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Year 2020, Volume: 33 Issue: 1, 152 - 162, 01.03.2020
https://doi.org/10.35378/gujs.542897

Abstract

References

  • [1] Amiraliyev, G.M. and Mamedov, Y.D., “Difference schemes on the uniform mesh for singularly perturbed pseudo-parabolic equations”, Tr. J. of Math., 19(1995):207-222, (1995). Google scholar.[2] Amiraliyev, G.M., “Difference method for a singularly perturbed initial value problem”, Turkish Journal of Mathematics, 22: 283-294, (1998). Google scholar.[3] Amiraliyev, G.M. and 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). Google scholar.[4] Amiraliyev, G.M. and Çakır, M., “Numerical solution of the singularly perturbed problem with nonlocal boundary condition”, Applied Mathematics and Mechanics, 23: 755 764, (2002). Google scholar. [5] Bakhvalov, N.S., “On optimization of methods for solving boundary-value problems in the presence of a boundary layer”, The use of special transformation the numerical solution of bounary-layer problems, Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki. 9(4): 841-859, (1969). Google scholar.
  • [6] Nayfeh, A.H., Introduction to Perturbation Techniques, Wiley, New York, (1993).[7] Bitsadze, A.V. and Samarskii, A.A., 1969. On some simpler generalization of linear elliptic boundary value problems. Doklady Akademii Nauk SSSR. 185: 739-740. Google scholar.[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). Google scholar.[9] Cakır, M. and Amiraliyev, G.M., “Numerical solution of a singularly perturbed three-point boundary value problem”, International Journal of Applied Mathematics, 84: 1465- 1481, (2007). Google scholar.[10] Cakır, M. and Amiraliyev, G.M., “A numerical method for a singularly perturbed three-point boundary value problem”, Hindawi Publishing Corporation Journal of Applied Math, 17 pages, (2010). Google scholar.[11] Cakır, M. and Arslan, D., “A numerical method for nonlinear singularly perturbed multi-point boundary value problem”, Journal of Applied Mathematics and Physics, 4: 1143-1156, (2016). Google scholar.[12] Cakır, M. and Arslan, D., “Finite difference method for nonlocal singularly perturbed problem”, International Journal of Modern Research in Engineering and Technology, 1(5): 25-39, (2016). Google scholar.[13] Cakır, M. and Arslan, D., “Numerical solution of the nonlocal singularly perturbed problem”, International Journal of Modern Research in Engineering and Technology, 1(5):13-24, (2016). Google scholar.[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). Google scholar. [15] Cimen, E. and Amiraliyev, G.M. “A uniform convergent method for singularly perturbed nonlinear differential-difference equation”, Journal of Informatics and Mathematical Sciences, 9: 191–199, (2017). Google scholar.[16] Cimen, E. and Cakir, M. “Numerical treatment of nonlocal boundary value problem with layer behaviour”, Bull. Belg. Math. Soc. Simon Stevin, 24, (2017). Google scholar.[17] Cimen, E. “A priori estimates for solution of singularly perturbed boundary value problem with delay in convection term”, J. Math. Anal., 8: 202–211, (2017). Google scholar.[18] Doolan, E.P., Miller, J.J.H. And Schilders, W.H.A., Uniform Numerical Methods for Problem with Initial and Boundary Layers, Boole Press, Dublin, (1980). Google scholar.
  • [19] Farrel, P.A., Miller, J.J.H., O’Riordan, E. and Shishkin, G.I., “A uniformly convergent finite difference scheme for a singularly perturbed semilinear equation”, SIAM Journal on Numerical Analysis, 33:1135-1149, (1996). Google scholar.[20] Gupta, C.P. and 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). Google scholar.[21] Miller, J.J.H., O’Riordan, E. and Shishkin, G.I., Fitted numerical methods for singular perturbation problems, World Scientific, Singapore, (1996). Google scholar.[22] Roos, H.G., Stynes, M. and Tobiska, L., Robust Numerical Methods for Singularly Perturbed Differential Equation: Convection-Diffusion and Flow Problems, Springer-Verlag, Berlin, 604, (2008). Google scholar.[23] Jankowski, T., “Existence of solutions of differential equations with nonlinear multipoint bondary conditions”, Comput. Math. Appl., 47:1095-1103, (2004). Google scholar.[24] O'Malley, R.E., Singular Perturbation Methods for Ordinary Differential Equations. Springer Verlag, New York (1991). Google scholar.[25] Stynes, M, Roos, H.G and Tobiska, L., Robust Numerical Methods for Singularly Perturbed Dif- ferential Equations, Springer-Verlag, Berlin (2008). Google scholar.[26] Geng, Z. and Tang, Q., Piecewise shooting reproducing kernel method for linear singularly perturbed boundary value problem, Applied Mathematics Letters, 62, 1-6, (2016). Google scholar.

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

Year 2020, Volume: 33 Issue: 1, 152 - 162, 01.03.2020
https://doi.org/10.35378/gujs.542897

Abstract

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.

References

  • [1] Amiraliyev, G.M. and Mamedov, Y.D., “Difference schemes on the uniform mesh for singularly perturbed pseudo-parabolic equations”, Tr. J. of Math., 19(1995):207-222, (1995). Google scholar.[2] Amiraliyev, G.M., “Difference method for a singularly perturbed initial value problem”, Turkish Journal of Mathematics, 22: 283-294, (1998). Google scholar.[3] Amiraliyev, G.M. and 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). Google scholar.[4] Amiraliyev, G.M. and Çakır, M., “Numerical solution of the singularly perturbed problem with nonlocal boundary condition”, Applied Mathematics and Mechanics, 23: 755 764, (2002). Google scholar. [5] Bakhvalov, N.S., “On optimization of methods for solving boundary-value problems in the presence of a boundary layer”, The use of special transformation the numerical solution of bounary-layer problems, Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki. 9(4): 841-859, (1969). Google scholar.
  • [6] Nayfeh, A.H., Introduction to Perturbation Techniques, Wiley, New York, (1993).[7] Bitsadze, A.V. and Samarskii, A.A., 1969. On some simpler generalization of linear elliptic boundary value problems. Doklady Akademii Nauk SSSR. 185: 739-740. Google scholar.[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). Google scholar.[9] Cakır, M. and Amiraliyev, G.M., “Numerical solution of a singularly perturbed three-point boundary value problem”, International Journal of Applied Mathematics, 84: 1465- 1481, (2007). Google scholar.[10] Cakır, M. and Amiraliyev, G.M., “A numerical method for a singularly perturbed three-point boundary value problem”, Hindawi Publishing Corporation Journal of Applied Math, 17 pages, (2010). Google scholar.[11] Cakır, M. and Arslan, D., “A numerical method for nonlinear singularly perturbed multi-point boundary value problem”, Journal of Applied Mathematics and Physics, 4: 1143-1156, (2016). Google scholar.[12] Cakır, M. and Arslan, D., “Finite difference method for nonlocal singularly perturbed problem”, International Journal of Modern Research in Engineering and Technology, 1(5): 25-39, (2016). Google scholar.[13] Cakır, M. and Arslan, D., “Numerical solution of the nonlocal singularly perturbed problem”, International Journal of Modern Research in Engineering and Technology, 1(5):13-24, (2016). Google scholar.[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). Google scholar. [15] Cimen, E. and Amiraliyev, G.M. “A uniform convergent method for singularly perturbed nonlinear differential-difference equation”, Journal of Informatics and Mathematical Sciences, 9: 191–199, (2017). Google scholar.[16] Cimen, E. and Cakir, M. “Numerical treatment of nonlocal boundary value problem with layer behaviour”, Bull. Belg. Math. Soc. Simon Stevin, 24, (2017). Google scholar.[17] Cimen, E. “A priori estimates for solution of singularly perturbed boundary value problem with delay in convection term”, J. Math. Anal., 8: 202–211, (2017). Google scholar.[18] Doolan, E.P., Miller, J.J.H. And Schilders, W.H.A., Uniform Numerical Methods for Problem with Initial and Boundary Layers, Boole Press, Dublin, (1980). Google scholar.
  • [19] Farrel, P.A., Miller, J.J.H., O’Riordan, E. and Shishkin, G.I., “A uniformly convergent finite difference scheme for a singularly perturbed semilinear equation”, SIAM Journal on Numerical Analysis, 33:1135-1149, (1996). Google scholar.[20] Gupta, C.P. and 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). Google scholar.[21] Miller, J.J.H., O’Riordan, E. and Shishkin, G.I., Fitted numerical methods for singular perturbation problems, World Scientific, Singapore, (1996). Google scholar.[22] Roos, H.G., Stynes, M. and Tobiska, L., Robust Numerical Methods for Singularly Perturbed Differential Equation: Convection-Diffusion and Flow Problems, Springer-Verlag, Berlin, 604, (2008). Google scholar.[23] Jankowski, T., “Existence of solutions of differential equations with nonlinear multipoint bondary conditions”, Comput. Math. Appl., 47:1095-1103, (2004). Google scholar.[24] O'Malley, R.E., Singular Perturbation Methods for Ordinary Differential Equations. Springer Verlag, New York (1991). Google scholar.[25] Stynes, M, Roos, H.G and Tobiska, L., Robust Numerical Methods for Singularly Perturbed Dif- ferential Equations, Springer-Verlag, Berlin (2008). Google scholar.[26] Geng, Z. and Tang, Q., Piecewise shooting reproducing kernel method for linear singularly perturbed boundary value problem, Applied Mathematics Letters, 62, 1-6, (2016). Google scholar.
There are 3 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Mathematics
Authors

