# Solution of some parabolic inverse problems by homotopy analysis method

## DOI:

https://doi.org/10.14419/ijamr.v3i1.1826## Published:

2014-02-25## Abstract

In this paper, three types of parabolic inverse problems are solved by homotopy analysis method (HAM). In order to solve these types of problems, an overspecified boundary condition is given. There are advantages to using HAM, firstly it is independent of small/large physical parameters, there is always a guarantee of convergence; there is flexibility on the choice of base function and initial guess of solution and lastly there is great generality. The numerical results obtained from this method indicate high accuracy and a strong rate of convergence.

**Keywords**: Control Function, Homotopy Analysis Method, One-Dimensional Heat Equation, Parabolic Inverse Problem.

## References

A.K. Alomari, “Modifications of Homotopy Analysis Method for Differential Equations: Modification of Homotopy analysis method, ordinary, fractional, delay and algebraic differential equations”, Lambert Academic Publishing, Germany, 2012.

A.M. Shahrezaee, “Solution of Some Parabolic Inverse Problems by Adomian Decomposition Method”, Applied Mathematical Sciences, 5 (2011), 3949 -3958.

A.M.Wazwaz, “Partial Differential Equations and Solitary Waves Theory”, Springer, Berlin Heidelberg, 2009.

A.M.Wazwaz, “Linear and Nonlinear Integral Equations”, Springer, Berlin Heidelberg, 2011.

E.G. Savateev, R. Riganti, “Inverse Problem for the Nonlinear Heat Equation with the Final Overdetermination”, Mathl. Comput. Modelling, 22 (1995), 29 – 43.

J.R.Cannon, Y.Lin, “Numerical Solutions of Some Parabolic Inverse Problems”, Numer. Math. Partial Diff. Eqns, 2(1990) 177 -191.

J.R.Cannon, H.M Yin, “On a class of non-classical parabolic problems”, J. Diff Eqs, 79 (1989), 266 -288.

J.R.Cannon, Y. Lin and S. Wang, “Determination of source parameter in parabolic equations”, Meccanica, 27 (1992), 85 -94.

J.R.Cannon, Y.Lin, “An inverse problem of finding a parameter in a semilinear heat equation”, J.Math.Anal.Appl, (1945), 470-484.

K. Hosseini, B. Daneshian, N. Amanifard and R. Ansari, “Homotopy Analysis Method for a Fin with Temperature Dependent Internal Heat Generation and Thermal Conductivity”, International Journal of Nonlinear Science, 14 (2012), 201 – 210.

M.Aylin Bayrak, Iclal Ulvi , “Identifying an Unknown Function in a Parabolic Equation by Homotopy Analysis Method and Comparison with the Adomian Decomposition Method”, First International Conference on Analysis and Applied Mathematics, 1470 (2012), 88 – 91.

M. Dehghan, “Parameter Determination in a Partial Differential Equation from the Overspecified Data”, Mathematical and Computer Modelling, 41(2005), 197-213.

M.Mousa, “Modifications of Homotopy perturbation and variational iteration methods: Convergence theorems and applications in fluid mechanics and mathematical physics”, Lambert Academic Publishing, Germany, 2011.

S.Aminsadrabad, “Solution for Inverse Space – Dependent Heat Source Problems by Homotopy Perturbation Method”, Applied Mathematical Sciences, 6 (2012), 575 – 578.

S. Liao, “Homotopy Analysis Method in Nonlinear Equations”, Springer, New York, 2012.

S. Liao, “Beyond Perturbation: Introduction to the Homotopy Analysis Method”, Chapman & Hall/CRC, 2004.

S. Liao, “Notes on the homotopy analysis method – Some definitions and theorems”, Commun. Nonlinear Sci. Numer. Simulat, 14(2009), 983-997.

S. Wang and Y.Lin, “A finite difference solution to an inverse problems determining a control function in a parabolic partial differential equation”, Inverse Problems,(1989) 631 – 640.

X. Li , S. Qian , “Numerical Solution of the Inverse Problem of Determining an Unknown Source Term in a Heat Equation”, Hindawi Publishing Corporation, Journal of Applied Mathematics, http://dx.doi:10.1155/2012/390876.