A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior

Musa ÇAKIR, (Van Yüzüncü Yıl Üniversitesi , Fen Fakültesi, Matematik Bölümü, Van, Türkiye)
R. Younis MASIHA, (Van Yüzüncü Yıl Üniversitesi , Fen Fakültesi, Matematik Bölümü, Van, Türkiye)
Derya ARSLAN (Bitlis Eren Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Bitlis, Türkiye)
Gazi University Journal of Science
2 0
Yıl: 2020 Cilt: 33 Sayı: 1 Sayfa Aralığı: 152 - 166 Metin Dili: İngilizce DOI: 10.35378/gujs.542897 İndeks Tarihi: 11-10-2020

A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior

Öz:
This study deals with a new approach method for solving singularly perturbed boundary value problem of convection-diffusion type. Firstly, bounds on the solution and its derivative of solution to be used later in the article are obtained. This robust method is constructed with fitted difference scheme on a uniform mesh. It is proved that the presented method is first-order convergent with respect to the perturbation parameter 𝜀 in the discrete maximum norm. Two examples are given to illustrate the efficiency of the method. The numerical results are presented in tables and figures.
Anahtar Kelime:

Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık
  • [1] Amiraliyev, G. M., Mamedov, Y. D., “Difference schemes on the uniform mesh for singularly perturbed pseudo-parabolic equations”, Turkish Journal of Mathematics, 19: 207-222, (1995).
  • [2] Cıbık, A., “The effect of a sparse grad–div stabilization on control of stationary Navier–Stokes equations”, Journal of Mathematical Analysis and Applications, 437(1): 613-628, (2016).
  • [3] Amiraliyev, G. M., Çakır, M., “A uniformily convergent difference scheme for singularly perturbed problem with convective term and zeroth order reduced equation”, International Journal of Applied Mathematics, 2(12): 1407-1419, (2000).
  • [4] Cıbık, A., Kaya, S., “Finite element analysis of a projection-based stabilization method for the Darcy–Brinkman equations in double-diffusive convection”, Applied Numerical Mathematics , 64: 35-49, (2013).
  • [5] Bakhvalov, N. S., “On optimization of methods for solving boundary-value problems in the presence of a boundary layer”, Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 9(4): 841-859, (1969).
  • [6] Nayfeh, A. H., “Introduction to Perturbation Techniques”, Wiley, New York, (1993).
  • [7] Bitsadze, A. V., Samarskii, A. A., “On some simpler generalization of linear elliptic boundary value problems”, Doklady Akademii Nauk SSSR, 185: 739-740, (1969).
  • [8] Cakır, M., “Uniform second-order difference method for a singularly perturbed three-point boundary value problem”, Hindawi Publishing Corporation Advances in Difference Equations, 13 pages, (2010).
  • [9] Cakır, M., Amiraliyev, G. M., “Numerical solution of a singularly perturbed three-point boundary value problem”, International Journal of Applied Mathematics, 84: 1465- 1481, (2007).
  • [10] Arslan, D., “Stability and convergence analysis on Shishkin mesh for a nonlinear singularly perturbed problem with three-point boundary condition", Quaestiones Mathematicae, (2019), Doi: 10.2989/16073606.2019.1636894.
  • [11] Cakır, M., Arslan, D., “A numerical method for nonlinear singularly perturbed multi-point boundary value problem”, Journal of Applied Mathematics and Physics, 4: 1143-1156, (2016).
  • [12] Cakır, M., Arslan, D., “Finite difference method for nonlocal singularly perturbed problem”, International Journal of Modern Research in Engineering and Technology, 1(5): 25-39, (2016).
  • [13] Cakır, M., Arslan, D., “Numerical solution of the nonlocal singularly perturbed problem”, International Journal of Modern Research in Engineering and Technology, 1(5): 13-24, (2016).
  • [14] Chegis, R., “The Numerical solution of problems with small parameter at higher derivatives and nonlocal conditions”, Lietuvos Matematikos Rinkinys (in Russian), 28: 144-152, (1988).
  • [15] Cimen, E., Amiraliyev, G. M., “A uniform convergent method for singularly perturbed nonlinear differential-difference equation”, Journal of Informatics and Mathematical Sciences, 9: 191–199, (2017).
  • [16] Cimen, E., Cakir, M., “Numerical treatment of nonlocal boundary value problem with layer behaviour”, Bulletin of the Belgian Mathematical Society-Simon Stevin, 24, (2017).
  • [17] Cimen, E., “A priori estimates for solution of singularly perturbed boundary value problem with delay in convection term”, Journal of Mathematical Analysis, 8: 202–211, (2017).
