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Introducing the windowed Fourier frames technique for obtaining the approximate solution of the coupled system of differential equations

  • M. M. Khader
  • M. AdelEmail author
Article
  • 54 Downloads

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).

Keywords

Coupled system of differential equations Windowed Fourier frames Sparse computations Fourth-order Runge–Kutta method 

References

  1. 1.
    Alligood, K.T., Sauer, T.D., Yorke, J.A.: An Introduction to Dynamical Systems. Springer, New York (1996)zbMATHGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Christensen, O.: An Introduction to Frames and Riesz Bases. ANHA, Birkhäuser, Boston (2004)zbMATHGoogle Scholar
  4. 4.
    Gröchenig, K.: Foundations of Time-Frequency Analysis. ANHA, Birkhäuser, Boston (2000)zbMATHGoogle Scholar
  5. 5.
    Khader, M.M.: On the numerical solutions for the fractional diffusion equation. Commun. Nonlinear Sci. Numer. Simul. 16, 2535–2542 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    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)Google Scholar
  7. 7.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Khader, M.M., Hendy, A.S.: A numerical technique for solving fractional variational problems. Math. Methods Appl. Sci. 36(10), 1281–1289 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Khader, M.M., Babatin, M.M.: Numerical treatment for solving fractional SIRC model and influenza A. Comput. Appl. Math. 33(3), 543–556 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mallat, S.: A Wavelet Tour of Signal Processing, 3rd edn. Academic Press, New York (2009)zbMATHGoogle Scholar
  11. 11.
    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)Google Scholar
  12. 12.
    Paraskevopolus, P.N., Saparis, P.D., Mouroutsos, S.G.: The Fourier series operational matrix of integration. Int. J. Syst. Sci. 16, 171–176 (1985)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ross, B. (ed.): Fractional Calculus and its Applications. Lecture Notes in Mathematiucs, vol. 457. Springer, Berlin (1975)Google Scholar
  14. 14.
    Scalas, E., Raberto, M., Mainardi, F.: Fractional calclus and continous-time finance. Phys. A Stat. Mech. Appl. 284(1–4), 376–384 (2000)CrossRefGoogle Scholar
  15. 15.
    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)CrossRefGoogle Scholar
  16. 16.
    Tan, Y., Abbasbandy, S.: Homotopy analysis method for quadratic Riccati differential equation. Commun. Nonlinear Sci. Numer. Simul. 13(3), 539–546 (2008)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, College of ScienceAl-Imam Mohammad Ibn Saud Islamic University (IMSIU)RiyadhSaudi Arabia
  2. 2.Department of Mathematics, Faculty of ScienceBenha UniversityBenhaEgypt
  3. 3.Department of Mathematics, Faculty of ScienceCairo UniversityGizaEgypt

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