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Numerical Solutions of Linear Fractional Order BVP by Galerkin Residual Method with Differentiable Polynomials

Received: 9 April 2020     Accepted: 3 May 2020     Published: 14 May 2020
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Abstract

The main objective of this paper is to solve numerically the differential equations of fractional order with homogeneous boundary conditions by the Galerkin weighted residual method. In this method, linear combinations of some types of functions are used to find the approximate solutions which must satisfy the homogeneous boundary conditions. Such type of functions should be differentiated and integrated easily, so the piecewise polynomials, namely, Bernstein, Bernoulli and Modified Legendre polynomials are used as basis functions in this paper. The fractional derivatives are used in the conjecture of Riemann-Liouville and Caputo sense. Thus, we develop the Galerkin weighted residual formulation, in matrix form, to the linear fractional order boundary value problems, in details, which is easy to understand. The accuracy and applicability of the present method are demonstrated through few numerical examples. We observe that the approximate results converge monotonically to the exact solutions. In addition, we compare the approximate results with the exact solutions, and also with the existing solutions which are available in the literature. The absolute errors are depicted in tabular form as well as graphical representations, a reliable accuracy is achieved. The proposed method may be applied to fractional order partial differential equations also.

Published in Applied and Computational Mathematics (Volume 9, Issue 2)
DOI 10.11648/j.acm.20200902.11
Page(s) 20-25
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Galerkin Method, Fractional Derivatives, Riemann-Liouville Derivative, Caputo Derivative, Fractional Order BVP

References
[1] I. Podlubny, “Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications,” vol. 198, Academic Press, San Diego, USA, 1999.
[2] M. Delkhosh, “Introduction of derivatives and integrals of fractional order and its applications,” Applied Mathematics and Physics, 1 (2013), 103–119.
[3] S. Momani and Z Odibat, “Numerical comparison of methods for solving linear differential equations of fractional order,” Chaos, Solutions and Fractals, 31 (2007), 1248–1255.
[4] Jin-Fa Cheng and Yu-Ming Chu, “Solution to the linear fractional differential equation using Adomian decomposition method,” Mathematical Problems in Engineering, Vol. 2011, Article ID 587068, 14 pages, doi: 10.1155/2011/587068.
[5] A. Secer, S. Alkan, M. A. Akinlar and M. Bayram, “Sinc-Galerkin method for approximate solutions of fractional order boundary value problems,” Boundary value problems, 2013: 281, 2013.
[6] Z. Odibat, S. Momani, V. S. Erturk, “Generalized differential transform method: Application to differential equations of fractional order,” Applied Mathematics and Computation, 197 (2008), 467–477.
[7] W. K. Zahra and S. M. Elkholy, “Cubic spline solution of fractional Bagley-Torvik equation, Electronic Journal of Mathematical Analysis and Applications, 1 (2013), 230–241.
[8] Xinxiu Li, “Numerical solution of fractional differential equations using cubic B-Spline wavelet collocation method,” Commun Nonlinear Sci Numer Simulat, 17 (2012) 3934–3946.
[9] H. Jafari, S. A. Yousefi, M. A. Firoozjaee, S. Momani, C. M. Khalique, “Application of Legendre wavelets for solving fractional differential equations,” Computers and Mathematics with Applications, 62 (2011), 1038–1045.
[10] Qasem M. Al-Mdallal, M. Syam and M. N. Anwar, “A collocation-shooting method for solving fractional boundary value problems,” Commun Nonlinear Sci Numer Simulat, 15 (2010), 3814–3822.
[11] E. Demirci and N. Ozalp, “A method for solving differential equations of fractional order”, Journal of Computational and Applied Mathematics, 236 (2012), 2754–2762).
[12] T. Zhang and C. Tong, “A remark on the fractional order differential equations,” Journal of Computational and Applied Mathematics, 340 (2018), 375-379.
[13] F. Mohammadi and C. Cattani, “A generalized fractional-order Legendre wavelet Tau method for solving fractional differential equations,” Journal of Computational and Applied Mathematics, 339 (2018), 306 – 316.
[14] P. E. Lewis and J. P. Ward, “The Finite Element Method, Principles and Applications,” Addison-Wesley, 1991.
[15] M. S. Islam and M. B. Hossain, “Numerical solutions of eighth order BVP by the Galerkin residual technique with Bernstein and Legendre polynomials”, Applied Mathematics and Computation, 261 (2015), 48–59.
[16] M. S. Islam and A. Shirin, “Numerical solutions of a class of second order boundary value problems on using Bernoulli Polynomials,” Applied Mathematics, 2 (2011), 1059–1067.
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  • APA Style

