The Homotopy Perturbation Method to Solve a Wave Equation

Authors

  • P.M. Gouder Department of Mathematics, KLE Dr. MS Sheshgiri College of Engg. & Technology, Belgavi 590008, Karnataka, India https://orcid.org/0000-0002-7526-2158
  • V.H. Kolli Department of Mathematics, KSS Arts, Science and Commerce College, Karnatak University, Gadag, Karnataka, India
  • Md. Hanif Page Department of Mathematics, KLE Technological University, Hubbalii 580031, Karnatak, India https://orcid.org/0000-0002-8007-6413
  • Krishna B. Chavaraddi Department of Mathematics, SS Government First Grade College and PG Studies Centre, Nargund 582207, Karnataka, India https://orcid.org/0000-0002-5292-3152
  • Praveen Chandaragi Department of Mathematics, SS Government First Grade College and PG Studies Centre, Nargund 582207, Karnataka, India https://orcid.org/0000-0001-6570-2503

DOI:

https://doi.org/10.26713/cma.v13i2.1764

Keywords:

Homotopy perturbation method, Wave equation, Non-local conditions, Exact solution

Abstract

In the paper, we discuss applications of Homotopy Perturbation Method (HPM) related to wave equations subjected to non-local conditions and the method is applied to two test problems in the paper. The method was introduced by J.-H. He (Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering 178(3-4) (1999), 257 – 262) and the solutions are matched against exact solutions as in the literature. The results indicate that the HPM produces accurate solutions and faster converging with less computational effort.

Downloads

Download data is not yet available.

References

S.D. Barforoushi, M. Rahimi and S. Danaee, Homotopy perturbation method for solving governing equation of nonlinear free vibration of systems with serial linear and nonlinear stiffness on a frictionless contact surface, University Politehnica of Bucharest Scientific Bulletin, Series A: Applied Mathematics and Physics 73(4) (2011), 107 – 118, URL: https://www.scientificbulletin.upb.ro/rev_docs_arhiva/full40544.pdf.

S.A. Beilin, Existence of solutions for one-dimensional wave equations with nonlocal conditions, Electronic Journal of Differential Equations 2001(76) (2001), 1 – 8, URL: http://emis.maths.tcd.ie/journals/EJDE/Volumes/2001/76/beilin.pdf.

J. Biazar and F. Azimi, He’s homotopy perturbation method for solving Helmholtz equation, International Journal of Contemporary Mathematical Sciences 3(15) (2008), 739 – 744.

R. Ezzati and S.M.R. Mousavi, Application of homotopy perturbation method for solving Brinkman momentum equation for fully developed forced convection in a porous saturated channel, Mathematical Sciences 5(2) (2011), 111 – 123, https://www.sid.ir/en/Journal/ViewPaper.aspx?ID=250627.

J. He, A new approach to nonlinear partial differential equations, Communications in Nonlinear Science and Numerical Simulation 2(4) (1997), 230 – 235, DOI: 10.1016/S1007-5704(97)90007-1.

J.H. He, Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation 135(1) (2003), 73 – 79, DOI: 10.1016/S0096-3003(01)00312-5.

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

J.H. He, The homotopy perturbation method for solving nonlinear wave equation, Chaos, Solitons & Fractals 26(3) (2005), 695 – 700, DOI: 10.1016/j.chaos.2005.03.006.

L. Jin, Analytical approach to the sine-Gordon equation using homotopy perturbation method, International Journal of Contemporary Mathematical Sciences 4(5) (2009), 225 – 231.

G.L. Karakostas and P.Ch. Tsamatos, Positive solutions for a nonlocal boundary-value problem with increasing response, Electronic Journal of Differential Equations 2000(73) (2000), 1 – 8, URL: https://ejde.math.txstate.edu/Volumes/2000/73/karakostas.pdf.

S. Liao, Comparison between the homotopy analysis method and homotopy perturbation method, Applied Mathematics and Computation 169(2) (2005), 1186 – 1194, DOI: 10.1016/j.amc.2004.10.058.

