Abstract
In this study, the Laguerre wavelets exact Parseval frame is introduced and proposed an effective numerical algorithm to get a numerical solution for the system of differential equations based on the Laguerre wavelets exact Parseval frame. This algorithm includes the collocation method and truncated Laguerre wavelet frames. Here, we reduce the system of differential equations into a set of algebraic equations which are having unknown Laguerre wavelet frame coefficients. Some numerical examples are given and compared to the numerical solution by the present method with the Adomian decomposition method. Moreover, the modeling of the spreading of a non-fatal disease in a population, which represents a system of an ordinary differential equation is numerically solved by the proposed technique and compared with the Adomina decomposed method. The obtained results reveal that the present algorithm provides a good approximation than existing methods.
Similar content being viewed by others
References
Akinfenwaa, O.A., Jator, S.N., Yaoa, N.M.: Continuous block backward differentiation formula for solving stiff ordinary differential equations. Comput. Math Appl. 65, 996–1005 (2013)
Benhammouda, B., Vazquez-Leal, H., Hernandez-Martinez, L.: Modified differential transform method for solving the model of pollution for a system of lakes. Discrete Dyn. Nat. Soc. (2014). https://doi.org/10.1155/2014/645726
Biazar, J.: Solution of the epidemic model by Adomian decomposition method. Appl. Math. Comput. 173, 1101–1106 (2006)
Biazar, J., Hosseini, K.: An effective modification of Adomian decomposition method for solving Emden–Fowler type systems. Natl. Acad. Sci. Lett. 40, 285–290 (2017)
Darvishi, M.T., Khani, F., Soliman, A.A.: The numerical simulation for stiff systems of ordinary differential equations. Comput. Appl. Math. 54, 1055–1063 (2007)
Gao, W., Rezazadeh, H., Pinar, Z., Baskonus, H.M., Sarwar, S., Ye, G.: Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique. Opt. Quant. Electron. (2020). https://doi.org/10.1007/s11082-019-2162-8
Gao, W., Veeresha, P., Prakasha, D.G., Baskonus, H.M., Yel, G.: A powerful approach for fractional Drinfeld–Sokolov–Wilson equation with Mittag–Leffler law. Alex. Eng. J. (2019). https://doi.org/10.1016/j.aej.2019.11.002
Hadi, R., Vahidi, J., Zafar, A., Bekir, A.: The functional variable method to find new exact solutions of the nonlinear evolution equations with dual-power-law nonlinearity. Int. J. Nonlinear Sci. Numer. Simul. (2019). https://doi.org/10.1515/ijnsns-2019-0064
Hadi, R., Korkmaz, A., Eslami, M., Mirhosseini-Alizamini, S.M.: A large family of optical solutions to Kundu–Eckhaus model by a new auxiliary equation method. Opt. Quant. Electron. (2019). https://doi.org/10.1007/s11082-019-1801-4
Hehenberger, M., Brandas, E., Elander N.: Weyl’s theory for a system of coupled second order differential equations. In: Proceedings of the International Symposium on Atomic, Molecular, and Solid-state Theory, Collision Phenomena, and Computational Methods, vol. 14 (S12), pp. 67–71 (1978)
Hehenberger, M., Brandas, E., Elander, N.: Matrix free methods for stiff systems of ODE’s. SIAM J. Numer. Anal. 23(3), 610–638 (1986)
Heil, C.: A Basis Theory Primer. Springer, Landon (1988)
Ngwane, F.F., Jator, S.N.: Block hybrid second derivative method for stiff systems. Int. J. Pure Appl. Math. 80(4), 543–559 (2012)
Ongun, M.Y.: The Laplace Adomian Decomposition Method for solving a model for HIV infection of CD4+ T cells. Math. Comput. Model. 53, 597–603 (2011)
Ozturk, Y.: Numerical solution of systems of differential equations using operational matrix method with Chebyshev polynomials. J. Taibah Univ. Sci. 12(2), 155–162 (2018)
Park, C., et al.: Dynamical analysis of the nonlinear complex fractional emerging telecommunication model with higher-order dispersive cubic–quintic. Alex. Eng. J. (2020). https://doi.org/10.1016/j.aej.2020.03.046
Prakasha, D.G., Veeresha, P.: Analysis of lakes pollution model with Mittag–Leffler kernel. J. Ocean Eng. Sci. (2020). https://doi.org/10.1016/j.joes.2020.01.004
Raza, N., Afzal, U., Butt, A.R., Rezazadeh, H.: Optical solitons in nematic liquid crystals with Kerr and parabolic law nonlinearities. Opt. Quant. Electron. (2019). https://doi.org/10.1007/s11082-019-1813-0
Shawagfeh, N., Kaya, D.: Comparing numerical methods for the solutions of systems of ordinary differential equations. Appl. Math. Lett. 17, 323–328 (2004)
Shiralashetti, S.C., Kumbinarasaiah, S.: Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems. Alex. Eng. J. 57, 2591–2600 (2018)
Shiralashetti, S.C., Kumbinarasaiah, S.: Theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear lane-Emden type equations. Appl. Math. Comput. 315, 591–602 (2017)
Shiralashetti, S.C., Kumbinarasaiah, S.: Cardinal B-spline wavelet-based numerical method for the solution of generalized Burgers–Huxley equation. Int. J. Appl. Comput. Math. 4, 73 (2018)
Shiralashetti, S.C., Kumbinarasaiah, S.: Some results on haar wavelets matrix through linear algebra. Wavelets Linear Algebra 4(2), 49–59 (2017)
Shiralashetti, S.C., Kumbinarasaiah, S., Hoogar, B.S.: Hermite wavelet-based numerical method for the solution of linear and nonlinear delay differential equations. Int. J. Eng. Sci. Math. 6(8), 71–79 (2017)
Shiralashetti, S.C., Kumbinarasaiah, S.: New generalized operational matrix of integration to solve nonlinear singular boundary value problems using Hermite wavelets. Arab J. Basic Appl. Sci. 26(1), 385–396 (2019)
Shiralashetti, S.C., Kumbinarasaiah, S.: CAS wavelets analytic solution and Genocchi polynomials numerical solutions for the integral and integrodifferential equations. J. Interdiscip. Math. 22(3), 201–218 (2019)
Shiralashetti, S.C., Kumbinarasaiah, S.: Hermite wavelets method for the numerical solution of linear and nonlinear singular initial and boundary value problems. Comput. Methods Differ. Equ. 7(2), 177–198 (2019)
Shiralashetti, S.C., Kumbinarasaiah, S.: Some results on shannon wavelets and wavelets frames. Int. J. Appl. Comput. Math. 5, 10 (2019)
Sweilama, N.H., Khaderb, M.M.: Exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method. Comput. Math Appl. 58, 2134–2141 (2009)
Veeresha, P., Prakasha, D.G.: Solution for fractional generalized Zakharov equations with Mittag–Leffler function. Results Eng. (2020). https://doi.org/10.1016/j.rineng.2019.100085
Widatalla, S., Koroma, M.A.: Approximation algorithm for a system of Pantograph equations. J. Appl. Math. (2012). https://doi.org/10.1155/2012/714681
Wu, X.Y., Xia, J.L.: Two low accuracy methods for stiff systems. Appl. Math. Comput. 123, 141–153 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Shiralashetti, S.C., Kumbinarasaiah, S. Laguerre Wavelets Exact Parseval Frame-based Numerical Method for the Solution of System of Differential Equations. Int. J. Appl. Comput. Math 6, 101 (2020). https://doi.org/10.1007/s40819-020-00848-9
Published:
DOI: https://doi.org/10.1007/s40819-020-00848-9