Geri Çekildi: Comparison of Multiple Scales Method and Finite Difference Method for Solving Singularly Perturbed Convection Diffusion Problem

Yıl 2020, , 1169 - 1181, 15.10.2020

Baransel Güneş , Afshin Barati Chianeh Mutlu Demirbaş

https://doi.org/10.17714/gumusfenbil.697534

Bu makale 15 Ekim 2020 tarihinde geri çekildi. https://dergipark.org.tr/tr/pub/gumusfenbil/issue/57242/1195843

Öz

In this study, multiple scale method is introduced for singularly perturbed convection-diffusion equation. In this context, the mentioned problem is transformed into partial differential equation. Besides exponentially fitted difference scheme is established by the method of integral identities with using linear basis functions and interpolating quadrature rules with weight functions and remainder term in integral form. Some numerical experiments have been carried out to validate the theoretical results. The main objective of this article is to compare the multiple scale method and finite difference method for singularly perturbed convection-diffusion problems.

Anahtar Kelimeler

Boundary Layer, Difference Scheme, Multiple Scales Method, Singular Perturbation, Uniform Convergent

Geri Çekildi: Singüler Pertürbe Özellikli Konveksiyon Difüzyon Problemleri İçin Çoklu Ölçekler Metodu ve Sonlu Fark Metodunun Karşılaştırılması

Bu makale 15 Ekim 2020 tarihinde geri çekildi. https://dergipark.org.tr/tr/pub/gumusfenbil/issue/57242/1195843

Öz

Bu çalışmada singüler pertürbe özellikli konveksiyon difüzyon problemi için çoklu ölçekler metodu tanıtılmıştır. Bu bağlamda, söz konusu problem kısmi diferansiyel denklemlere dönüştürülmüştür. Ayrıca ağırlık fonksiyonu içeren ve kalan terimi integral biçiminde olan interpolasyon kuadratür kuralları ve lineer baz fonksiyonlarının kullanımı ile üstel katsayılı fark şeması kurulmuştur. Teorik sonuçları doğrulamak için bazı nümerik çalışmalara yer verilmiştir. Bu makalenin temel amacı, singüler pertürbe özellikli konveksiyon-difüzyon problemleri için çoklu ölçekler metodu ile sonlu fark metodunu karşılaştırmaktır.

Anahtar Kelimeler

Sınır Katmanı, Fark Şeması, Çoklu Ölçekler Metodu, Singüler Pertürbasyon, Düzgün Yakınsaklık

