### Variational homotopy perturbation method for solving systems of homogeneous linear and nonlinear partial differential equations

Atika Faradilla, Aang Nuryaman, Asmiati Asmiati, Dewi Rakhmatia Nur

#### Abstract

The variational homotopy perturbation method is developed by combining variational iteration method and homotopy perturbation method. Variational iteration method has an efficient process in solving a wide variety of equations and systems of equations. Meanwhile, homotopy perturbation method yields a very rapid convergence of the solution series in most cases. The developed method, variational homotopy perturbation method, took full advantage of both methods. In this study, we described an application of the variational homotopy perturbation method to solve systems of homogeneous partial differential equations. Here we consider some initial value problems of homogeneous partial differential equation systems with two and three variables. The results show that the obtained solution using this method was in agreement with the solution using the homotopy analysis method and variational iteration method, which prove the validity of the variational homotopy perturbation method when applied to systems of partial differential equations.

#### Keywords

Variational Homotopy Perturbation Method; Variational Iteration Method; Homotopy Perturbation Method; Partial Differential Equation System.

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#### References

Akbarzade, M., & Langari, J. (2011). Application of variational iteration method to partial differential equation systems. International Journal of Mathematical Analysis, 5(17–20), 863–870.

Allahviranloo, T., Armand, A., & Pirmuhammadi, S. (2014). Variational homotopy perturbation method: An efficient scheme for solving partial differential equations in fluid mechanics. Journal of Mathematics and Computer Science, 09(04), 362–369. https://doi.org/10.22436/jmcs.09.04.12

Ateş, I., & Yildirim, A. (2009). Application of variational iteration method to fractional initialvalue problems. International Journal of Nonlinear Sciences and Numerical Simulation, 10(7), 877–883. https://doi.org/10.1515/IJNSNS.2009.10.7.877

Biazar, J. (2008). He’s homotopy perturbation method homotopy perturbation method for hemholtz. Int. J. Contemp. Math. Sciences, 3(15), 739–744.

Biazar, J., Badpeima, F., & Azimi, F. (2009). Application of the homotopy perturbation method to zakharov-kuznetsov equations. Computers and Mathematics with Applications, 58(11–12), 2391–2394. https://doi.org/10.1016/j.camwa.2009.03.102

Biazar, J., Eslami, M., & Ghazvini, H. (2007). Homotopy perturbation method for systems of partial differential equations. International Journal of Nonlinear Sciences and Numerical Simulation, 8(3), 413–418. https://doi.org/10.1515/IJNSNS.2007.8.3.413

Daga, A., & Pradhan, V. H. (2013). Variational homotopy perturbation method for solving nonlinear reaction – diffusion – convection problems. International Journal of Advanced Engineering Research and Studies, II(II), 11–14.

Gepreel, K. A. (2011). The homotopy perturbation method applied to the nonlinear fractional kolmogorov petrovskii piskunov equations. Applied Mathematics Letters, 24(8), 1428–1434. https://doi.org/10.1016/j.aml.2011.03.025

He, J. H. (1999a). Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178(3–4), 257–262. https://doi.org/dx.doi.org/10.1016/S0045-7825(99)00018-3

He, J. H. (1999b). Variational iteration method - a kind of non-linear analytical technique: Some examples. International Journal of Non-Linear Mechanics, 34(4), 699–708. https://doi.org/10.1016/s0020-7462(98)00048-1

He, J. H. (2005). Application of homotopy perturbation method to nonlinear wave equations. Chaos, Solitons & Fractals, 26(3), 695–700. https://doi.org/10.1016/j.chaos.2005.03.006

Hendi, F. A., Kashkari, B. S., & Alderremy, A. A. (2013). The variational homotopy perturbation method for solving ((n × n) + 1) dimensional burgers’ equations. Journal of Applied Mathematics, 2(2). https://doi.org/10.14419/ijamr.v2i2.899

Jin, L. (2009). Application of variational iteration method and homotopy perturbation method to the modified kawahara equation. Mathematical and Computer Modelling, 49(3–4), 573–578. https://doi.org/10.1016/j.mcm.2008.06.017

Matinfar, M., Mahdavi, M., & Raeisy, Z. (2010). The implementation of variational homotopy perturbation method for fisher’s equation. International Journal of Nonlinear Science, 9(2), 188–194.

Matinfar, M., & Saeidy, M. (2009). The homotopy perturbation method for solving higher dimensional initial boundary value problems of variable coefficients. World Journal of Modelling and Simulation, 5(1), 72–80.

Nofel, T. A. (2014). Application of the homotopy perturbation method to nonlinear heat conduction and fractional van der pol damped nonlinear oscillator. Applied Mathematics, 05(06), 852–861. https://doi.org/10.4236/am.2014.56081

Noor, M. A., & Mohyud-Din, S. T. (2008). Variational homotopy perturbation method for solving higher dimensional initial boundary value problems. Mathematical Problems in Engineering, 2008. https://doi.org/10.1155/2008/696734

Nuryaman, A. (2019). An analytical solution of 1-D pseudo homoneneous model for oxidation reaction using homotopy perturbation method. Journal of Research in Mathematics Trends and Technology, 1(1), 7–12. https://doi.org/10.32734/jormtt.v1i1.751

Olayiwola, M. (2016). Application of variational iteration method to the solution of convection-diffusion equation. Journal of Applied & Computational Mathematics, 05(02), 2–5. https://doi.org/10.4172/2168-9679.1000299

Sami Bataineh, A., Noorani, M. S. M., & Hashim, I. (2008). Approximate analytical solutions of systems of PDEs by homotopy analysis method. Computers and Mathematics with Applications, 55(12), 2913–2923. https://doi.org/10.1016/j.camwa.2007.11.022

Shang, X., & Han, D. (2010). Application of the variational iteration method for solving nth-order integro-differential equations. Journal of Computational and Applied Mathematics, 234(5), 1442–1447. https://doi.org/10.1016/j.cam.2010.02.020

Wazwaz, A. M. (2007). The variational iteration method for solving linear and nonlinear systems of PDEs. Computers and Mathematics with Applications, 54(7–8), 895–902. https://doi.org/10.1016/j.camwa.2006.12.059

Wu, Y., & He, J. H. (2018). Homotopy perturbation method for nonlinear oscillators with coordinate-dependent mass. Results in Physics, 10(June), 270–271. https://doi.org/10.1016/j.rinp.2018.06.015

DOI: http://dx.doi.org/10.24042/djm.v4i2.7825

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