0
Research Papers

# A Piecewise Nonpolynomial Collocation Method for Fractional Differential Equations

[+] Author and Article Information
Shahrokh Esmaeili

Department of Applied Mathematics,
University of Kurdistan,
P.O. Box 416,
Sanandaj 66177-15177, Iran
e-mail: sh.esmaeili@uok.ac.ir

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 12, 2015; final manuscript received May 4, 2017; published online June 16, 2017. Assoc. Editor: Brian Feeny.

J. Comput. Nonlinear Dynam 12(5), 051020 (Jun 16, 2017) (7 pages) Paper No: CND-15-1335; doi: 10.1115/1.4036710 History: Received October 12, 2015; Revised May 04, 2017

## Abstract

Since the solutions of the fractional differential equations (FDEs) have unbounded derivatives at zero, their numerical solutions by piecewise polynomial collocation method on uniform meshes will lead to poor convergence rates. This paper presents a piecewise nonpolynomial collocation method for solving such equations reflecting the singularity of the exact solution. The entire domain is divided into several small subdomains, and the nonpolynomial pieces are constructed using a block-by-block scheme on each subdomain. The method is applied to solve linear and nonlinear fractional differential equations. Numerical examples are given and discussed to illustrate the effectiveness of the proposed approach.

<>

## References

Atanacković, T. M. , Pilipović, S. , Stanković, B. , and Zorica, D. , 2014, Fractional Calculus With Applications in Mechanics: Vibrations and Diffusion Processes, ISTE, London/Wiley, Hoboken, NJ.
Baleanu, D. , Diethelm, K. , Scalas, E. , and Trujillo, J. J. , 2016, Fractional Calculus: Models and Numerical Methods, 2nd ed., World Scientific, Singapore.
Hilfer, R. , ed., 2000, Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ.
Podlubny, I. , 1999, Fractional Differential Equations, Academic Press, San Diego, CA.
Diethelm, K. , 2010, The Analysis of Fractional Differential Equations, Springer, Berlin.
Li, C. , and Zeng, F. , 2015, Numerical Methods for Fractional Calculus, Chapman and Hall/CRC, London.
Garrappa, R. , and Popolizio, M. , 2011, “ On Accurate Product Integration Rules for Linear Fractional Differential Equations,” J. Comput. Appl. Math., 235(5), pp. 1085–1097.
Garrappa, R. , and Popolizio, M. , 2011, “ Generalized Exponential Time Differencing Methods for Fractional Order Problems,” Comput. Math. Appl., 62(3), pp. 876–890.
Doha, E. H. , Bhrawy, A. H. , and Ezz-Eldien, S. S. , 2011, “ A Chebyshev Spectral Method Based on Operational Matrix for Initial and Boundary Value Problems of Fractional Order,” Comput. Math. Appl., 62(5), pp. 2364–2373.
Esmaeili, S. , Shamsi, M. , and Luchko, Y. , 2011, “ Numerical Solution of Fractional Differential Equations With a Collocation Method Based on Müntz Polynomials,” Comput. Math. Appl., 62(3), pp. 918–929.
Yan, Y. , Pal, K. , and Ford, N. J. , 2014, “ Higher Order Numerical Methods for Solving Fractional Differential Equations,” BIT Numer. Math., 54(2), pp. 555–584.
Ford, N. J. , Morgado, M. L. , and Rebelo, M. , 2015, “ A Nonpolynomial Collocation Method for Fractional Terminal Value Problems,” J. Comput. Appl. Math., 275, pp. 392–402.
Firoozjaee, M. A. , Yousefi, S. A. , Jafari, H. , and Baleanu, D. , 2015, “ On a Numerical Approach to Solve Multi-Order Fractional Differential Equations With Initial/Boundary Conditions,” ASME J. Comput. Nonlinear Dyn., 10(6), p. 061025.