    Umme Ruman, Md. Shafiqul Islam. (2020). Numerical Solutions of Linear Fractional Order BVP by Galerkin Residual Method with Differentiable Polynomials. Applied and Computational Mathematics, 9(2), 20-25. https://doi.org/10.11648/j.acm.20200902.11

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    ACS Style

    Umme Ruman; Md. Shafiqul Islam. Numerical Solutions of Linear Fractional Order BVP by Galerkin Residual Method with Differentiable Polynomials. Appl. Comput. Math. 2020, 9(2), 20-25. doi: 10.11648/j.acm.20200902.11

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    AMA Style

    Umme Ruman, Md. Shafiqul Islam. Numerical Solutions of Linear Fractional Order BVP by Galerkin Residual Method with Differentiable Polynomials. Appl Comput Math. 2020;9(2):20-25. doi: 10.11648/j.acm.20200902.11

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  • @article{10.11648/j.acm.20200902.11,
      author = {Umme Ruman and Md. Shafiqul Islam},
      title = {Numerical Solutions of Linear Fractional Order BVP by Galerkin Residual Method with Differentiable Polynomials},
      journal = {Applied and Computational Mathematics},
      volume = {9},
      number = {2},
      pages = {20-25},
      doi = {10.11648/j.acm.20200902.11},
      url = {https://doi.org/10.11648/j.acm.20200902.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20200902.11},
      abstract = {The main objective of this paper is to solve numerically the differential equations of fractional order with homogeneous boundary conditions by the Galerkin weighted residual method. In this method, linear combinations of some types of functions are used to find the approximate solutions which must satisfy the homogeneous boundary conditions. Such type of functions should be differentiated and integrated easily, so the piecewise polynomials, namely, Bernstein, Bernoulli and Modified Legendre polynomials are used as basis functions in this paper. The fractional derivatives are used in the conjecture of Riemann-Liouville and Caputo sense. Thus, we develop the Galerkin weighted residual formulation, in matrix form, to the linear fractional order boundary value problems, in details, which is easy to understand. The accuracy and applicability of the present method are demonstrated through few numerical examples. We observe that the approximate results converge monotonically to the exact solutions. In addition, we compare the approximate results with the exact solutions, and also with the existing solutions which are available in the literature. The absolute errors are depicted in tabular form as well as graphical representations, a reliable accuracy is achieved. The proposed method may be applied to fractional order partial differential equations also.},
     year = {2020}
    }
    

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    T1  - Numerical Solutions of Linear Fractional Order BVP by Galerkin Residual Method with Differentiable Polynomials
    AU  - Umme Ruman
    AU  - Md. Shafiqul Islam
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    N1  - https://doi.org/10.11648/j.acm.20200902.11
    DO  - 10.11648/j.acm.20200902.11
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
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    PB  - Science Publishing Group
    SN  - 2328-5613
    UR  - https://doi.org/10.11648/j.acm.20200902.11
    AB  - The main objective of this paper is to solve numerically the differential equations of fractional order with homogeneous boundary conditions by the Galerkin weighted residual method. In this method, linear combinations of some types of functions are used to find the approximate solutions which must satisfy the homogeneous boundary conditions. Such type of functions should be differentiated and integrated easily, so the piecewise polynomials, namely, Bernstein, Bernoulli and Modified Legendre polynomials are used as basis functions in this paper. The fractional derivatives are used in the conjecture of Riemann-Liouville and Caputo sense. Thus, we develop the Galerkin weighted residual formulation, in matrix form, to the linear fractional order boundary value problems, in details, which is easy to understand. The accuracy and applicability of the present method are demonstrated through few numerical examples. We observe that the approximate results converge monotonically to the exact solutions. In addition, we compare the approximate results with the exact solutions, and also with the existing solutions which are available in the literature. The absolute errors are depicted in tabular form as well as graphical representations, a reliable accuracy is achieved. The proposed method may be applied to fractional order partial differential equations also.
    VL  - 9
    IS  - 2
    ER  - 

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Author Information
  • Department of Computer Science & Engineering, Green University of Bangladesh, Dhaka, Bangladesh

  • Department of Applied Mathematics, University of Dhaka, Dhaka, Bangladesh

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