Troesch denkleminin çözümü için Laguerre dalgacık yöntemi

Abstract

Bu makalenin amacı lineer olmayan Troesch denklemini Laguerre dalgacık yöntemini kullanarak çözmektir.  Bilinmeyen fonksiyon Laguerre dalgacıkları ile yaklaştırılarak denklem bir cebirsel denklem sistemine dönüştürülür.  Bu yöntemin avantajlarından biri, lineer olmayan terimin lineer hale dönüştürülmesine gerek kalmamasıdır.  Denklem Troesch parametresinin farklı değerleri için çözülmüştür. Yöntemin etkin olduğunu göstermek için elde edilen sonuçlar gerek gerçek gerekse literatürdeki diğer sayısal sonuçlar ile karşılaştırılmıştır. 

Keywords

• Laguerre dalgacık yöntemi,Troesch denklemi,Laguerre polinomu,Lineer olmayan diferensiyel denklem

Laguerre wavelet method for solving Troesch equation

Abstract

The purpose of this paper is to illustrate the use of the Laguerre wavelet method in the solution of Troesch’s equation, which is a stiff nonlinear equation. The unknown function is approximated by Laguerre wavelets and the equation is transformed into a system of algebraic equations. One of the advantages of the method is that it does not require the linearization of the nonlinear term. The problem is solved for different values of Troesch’s parameter (μ) and the results are compared with both the analytical and other numerical results to validate the accuracy of the method.

Keywords

• Laguerre Wavelet method,Troesch equation,Laguerre polynomial,Nonlinear differential equation

References

1. Temimi, H., Ben-Romdhane, M., Ansari, A.R. and Shishkin, G.I., Finite difference numerical solution of Troesch’s problem on a piecewise uniform Shishkin mesh, Calcolo, 54, 225–242, (2017).
2. Kazemi Nasab, A., Pashazadeh Atabakan, Z. and Kılıçman, A., An Efficient Approach for Solving Nonlinear Troesch’s and Bratu’s Problems by Wavelet Analysis Method, Mathematical Problems in Engineering, 2013, 10 pages, (2013).
3. El-Gamel, M. and Sameeh, M., A Chebyshev collocation method for solving Troesch’s problem, International Journal of Mathematics and Computer Applications Research, 3(2), 23-32, (2013).
4. Khuri, S.A. and Sayfy, A., Troesch’s problem: A B-spline collocation approach, Mathematical and Computer Modelling, 54, 1907–1918, (2011).
5. Temimi, H. and Kürkçü, H., An accurate asymptotic approximation and precise numerical solution of highly sensitive Troesch’s problem, Applied Mathematics and Computation, 235, 253–260, (2014).
6. Geng, F. and Cui, M., A novel method for nonlinear two-point boundary value problems: Combination of ADM and RKM, Applied Mathematics and Computation, 217, 4676–4681, (2011).
7. Saadatmandi, A. and Abdolahi-Niasar, T., Numerical solution of Troesch's problem using Christov rational functions, Computational Methods for Differential Equations, 3(4), 247-257, (2015).
8. Deeba, E., Khuri, S.A. and Xie, S., An Algorithm for Solving Boundary Value Problems, Journal of Computational Physics, 159, 125–138, (2000).

