A Hybrid Homotopy Perturbation Method with Natural Transform to Solve Partial Differential equations

Bachir N. Kharrat¹ , George A. Toma²

Section:Research Paper, Product Type: Journal-Paper
Vol.7 , Issue.4 , pp.14-19, Aug-2020

Online published on Aug 31, 2020

Copyright © Bachir N. Kharrat, George A. Toma . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

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Abstract :
In this paper, we present a new combination between the natural transform and the homotopy perturbation method. This hybrid technique allows to obtain numerical and analytical solutions for initial value problems represented by non - linear partial differential equations of a various normal orders. This presented method depends on applying the natural transform to solve the non - linear partial differential equation, then we construct the Homotopy on the transformed equation that including the initial conditions, after that we apply the inverse transform of the natural transform to find the solution of the equation. Obviously that the solution of the equation depends on the initial conditions. To show the power and accuracy of our method we present three initial value problems where we obtained exact or approximate solutions with high accuracy. This work also showed that this proposed hybrid method is suitable scheme for solving non - linear partial differential equations

Key-Words / Index Term :
Approximate solution, Exact solution, Homotopy perturbation method, Initial value problems, Natural Transform, Non-linear partial differential equations

References :
[1] J. H. He, ?A coupling method of a homotopy technique and a perturbation technique for non-linear problems,? International Journal of Non-Linear Mechanic, Vol. 35, No. 1, pp. 37-43, 2000.
[2] J. H. He, ?Homotopy perturbation technique,? Computer Methods in Applied Mechanics and Engineering, vol. 178, pp. 257-262, 1999.
[3] J. H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities,? Applied Mathematics and Computation, vol. 151, no. 1, pp. 287?292, 2004.
[4] J. H. He, ?Homotopy perturbation method: A new nonlinear analytical technique,? Applied Mathematics Computation, vol. 135 (1), pp. 73-79, 2003.
[5] J. H. He, ?Application of homotopy perturbation method to nonlinear wave Equations,? Chaos Solitons Fractals, vol. 26 (3), pp. 695-700, 2005.
[6] B. N. Kharrat, G. A.Toma, ?Development of Homotopy Perturbation Method for Solving Nonlinear Algebraic Equations,? International Journal of Scientific Research in Mathematical and Statistical Sciences, vol. 7 (2), pp. 47-50, 2020.
[7] M. A. Tahmina, A. M. Mansur, ?Homotopy Perturbation Method for Solving Highly Nonlinear Reaction-Diffusion-Convection Problem,? , American Journal of Mathematics and Statistics Vol. 9 (3). pp. 136-141, 2019.
[8] S. Iqbal, A. Javed, ?Application of optimal homotopy asymptotic method for the analytic solution of singular Lane?Emden type equation,? Applied Mathematics And Computation, Elsevier, 2011.
[9] A. Yildirim, D. Agirseven, ?The Homotopy Perturbation Method for Solving Singular Initial Value Problems,? International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 10 (2). 235, 2009.
[10] J. H. He, ?Application of homotopy perturbation method to nonlinear wave Equations,? Chaos Solitons Fractals, Vol. 26 (3), pp. 695-700, 2005.
[11] M. A. Tahmina, M. S. A. Moinuddin, A. M. Mansur, ?Semi-Analytical Approach to Solve Non-linear Differential Equations and Their Graphical Representations,? International Journal of Applied Mathematics & Statistical Sciences, Vol. 3, Issue 1, pp. 35-56, 2014.
[12] M. Ghoreishi, M. B. Ismail, ?The Homotopy Perturbation Method (HPM) for Nonlinear Parabolic Equation with Nonlocal Boundary Conditions,? Applied Mathematical Sciences, Vol. 5 (3), pp. 113 ? 123, 2011.
[13] S. H. Javeed, D. Baleanu, A. Waheed, M. Khan, H. Affan, ?Analysis of Homotopy Perturbation Method for Solving Fractional Order Differential Equations,? MDPI - Mathematics, Vol. 7, 40, 2019.
[14] H. Z. Khan, A. W. Khan, ?N-transform properties and applications,? NUST Journal of Engineering Sciences, Vol. 1, pp. 127-133, 2008.
[15] M. B. F. Belgacem, R. Silambarasan, ?Theory of natural transform,? Mathematics in Engineering Science and Aerospace, Vol. 3, No. 1, pp. 105-135, 2012.
