×

Application of weighted homotopy analysis method to solve an inverse source problem for wave equation. (English) Zbl 1471.65125

Summary: In this paper, inverse source problem for the wave equation is considered in one dimension with overdetermination condition. To solve this problem, a version of homotopy analysis method, called weighted homotopy analysis method, is introduced and applied. To show accuracy and reliability of the mentioned method, three numerical examples are given.

MSC:

65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
35M99 Partial differential equations of mixed type and mixed-type systems of partial differential equations
35R30 Inverse problems for PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L05 Wave equation
35C10 Series solutions to PDEs

Software:

BVPh
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Kabanikhin SI . Definitions and examples of inverse and ill-posed problems. J Inverse Ill-posed Probl. 2008;16:317-357. · Zbl 1170.35100
[2] Yamamoto M . Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method. Inverse Probl. 1995;11:481-496. · Zbl 0822.35154
[3] Yamamoto M . Uniqueness and stability in multidimensional hyperbolic inverse problems. J Math Pures Appl. 1999;9(78):65-98. · Zbl 0923.35200
[4] Kristensson G , Krueger R . Direct and inverse scattering in the time domain for a dissipative wave equation. I. Scattering operators. J Math Phys. 1986;27:1667-1682. · Zbl 0595.45018
[5] Weston V . Invariant imbedding for the wave equation in three dimensions and the applications to the direct and inverse problems. Inverse Probl. 1990;6(6):1075-1105. · Zbl 0734.35048
[6] Chen Y , Liu JQ . numerical algorithm for solving inverse problems of two-dimensional wave equations. Comput Phys. 1983;50:193-208. · Zbl 0515.65088
[7] He S , Weston VH . Inverse problem for the dissipative wave equation in a stratified half-space and linearization of the imbedding equations. Inverse Probl. 1992;8(3):435-455. · Zbl 0748.35056
[8] Oksanen L . Solving an inverse problem for the wave equation by using a minimization algorithm and time-reversed measurements. 2011:22, preprint arXiv:1101.4836. · Zbl 1230.35145
[9] Xie G , Chen Y . A modified pulse-spectrum technique for solving inverse problems of two-dimensional elastic wave equation. Appl Numer Math. 1985;1(3):217-237. · Zbl 0566.65092
[10] Alves C , Silvestre AL , Takahashi T , et al . Solving inverse source problems using observability applications to the Euler-Bernoulli plate equation. SIAM. 2009;48(3):1632-1659. · Zbl 1282.93059
[11] Hong-Sun F , Bo H . A regularization homotopy method for the inverse problem of 2-d wave equation and well log constraint inversion. Chin J Geophys. 2005;48(6):1509-1517.
[12] Moireau P , Chapelle D , Tallec PL . Joint state and parameter estimation for distributed mechanical systems. Elsevier. 2008;T136:659-677. · Zbl 1169.74439
[13] Ramdani K , Tucsnak M , Weiss G . Recovering the initial state of an infinite-dimensional system using observers. Automatica. 2010;46:1616-1625. · Zbl 1204.93023
[14] Guo W , Guo B-Z . Parameter estimation and stabilisation for a one dimensional wave equation with boundary output constant disturbance and noncollocated control. Int J Control. 2011;84(2):381-395. · Zbl 1222.93118
[15] Chapouly M , Mirrahimi M . Distributed source identification for wave equations: an observer-based approach. In: 19th International Symposium on Mathematical Theory of Networks and Systems. Budapest: Hungary; 2010. p. 389-394. · Zbl 1369.93156
[16] Guo B-Z , Xu C-Z , Hammouri H . Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation. ESAIM: Control, Optim Calc Var. 2012;18:22-35. · Zbl 1246.35120
[17] Liao SJ . The proposed homotopy analysis technique for the solution of nonlinear problems [PhD dissertation]. Shanghai: Shanghai Jiao Tong University; 1992. English.
[18] Liang S , Jeffrey DJ . Approximate solutions to a parameterized sixth order boundary value problem. Comput Math Appl. 2010;59:247-253. · Zbl 1189.65147
[19] Liao SJ . Beyond perturbation: introduction to the homotopy analysis method. Boca Raton (FL): Chapman & Hall/CRC Press; 2003. · Zbl 1051.76001
[20] Liao SJ . On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet. J Fluid Mech. 2003;488:189-212. · Zbl 1063.76671
[21] Liao SJ . On the homotopy analysis method for nonlinear problems. Appl Math Comput. 2004;147:499-513. · Zbl 1086.35005
[22] Liao SJ . Notes on the homotopy analysis method: some definitions and theorems. Commun Nonlinear Sci Numer Simul. 2009;14:983-997. · Zbl 1221.65126
