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Approximate Solutions for Fractional Boundary Value Problems via Green-CAS Wavelet Method

Author

Listed:
  • Muhammad Ismail

    (School of Natural Sciences, National University of Sciences and Technology, Islamabad 44000, Pakistan)

  • Umer Saeed

    (NUST Institute of Civil Engineering, School of Civil and Environmental Engineering, National University of Sciences and Technology, Islamabad 44000, Pakistan)

  • Jehad Alzabut

    (Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia)

  • Mujeeb ur Rehman

    (School of Natural Sciences, National University of Sciences and Technology, Islamabad 44000, Pakistan)

Abstract

In this study, we present a novel numerical scheme for the approximate solutions of linear as well as non-linear ordinary differential equations of fractional order with boundary conditions. This method combines Cosine and Sine (CAS) wavelets together with Green function, called Green-CAS method. The method simplifies the existing CAS wavelet method and does not require conventional operational matrices of integration for certain cases. Quasilinearization technique is used to transform non-linear fractional differential equations to linear equations and then Green-CAS method is applied. Furthermore, the proposed method has also been analyzed for convergence, particularly in the context of error analysis. Sufficient conditions for the existence of unique solutions are established for the boundary value problem under consideration. Moreover, to elaborate the effectiveness and accuracy of the proposed method, results of essential numerical applications have also been documented in graphical as well as tabular form.

Suggested Citation

  • Muhammad Ismail & Umer Saeed & Jehad Alzabut & Mujeeb ur Rehman, 2019. "Approximate Solutions for Fractional Boundary Value Problems via Green-CAS Wavelet Method," Mathematics, MDPI, vol. 7(12), pages 1-20, December.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:12:p:1164-:d:293216
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    References listed on IDEAS

    as
    1. Saeed, Umer, 2017. "CAS Picard method for fractional nonlinear differential equation," Applied Mathematics and Computation, Elsevier, vol. 307(C), pages 102-112.
    2. Li, Xian-Fang, 2007. "Approximate solution of linear ordinary differential equations with variable coefficients," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 75(3), pages 113-125.
    3. Hassan Khan & Muhammad Arif & Syed Tauseef Mohyud-Din, 2019. "Numerical Solution Of Fractional Boundary Value Problems By Using Chebyshev Wavelet," Matrix Science Mathematic (MSMK), Zibeline International Publishing, vol. 3(1), pages 13-16, March.
    4. Gholami, Saeid & Babolian, Esmail & Javidi, Mohammad, 2019. "Fractional pseudospectral integration/differentiation matrix and fractional differential equations," Applied Mathematics and Computation, Elsevier, vol. 343(C), pages 314-327.
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