Goursat problem in Hyperbolic partial differential equations with variable coefficients solved by Taylor collocation method

Document Type : Research Article

Authors

• F. Birem
• A. Boulmerka
• H. Laib
• C. Hennous

Laboratory of Mathematics and their interactions, University Center Abdelhafid Boussouf, Mila, Algeria.

10.22067/ijnao.2024.85895.1364

Abstract

The hyperbolic partial differential equation (PDE) has important practical uses in science and engineering. This article provides an estimate for solving the Goursat problem in hyperbolic linear PDEs with variable coefficients. The Goursat PDE is transformed into a second kind of linear Volterra in-tegral equation. A convergent algorithm that employs Taylor polynomials is created to generate a collocation solution, and the error using the maxi-mum norm is estimated. The paper includes numerical examples to prove the method’s effectiveness and precision.

Keywords

• Hyperbolic partial differential equations
• Goursat problem
• Volterra integral equation
• Collocation method
• Taylor polynomials

Main Subjects

• Numerical methods for integral equations and its applications

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References

[1] Asaduzzaman, M., Özger, F. and Kilicman, A. Analytical approximate solutions to the nonlinear Fornberg –Whitham type equations via modified variational iteration method, Partial Differ. Equ. Appl. Math. 9 (2024), 100631.
[2] Bai, D. and Zhang, L. The quadratic B-spline finite-element method for the coupled schrödinger –Boussinesq equations, Int. J. Comput. Math. 88 (8) (2011), 1714–1729.
[3] Bellour, A., Bousselsal, M. and Laib, H. Numerical solution of second-order linear delay differential and integro-differential equations by using Taylor collocation method, Int. J. Comput. Methods, 17 (09) (2020), 1950070.
[4] Boutayeb, A. and Twizell, E. Finite-difference methods for the solution of special eighth-order boundary-value problems, Int. J. Comput. Math. 48 (1-2) (1993), 63–75.
[5] Brunner, H. Collocation methods for Volterra integral and related func-tional differential equations, Cambridge university press, 15, 2004.
[6] Bülbül, B. and Sezer, M. Taylor polynomial solution of hyperbolic type partial differential equations with constant coefficients, Int. J. Comput. Math. 88 (3) (2011), 533–544.
[7] Çelik, A. and Düzgün, A. Approximate solution of ordinary linear differ-ential equations with analytical complex functions coefficient by means of Taylor matrix method, Int. J. Comput. Math. 82 (6) (2005), 765–775.
[8] Datta, M., Alam, M.S., Hahiba, U., Sultana, N. and Hossain, M.B. Exact solution of Goursat Problem with Linear and Non-linear Partial Differential Equations by Double Elzaki Decomposition Method, Applied Mathematics 11 (1) (2021), 5–11.
[9] Day, J.T. A Runge-Kutta method for the numerical solution of the Gour-sat problem in hyperbolic partial differential equations, Comput. J. 9 (1) (1966), 81–83.

[10] Deniz, S., Ozger, F., Ozger, Z.O., Mohiuddine, S.A. and Ersoy, M.T. Numerical solution of fractional Volterra integral equations based on ra-tional chebyshev approximation, Miskolc Math. Notes, 24 (3) (2023),1287–1305.
[11] Drignei, M.C. A numerical algorithm for a coupled hyperbolic boundary value problem, Int. J. Comput. Methods 19 (10) (2022), 2250027.
[12] Evans, D. and Sanugi, B. Numerical solution of the Goursat problem by a nonlinear trapezoidal formula, Applied Mathematics Letters 1 (3) (1988), 221–223.
[13] Heydari, M.H., Hooshmandasl, M. and Ghaini, F.M. A new approach of the Chebyshev wavelets method for partial differential equations with boundary conditions of the telegraph type, Appl. Math. Model. 38 (5-6) (2014), 1597–1606.
[14] Ismail, M. and Alamri, S. Highly accurate finite difference method for coupled nonlinear schrödinger equation, Int. J. Comput. Math. 81 (3) (2004), 333–351.
[15] Jeffrey, A. and Taniuti, T. Non-Linear Wave Propagation With Applica-tions to Physics and Magnetohydrodynamics by A Jeffrey and T Taniuti, Elsevier, 2000.
[16] Laib, H., Bellour, A. and Boulmerka, A. Taylor collocation method for a system of nonlinear Volterra delay integro-differential equations with application to COVID-19 epidemic, Int. J. Comput. Math. (2021), 1–25.
[17] Laib, H., Bellour, A. and Boulmerka, A. Taylor collocation method for high-order neutral delay Volterra integro-differential equations, J. Innov. Appl. Math. Comput. Sci. 2 (1) (2022), 53–77.
[18] Laib, H., Bellour, A. and Bousselsal, M. Numerical solution of high-order linear Volterra integro-differential equations by using Taylor collocation method, Int. J. Comput. Math. 96 (5) (2019), 1066–1085.
[19] Laib, H., Boulmerka, A., Bellour, A. and Birem, F. Numerical solution of two-dimensional linear and nonlinear Volterra integral equations using Taylor collocation method, J. Comput. Appl. Math. 417 (2023), 114537.

[20] Mahdy, A.M.S., Shokry, D. and Lotfy, K. Chelyshkov polynomials strat-egy for solving 2-dimensional nonlinear Volterra integral equations of the first kind, Comput. Appl. Math. 41 (2022), 257.
[21] Maleknejad, K. and Shahabi, M. Numerical solution of two-dimensional nonlinear Volterra integral equations of the first kind using operational matrices of hybrid functions, Int. J. Comput. Math. (2022), 1–18.
[22] Maleknejad, K., Sohrabi, S. and Baranji, B. Application of 2d-bpfs to nonlinear integral equations, Commun. Nonlinear Sci. Numer. Simul. 15(3) (2010), 527–535.
[23] Mckee, S., Tang, T. and Diogo, T. An Euler-type method for two-dimensional Volterra integral equations of the first kind, IMA J. Numer. Anal. 20 (3) (2000), 423–440.
[24] Pandey, P.K. A finite difference method for numerical solution of Goursat problem of partial differential equation, Open Access Libr J. 1 (2014), 1–6.
[25] Scott, E. Determination of the Riemann function, Amer. Math. Monthly 80 (8) (1973), 906–909.
[26] Wollkind, D.J. Applications of linear hyperbolic partial differential equa-tions: Predator-prey systems and gravitational instability of nebulae, Math. Model. 7 (2-3) (1986), 413–428.
[27] Zheng, X. and Wei, Z. Discontinuous Legendre wavelet Galerkin method for reaction –diffusion equation, Int. J. Comput. Math. 94 (9) (2017), 1806–1832.
[28] Zheng, X., Yang, X., Su, H. and Qiu, L. Mixed discontinuous Legendre wavelet Galerkin method for solving elliptic partial differential equations, Int. J. Comput. Math. 88 (17) (2011), 3626–3645.