International Journal of Nonlinear Analysis and Applications
Solving partial differential equations via a hybrid method between homotopy analytical method and Harris hawks optimization algorithm
Document Type : Research Paper
Author
Department of Mathematics, College of Computer Sciences and Mathematics, University of Mosul, Iraq
Abstract
In this article, a modern analytical technique called Homotopy Analysis Method (HAM) has been applied to several partial differential equations to obtain their sequential solution. The performance of the method was analysed on partial differential equations and found to be computationally efficient. The homotopy analysis method (HAM) has been found to be effective. The numerical results have been improved by proposing a new technique that combines the numerical method with the Harris Hawks Optimization algorithm. The current numerical results are compared to the exact solutions and it's found to be in very good results.
Keywords
[1] S. Abbasbandy, The application of homotopy analysis method to nonlinear equations arising in heat transfer,
Phys. Lett. A 360 (2006) 109–113.
[2] S. Abbasbandy, The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KdV
equation, Phys. Lett. A 361 (2007) 478–483.
[3] K.A. Abed, Solving Kuramoto–Sivashinsky equation by the new iterative method and estimate the optimal parameters by using PSO algorithm, Indonesian J. Elect. Engin. Comput. Sci. 19 (2020) 709.
[4] K.A. Abed and A.A. Ahmad, The best parameters selection using pso algorithm to solving for ito system by new
iterative technique, Indonesian J. Elect. Engin. Comput. Sci. 18 (2020) 1638.
[5] K.A. Abed, Controlling of jerk chaotic system via linear feedback control strategies, Indonesian J. Elect. Engin.
Comput. Sci. 20 (2020) 370–378.
[6] Q. AF and B.J. Salim, Application new iterative method for solving modified Korteweg-Devries (MKdV) system
from three equations, J. Adv. Res. Dyn. Cont. Syst. 11 (2019) 2019–2026.
[7] H. Ahmad, T.A. Khan, P.S. Stanimirovic and I. Ahmad, Modified variational iteration technique for the numerical
solution of fifth order KdV-type equations, J. Appl. Comput. Mech. 6(SI) (2020) 1220–1227.
[8] M.M. Aziz and S.F. Al-Azzawi, A modification of nonlinear feedback controller, Int. J. Comput. Sci. Math. 13
(2021) 64–79.
[9] S. Al-Azzawi, Mujiarto, L. Patria, A. Sambas and W.S.M. Sanjaya, Stability of Lorenz system at the second
equilibria point based on Gardano’s method, J. Phys.: Conf. Ser. 1477 (2020) 022009.
[10] Z.S. Al-Talib and S. F. AL-Azzawi, Projective synchronization for 4D hyperchaotic system based on adaptive
nonlinear control strategy, Indonesian J. Elect. Engin. Comput. Sci. 19 (2020) 715–722.
[11] A. Fallahzadeh and K. Shakibi, A method to solve Convection-Diffusion equation based on homotopy analysis
method, J. Interpolat. App. Sci. Comput. 8 (2015).
[12] N.A. Golilarz, A. Addeh, H. Gao, L. Ali, A.M. Roshandeh, H.M. Munir and R. Ullah Khan, A new automatic
method for control chart patterns recognition based on ConvNet and harris hawks meta heuristic optimization
algorithm, IEEE Access 7 (2019) 149398–149405.
[13] V. Gupta and S. Gupta, Application of homotopy analysis method for solving nonlinear Cauchy problem, Surv.
Math. Appl. 7 (2012) 105–116.
[14] J.-H. He, Some asymptotic methods for strongly nonlinear equations, Int. J. Modern Phys. B 20 (2006) 1141–1199.
[15] J.-H. He, Homotopy perturbation method for solving boundary value problems, Phys. Lett. A 350 (2006) 87–88.
[16] J.-H. He, Comparison of homotopy perturbation method and homotopy analysis method, Appl. Math. Comput.
156 (2004) 527–539.
[17] A.A. Heidari, S.A. Mirjalili, H. Farisd, I. Aljarah, M. Mafarja and H. Chen, Harris hawks optimization: Algorithm
and applications, Future Gener. Computer Syst. 97 (2019) 849–872.
[18] M. Ismail and H. Ashi, A numerical solution for Hirota-Satsuma coupled KdV equation, Abst. Appl. Anal. 2014
(2014).
[19] H. Jafari and S. Seifi, Solving a system of nonlinear fractional partial differential equations using homotopy
analysis method, Commun. Nonlinear Sci. Numerical Simul.14 (2009) 1962–1969.
[20] S.-J. Liao, The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems, Ph. D. Thesis,
Shanghai Jiao Tong University Shanghai, 1992.
[21] S. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC press, 2003.
[22] S. Liao, Comparison between the homotopy analysis method and homotopy perturbation method, Applied Mathematics and Computation, 169 (2005) 1186-1194.
[23] A.F. Qasim, K. Adel Abed and O.S. Qasim, Optimal parameters for nonlinear Hirota-Satsuma coupled KdV
system by using hybrid firefly algorithm with modified Adomian decomposition, J. Math. Fund. Sci. 52 (2020)
339–352.
[24] M. Rasheed, R. Omar, M. Sulaiman and W. Abd Halim, Particle swarm optimisation (PSO) algorithm with
reduced number of switches in multilevel inverter (MLI), Indonesian J. Elect. Engin. Computer Sci. 14 (2019)
1114–1124.[25] J. Vahidi,The combined Laplace-homotopy analysis method for partial differential equations, J. Math. Computer
Sci. 16 (2016) 88–102.
