Numerical Solutions of the Singularly Perturbed Semilinear Delay Differential Equations

Yıl 2022, Cilt: 27 Sayı: 2, 330 - 343, 30.08.2022

Hakkı Duru , Bahar Gürbüz

https://doi.org/10.53433/yyufbed.1085501

Öz

In this study, the numerical solution of the singularly perturbed quasilinear differential equations with constant delay is investigated by the method of integral identities with use of linear basis functions and interpolating quadrature formulas, a finite difference scheme is established on Boglaev-Bakhvalov type mesh. The error approximations are obtained in the discrete maximum norm. A numerical example is solved to clarify the theoretical analysis.

Anahtar Kelimeler

Bakhvalov mesh, Difference scheme, Singular perturbation, Uniform convergence

Singüler Pertürbe Özellikli Yarılineer Gecikmeli Diferansiyel Denklemlerin Nümerik Çözümleri

Öz

Bu çalışmada sabit gecikme içeren singüler pertürbe özellikli kuasilineer diferansiyel denklemlerin nümerik çözümleri araştırılmıştır. Lineer baz fonksiyonları ve interpolasyon kuadratür kurallarını kullanarak Boglaev-Bakhvalov tipli şebeke üzerinde sonlu fark şeması kurulmuştur. Ayrık maksimum normda hata yaklaşımları elde edilmiştir. Teorik analizi doğrulamak için bir adet nümerik örnek çözülmüştür.

Anahtar Kelimeler

Bakhvalov şebeke, Düzgün yakınsaklık, Fark şeması, Singüler pertürbasyon

Kaynakça

• Amiraliyev, G. M., & Mamedov, Y. D. (1995). Difference schemes on the uniform mesh for singularly perturbed pseudo-parabolic equations. Turkish Journal of Mathematics, 19(3), 207-222.
• Amiraliyev, G. M., & Duru, H. (2003). A uniformly convergent difference method for the periodical boundary value problem. Computers & Mathematics with Applications, 46(5-6), 695-703.
• Amiraliyev, G. M., & Duru, H. (2005). A note on a parameterized singular perturbation problem. Journal of Computational and Applied Mathematics, 182(1), 233-242.
• Amiraliyev, G. M., & Cimen, E. (2010). Numerical method for a singularly perturbed convection–diffusion problem with delay. Applied Mathematics and Computation, 216(8), 2351-2359.
• Amiraliyeva, I. G., Erdogan, F., & Amiraliyev, G. M. (2010). A uniform numerical method for dealing with a singularly perturbed delay initial value problem. Applied Mathematics Letters, 23(10), 1221-1225.
• 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.
• Cakir, M., & Amiraliyev, G. M. (2005). A finite difference method for the singularly perturbed problem with nonlocal boundary condition. Applied Mathematics and Computation, 160(2), 539-549.
• Doolan, E. R., Miller J. J. H., & Schilders, W. H. A. (1980). Uniform Numerical Methods for Problems with Initial and Boundary Layers. Dublin: Boole Press,.
• Duru, H., & Gunes, B., (2019). Numerical solutions for singularly perturbed nonlinear reaction diffusion problems on the piecewise equidistant mesh. Erzincan University Journal of Science and Technology, 12 (1), 425-436.
• Duru, H., & Güneş, B. (2020). 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).
• Erdogan, F., & Amiraliyev, G. M. (2012). Fitted finite difference method for singularly perturbed delay differential equations. Numerical Algorithms, 59(1), 131-145.
• Erdogan, F., Sakar, M. G., & Saldır, O. (2020). A finite difference method on layer-adapted mesh for singularly perturbed delay differential equations. Applied Mathematics and Nonlinear Sciences, 5(1), 425-436.
• Farrell, P. A., Hegarty, A. F., Miller, J. J. H., O'Riordan E., & Shishkin, G. I. (2000). Robust Computational Techniques for Boundary Layers. New York : Chapman-Hall/CRC.
• Gunes, B., Chianeh, A. B., & Demirbas, M., (2020). 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), 1169-1181.
• Kumar, S. (2014). Layer-adapted methods for quasilinear singularly perturbed delay differential problems. Applied Mathematics and Computation, 233(1), 214-221.
• Miller, J. J. H., O'Riordan, E., & Shishkin, G. I. (1996). Fitted Numerical Methods for Singular Perturbation Problems. Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions. Singapore: World Scientific.
• Roos, H. G., Stynes, M., & Tobiska, L. (1996). Numerical Methods for Singularly Perturbed Differential Equations, Convection-Diffusion and Flow Problems. Berlin: Springer Verlag.
• Samarski, A. A., (2001). The Theory of Difference Schemes. Moscow, Russia: M.V. Lomonosov State University.
• Zheng, Q., & Ye, F., (2020). Numerical solution of quasilinear singularly perturbed problems by the principle of equidistribution. Journal of Applied Mathematics and Physics, 8(10), 103603.