On Solution of Complex Equations with the Homotopy Perturbation Method

Year 2024, Volume: 14 Issue: 2, 157 - 167, 31.12.2024

Murat Düz , Şeyda Çamlıca

https://doi.org/10.37094/adyujsci.1484685

Abstract

In this study, complex differential equations are solved using the homotopy perturbation method (HPM). Firstly, we separated the real and imaginary parts of the equation. Thus, from one unknown equation is obtained two unknown equation systems. Later, we applied HPM real and imaginary to parts of the equation. So, we have obtained real and imaginary parts of the solution.

Keywords

Homotopy Perturbation method, Complex equations, Equation system

Homotopi Pertürbasyon Metodu ile Kompleks Denklemlerin Çözümü Üzerine

Abstract

Bu çalışmada kompleks denklemler homotopi pertürbasyon metodu(HPM) kullanılarak çözüldü. İlk önce denklemi reel ve imajiner kısımlarına ayırdık. Böylece bir bilinmeyenli denklemden iki bilnmeyenli denklem sistemi elde edildi. Daha sonra HPM’yi denklemin reel ve imajiner kısımlarına uyguladık. Böylece çözümün reel ve imajiner kısımlarını elde ettik.

Keywords

Homotopi pertürbasyon metodu, Kompleks denklemler, Denklem sistemi

References

• [1] Bers, L., Theory of Pseudo-Analytic Functions. New York University Institute for Mathematics and Mechanics, 1953.
• [2] Vekua, I. N., Verallgemeinerte Analytische Funktionen. Berlin, Akademie Verlag, 1st edition, 538p, 1959.
• [3] Merdan, M., Merdan, M., Şahin, R., Investigation of Behavior on Solutions of Lane–Emden Complex Differential Equations by a Random Differential Transformation Method, Complexity, 2023, 1-12, 713454, 2023.
• [4] Sezer, M., Tanay, B., Gülsu, M.A., Polynomial approach for solving high-order linear complex differential equations with variable coefficients, Erciyes Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 25(1-2), 374 – 389, 2009.
• [5] Gülsu, M., Sezer, M., Approximate solution to linear complex differential equation by a new approximate approach, Applied Mathematics and Computation, 185, 636-645, 2007
• [6] Düz, M., On an application of Laplace transforms, New Trends in Mathematical Sciences, 5, 193-198, 2017.
• [7] Düz, M., Application of Elzaki transform to first order constant coefficients complex equations, Bulletin of International Mathematical Virtual Institute, 7, 387-393, 2017.
• [8] Düz, M., Solution of complex differential equations by using Fourier Transform, International Journal of Applied Mathematics, 31(1), 23-32, 2018.
• [9] Kayabaşı, H., Murat Düz, M., Issa, A., Solution of complex differential equations by using variational iteration method, The Aligarh Bulletin of Mathematics, 43(1), 57 – 63, 2024.
• [10] Düz, M., Solutions of complex equations with adomian decomposition method, TWMS Journal of Applied and Engineering Mathematics, 7, 66-73, 2017.
• [11] He, J., Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation, 135, 73–79, 2003.
• [12] He, J., Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178(3-4), 257-262, 1999.
• [13] He, J., A coupling method of a homotopy technique and a perturbation technique for non linear problems, International Journal of Non-Linear Mechanics, 35(1), 37-43, 2000.
• [14] Abbasbandy, S., Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition method, Applied Mathematics and Computation, 172(1), 485-490, 2006.
• [15] Abbasbandy, S., Numerical solutions of the integral equations: Homotopy perturbation method and Adomian’s decomposition method, Applied Mathematics and Computation, 173(1), 493-500, 2006.
• [16] Babolian, E., Azizi, A., Saeidia, J., Some notes on using the homotopy perturbation method for solving time-dependent differential equations, Mathematical and Computer Modelling, 50(1-2), 213-224, 2009.
• [17] Türkyılmazoglu, M., An analytic shooting-like approach for the solution of nonlinear boundary value problems, Mathematical and Computer Modelling, 53(9-10), 1748-1755, 2011.
• [18] Merdan, M., Homotopy perturbation method for solving modelling the pollution of a system of lakes, Süleyman Demirel University Faculty of Arts and Science Journal of Science, 4 (1) 99-111, 2009.
• [19] Merdan., M., A multistage homotopy perturbation method for solving human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells model, Mathematical and Computational Applications, 14(2), 85-96, 2009.
• [20] Merdan, M., Homotopy perturbation method for solving a model for HIV infection of CD4+ T cells, İstanbul Commerce University Journal of Science, 6(12), 39-52, 2007.
• [21] Türkyılmazoglu., M., Is homotopy perturbation method the traditional Taylor series expansion, Hacettepe Journal of Mathematics and Statistics, 44(3), 651 – 657, 2015.