Abstract
This paper is devoted to introduce an efficient solver using a combination of the symbol of the operator and the windowed Fourier frames (WFFs) of the coupled system of second order ordinary differential equations. The given system has a basic importance in modeling various phenomena like, Cascades and Compartment Analysis, Pond Pollution, Home Heating, Chemostats and Microorganism Culturing, Nutrient Flow in an Aquarium, Biomass Transfer and others. The proposed method reduces the system of differential equations to a system of algebraic equations in the coefficients of WFFs. The introduced method is computer oriented with highly accurate solution. To demonstrate the efficiency of the proposed method, two examples are presented and the results are displayed graphically. Finally, we convert the presented coupled systems of BVPs to a first order system of ODEs to compare the obtained numerical solution with those solutions using the fourth-order Runge–Kutta method (RK4).
Similar content being viewed by others
References
Alligood, K.T., Sauer, T.D., Yorke, J.A.: An Introduction to Dynamical Systems. Springer, New York (1996)
Bhowmik, S.K., Stolk, C.C.: Preconditioners based on windowed Fourier frames applied to elliptic partial differential equations. J. Pseudodiffer. Oper. Appl. 2(3), 317–342 (2011)
Christensen, O.: An Introduction to Frames and Riesz Bases. ANHA, Birkhäuser, Boston (2004)
Gröchenig, K.: Foundations of Time-Frequency Analysis. ANHA, Birkhäuser, Boston (2000)
Khader, M.M.: On the numerical solutions for the fractional diffusion equation. Commun. Nonlinear Sci. Numer. Simul. 16, 2535–2542 (2011)
Khader, M.M.: The use of generalized Laguerre polynomials in spectral methods for fractional-order delay differential equations. J. Comput. Nonlinear Dyn. 8, 041018:1-5 (2013)
Khader, M.M., Sweilam, N.H.: On the approximate solutions for system of fractional integro-differential equations using Chebyshev pseudo-spectral method. Appl. Math. Model. 37, 9819–9828 (2013)
Khader, M.M., Hendy, A.S.: A numerical technique for solving fractional variational problems. Math. Methods Appl. Sci. 36(10), 1281–1289 (2013)
Khader, M.M., Babatin, M.M.: Numerical treatment for solving fractional SIRC model and influenza A. Comput. Appl. Math. 33(3), 543–556 (2014)
Mallat, S.: A Wavelet Tour of Signal Processing, 3rd edn. Academic Press, New York (2009)
D. Matignon, Stability results for fractional differential equations with applications to control processing, Computational Engineering in Systems and Application. In: Multiconference, IMACS, IEEE-SMC, Lille, France, vol. 2, pp. 963–968 (1996)
Paraskevopolus, P.N., Saparis, P.D., Mouroutsos, S.G.: The Fourier series operational matrix of integration. Int. J. Syst. Sci. 16, 171–176 (1985)
Ross, B. (ed.): Fractional Calculus and its Applications. Lecture Notes in Mathematiucs, vol. 457. Springer, Berlin (1975)
Scalas, E., Raberto, M., Mainardi, F.: Fractional calclus and continous-time finance. Phys. A Stat. Mech. Appl. 284(1–4), 376–384 (2000)
Sweilam, N.H., Khader, M.M., Adel, M.: On the numerical solution for the fractional wave equation using pseudo-spectral method based on the generalized Laguerre polynomials. Appl. Math. 6, 647–654 (2015)
Tan, Y., Abbasbandy, S.: Homotopy analysis method for quadratic Riccati differential equation. Commun. Nonlinear Sci. Numer. Simul. 13(3), 539–546 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Khader, M.M., Adel, M. Introducing the windowed Fourier frames technique for obtaining the approximate solution of the coupled system of differential equations. J. Pseudo-Differ. Oper. Appl. 10, 241–256 (2019). https://doi.org/10.1007/s11868-018-0240-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11868-018-0240-5