9. Chang , S.H. and Chang, I.L., A new algorithm for calculating one-dimensional differential transform of nonlinear functions, Applied Mathematics and Computation, 195, 799–808, (2008).
10. Bisheh-Niasar, M., Saadatmandi, A. and Akrami-Arani, M., A New Family of High-Order Difference Schemes for the Solution of Second Order Boundary Value Problems, Iranian Journal of Mathematical Chemistry, 9(3), 187 – 199, (2018).
11. Mirmoradi, S.H., Hosseinpour, I. , Ghanbarpour, S., Barari, A., Application of an Approximate Analytical Method to Nonlinear Troesch’s Problem, Applied Mathematical Sciences, 3, 32, 1579 – 1585, (2009).
12. Malik, S.A., Qureshi, I.M., Zubair, M. and Amir, M., Numerical Solution to Troesch’s Problem Using Hybrid Heuristic Computing, Journal of Basic and Applied Scientific Research, 3(7), 10-16, (2013).
13. Doha, E.H., Baleanu, D., Bhrawi, A.H. and Hafez, R.M., A Jacobi collocation method for Troesch’s problem in plasma physics, Proceedings of the Romanian Academy, Series A, 15(2), 130–138, (2014).
14. Khuri, S.A., A numerical algorithm for solving Troesc’s problem, International Journal of Computer Mathematics, 80(4), 493–498, (2003).
15. Feng, X., Mei, L. and He, G., An efficient algorithm for solving Troesch’s problem, Applied Mathematics and Computation, 189, 500–507, (2007).
16. Ben-Romdhane, M. and Temimi, H., A novel computational method for solving Troesch’s problem with high-sensitivity parameter, International Journal for Computational Methods in Engineering Science and Mechanics, 18(4-5), 230-237, (2017).
17. Khalid, M., Zaidi, F., Sultana, M. and Aurangzaib, A Numerical Solution of Troesch’s Problem via Optimal Homotopy Asymptotic Method, International Journal of Computer Applications, 140(5), 1-5, (2016).
18. Filobello-Nino, U., Vázquez-Leal, H., Benhammouda, B., Pérez-Sesma, A., Cervantes-Pérez, J., Jiménez-Fernández, V.M., Díaz-Sánchez, A., Herrera-May, A., Pereyra-Díaz, D., Marín-Hernández, A., Huerta-Chua, J. and Sánchez-Orea, J., Perturbation Method and Laplace-Padé Approximation as a novel tool to find approximate solutions for Troesch,s problem, Revista Electrónica Nova Scientia, 14, 7, 2, 57 – 73, (2015).
19. Scott, M.R. and Vandevender, W.H., A comparison of several invariant imbedding algorithms for the solution of two-point boundary-value problems, Applied Mathematics and Computation, 1, 187-218, (1975).
20. Alias, N., Manaf, A., Ali, A. and Habib, M., Solving Troesch’s problem by using modified nonlinear shooting method, Jurnal Teknologi, 78, 4–4, 45–52, (2016).
21. El-Gamel, M., Numerical Solution of Troesch’s Problem by Sinc-Collocation Method, Applied Mathematics, 4, 707-712, (2013).
22. Zarebnia, M. and Sajjadian, M., The sinc–Galerkin method for solving Troesch’s problem, Mathematical and Computer Modelling, 56, 218–228, (2012).
23. Chang, S.H., A variational iteration method for solving Troesch's problem, Journal of Computational and Applied Mathematics, 234, 3043-3047, (2010).
24. Momani, S., Abuasad, S. and Odibat, Z., Variational iteration method for solving nonlinear boundary value problems, Applied Mathematics and Computation, 183, 1351–1358, (2006).
25. Savasaneril, N., Laguerre Series Solutions of the Delayed Single Degree-of-Freedom Oscillator Excited by an External Excitation and Controlled by a Control Force, Journal of Computational and Theoretical Nanoscience, 15, 1–5, (2018).
26. Savasaneril, N. and Sezer, M., Laguerre Polynomial Solution of High- Order Linear Fredholm Integro-Differential Equations, New Trends in Mathematical Sciences, 4, 2, 273-284, (2016).
27. Yüzbaşı, Ş., Laguerre approach for solving pantograph-type Volterra integro-differential equations, Applied Mathematics and Computation, 232, 1183–1199, (2014).
28. Goswami, J.C. and Chan, A.K., Fundamentals of Wavelets, Theory, Algorithms and Applications, 2nd edition, John Wiley and Sons Inc., New York, 72-97, (2011).
29. Gu, J.S. and Jiang, W.S., The Haar wavelets operational matrix of integration, International Journal of Systems Science, 27, 7, 623-628, (1996).
30. Babolian, E. and Fattahzadeh, F., Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration, Applied Mathematics and Computation, 188, 417-426, (2007).
31. Razzaghi, M. and Yousefi, S., Legendre wavelets operational matrix of integration, International Journal of Systems Science, 32, 495-502, (2001).
32. Mohammadi, F. and Hosseini, M.M., Legendre wavelet method for solving linear stiff systems, Journal of Advanced Research in Differential Equations, 2, 47-57, (2010).
33. Mohammadi, F., Hosseini, M.M. and Mohyud-Din, S.T., Legendre wavelet Galerkin method for solving ordinary differential equations with nonanalytic solution, International Journal of Systems Science, 42, 579-585, (2011).
34. Mohammadi, F. and Hosseini, M.M., A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations. Journal of Franklin Institute, 348, 1787-1796, (2011).
35. Arfken, G.B. and Weber, H.J., Mathematical Methods for Physicists, 6th edition, Elsevier Academic Press, London, 837-845, (2005).
36. Roberts, S.M. and Shipman, J.S., On the closed form solution of Troesch’s problem, Journal of Computational Pyhsics, 21, 291-304, (1976).