NUMERICAL SOLUTION OF HYPERBOLIC GOURSAT PARTIAL DIFFERENTIAL EQUATIONS WITH HYBRID CENTRAL DIFFERENCE - TAYLOR SERIES EXPANSIONS METHOD
DOI:
https://doi.org/10.24191/mjoc.v9i1.26051Keywords:
Central Finite Difference Method, Goursat Problem, Hyperbolic Partial Differential Equation, Numerical Differentiation, Taylor Series ExpansionsAbstract
This paper investigates a new method for solving the Goursat partial differential equation (PDE) using a combination of the central finite difference method (FDM) and Taylor series expansion. The study evaluates the effectiveness and accuracy of this new approach, analyzing linear Goursat problems and conducting multiple numerical experiments. The simulation study demonstrates that the suggested approach surpasses the existing method in terms of performance and accuracy. Applying this proposed scheme will minimize the cost, especially for engineers that might apply this model in solving their real-life problems.
References
Ahmad, J., & Mushtaq, M. (2015). Exact solution of linear and non-linear Goursat problems. Universal Journal of Computational Mathematics, 3, 14-17.
Datta, M., Alam, M. S., Hahiba, U., Sultana, N., & Hossain, M. B. (2021). Exact Solution of Goursat Problem with Linear and Non-linear Partial Differential Equations by Double Elzaki Decomposition Method. Applied Mathematics, 11(1), 5-11.
Deraman, R. F., & Nasir, M. A. S. (2015). Goursat Partial Differentian Equation (Numerical Integration Method). Germany: LAP Lambert.
Fadugba, S. E., Ogunrinde, R. B., & Ogunrinde, R. R. (2021). Stability analysis of a proposed scheme of order five for first order ordinary differential equations. Malaysian Journal of Computing (MJoC), 6(2), 898-912.
Jacquemin, T., Tomar, S., Agathos, K., Mohseni-Mofidi, S., & Bordas, S. P. (2020). Taylor-series expansion based numerical methods: A primer, performance benchmarking and new approaches for problems with non-smooth solutions. Archives of Computational Methods in Engineering, 27(5), 1465-1513.
Karimov, S. T., & Yulbarsov, K. A. (2023). The Goursat problem for a third-order pseudoparabolic equation with the Bessel operator. In AIP Conference Proceedings 2781(1). AIP Publishing.
Kim Son, N. T., Long, H. V., & Dong, N. P. (2021). On Goursat problem for fuzzy random partial differential equations under generalized Lipschitz conditions. Iranian Journal of Fuzzy Systems, 18(2), 31-49.
Meziani, M. S. E., Boussetila, N., Rebbani, F., & Benrabah, A. (2021). Iterative regularization method for an abstract inverse Goursat problem. Khayyam Journal of Mathematics, 1-19.
Mokdad, M. (2021). Conformal Scattering and the Goursat Problem for Dirac Fields in the Interior of Charged Spherically Symmetric Black Holes. arXiv preprint arXiv:2101.04166.
Naseem, T. (2022). Novel techniques for solving Goursat partial differential equations in the linear and nonlinear regime. Mathematics (ISSN: 2790-1998 Print, 2790-3257 Online), 1(1), 17-37.
Nasir, M.A.S., & Md Ismail, A.I. (2012). Application of high-order compact finite difference scheme to nonlinear Goursat problem. World Academy of Science, Engineering and Technology (WASET), 68, 1605-1615.
Nasir, M.A.S., & Md Ismail, A.I. (2013). A fourth order compact finite difference scheme for Goursat problem. Sains Malaysiana, 42(3), 341-346.
Pandey, P.K. (2014a). A Finite Difference Method for Numerical Solution of Goursat Problem of Partial Differential Equation. OALib, 01(06), 1–6.
Pandey, P.K. (2014b). A Fourth Order Finite Difference Method for Numerical Solution of the Goursat Problem. Acta Technica Jaurinensis, 7(3), 319–327.
Pantaleón, C., & Ghosh, A. (2015). Taylor series expansion using matrices: An implementation in MATLAB®. Computers & Fluids, 112, 79-82.
Saharizan, N. S., & Zamri, N. (2019). Numerical solution for a new fuzzy transform of hyperbolic Goursat partial differential equation. Indonesian Journal of Electrical Engineering and Computer Science, 16(1), 292-298.
Salvi, C., Cass, T., Foster, J., Lyons, T., & Yang, W. (2021). The Signature Kernel is the solution of a Goursat PDE. SIAM Journal on Mathematics of Data Science, 3(3), 873-899.
Sitnik, S. M., & Karimov, S. T. (2023). Solution of the Goursat problem for a fourth-order hyperbolic equation with singular coefficients by the method of transmutation operators. Mathematics, 11(4), 951.
Son, N. T. K., & Thao, H. T. P. (2019). On Goursat problem for fuzzy delay fractional hyperbolic partial differential equations. Journal of Intelligent & Fuzzy Systems, 36(6), 6295-6306.
Tailor, M. R., & Bhathawala, P. H. (2011). Linearization of nonlinear differential equation by Taylor’s series expansion and use of Jacobian linearization process. International Journal of Theoretical and Applied Science, 4(1), 36-38.
Tian, C., Chang, K. C., & Chen, J. (2020). Application of hyperbolic partial differential equations in global optimal scheduling of UAV. Alexandria Engineering Journal, 59(4), 2283-2289.
Twizell, E.H. (1984). Computational method for partial differential equations. New York: Ellis Horwood.
Wazwaz, A.M. (1995). The decomposition method for approximate solution of Goursat problem. Applied Mathematics and Computation, 69(2), 299-311.
Wazwaz, A.M. (2009). Partial Differential Equations and Solitary Waves Theory. Beijing: Higherof Education Press and Springer-Verlag Berlin Heidelberg.
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