Method for solving an inverse term source problem based on the Laplace transform

N. M. Yaparova

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ Vyčislitelʹnaâ matematika i informatika, Tome 5 (2016) no. 3, pp. 20-35

Résumé

In this contribution, the method for solving the inverse source problem for parabolic partial differential equations with Dirichlet and Neumann boundary conditions is proposed. We reduce this problem to solving Volterra integral equation of the first kind via applications of the Laplace transforms. Then we use the regularization technique for numerical solving for the obtained equation. The integral equation describe the explicit dependence of the unknown source term on the Neumann boundary condition. The proposed approach allows to eliminate the unstable inverse Laplace transform from numerical scheme and simplify the computational procedure provided the basis to develop method for solving an inverse source problem. This approach to solving the inverse source problem is used for the first time. The efficiency of method and accuracy of the numerical solutions were evaluated by means of computational experiment.

Détail

Keywords: inverse source problem, Volterra integral equation, numerical method.
Mots-clés : Laplace transform

Bibliographie

[1] O.M. Alifanov, Inverse Heat Transfer Problems, Mashinostroenie, Moscow, 1988, 280 pp.

[2] A.S. Erdogan, A.U. Sazaklioglu, “A Note on the Numerical Solution of an Identification Problem for Observing Two-Phase Flow in Capillaries”, Mathematical Method in the Applied Sciences, 37:16 (2014), 2393–2405 | DOI

[3] A.M. Zenkour, A.E. Abouelregal, “Vibration of FG Nanobeams Induced by Sinusoidal Pulse-Heating via a Nonlocal Thermoelastic Model”, ACTA Mechanica, 225:12 (2014), 3409–3421 | DOI

[4] P.N. Vabishchevich, “Numerical Solution of the Problem Identifying the Right Part a Parabolic Equation”, Russian Mathematics, 47:1 (2003), 27–35

[5] N.L. Gol’dman, “Uniqueness of Determination of a Source Function in a Quasilinear Inverse Stefan Problem with Final Observation”, Doklady Mathematics, 444:6 (2012), 597–601 | DOI

[6] A.I. Prilepko, D.S. Tkachenko, “Well-Posedness of the Inverse Source Problem for Parabolic Systems”, Differential Equations, 40:11 (2004), 1619–1626 | DOI

[7] O.N. Cherepanova, T.N. Shipina, “On a Problem of Identification of the Source Function in the Parabolic Equation”, J. Noth. Federal Univ., 2:3 (2009), 370–375

[8] A. Hasanov, B. Pektas, “Identification of an Unknown Time-Dependent Heat Source Term from Overspecified Dirichlet Boundary Data by Conjugate Gradient Method”, Computers and Mathematics with Applications, 65:1 (2013), 42–57 | DOI

[9] M. Cialkowski, K. Grysa, “A Sequential and Global Method of Solving an Inverse Problem of Heat Conduction Equation”, Journal of Theoretical and Applied Mechanics, 48:1 (2010), 111–134

[10] M. Monde, H. Arima, W. Liu, Y. Mitutake, J.A. Hammad, “An Analytical Solution for Two-Dimensional Inverse Heat Conduction Problems Using Laplace Transform”, International Journal of Heat and Mass Transfer, 46 (2003), 2135–2148 | DOI

[11] N.M. Yaparova, “A Method for Solving the Inverse Problem of Identifying the Source Function for System with Distributed Parameters”, Tambov University Reports. Series: Natural and Technical Sciences, 20:5 (2015), 1549–1552

[12] S.V. Solodusha, N.M. Yaparova, “Numerical Solving an Inverse Boundary Problem of Heat Conduction Using Volterra Equations of the First Kind”, Numerical Analysis and Applications, 8:3 (2015), 267–274 | DOI

[13] N. Yaparova, “Numerical Methods for Solving a Boundary Value Inverse Heat Conduction Problem”, Inverse Problems in Science and Engineering, 22:5 (2014), 832–847 | DOI

[14] A.S. Apartsyn, A.B. Bakushinskii, “Approximate Solution of Volterra Integral Equations of the First Kind by the Quadratures Method”, Differential and Integral Equations, 1 (1972), 248–258

[15] V.V. Vasin, T.I. Serezhnikova, “A Regularizing Algorithm for Approximation of a Non-Smoth Solution of Fredholm Integral Equations of the First Kind”, Computatuional Technologies, 15:2 (2010), 15–23

[16] S.I. Kabanikhin, Inverse and Ill-Posed Problems. Theory and Applications, De Gruyter, Germany, 2011, 457 pp. | DOI

[17] Yu.M. Korolev, A.G. Yagola, “Error Estimation in Linear Inverse Problems with Prior Information”, Numerical Methods and Programming, 13:1(25) (2012), 14–18

[18] I. Bushuev, “Global Uniqueness for Inverse Parabolic Problems with Final Observation”, Inverse Problems, 11:4 (1995), L11–L16 | DOI

[19] G.Doetsch, Guide to the Applications of the Laplace and Z Transforms, Nauka, Moscow, 1971, 291 pp.

[20] M.L. Krasnov, Integral Equations. Introduction to the Theory, Nauka, Moscow, 1975, 302 pp.

[21] M.M. Lavrentiev, Conditionally Correct Problems for Differential Equations, Publishing of Novosibirsk Goverment University, Novosibirsk, 1973, 71 pp.

[22] A.S. Leonov, “On Quasioptimum Selection of the Regularization Parameter in M.M. Lavreny’ev’s Method”, Siberian Mathemathical Journal, 34:4 (1993), 695–703 | DOI