Esmaeili, S. , Shamsi, M. , and Dehghan, M. , 2013, “ Numerical Solution of Fractional Differential Equations Via a Volterra Integral Equation Approach,” Cent. Eur. J. Phys., 11(10), pp. 1470–1481.
Zayernouri, M. , and Karniadakis, G. E. , 2013, “ Fractional Sturm–Liouville Eigen-Problems: Theory and Numerical Approximation,” J. Comput. Phys., 252, pp. 495–517.
Ford, N. J. , Morgado, M. L. , and Rebelo, M. , 2013, “ Nonpolynomial Collocation Approximation of Solutions to Fractional Differential Equations,” Fractional Calculus Appl. Anal., 16(4), pp. 874–891.
Eslahchi, M. R. , Dehghan, M. , and Parvizi, M. , 2014, “ Application of the Collocation Method for Solving Nonlinear Fractional Integro-Differential Equations,” J. Comput. Appl. Math., 257, pp. 105–128.
Doha, E. H. , Bhrawy, A. H. , Baleanu, D. , and Hafez, R. M. , 2014, “ A New Jacobi Rational-Gauss Collocation Method for Numerical Solution of Generalized Pantograph Equations,” Appl. Numer. Math., 77, pp. 43–54.
Huang, C. , Jiao, Y. , Wang, L.-L. , and Zhang, Z. , 2016, “ Optimal Fractional Integration Preconditioning and Error Analysis of Fractional Collocation Method Using Nodal Generalized Jacobi Functions,” SIAM J. Numer. Anal., 54(6), pp. 3357–3387.
Baffet, D. , and Hesthaven, J. S. , 2017, “ A Kernel Compression Scheme for Fractional Differential Equations,” SIAM J. Numer. Anal., 55(2), pp. 496–520.
Brunner, H. , 2004, Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, Cambridge, UK.
Cao, Y. , Herdman, T. , and Xu, Y. , 2003, “ A Hybrid Collocation Method for Volterra Integral Equations With Weakly Singular Kernels,” SIAM. J. Numer. Anal., 41(1), pp. 364–381.
Pedas, A. , and Tamme, E. , 2014, “ Numerical Solution of Nonlinear Fractional Differential Equations by Spline Collocation Methods,” J. Comput. Appl. Math., 255, pp. 216–230.
Kolk, M. , Pedas, A. , and Tamme, E. , 2015, “ Modified Spline Collocation for Linear Fractional Differential Equations,” J. Comput. Appl. Math., 283, pp. 28–40.
Abramowitz, M. , and Stegun, I. A. , 1972, Handbook of Mathematical Functions, Dover, New York.
Esmaeili, S. , and Milovanović, G. V. , 2014, “ Nonstandard Gauss–Lobatto Quadrature Approximation to Fractional Derivatives,” Fractional Calculus Appl. Anal., 17(4), pp. 1075–1099.
Zayernouri, M. , and Karniadakis, G. E. , 2014, “ Fractional Spectral Collocation Method,” SIAM J. Sci. Comput., 36(1), pp. 40–62.
Garrappa, R. , 2015, “ Numerical Evaluation of Two and Three Parameter Mittag–Leffler Functions,” SIAM J. Numer. Anal., 53(3), pp. 1350–1369.
Rossikhin, Y. A. , and Shitikova, M. V. , 2010, “ Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Result,” ASME Appl. Mech. Rev., 63(1), p. 010801.

## Figures

Fig. 1

Nonpolynomial basis ϕjk with Sα={0.3,0.6,1,1.3,1.6} and n = 4

Fig. 2

Indicator function (top) and fractional integral of indicator function (bottom)

Fig. 3

Analytical and numerical solution of problem (20); n = 100, for α=0.3 (left) and α=1.8 (right) (Example 2)

Fig. 4

Analytical and numerical solution of problem (21); n = 10, for α=0.3 (left) and α=1.8 (right) (Example 3)

Fig. 5

Numerical solution of Duffing equation for α=1.95 and a = 1 with n = 200 (Example 5)

## Discussions

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections