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The finite difference method on adaptive mesh for singularly perturbed nonlinear 1D reaction diffusion boundary value problems

Treść / Zawartość
Identyfikatory
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this paper, we study singularly perturbed nonlinear reaction-diffusion equations. The asymptotic behavior of the solution is examined. The difference scheme which is accomplished by the method of integral identities with using of interpolation quadrature rules with weight functions and remainder term integral form is established on adaptive mesh. Uniform convergence and stability of the difference method are discussed in the discrete maximum norm. The discrete scheme shows that orders of convergent rates are close to 2. An algorithm is presented, and some problems are solved to validate the theoretical results.
Rocznik
Strony
45--56
Opis fizyczny
Bibliogr. 25 poz., rys., tab.
Twórcy
autor
  • Department of Mathematics, Van Yuzuncu Yil University Van, Turkey
  • Department of Mathematics, Van Yuzuncu Yil University Van, Turkey
Bibliografia
  • [1] Chakravarthy, P.P., & Kumar, K. (2017). A novel method for singularly perturbed delay differential equations of reaction-diffusion type. Differential Equations and Dynamical Systems. https://doi.org/10.1007/s12591-017-0399-x.
  • [2] Zeng, F., Lu, F., Li, C., Burrage, K., Tuner, I., & Anh, V. (2014). A Crank-Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion equation. SIAM Journal on Numerical Analysis, 52(6), 2599-2622.
  • [3] Mitchell, A.R., & Bruch, Jr. J.C. (1985). A numerical study of chaos in a reaction-diffusion equation. Numerical Methods for Partial Differential Equations, 1(1), 13-23.
  • [4] Baeumer, B., Kovacs, M., & Meerschaert, M.M. (2008). Numerical solutions for fractional reaction-diffusion equations. Computer and Mathematics with Applications, 55, 2212-2226.
  • [5] Das, S., Gupta, P.K., & Ghosh, P. (2011). An approximate solution of nonlinear fractional reaction-diffusion equation. Applied Mathematical Modelling, 35, 4071-4076.
  • [6] Pulkkinen, A. (2011). Blow-up profiles of solutions for the exponential reaction-diffusion equations. Mathematical Methods in the Applied Sciences. https://doi.org/10.1002/mma.1501.
  • [7] Liao, W., & Yan, Y. (2010). Singly diagonally ımplicit Runge-Kutta method for time-dependent reaction-diffusion equation. Numerical Methods for Partial Differential Equations, 27(6), 1423-1441.
  • [8] Kopteva, N., & Stynes, M. (2011). Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction-diffusion problem. Numerische Mathematik, 119(4), 787-810.
  • [9] Cheng, R. (2006). Singularly perturbed problem for non-local reaction-diffusion equations involving two small parameters. Journal of Shanghai University (English Edition), 10(6), 479-483.
  • [10] Zhu, G., & Chen, S. (2010). Convergence and superconvergence analysis of an anisotropic noncoforming finite element methods for singularly perturbed reaction-diffusion problems. Computational and Applied Mathematics, 234(10), 3048-3063.
  • [11] Shishkin, G.I., & Shishkina, L.P. (2011). Improved approximations of the solution and derivatives to a singularly perturbed reaction-diffusion equation based on the solution deceomposition method. Computational Mathematics and Mathematical Physics, 51(6), 1020-1049.
  • [12] Siraj, M.K., Duressa, G.F., & Bullo, T.A. (2019). Fourth-order stable central difference with Richardson extrapolation method for second-order self-adjoint singularly perturbed boundary value problems. Journal of Egyptian Mathematical Society, 27, 50.
  • [13] Gelu, F.W., Duressa, G.F., & Bullo, T.A. (2017). Sixth-order compact finite difference method for singularly perturbed 1D reaction-diffusion problems. Journal of Taibah University for Science, 11, 302-308.
  • [14] Clavero, C., & Gracia, J.L. (2012). A high order HODIE finite difference scheme for 1D parabolic singularly perturbed reaction-diffusion problems. Applied Mathematics and Computation, 218(9), 5067-5080.
  • [15] Raza, N., & Butt, A.R. (2013). Numerical solutions of singularly perturbed reaction diffusion equation with Sobolev gradients. Journal of Function Spaces. https://doi.org/10.1155/2013/ 542897.
  • [16] Srivastava, A. (2017). Numerical simulation of singularly perturbed reaction-diffusion equations using finite element method. Computational Mathematics and Modelling, 28, 3.
  • [17] Cen, Z., Le, A., & Xu, A. (2017). A high-order finite difference scheme for a singularly perturbed reaction-diffusion problem with an interior layer. Advences in Difference Equations, 202.
  • [18] Hong, Y., Jung, C., & Laminie, J. (2013). Singularly perturbed reaction-diffusion equations in a circle with numerical applications. International Journal of Computer Mathematics, 90, 11, 2308-2325.
  • [19] He, X., & Wang, K. (2019). Uniformly convergent novel finite difference methods for singularly perturbed reaction-diffusion equations. Numerical Methods for Partial Differential Equations. https://doi.org/10.1002/num.22405.
  • [20] Singh, J., Kumar, S., & Kumar, M. (2018). A domain decomposition method for solving singularly perturbed parabolic reaction-diffusion problesm with time delay. Numerical Methods for Partial Differential Equation, 34(5), 1849-1866.
  • [21] Sekar, E., & Tamilselvan, A. (2019). Parameter uniform method for a singularly perturbed system of delay differential equations of reaction-diffusion type wtih integral boundary conditions. International Journal of Applied and Computational Mathematics, 5, 85.
  • [22] Amiraliyev, G.M., & Duru, H. (2002). Nümerik Analiz. Pegem Yayıncılık.
  • [23] Boglaev, I.P. (1984). Approximate solution of a nonlinear boundary value problem with a small parameter for the highest-order differential. USSR Computational Mathematics and Mathematical Physics, 24(6), 30-35.
  • [24] Amiraliyev, G.M., & Mamedov, Y.D. (1995). Difference schemes on the uniform mesh for singularly perturbed pseudo-parabolic equations. Turkish Journal of Mathematics, 19, 207-222.
  • [25] Samarskii, A.A. (2001). The Theory of Difference Schemes. Moscow; M.V. Lomonosov State University.
Uwagi
Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-24379b62-3294-4712-a750-9dce6fdad424
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