Derya Arslan 0000-0001-6138-0607

Musa Çakır This is me 0000-0002-1979-570X

Younis Masiha This is me 0000-0002-5405-0594

Publication Date March 1, 2020
Published in Issue Year 2020 Volume: 33 Issue: 1

Cite

APA Arslan, D., Çakır, M., & Masiha, Y. (2020). A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior. Gazi University Journal of Science, 33(1), 152-162. https://doi.org/10.35378/gujs.542897
AMA Arslan D, Çakır M, Masiha Y. A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior. Gazi University Journal of Science. March 2020;33(1):152-162. doi:10.35378/gujs.542897
Chicago Arslan, Derya, Musa Çakır, and Younis Masiha. “A Novel Numerical Approach for Solving Convection-Diffusion Problem With Boundary Layer Behavior”. Gazi University Journal of Science 33, no. 1 (March 2020): 152-62. https://doi.org/10.35378/gujs.542897.
EndNote Arslan D, Çakır M, Masiha Y (March 1, 2020) A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior. Gazi University Journal of Science 33 1 152–162.
IEEE D. Arslan, M. Çakır, and Y. Masiha, “A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior”, Gazi University Journal of Science, vol. 33, no. 1, pp. 152–162, 2020, doi: 10.35378/gujs.542897.
ISNAD Arslan, Derya et al. “A Novel Numerical Approach for Solving Convection-Diffusion Problem With Boundary Layer Behavior”. Gazi University Journal of Science 33/1 (March 2020), 152-162. https://doi.org/10.35378/gujs.542897.
JAMA Arslan D, Çakır M, Masiha Y. A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior. Gazi University Journal of Science. 2020;33:152–162.
MLA Arslan, Derya et al. “A Novel Numerical Approach for Solving Convection-Diffusion Problem With Boundary Layer Behavior”. Gazi University Journal of Science, vol. 33, no. 1, 2020, pp. 152-6, doi:10.35378/gujs.542897.
Vancouver Arslan D, Çakır M, Masiha Y. A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior. Gazi University Journal of Science. 2020;33(1):152-6.