A Numerical Method for Solving Singularly Perturbed Quasilinear Boundary Value Problems on Shishkin Mesh

Yıl 2022, Cilt: 14 Sayı: 1, 145 - 154, 30.06.2022

Hakkı Duru Mutlu Demirbaş

https://doi.org/10.47000/tjmcs.1010528

Öz

In this paper, singularly perturbed quasilinear boundary value problems are taken into account. With this purpose, a finite difference scheme is proposed on Shishkin-type mesh (S-mesh). Quasilinearization technique and interpolating quadrature rules are used to establish the numerical scheme. Then, an error estimate is derived. A numerical experiment is demonstratedto verify the theory.

Anahtar Kelimeler

Boundary value problem, error estimate, difference scheme, Shishkin mesh

Kaynakça

• Aga Bullo, T., Degla, G.A., Duressa, G.F., Uniformly convergent higher-order finite difference scheme for singularly perturbed parabolic problems with non-smooth data, Journal of Applied Mathematics and Computational Mechanics, 20(1)(2021), 5-16.
• Alquran, M.T., DoĞan, N., Variationaliteration method for solving two-parameter singularly perturbed two point boundary value problem, Applications and Applied Mathematics: An International Journal (AAM), 5(1)(2010), 81-95.
• Amiraliyev, G.M., Mamedov, Y.D., Difference schemes on the uniform mesh for singularly perturbed pseudo-parabolic equations, Turkish Journal of Mathematics, 19(1995), 207-222.
• Cakir, M., Gunes, B., Duru, H., A novel computational method for solving nonlinear Volterra integro-differential equation, Kuwait Journal of Science, 48(1)(2021), 1-9.
• Cakir, M., Gunes, B., Exponentially fitteddifference scheme for singularly perturbed mixed integro-differential equations, Georgian Mathematical Journal, (2022).
• Cassani, D., Wang, Y., Zhang, J., A unifed approach to singularly perturbed quasilinear Schrödinger equations, Milan Journal of Mathematics, 88(2020), 507-534.
• Chen, S-B., Soradi-Zeid, S., Dutta, H., Mesrizadeh, M., Jahanshahi, H. et al., Reproducing kernel Hilbert space method for nonlinear second order singularly perturbed boundary value problems with time-delay, Chaos, Solitons and Fractals, 144(2021), 110674.
• Çakır, M., Güneş, B., A new difference method for the singularly perturbed Volterra-Fredholm integro-differential equations on a Shishkin mesh, Hacettepe Journal of Mathematics and Statistics, 51(3)(2022), 787-799.
• Doğan, N., Ertürk, V.S., Momani, S., Akın, Ö., Yıldırım, A., Differential transform method for solving singularly perturbed Volterra integral equations, Journal of King Saud University-Science, 23(2011), 223-228.
• Doğan, N., Ertürk, V.S., Momani, S., He's variational iteration method for solving the singularly perturbed Volterra integral equations, World Applied Sciences Journal, 22(11)(2013), 1657-1661.
• Duru, H., Gunes, B., Numerical solutions for singularly perturbed nonlinear reaction diffusion problems on the piecewise equidistant mesh, Erzincan University Journal of Science and Technology, 12(1)(2019), 425-436.
• Duru, H., Gunes, B., The finite difference method on adaptive mesh for singularly perturbed nonlinear 1D reaction-diffusion boundary value problems, Journal of Applied Mathematics and Computational Mechanics, 19(4)(2020), 45-56.
• Erdogan, F., Sakar, M.G., A quasilinearization technique for the solution of singularly perturbed delay differential equation, Mathematics in Natural Science, 2(2018), 1-7.
• Erdogan, F., Sakar, M.G., Saldır, O., A finite difference method on layer-adapted mesh for singularly perturbed delay differential equations, Applied Mathematics and Nonlinear Sciences, 5(1)(2020), 425-436.
• Gunes, B., Chianeh, A.B., Demirbas, M., Comparison of multiple scales method and finite difference method for solving singularly perturbed convection diffusion problem}, Gumushane University Journal of Science and Technology Institute, 10(4)(2020), 1169-1181.
• Konyaev, Y.A., Workneh, A.Z., Estimating the norm of solution of singularly perturbed quasilinear problems for ODE systems with nonlinear normal matrices on the semiaxis, Discrete and Continuous Models and Applied Computational Science, 4(2013), 5-10.
• Kumar, S., Layer-adapted methods for quasilinear singularly perturbed delay differential problems, Applied Mathematics and Computation, 233(1)(2014), 214-221.
• Lin, R., Ye, X., Zhang, S., Zhu, P., A weak Galerkin finite element method for singularly perturbed convection-diffusion-reaction problems, SIAM J. Numer. Anal., 56(3)(2018), 1482-1497.
• Nefedov, N.N., Davydova, M.A., Constrast structures in singularly perturbed quasilinear reactions-diffusion-advection equations, Differential Equations, 49(2013), 688-706.
• Ni, M., Pang, Y., Levashova, N.T., Nikolaeva, O.A., Internal layers for a singularly perturbed second-order quasilinear differential equation with discontinuous right-hand side, Differential Equations, 53(2017), 1567-1577.
• Samarski, A.A., The Theory of Difference Schemes, Moscow M.V. Lomonosov State University, Russia, 2001.
• Sekar, E., Tamilselvan, A., Third order singularly perturbed delay differential equation of reaction diffusion type with integral boundary condition, Journal of Applied Mathematics and Computational Mechanics, 18(2)(2019), 99-110.
• Selvakumar, K., Lazarus, G.P., A fitted operator and fitted mesh method for singularly perturbed convection diffusion problem, International Journal of Recent Research Aspects, (2018), 544550.
• Shishkin, G.I., Higher-order accurate method for a quasilinear singularly perturbed elliptic convection-diffusion equation, Sib. Zh. Vychisl. Mat., 9(1)(2006), 81-108.
• Voevodin, A.F., Factorization method for linear and quasilinear singularly perturbed boundary value problems for ordinary differential equations, Numerical Analysis and Applications, 2(2009), 1-12.
• Xie, S., Zhu, P., Wang, X., Error analysis of weak Galerkin finite element methods for time-dependent convection-diffusion equations, Applied Numerical Mathematics, 137(2019), 19-33.
• Zheng, Q., Ye, F., Numerical solution of quasilinear singularly perturbed problems by the principle of equidistribution, Journal of Applied Mathematics and Physics, 8(10)(2020), 103603.