability of the inverse boundary value problem of finding a coefficient of a lower order term in a parabolic equation, J. Differential Equations 23 (1987), no. 1, 136-143.
• [18] H. L. Royden, Real Analysis, 3rd ed., Macmillan, New York, 1988.
• [19] S. Wang, Numerical solutions of two inverse problems for identifying control parameters in 2-dimensional parabolic partial differential equations, Modern Dev. Numer. Simul. Flow Heat Transfer 194 (1992), 11-16.
• [20] S. Wang and Y. Lin, A finite difference solution to an inverse problem determining a control function in a parabolic partial differential equations, Inverse Problems 5 (1989), 631-640.
• [21] S. A. Yousefi, Finding a control parameter in a one-dimensional parabolic inverse problem by using the Bernstein Galerkin method, Inverse Probl. Sci. Eng. 17 (2009), no. 6, 821-828.
• [22] S. A. Yousefi, D. Lesnic and Z. Barikbin, Satisfier function in Ritz-Galerkin method for the identification of a time-dependent diffusivity, J. Inverse Ill-Posed Probl. 20(2012), 701-722.