[23] Molabahrami A , Shidfar A . A study on the PDEs with power-law nonlinearity. Nonlinear Anal RWA. 2010;11:1258-1268. · Zbl 1189.35012
[24] Sajid M , Awais M , Nadeem S , et al . The influence of slip condition on thin film flow of a fourth grade fluid by the homotopy analysis method. Comput Math Appl. 2008;56:2019-2026. · Zbl 1165.76312
[25] Sami Bataineh A , Noorani MSM , Hashim I . Approximate analytical solutions of systems of PDEs by homotopy analysis method. Comput Math Appl. 2008;55:2913-2923. · Zbl 1142.65423
[26] Shidfar A , Babaei A , Molabahrami A , et al. Approximate analytical solutions of the nonlinear reaction-diffusion-convection problems. Math Comput Model. 2011;53:261-268. · Zbl 1211.65141
[27] Shidfar A , Molabahrami A , Babaei A , et al . A series solution of the nonlinear Volterra and Fredholm integro-differential equations. Commun Nonlinear Sci Numer Simul. 2010;15:205-215. · Zbl 1221.65343
[28] Yücel U . Homotopy analysis method for the sine-Gordon equation with initial conditions. Appl Math Comput. 2008;203:387-395. · Zbl 1157.65464
[29] Hetmaniok E , Slota D , Witula R , et al . An analytical method for solving the two-phase inverse Stefan problem. Bull Pol Acad Sci Tech Sci. 2015;63:583-590.
[30] Hetmaniok E , Slota D , Witula R , et al . Solution of the one-phase inverse Stefan problem by using the homotopy analysis method. Appl Math Model. 2015;39:6793-6805. · Zbl 1443.65272
[31] Shidfar A , Babaei A , Molabahrami A . Solving the inverse problem of identifying an unknown source term in a parabolic equation. Comput Math Appl. 2010;60:1209-1213. · Zbl 1201.65175
[32] Shidfar A , Molabahrami A . A weighted algorithm based on the homotopy analysis method: application to inverse heat conduction problems. Commun Nonlinear Sci Numer Simul. 2010;15:2908-2915. · Zbl 1222.65105
[33] Liao SJ . Homotopy analysis method in nonlinear differential equations. Beijing: Higher Education Press, Springer-Verlag Berlin Heidelberg; 2012. · Zbl 1253.35001
[34] Abbasbandy S , Magyari E , Shivanian E . The homotopy analysis method for multiple solutions of nonlinear boundary value problems. Commun Nonlinear Sci Numer Simul. 2009;14:3530-3536. · Zbl 1221.65170
[35] Abbasbandy S , Shivanian E . Predictor homotopy analysis method and its application to some nonlinear problems. Commun Nonlinear Sci Numer Simul. 2011;16:2456-2468. · Zbl 1221.65190
[36] Motsa SS , Sibanda P , Shateyi S . A new spectral homotopy analysis method for solving a nonlinear second order BVP. Commun Nonlinear Sci Numer Simulat. 2010;15:2293-2302. · Zbl 1222.65090
[37] Motsa SS , Sibanda P , Auad FG , et al . A new spectral homotopy analysis method for the MHD Jeffery-Hamel problem. Comput Fluids. 2010;39:1219-1225. · Zbl 1242.76363
[38] Marinca V , Herişnu N . Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer. Int Commun Heat Mass. 2008;35:710-715.
[39] Marinca V , Herisşanu N . An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plat. Appl Math Lett. 2009;22:245-251. · Zbl 1163.76318
[40] Watson LT . Probability-one homotopies in computational science. J Comput Appl Math. 2002;140:785-807. · Zbl 0996.65055
[41] Watson LT . Globally convergent homotopy methods: a tutorial. Appl Math Comput. 1989;13BK:369-396. · Zbl 0689.65033
[42] Watson LT , Scott MR . Solving spline-collocation approximations to nonlinear two-point-value problems by a homotopy method. Appl Math Comput. 1987;24:333-357. · Zbl 0635.65099
[43] Watson LT , Haftka RT . Modern homotopy methods in optimization. Comput Methods Appl Mech Eng. 1989;74:289-305. · Zbl 0693.65046
[44] Shidfar A , Garshasbi M . A weighted algorithm based on Adomian decomposition method for solving an special class of evolution equations. Commun Nonlinear Sci Numer Simul. 2009;14:1146-1151. · Zbl 1221.35202
[45] Lesnic D , Elliott L . The decomposition approach to inverse heat conduction. J Math Anal Appl. 1999;232:82-98. · Zbl 0922.35189
[46] Akewe H , Okeke GA . Convergence and stability theorems for the Picard-Mann hybrid iterative scheme for a general class of contractive-like operators. Fixed Point Theory Appl. 2015;2015:66-73. · Zbl 1312.47078
[47] Berinde V . Iterative approximation of fixed points. Baia Mare: Springer, Editura Efemeride; 2002. · Zbl 1036.47037
[48] Qing Y , Rhoades BE . T-stability of Picard iteration in metric spaces. Fixed Point Theory Appl. 2008;2008. Article ID 418971. 4 p. · Zbl 1145.54328
[49] Atkinson K , Han W . Theoretical numerical analysis: a functional analysis framework. 3rd ed. New York (NY): Springer; 2009. · Zbl 1181.47078
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.