Soner AYDINLIK, Ahmet KIRIŞ

Doğrusal Olmayan Tekil Sınır Değer Problemlerinin Chebyshev Sonlu Farklar Yöntemi ile Çözümü

Chebyshev Sonlu Farklar Yöntemi (CSFY) ile doğrusal olmayan tekil sınır değer problemleri çözülmüştür. Ayrıca yöntemin hata ve yakınsama analizi verilmiştir. Yöntemin yakınsaklığını ve etkinliğini göstermek için termal patlamada, küresel bir hücrede kararlı hal oksijen difüzyonunda ve izotermal gaz kürelerinin dengesinde ortaya çıkan gerçek mühendislik problemleri incelenmiştir. Sonuçlar, sunulan yöntemin problemi alt aralıklara bölmeksizin ve farklı sınır koşulları için farklı değişiklikler yapmaksızın doğruluğu yüksek ve çözüme oldukça hızlı yakınsadığını göstermektedir.

Anahtar Kelimeler:

Chebyshev Polinomları, Chebyshev Sonlu Farklar Yöntemi, Doğrusal olmayan Tekil Sınır Değer Problemi, Lane-Emden denklemi.

PDF

___

[1] Caglar, H., Caglar, N., Ozer, M. (2009). B-spline solution of non-linear singular boundary value problems arising in physiology. Chaos Solitons Fractals, 39, 1232-1237.
[2] Abukhaled, M., Khuri, S.A., Sayef, A. (2011). A numerical approach for solving a class of singular boundary value problems arising in physiology. International Journal of Numerical Analysis and Modeling, 8(2), 353-363.
[3] Thula, K., Roul, P. (2018). A High-Order B-Spline Collocation Method for solving nonlinear singular boundary value problems arising in engineering and applied science. Mediterranean Journal of Mathematics, 15(76), 1-24
[4] Yucel, U., Sari, M. (2009). Differential quadrature method (DQM) for a class of singular two-point boundary value problem. International Journal of Computer Mathematics, 86(3), 465-475.
[5] Danish, M., Kumar, S. (2012). A note on the solution of singular boundary value problems arising in engineering and applied sciences: use of OHAM. Computers & Chemical Engineering, 36, 57-67.
[6] Roul, P., Warbhe, U. (2016). New approach for solving class of singular boundary value problem arising in various physical models. Journal of Mathematical Chemistry, 54, 1255-1285.
[7] Ravi Kanth, A., Aruna, K. (2010). He’s varitional iteration method for treating nonlinear singular boundary value problem. Computers & Mathematics with Applications, 60(3), 821-829.
[8] Chawla, M. M., Katti, C.P. (1982). Finite difference methods and their convergence for a class of singular two-point boundary value problems. Numerische Mathematik, 39, 341-350.
[9] Roul, P. (2017). On the numerical solution of singular two-point boundary value problems: A domain decomposition homotopy perturbation approach. Mathematical Methods in the Applied Sciences, 40, 7396-7409.
[10] Pandey, R. K., Singh, A. K. (2006). A new high-accuracy difference method for a class of weakly nonlinear singular boundary-value problems. International Journal of Computer Mathematics, 83(11), 809-817.
[11] Schreiber, R. (1980). Finite element methods of high order accuracy for singular two-point boundary value problems with non-smooth solutions. SIAM Journal on Numerical Analysis, 17, 547-566.
[12] Khaleghi, M., Moghaddam, M. T., Babolian, E., Abbasbandy, S. (2018). Solving a class of singular two-point boundary value problems using new effective reproducing kernel technique. Applied Mathematics and Computation, 331, 264-273.
[13] Fox, L., Parker, I. B. (1968). Chebyshev Polynomials in Numerical Analysis, Oxford University Press, London.
[14] Clenshaw, C. W., Curtis, A. R. (1960). A method for numerical integration on an automatic computer. Numerische Mathematik, 2, 197-205.
[15] Elbarbary, E. M. E., El-Kady, M. (2003). Chebyshev finite difference approximation for the boundary value problems, Applied Mathematics and Computation,139, 513-523.
[16] Aydinlik, S., Kiris, A. (2018). A High-Order Numerical Method for Solving Fractional Nonlinear Lane-Emden Type Equations Arising in Astrophysics. Astrophysics and Space Science, 363, 264.
[17] Elbarbary, E. M. E., El-Sayed, S. M. (2005). Higher order pseudospectral differentiation matrices. Applied Numerical Mathematics, 55, 425-438.
[18] Zhu, H., Niu, J., Zhang, R., Lin, Y. (2018). A new approach for solving nonlinear singular boundary value problems. Mathematical Modelling and Analysis, 23(1), 33-43.
[19] Singh, R., Kumar. J. (2016). An efficient numerical technique for the solution of nonlinear singular boundary value problems. Computer Physics Communications, 185(4), 466-477.
[20] Xie, L. J., Zhou, C. L., Xu, S. (2016). An effective numerical method to solve a class of nonlinear singular boundary value problems using improved differential transform method. SpringerPlus, 5, 1066-1084.
[21] Singh, R., Garg, H., Guleria, V. (2019). Haar wavelet collocation method for Lane–Emden equations with Dirichlet, Neumann and Neumann–Robin boundary conditions. Journal of Computational and Applied Mathematics, 346, 150–161.
[22] Pandey, R. K., Singh, A. K. (2009). On the convergence of a fourth-order method for a class of singular boundary value problems. Journal of Computational and Applied Mathematics, 224, 734–742.