Homotopy Perturbation Elzaki Transform Method for Obtaining the Approximate Solutions of the Random Partial Differential Equations

Abstract

The series solutions of the random nonlinear partial differential equations have been examined by a hybrid method. The random nonlinear partial differential equations are studied by both normal and uniform distributions. Two initial-value problems are indicated to exemplify the influence of the solutions acquired by the hybrid method. Also, the functions for the first and second moments of the approximate solutions of random nonlinear partial differential equations are acquired in the MAPLE software. The hybrid method is implemented to analyze the solutions of the random nonlinear partial differential equations. MAPLE software is used to find the solutions. Besides, MAPLE software is used for the drawing the figures.

Keywords

• Expected value
• Homotopy perturbation elzaki transform method
• Variance

References

1. [1] Lü, Q., Zuazua, E., “Averaged controllability for random evolution partial differential equations”, Journal de Mathématiques Pures et Appliquées, 105(3): 367-414, (2016).
2. [2] Nabian, M.A., Meidani, H., “A deep learning solution approach for high-dimensional random differential equations”, Probabilistic Engineering Mechanics, 57: 14-25, (2019).
3. [3] Guignard, D., “Partial differential equations with random input data: A perturbation approach”, Archives of Computational Methods in Engineering, 26(5): 1313-1377, (2019).
4. [4] Egorova, V.N., Jódar, L., “Quadrature Integration Techniques for Random Hyperbolic PDE Problems”, Mathematics, 9(2): 160, (2021).
5. [5] Anaç, H., Merdan, M., and Kesemen, T., “Solving for the random component time-fractional partial differential equations with the new Sumudu transform iterative method”, SN Applied Sciences, 2: 1-11, (2020).
6. [6] Khudair, A. R., Ameen, A.A., and Khalaf, S.L., “Mean square solutions of second-order random differential equations by using adomian decomposition method”, Applied Mathematical Sciences, 5: 2521-2535, (2011).
7. [7] He, J.H., “Homotopy perturbation method: a new nonlinear analytical technique”, Applied Mathematics and Computation, 135(1): 73-79, (2003).
8. [8] He, J.H., “Homotopy perturbation method for solving boundary value problems”, Physics Letters A, 350(1-2): 87-88, (2006).

9. [9] He, J.H., “Addendum: new interpretation of homotopy perturbation method”, International Journal of Modern Physics B, 20(18): 2561-2568, (2006).
10. [10] Odibat, Z., Momani, S., “Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order”, Chaos, Solitons & Fractals, 36(1): 167-174, (2008).
11. [11] Anaç, H., Merdan, M., Bekiryazıcı, Z., and Kesemen, T., “Bazı Rastgele Kısmi Diferansiyel Denklemlerin Diferansiyel Dönüşüm Metodu ve Laplace-Padé Metodu Kullanarak Çözümü”, Gümüşhane Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 9(1): 108-118, (2019).
12. [12] Ayaz, F., “Solutions of the system of differential equations by differential transform method”, Applied Mathematics and Computation, 147(2): 547-567, (2004).
13. [13] Kangalgil, F., Ayaz, F., “Solitary wave solutions for the KdV and mKdV equations by differential transform method”, Chaos, Solitons & Fractals, 41(1): 464-472, (2009).
14. [14] Merdan, M., “A new application of modified differential transformation method for modeling the pollution of a system of lakes”, Selçuk Journal of Applied Mathematics, 11(2): 27-40, (2010).
15. [15] Yüzbaşi, Ş., Ismailov, N., “Differential Transform Method to Solve Two-Dimensional Volterra Integral Equations with Proportional Delays”, New Trends in Mathematical Sciences, 5(4): 65-71, (2017).
16. [16] Zhou, J.K. “Differential Transform and Its Applications for Electrical Circuits”, China: Huazhong University Press, (1986).
17. [17] He, J.H., “Variational iteration method-a kind of non-linear analytical technique: some examples”, International Journal of Non-Linear Mechanics, 34(4): 699-708, (1999).
18. [18] Ekolin, G., “Finite difference methods for a nonlocal boundary value problem for the heat equation”, BIT, 31: 245-261, (1991).
19. [19] Smith, G.D. “Numerical Solution of Partial Differential Equations”, UK: Oxford University Press, (1965).
20. [20] Cortés, J.C., Jódar, L., Villafuerte, L., and Villanueva, R.J., “Computing mean square approximations of random diffusion models with source term”, Mathematics and Computers in Simulation, 76(1-3): 44-48, (2007).
21. [21] El-Tawil, M.A., Sohaly, M.A., “Mean square convergent three points finite difference scheme for random partial differential equations”, Journal of the Egyptian Mathematical Society, 20(3): 188-204, (2012).
22. [22] Jena, R.M., Chakraverty, S., “Solving time-fractional Navier–Stokes equations using homotopy perturbation Elzaki transform”, SN Applied Sciences, 1(1): 1-13, (2019).
23. [23] Feller, W. “An introduction to probability theory and its applications”, USA: John Wiley & Sons, (1971).
24. [24] Elzaki, T.M., “Applications of new transform “Elzaki transform” to partial differential equations”, Global Journal of Pure and Applied Mathematics, 7(1): 65-70, (2011).
25. [25] Elzaki, T.M., Hilal, E.M.A., “Homotopy perturbation and Elzaki transform for solving nonlinear partial differential equations”, Mathematical Theory and Modeling, 2(3): 33-42, (2012).