ed63674f-6c62-4c41-a897-23364bb13a0f20230323103035621wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON SYSTEMS2224-26781109-277710.37394/23202http://wseas.org/wseas/cms.action?id=4067125202312520232210.37394/23202.2023.22https://wseas.com/journals/systems/2023.phpNecibAbdelhalimDepartment of Mathematics and Computer Science University of Oum El Bouaghi, Algeria. Laboratory of Dynamics systems and control ALGERIARezzougImadDepartment of Mathematics and Computer Science University of Oum El Bouaghi, Algeria. Laboratory of Dynamics systems and control ALGERIABenbrahimAbdelouahabDepartment of Mathematics and Computer Science University of Oum El Bouaghi, Algeria. Laboratory of Dynamics systems and control ALGERIAIn this paper and in the first part of it, homotopy perturbation method is applied to solve second order differential equation with non-constant coefficients. The method yields solutions in convergent series forms with easily computable terms (the convergence of this series is demonstrated in this paper). The result shows that this method is very convenient and can be applied to large class of problems. As for the second part, we found a solution of Telegraph equation using the Laplace transform and Stehfest algorithm method. Next, we used method of Homotopy perturbation. Finally, we give some examples for illustration.3232023323202329530332https://wseas.com/journals/systems/2023/a645102-019(2023).pdf10.37394/23202.2023.22.32https://wseas.com/journals/systems/2023/a645102-019(2023).pdf10.46793/kgjmat2002.251nA. Necib and A. Merad. Laplace transform and Homotopy perturbation methods for solving the pseudohyperbolic integro-differential problems with purely integral conditions. Kragujevac Journal of Mathematics. Volume 44(2), Pages 251–272. 2020. A. Necib and A. Merad. Problèmes aux limites avec conditions non locales. Thèse de Doctorat. Université Oum El Bouaghi. http://hdl.handle.net/123456789/6591. 2018. 10.5802/jedp.343A. Necib and M. Bouzit. Problème mixte avec condition non locale pour une équation différenielle aux dérivées partielle d’ordre cinq de type mixte. Mémoire de Magister. Université Oum El Bouaghi. 2009. 10.1142/s0217979206034819He, J.H. New Interpretation of Homotopy Perturbation Method. International Journal of Modern Physics B, 20, 2561-2568, 2006. 10.1016/s0096-3003(01)00312-5He, J.H. Homotopy perturbation method: a new nonlinear analytical technique. Applied Mathematics and Computation 135 (1): 73–79, 2003. H. Stehfest. Numerical Inversion of the Laplace Transform, Comm. ACM 13, 47-49, 1970. 10.1016/j.chaos.2005.03.007J.H. He. Limit cycle and bifurcation of nonlinear problems, Chaos Soliton. Fract.26 827-833, 2005. 10.1016/s0096-3003(03)00341-2J.H. He. The homotopy perturbation method for non-linear oscillators with discontinuities. Appl. Math. Comput. 151(1), pp. 287-292, 2004. J.H. He. A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. Nonlinear Mech. 35 (1)37-43, 2000. 10.1016/s0045-7825(99)00018-3J. H. He. Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering.178:257-262, 1999. Liu, G.L. New research directions in singular perturbation theory: artificial parameter approach and inverse-perturbation technique. Proceedings of the 7th Conference of moder, 1997. 10.1016/s0955-7997(97)00043-xLiao, S.J. Boundary element method for general nonlinear differential operators. Engineering Analysis with Boundary Elements 20 (2): 91-99, 1997. 10.1016/0020-7462(94)00054-eLiao, S.J. An approximate solution technique not depending on small parameters: a special example. International Journal of Non-Linear Mechanics 30 (3): 371-380, 1995.