S. Liao, Notes on the homotopy analysis method: Some definitions and theorems, Communications in Nonlinear Science and Numerical Simulation 14(4) (2009), 983 – 997, DOI: 10.1016/j.cnsns.2008.04.013.

G. L. Liu, New research directions in singular perturbation theory: artificial parameter approach and inverse perturbation technique, Proceeding of the 7th Conference of the Modern Mathematics and Mechanics, Shanghai, September 1997, pp. 47 – 53.

R. Ma, A survey on nonlocal boundary value problems, Applied Mathematics E-Notes 7 (2007), 257 – 279, http://www.kurims.kyoto-u.ac.jp/EMIS/journals/AMEN/2007/070907-5.pdf.

A. Rezania, A.R. GHorbali, D.D. Ganji and H. Bararnia, Application on Homotopy-perturbation and Variational Iteration Methods for Heat Equation, Australian Journal of Basic and Applied Sciences 3(3) (2009), 1863 – 1874, URL: http://www.ajbasweb.com/old/ajbas/2009/1863-1874.pdf.

A. Roozi, E. Alibeiki, S.S. Hosseini, S.M. Shafiof and M. Ebrahimi, Homotopy perturbation method for special nonlinear partial differential equations, Journal of King Saud University - Science 23(1) (2011), 99 – 103, DOI: 10.1016/j.jksus.2010.06.014.

M. Waqas, M. Farooq, M.I. Khan, A. Alsaedi, T. Hayat and T. Yasmeen, Magnetohydrodynamic (MHD) mixed convection flow of micropolar liquid due to nonlinear stretched sheet with convective condition, International Journal of Heat and Mass Transfer 102 (2016), 766 – 772, DOI: 10.1016%2Fj.ijheatmasstransfer.2016.05.142.

M. Waqas, M.I. Khan, T. Hayat, A. Alsaedi and M.I. Khan, On Cattaneo-Christov heat ßux impact for temperature-dependent conductivity of Powell-Eyring liquid, Chinese Journal of Physics 55(3) (2017), 729 – 737, DOI: 10.1016/j.cjph.2017.02.003.

M. Waqas, S.A. Shehzad, T. Hayat, M.I. Khan and A. Alsaedi, Simulation of magnetohydrodynamics and radiative heat transportation in convectively heated stratified flow of Jeffrey nanomaterial, Journal of Physics and Chemistry of Solids 133 (2019), 45 – 51, DOI: 10.1016/j.jpcs.2019.03.031.

M. Waqas, T. Hayat, M. Farooq, S.A. Shehzad and A. Alsaedi, Cattaneo-Christov heat flux model for flow of variable thermal conductivity generalized Burgers fluid, Journal of Molecular Liquids 220 (2016), 642 – 648, DOI: 10.1016/j.molliq.2016.04.086.

M. Waqas, T. Hayat, S.A. Shehzad and Alsaedi, Analysis of forced convective modified Burgers liquid flow considering Cattaneo-Christov double diffusion, Results in Physics 8 (2018), 908 – 913, DOI: 10.1016/j.rinp.2017.12.069.

L.-N. Zhang and J.-H. He, Homotopy perturbation method for the solution of electrostatic potential differential equation, Mathematical Problems in Engineering 2006 (2006), Article ID 083878, DOI: 10.1155/MPE/2006/83878.

Downloads

Published

17-08-2022
CITATION

How to Cite

Gouder, P., Kolli, V., Page, M. H., Chavaraddi, K. B., & Chandaragi, P. (2022). The Homotopy Perturbation Method to Solve a Wave Equation. Communications in Mathematics and Applications, 13(2), 691–701. https://doi.org/10.26713/cma.v13i2.1764

Issue

Vol. 13 No. 2 (2022)

Section

Research Article

License

Creative Commons Licence 4.0 by  

Authors who publish with this journal agree to the following terms:

  • Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  • Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  • Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.