[26] A.R. Yıldız, B.S. Yıldız, S.M. Sait, S. Bureerat and N. Pholdee, A new hybrid Harris hawks-Nelder-Mead optimization algorithm for solving design and manufacturing problems, Mater. Test. 61 (2019) 735–743.
Phys. Lett. A 360 (2006) 109–113.
[2] S. Abbasbandy, The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KdV
equation, Phys. Lett. A 361 (2007) 478–483.
[3] K.A. Abed, Solving Kuramoto–Sivashinsky equation by the new iterative method and estimate the optimal parameters by using PSO algorithm, Indonesian J. Elect. Engin. Comput. Sci. 19 (2020) 709.
[4] K.A. Abed and A.A. Ahmad, The best parameters selection using pso algorithm to solving for ito system by new
iterative technique, Indonesian J. Elect. Engin. Comput. Sci. 18 (2020) 1638.
[5] K.A. Abed, Controlling of jerk chaotic system via linear feedback control strategies, Indonesian J. Elect. Engin.
Comput. Sci. 20 (2020) 370–378.
[6] Q. AF and B.J. Salim, Application new iterative method for solving modified Korteweg-Devries (MKdV) system
from three equations, J. Adv. Res. Dyn. Cont. Syst. 11 (2019) 2019–2026.
[7] H. Ahmad, T.A. Khan, P.S. Stanimirovic and I. Ahmad, Modified variational iteration technique for the numerical
solution of fifth order KdV-type equations, J. Appl. Comput. Mech. 6(SI) (2020) 1220–1227.
[8] M.M. Aziz and S.F. Al-Azzawi, A modification of nonlinear feedback controller, Int. J. Comput. Sci. Math. 13
(2021) 64–79.
[9] S. Al-Azzawi, Mujiarto, L. Patria, A. Sambas and W.S.M. Sanjaya, Stability of Lorenz system at the second
equilibria point based on Gardano’s method, J. Phys.: Conf. Ser. 1477 (2020) 022009.
[10] Z.S. Al-Talib and S. F. AL-Azzawi, Projective synchronization for 4D hyperchaotic system based on adaptive
nonlinear control strategy, Indonesian J. Elect. Engin. Comput. Sci. 19 (2020) 715–722.
[11] A. Fallahzadeh and K. Shakibi, A method to solve Convection-Diffusion equation based on homotopy analysis
method, J. Interpolat. App. Sci. Comput. 8 (2015).
[12] N.A. Golilarz, A. Addeh, H. Gao, L. Ali, A.M. Roshandeh, H.M. Munir and R. Ullah Khan, A new automatic
method for control chart patterns recognition based on ConvNet and harris hawks meta heuristic optimization
algorithm, IEEE Access 7 (2019) 149398–149405.
[13] V. Gupta and S. Gupta, Application of homotopy analysis method for solving nonlinear Cauchy problem, Surv.
Math. Appl. 7 (2012) 105–116.
[14] J.-H. He, Some asymptotic methods for strongly nonlinear equations, Int. J. Modern Phys. B 20 (2006) 1141–1199.
[15] J.-H. He, Homotopy perturbation method for solving boundary value problems, Phys. Lett. A 350 (2006) 87–88.
[16] J.-H. He, Comparison of homotopy perturbation method and homotopy analysis method, Appl. Math. Comput.
156 (2004) 527–539.
[17] A.A. Heidari, S.A. Mirjalili, H. Farisd, I. Aljarah, M. Mafarja and H. Chen, Harris hawks optimization: Algorithm
and applications, Future Gener. Computer Syst. 97 (2019) 849–872.
[18] M. Ismail and H. Ashi, A numerical solution for Hirota-Satsuma coupled KdV equation, Abst. Appl. Anal. 2014
(2014).
[19] H. Jafari and S. Seifi, Solving a system of nonlinear fractional partial differential equations using homotopy
analysis method, Commun. Nonlinear Sci. Numerical Simul.14 (2009) 1962–1969.
[20] S.-J. Liao, The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems, Ph. D. Thesis,
Shanghai Jiao Tong University Shanghai, 1992.
[21] S. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC press, 2003.
[22] S. Liao, Comparison between the homotopy analysis method and homotopy perturbation method, Applied Mathematics and Computation, 169 (2005) 1186-1194.
[23] A.F. Qasim, K. Adel Abed and O.S. Qasim, Optimal parameters for nonlinear Hirota-Satsuma coupled KdV
system by using hybrid firefly algorithm with modified Adomian decomposition, J. Math. Fund. Sci. 52 (2020)
339–352.
[24] M. Rasheed, R. Omar, M. Sulaiman and W. Abd Halim, Particle swarm optimisation (PSO) algorithm with
reduced number of switches in multilevel inverter (MLI), Indonesian J. Elect. Engin. Computer Sci. 14 (2019)
1114–1124.[25] J. Vahidi,The combined Laplace-homotopy analysis method for partial differential equations, J. Math. Computer
Sci. 16 (2016) 88–102.
[26] A.R. Yıldız, B.S. Yıldız, S.M. Sait, S. Bureerat and N. Pholdee, A new hybrid Harris hawks-Nelder-Mead optimization algorithm for solving design and manufacturing problems, Mater. Test. 61 (2019) 735–743.
)
Volume 13, Issue 1
March 2022Pages 663-671
- Receive Date: 11 August 2021
- Revise Date: 16 September 2021
- Accept Date: 23 September 2021