Thenumerical results demonstrate that the method is accurate, effectual and simple to apply
📌 3 Key Takeaways
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Objective & Hypothesis:The purpose of this paper is to illustrate the use of the Legendre wavelet method in the solution of high-ordernonlinear ordinary differential equations with variable and proportional delays.
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Key Findings & Data:The main advantage of using Legendrepolynomials lies in the orthonormality property, which enables a decrease in the computational cost and runtime.
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Conclusion & Impact:Thenumerical results demonstrate that the method is accurate, effectual and simple to apply
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters
Abstract
The purpose of this paper is to illustrate the use of the Legendre wavelet method in the solution of high-ordernonlinear ordinary differential equations with variable and proportional delays. The main advantage of using Legendrepolynomials lies in the orthonormality property, which enables a decrease in the computational cost and runtime. Themethod is applied to five differential equations up to sixth order, and the results are compared with the exact solutionsand other numerical solutions when available. The accuracy of the method is presented in terms of absolute errors. Thenumerical results demonstrate that the method is accurate, effectual and simple to apply
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Cite This Article
GÜMGÜM, S., ÖZDEK, D. E., ÖZALTUN, G. (2022). Legendre wavelet solution of high order nonlinear ordinary delay differential equations. Turkish Journal of Mathematics, Cilt: 43 - Sayı: 3, 1339–1352. https://ajindex.com/articles/607700
Abdullah ZA. Solution of nonlinear ordinary delay differential equations using variational approach. Journal of Al-Nahrain University 2012
15(4): 183-187. doi:10.22401/JNUS.15.4.26 [2] Aboodh KS, Farah RA, Almardy IA, Osman AK. Solving delay differential equations by Aboodh transformation method. International Journal of Applied Mathematics & Statistical Sciences 2018
7(2): 55-64. [3] Ali I, Brunner H, Tang T. A Spectral method for pantograoh-type delay differential equations and its convergence analysis. Journal of Computational Mathematics 2009
27(2-39): 254-265. [4] Arfken GB, Weber HJ. Mathematical Methods for Physicists. 6th ed. London, UK: Elsevier Academic Press, 2005. [5] Babolian E, Fattahzadeh F. Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration. Applied Mathematics and Computation 2007
doi:10.1016/j.amc.2006.10.008 [6] Benhammouda B, Vazquez-Leal H. A new multi-step technique with differential transform method for analytical solution of some nonlinear variable delay differential equations. SpringerPlus 2016
5(1): 1723. doi:10.1186/s40064- 016-3386-8 [7] Benhammouda B, Vazquez-Leal H, Hernandez-Martinez L. Procedure for exact solutions of nonlinear Pantograph delay differential equations. British Journal of Mathematics & Computer Science 2014
4(19): 2738-2751. doi: 10.9734/BJMCS/2014/11839 [8] Blanco-Cocom L, Estrella AG, Avila-Vales E. Solving delay differential systems with history functions by the Adomian decomposition method. Applied Mathematics and Computation 2012
doi:10.1016/j.amc.2011.11.082 [9] Bocharova GA, Rihanb FA. Numerical modelling in biosciences using delay differential equations. Journal of Computational and Applied Mathematics 2000
doi:10.1016/S0377-0427(00)00468-4 [10] Boggess A, Narcowich FJ. A First Course in Wavelets With Fourier Analysis. New York, NY, USA: John Wiley and Sons Inc, 2001. [11] Canuto C, Hussaini M, Quarteroni A, Zang T. Spectral Methods in Fluid Dynamics. Berlin, Germany: Springer, 1988. [12] Caruntu B, Bota C. Analytical approximate solutions for a general class of nonlinear delay differential equations. Scientific World Journal 2014
doi: 10.1155/2014/631416 [13] Davaeifar S, Rashidinia J. Solution of a system of delay differential equations of multipantograph type. Journal of Taibah University for Science 2017