Kaynakça

• Amiraliyev, G. M. and Mamedov, Y. D., 1995. Difference Schemes on the Uniform Mesh for Singularly Perturbed Pseudo-Parabolic Equations, Tr. J. of Math., 19, 207-222.
• Amiraliyev, G. M. and Duru, H., 2002. Nümerik Analiz, Pegem Yayıncılık, Ankara.
• Amiraliyev, G. M. and Cimen, E., 2010. Numerical Method for a Singularly Perturbed Convection-Diffusion Problem with Delay, Applied Mathematics and Computation, 216, 2351-2359.
• Amiraliyeva, I. G., Erdogan, F. and Amiraliyev, G. M., 2010. A Uniform Numerical Method for Dealing with a Singularly Perturbed Delay Initial Value Problem, Applied Mathematics Letters, 23, 1221-1225.
• Bellew, S. and Riordan, E. O., 2004. A Parameter Robust Numerical Method For A System of Two Singularly Perturbed Convection-Diffusion Equations, Applied Numerical Mathematics, 51, 171-186.
• Boyacı, H. and Pakdemirli, M., 1997. A Comparison of Different Versions of the Method of Multiple Scales for Partial Differential Equations, Journal of Sound and Vibration, 204(4), 595-607.
• Çakır, M. and Amiraliyev, G. M., 2005. A Finite Difference Method for Singularly Perturbed Problem with Nonlocal Boundary Condition, Applied Mathematics and Computation, 160, 539-549.
• El-Gamel, M., 2006. A Wavelet-Galerkin Method for a Singularly Perturbed Convection-Dominated Diffusion Equation, Applied Mathematics and Computation, 181, 1635-1644.
• Farrell, P. A., Hegarty, A. F., Miller, J. J. H., Riordan, E. O. and Shishkin, G. I., 2004. Singularly Perturbed Convection-Diffusion Problems with Boundary and Weak Interior Layers, Journal of Computational and Applied Mathematics, 166, 133-151.
• Gupta, P. and Kumar, M., 2016. Multiple Scales Method and Numerical Simulation of Singularly Perturbed Boundary Layer Problem, Appl. Math. Inf. Sci., 10(3), 1119-1127.
• Jager, E. M. D. and Furu, J., 1966. The Theory of Singular Perturbations, North Holland Series in Applied Mathematics and Mechanics, Elsevier.
• Janowicz, M., 2003. Method of Multiple Scales in Quantum Optics, Physics Reports, 375, 327-410.
• Kevorkian, J. and Cole, J. D., 1981. Perturbation Methods in Applied Mathematics, Springer Verlag, New York.
• Lakrad, F. and Belhaq, M., 2002. Periodic Solutions of Strongly Non-Linear Oscillators By The Multiple Scales Method, Journal of Sound and Vibration, 258(4), 677-700.
• Linβ, T., 2010. Layer Adapted Meshes for Reaction-Convection-Diffusion Problems, Springer-Verlag, Berlin Heidelberg.
• Liu, C. S. and Wen, C. C., 2019. Collocation Method with Fractional Powers Exponential Trial Functions for Singularly Perturbed Reaction-Convection-Diffusion Equation, International Journal of Thermal Sciences, 146, 106070.
• Malley, R. E. O., 1974. Introduction to Singular Perturbations, Academic Press, New York.
• Nayfeh, A. H., 1973. Perturbation Methods, Wiley-VCH Verlag GmbH&CoKGaA.
• Pakdemirli, M., Karahan, M. M. F. and Boyacı, H., 2009. A New Perturbation Algorithm with Better Convergence Properties: Multiple Scales Lindstedt Poincare Method, Mathematical and Computational Applications, 14(1), 31-44.
• Park, P. J. and Hou, T. Y., 2004. Multiscale Numerical Methods for Singularly Perturbed Convection-Diffusion Equations. International Journal of Computational Methods, 1(1), 17-65.
• Reddy, Y. and Chakravarthy, P., 2003. Method of Reduction of Order for Solving Singularly Perturbed Two Point Boundary Value Problem, Appl. Math. Comput., 136(1), 27-45.
• Ren, Z. F., Yao, S. W. and He, J. H., 2019. He’s Multiple Scales Method for Nonlinear Vibrations, Journal of Law Frequency Noise, Vibration and Active Control, 38(3-4), 1708-1712.
• Romanazzi, P., Bruna, M. and Howey, D. A., 2017. Thermal Homogenization of Electrical Machine Windings Applying The Multiple Scales Method, Journal of Heat Transfer, 139, 02101-7.
• Roos, H. G., Stynes, M. and Tobiska, L., 2008. Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems, vol. 24, Springer Science&Business Media.
• Sakar, M. G., Saldir, O. and Erdogan, F., 2019. A Hybrid Method for Singularly Perturbed Convection-Diffusion Equation, Int. J. Appl. Comput. Math., 5, 135.
• Salahshoor, E., Ebrahimi, S. and Maasoomi, M., 2016. Nonlinear Vibration Analysis of Mechanical Systems with Joint Clearances Using The Method of Multiple Scales, Mechanism and Machine Theory, 105, 495-509.
• Samarskii, A.A., 2001. The Theory of Difference Schemes. Moscow M.V. Lomonosov State University, Russia.
• Sekar, E. and Tamilselvan, A., 2019. Finite Difference Scheme for Third Order Singularly Perturbed Delay Differential Equation of Convection Diffusion Type with Integral Boundary Condition, Journal of Applied Mathematics and Computing, //https://doi.org./10.1007/s12190-019-01239-0.
• Shishkin, G. I. and Shishkina, L. P., 2019. Difference Schemes on Uniform Grids for an Initial-Boundary Value Problem for A Singularly Perturbed Parabolic Convection-Diffusion Equation, Comput. Methods Appl. Math., https://doi.org/10.1515/cmam-2019-0023.
• Shishkin, G. I., 2004. Discrete Approximations of Solutions and Derivatives for a Singularly Perturbed Parabolic Convection-Diffusion Equation, Journal of Computational and Applied Mathematics, 166, 247-266.
• Subburayan, V. and Ramanujam, N., 2013. An Initial Value Technique for Singularly Perturbed Convection-Diffusion Problems with a Negative Shift, J. Optim. Theory Appl., 158, 234-250.
• Wu, C. P. and Tsai, Y. H., 2010. Dynamic Responses of Functionally Graded Magneto-Electro-Elastic Shells with Closed-Circuit Surface Conditions Using The Method of Multiple Scales, European Journal of Mechanics A/ Solids, 29, 166-181.