  • [18] Doolan, E. P., Miller, J. J. H., Schilders, W. H. A., “Uniform Numerical Methods for Problem with Initial and Boundary Layers”, Boole Press, Dublin, (1980).
  • [19] Farrel, P. A., Miller, J. J. H., O’Riordan, E., Shishkin, G. I., “A uniformly convergent finite difference scheme for a singularly perturbed semilinear equation”, SIAM Journal on Numerical Analysis, 33: 1135-1149, (1996).
  • [20] Gupta, C. P., Trofimchuk, S. I., “A sharper condition for the solvability of a three-point second order boundary value problem”, Journal of Mathematical Analysis and Applications, 205: 586– 597, (1997).
  • [21] Miller, J. J. H., O’Riordan, E., Shishkin, G. I., “Fitted numerical methods for singular perturbation problems”, World Scientific, Singapore, (1996).
  • [22] Roos, H. G., Stynes, M., “Tobiska, L., Robust Numerical Methods for Singularly Perturbed Differential Equation: Convection-Diffusion and Flow Problems”, Springer-Verlag, Berlin, 604, (2008).
  • [23] Cıbık, A. B., Yılmaz, F. N., “Variational multiscale method for the optimal control problems of convection-diffusion-reaction equations”, Turkish Journal of Mathematics, 42: 164-180, (2018).
  • [24] O'Malley, R. E., “Singular Perturbation Methods for Ordinary Differential Equations”, Springer Verlag, New York, (1991).
  • [25] Stynes, M., Roos, H. G., Tobiska, L., “Robust Numerical Methods for Singularly Perturbed Differential Equations”, Springer-Verlag, Berlin, (2008).
  • [26] Jankowski, T., “Existence of solutions of differential equations with nonlinear multipoint boundary conditions”, Computers & Mathematics with Applications, 47: 1095-1103, (2004).
  • [27] Geng, Z., Tang, Q., “Piecewise shooting reproducing kernel method for linear singularly perturbed boundary value problem”, Applied Mathematics Letters, 62: 1-6, (2016).
APA Çakır M, Masiha R, Arslan D (2020). A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior. , 152 - 166. 10.35378/gujs.542897
Chicago Çakır Musa,Masiha R.YOUNİS,Arslan Derya A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior. (2020): 152 - 166. 10.35378/gujs.542897
MLA Çakır Musa,Masiha R.YOUNİS,Arslan Derya A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior. , 2020, ss.152 - 166. 10.35378/gujs.542897
AMA Çakır M,Masiha R,Arslan D A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior. . 2020; 152 - 166. 10.35378/gujs.542897
Vancouver Çakır M,Masiha R,Arslan D A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior. . 2020; 152 - 166. 10.35378/gujs.542897
IEEE Çakır M,Masiha R,Arslan D "A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior." , ss.152 - 166, 2020. 10.35378/gujs.542897
ISNAD Çakır, Musa vd. "A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior". (2020), 152-166. https://doi.org/10.35378/gujs.542897
APA Çakır M, Masiha R, Arslan D (2020). A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior. Gazi University Journal of Science, 33(1), 152 - 166. 10.35378/gujs.542897
Chicago Çakır Musa,Masiha R.YOUNİS,Arslan Derya A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior. Gazi University Journal of Science 33, no.1 (2020): 152 - 166. 10.35378/gujs.542897
MLA Çakır Musa,Masiha R.YOUNİS,Arslan Derya A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior. Gazi University Journal of Science, vol.33, no.1, 2020, ss.152 - 166. 10.35378/gujs.542897
AMA Çakır M,Masiha R,Arslan D A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior. Gazi University Journal of Science. 2020; 33(1): 152 - 166. 10.35378/gujs.542897
Vancouver Çakır M,Masiha R,Arslan D A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior. Gazi University Journal of Science. 2020; 33(1): 152 - 166. 10.35378/gujs.542897
IEEE Çakır M,Masiha R,Arslan D "A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior." Gazi University Journal of Science, 33, ss.152 - 166, 2020. 10.35378/gujs.542897
ISNAD Çakır, Musa vd. "A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior". Gazi University Journal of Science 33/1 (2020), 152-166. https://doi.org/10.35378/gujs.542897
Sayfa Oluşturulma Tarihi
23-04-2024 13:47

İLETİŞİM

TÜBİTAK ULAKBİM TR Dizin

Yüzüncüyıl, İşçi Blokları Mahallesi

Muhsin Yazıcıoğlu Caddesi No:51/C

06530 Çankaya / ANKARA +90 (312) 298 92 00

trdizin@tubitak.gov.tr

TÜBİTAK ULAKBİM Ulusal Akademik Ağ ve Bilgi Merkezi Cahit Arf Bilgi Merkezi © Tüm Hakları Saklıdır. Gizlilik Politikası Kullanım Şartları