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

  • M. M. Khader
  • M. AdelEmail author


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).


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


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© 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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