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Analytical Solution for the Time-Dependent Emden-Fowler Type of Equations by Homotopy Analysis Method with Genetic Algorithm
Department of Mathematics, College of Computer Science and Mathematics, University of Mosul, Mosul, Iraq
Department of Software Engineering, College of Computer Science and Mathematics, University of Mosul, Mosul, Iraq
Department of Mathematics, College of Computer Science and Mathematics, University of Mosul, Mosul, Iraq
- 1 Department of Mathematics, College of Computer Science and Mathematics, University of Mosul, Mosul, Iraq
- 2 Department of Software Engineering, College of Computer Science and Mathematics, University of Mosul, Mosul, Iraq
- 3 Department of Mathematics, College of Computer Science and Mathematics, University of Mosul, Mosul, Iraq
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Abstract
In this paper, Homotopy Analysis method with Genetic Algorithm is presented and used to obtain an analytical solution for the time-dependent Emden-Fowler type of equations and wave-type equation with singular behavior at x = 0. The advantage of this single global method employed to present a reliable framework is utilized to overcome the singularity behavior at the point x = 0 for both models. The method is demonstrated for a variety of problems in one and higher dimensional spaces where approximate-exact solutions are obtained. The results obtained in all cases show the reliability and the efficiency of this method.
KeywordsHomotopy Analysis MethodGenetic AlgorithmEmden-Fowler EquationWave-Type EquationAdomian PolynomialsNoise TermsPadéApproximantsSimpson Rule
- Davis, H.T. (1962) Introduction to Nonlinear Differential and Integral Equations. Dover Publications, New York.
- Chandrasekhar, S. (1967) Introduction to the Study of Stellar Structure. Dover Publications, New York.
- Richardson, O.U. (1921) The Emission of Electricity from Hot Bodies. London.
- Adomian, G., Rach, R. and Shawagfeh, N.T. (1995) On the Analytic Solution of Lane-Emden Equation, Found. Physics Letters, 8, 161-181. https://doi.org/10.1007/BF02187585
- Shawagfeh, N.T. (1993) Nonperturbative Approximate Solution for Lane-Emden Equation. Journal of Mathematical Physics, 34, 4364-4369. https://doi.org/10.1063/1.530005
- Wazwaz, A.M. (2001) A New Method for Solving Differential Equations of the Lane-Emden Type. Applied Mathematics and Computation, 118, 287-310.
- Wazwaz, A.M. (2005) Adomian Decomposition Method for a Reliable Treatment of the Emden-Fowler Equation. Applied Mathematics and Computation, 161, 543-560.
- Wazwaz, A.M. (2005) Analytical Solution for the Time-Dependent Emden-Fowler Type of Equations by Adomian Decomposition Method. Applied Mathematics and Computation, 166, 638-651.
- Yildirim, A. and Ozis, T. (2009) Solutions of Singular IVPs of Lane-Emden Type by the Variational Iteration Method. Nonlinear Analysis, 70, 2480-2484.
- Dehghan, M. and Shakeri, F. (2008) Approximate Solution of a Differential Equation Arising in Astrophysics Using the Variational Iteration Method. New Astronomy, 13, 53-59.
- Liao, S.J. (2003) Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman and Hall, CRC Press, Boca Raton. https://doi.org/10.1201/9780203491164
- He, J.H. (2003) Homotopy Perturbation Method: A New Nonlinear Analytical Technique. Applied Mathematics and Computation, 135, 73-79.
- Gen, M. and Cheng, R. (1997) Genetic Algorithms and Engineering Design. John Wiley & Sons, Hoboken.
- Melanie, M. (1998) An Introduction to Genetic Algorithms. MIT Press, Cambridge.
- Adomian, G. (1989) Nonlinear Stochastic Systems Theory and Applications to Physics. Kluwer Academic Publishers, Dordrecht. https://doi.org/10.1007/978-94-009-2569-4
- Wazwaz, A.M. (2000) A New Algorithm for Calculating Adomian Polynomials for Nonlinear Operators. Applied Mathematics and Computation, 111, 53-69.