[16] G. A. Toma, ?A Hybrid Semi-Analytical Method with Natural Transform for Solving Integro ? Differential Equations,? International Journal of Scientific Research in Mathematical and Statistical Sciences, vol. 7, Issue. 3, pp.1?7,2020.
[17] M. N. L. Tawfiqa, K. A. Jabberb, ?Solve the groundwater model equation using fourier transforms method,? International Journal of Applied Mathematics and Mechanics, vol. 5, No. 1, pp.75?80,2017.
[18] D. A. Polyanin, M. V. Alexander, ?Handbook of Mathematics for Engineers and Scientists,? CRC Press, 2006.
[19] F. Bulnes, ?Mathematical electrodynamics: groups, cohomology classes, unitary representations, orbits and integral transforms in electro-physics,? American Journal of Electromagnetics and Applications, Vol. 3, No. 6, pp. 43?52, 2015.
[20] D. Bolukbas, A. Arifergin, ?A radon transform interpretation of the physical optics integral,? Microwave and Optical Technology Letters, Vol. 44, No. 3, pp. 284?288, 2005.
[21] E. W. Higgins, D. C. Munson, ?A hankel transform approach to tomographic image reconstruction,? IEEE Transactions on Medical Imaging ,Vol. 7, No. 1, pp. 59?72, 1988.
[22] B. F. Belgacem, A. A. Karaballi, ?Sumudu transform fundamental properties,? Journal of Applied Mathematics and Stochastic Analysis, 2006.
[23] C. Yang, M. A. Wazwaz, Q. Zhou, W. Liu, ?Transformation of soliton states for a (2+1) dimensional fourth-order nonlinear schrdinger equation in the Heisenberg ferromagnetic spin chain,? Laser Physics, Vol. 29, No. 3, 2019.
[24] X. Liu, H. Triki, Q. Zhou, ?Generation and control of multiple solitons under the influence of parameters,? Nonlinear Dynamics, Vol. 95, No. 1, pp. 143?150, 2019.
[25] L. W. Yang, Q. Zhou, D. Mihalache, B. A. Malomed, ?One soliton shaping and two-soliton interaction in the fifth-order variable-coefficient nonlinear schrdinger equation,? Nonlinear Dynamics, Vol. 95, No. 1, pp. 369?380, 2019.
[26] A. Entesar, O. , W. Al-Hayani, "Hybridization of Genetic Algorithm with Homotopy Analysis Method for Solving Fractional Partial Differential Equations ", Eurasian Journal of Science & Engineering, 2019.
[27] N. Panagant, S. Bureerat, "Solving Partial Differential Equations Using a New Differential Evolution Algorithm", Mathematical Problems in Engineering, 2014.
[28] J. Biazar a, H. Ghazvini , "Convergence of the homotopy perturbation method for partial differential equations". Nonlinear Analysis: Real World Applications, Elsevier, 2009.
[29] L. F. Spevaka and O. A. Nefedova , "Solving a Two- Dimensional Nonlinear Heat Conduction Equation with Nonzero Boundary Conditions by the Boundary Element Method", AIP Conference Proceedings , 2017.
[30] Ming Li , C.S. Chen , A. Karageorghis, " The MFS for the solution of harmonic boundary value problems with non-harmonic boundary conditions ", Computers and Mathematics with Applications, Elsevier, 2013.
[31] S. Afshar, B. Soltanalizadeh ," Solution Of The Two- Dimensional Second-Order Diffusion Equation With Nonlocal Boundary Condition", International Journal of Pure and Applied Mathematics, 2014.
[32] M. S. Rawashdeh, Ha. Al-Jammal, "New approximate solutions to fractional nonlinear systems of partial differential equations using the FNDM", Advances in Difference Equations, Springer, 2016.
[33] A. Cheniguel, " Numerical Simulation of Two-Dimensional Diffusion Equation with Non Local Boundary Conditions", International Mathematical Forum, 2012.
[34] M. Tahmina Akter, M. A. Chowdhury, "Homotopy Perturbation Method for Solving Highly Nonlinear Reaction-Diffusion-Convection Problem", American Journal of Mathematics and Statistics, 2019.
[35] E. E. Eladdad and E. A. Tarif ," On the Coupling of the Homotopy Perturbation Method and New Integral Transform for Solving Systems of Partial Differential Equations", Advances in Mathematical Physics, 2019.
[36] C. Jesuraj, A. Rajkumar, "A New Modified Sumudu Transform Called Raj Transform to Solve Differential Equations and Problems in Engineering and Science", International Journal on Emerging Technologies, 2020.
[37] B. N. Kharrat and G. Toma, " A New Hybrid Sumudu Transform With Homotopy Perturbation Method For Solving Boundary Value Problems", Middle-East Journal of Scientific Research, 28 (2): 142-149, 2020.
[38] B. N. Kharrat and G. Toma, "Development of Homotopy Perturbation Method for Solving Nonlinear Algebraic Equations", International Journal of Scientific Research in Mathematical and Statistical Sciences. V 7, Issue 2, pp. 47-50, 2020.