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A Hybrid Spectral Element Method for Fractional Two-Point Boundary Value Problems

Part of: Real functions General theory in ordinary differential equations Miscellaneous topics of analysis in the complex domain Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  09 May 2017

Changtao Sheng  and
Jie Shen
Changtao Sheng*
Affiliation:
Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing and School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China
Jie Shen*
Affiliation:
Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing and School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China Department of Mathematics, Purdue University, West Lafayette, IN 47907-1957, USA
*
*Corresponding author. Email addresses:shen7@purdue.edu (J. Shen), ctsheng@stu.xmu.edu.cn (C. T. Sheng)
*Corresponding author. Email addresses:shen7@purdue.edu (J. Shen), ctsheng@stu.xmu.edu.cn (C. T. Sheng)
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Abstract

We propose a hybrid spectral element method for fractional two-point boundary value problem (FBVPs) involving both Caputo and Riemann-Liouville (RL) fractional derivatives. We first formulate these FBVPs as a second kind Volterra integral equation (VIEs) with weakly singular kernel, following a similar procedure in [16]. We then design a hybrid spectral element method with generalized Jacobi functions and Legendre polynomials as basis functions. The use of generalized Jacobi functions allow us to deal with the usual singularity of solutions at t = 0. We establish the existence and uniqueness of the numerical solution, and derive a hptype error estimates under L2(I)-norm for the transformed VIEs. Numerical results are provided to show the effectiveness of the proposed methods.

Keywords

Boundary value problemsgeneralized Jacobi functionsspectral element methodfractional differential equationserror bounds

MSC classification

Secondary: 26A33: Fractional derivatives and integrals 30E25: Boundary value problems 34A08: Fractional differential equations 65L60: Finite elements, Rayleigh-Ritz, Galerkin and collocation methods 65L70: Error bounds
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 2 , May 2017 , pp. 437 - 464
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Brunner, H., Nonpolynomial spline collocation for Volterra equations with weakly singular kernels, SIAM J. Numer. Anal., 20 (1983), pp. 1106–1119.CrossRefGoogle Scholar
[2] Brunner, H., Collocation methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge, 2004.Google Scholar
[3] Brunner, H., Pedas, A. and Vainikko, G., The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations, Math. Comput., 227 (1999), pp. 1079–1095.Google Scholar
[4] Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A., Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006.Google Scholar
[5] Cao, Y., Herdman, T. and Xu, Y., A hybrid collocation method for Volterra integral equations with weakly singular kernels, SIAM J. Numer. Anal., 41 (2003), pp. 364–381.Google Scholar
[6] Chen, S., Shen, J. and Wang, L., Generalized Jacobi functions and their applications to fractional differential equations, Math. Comput., 85 (2016), pp. 1603–1638.Google Scholar
[7] Chen, Y. and Tang, T., Spectral methods for weakly singular Volterra integral equations with smooth solutions, J. Comput. Appl. Math., 233 (2009), pp. 938–950.Google Scholar
[8] Chen, Y. and Tang, T., Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel, Math. Comput., 79 (2010), pp. 147–167.Google Scholar
[9] Diethelm, K., The Analysis of Fractional Differential Equations, Lecture Notes in Math., Vol. 2004. Springer, Berlin, 2010.Google Scholar
[10] Funaro, D., Polynomial Approximations of Differential Equations, Springer-Verlag, Berlin, 1992.Google Scholar
[11] Gracia, J. and Stynes, M., Upwind and central difference approximation of convection in Caputo fractional derivative two-point boundary value problems, J. Comput. Appl. Math., to appear.Google Scholar
[12] Guo, B., Spectral Methods and Their Applications, World Scientific, Singapore, 1998.Google Scholar
[13] Jin, B., Lazarov, R. and Pasciak, J., Variational formulation of problems involving fractional order differential operators, Math. Comput., 86 (2015), pp. 2665–2700.CrossRefGoogle Scholar
[14] Khader, M., On the numerical solutions for the fractional diffusion equation, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), pp. 2535–2542.Google Scholar
[15] Khader, M. and Hendy, A., The approximate and exact solutions of the fractional-order delay differential equations using Legendre pseudospectral method, Int. J. Pure Appl. Math., 74 (2012), pp. 287–297.Google Scholar
[16] Kopteva, N. and Stynes, M., An efficient collocation method for a Caputo two-point boundary value problem, BIT, 55 (2015), pp. 1105–1123.Google Scholar
[17] Larsson, S., Thomée, V. and Wahlbin, L., Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method, Math. Comput., 67 (1998), pp. 45–71.Google Scholar
[18] Li, X., Tang, T. and Xu, C., Parallel in time algorithm with spectral-subdomain enhancement for Volterra integral equations, SIAM J. Numer. Anal., 51 (2013), pp. 1735–1756.Google Scholar
[19] Li, X. and Xu, C., A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), pp. 2108–2131.CrossRefGoogle Scholar
[20] Lin, Y. and Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), pp. 1533–1552.CrossRefGoogle Scholar
[21] Machado, J. T., Kiryakova, V. and Mainardi, F., Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), pp. 1007–5704.CrossRefGoogle Scholar
[22] Podlubny, I., Fractional Differential Equations, Academic Press Inc., San Diego, CA, 1999.Google Scholar
[23] Shen, J., Sheng, C. and Wang, Z., Generalized Jacobi spectral-Galerkin method for non-linear Volterra integral equations with weakly singular kernels, J. Math. Study, 48 (2015), pp. 315–329.Google Scholar
[24] Shen, J. and Tang, T., Spectral and High-Order Methods with Applications, Science Press, Beijing, 2006.Google Scholar
[25] Shen, J., Tang, T. and Wang, L., Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, Vol. 41, Springer-Verlag, Berlin, Heidelberg, 2011.Google Scholar
[26] Sheng, C. and Shen, J., A hybrid spectral element method for Volterra integral equations with weakly singular kernel, Sci. Sin. Math., 46 (2016), pp. 1017–1036.Google Scholar
[27] Sheng, C., Wang, Z. and Guo, B., A multistep Legendre-Gauss spectral collocation method for nonlinear Volterra integral equations, SIAM J. Numer. Anal., 52 (2014), pp. 1953–1980.CrossRefGoogle Scholar
[28] Stynes, M. and Gracia, J. L., A finite difference method for a two-point boundary value problem with a Caputo fractional derivative, IMA J. Numer. Anal., 35 (2015), pp. 698–721.Google Scholar
[29] Wang, Z. and Sheng, C., An hp-spectral collocation method for nonlinear Volterra integral equations with vanishing variable delays, Math. Comput., 85 (2016), pp. 635–666.Google Scholar
[30] Zayernouri, M. and Karniadakis, G. E., Fractional spectral collocation methods for linear and nonlinear variable order FPDEs, J. Comput. Phys., (2014), pp. 312–338.Google Scholar
[31] Zayernouri, M. and Karniadakis, G. E., Discontinuous spectral element methods for time and space-fractional advection equations, SIAM J. Sci. Comput., 36 (2014), pp. 684–707.Google Scholar
[32] Zayernouri, M. and Karniadakis, G. E., Exponentially accurate spectral and spectral element methods for fractional ODEs, J. Comput. Phys., 257 (2014), pp. 460–480.Google Scholar
[33] Zayernouri, M. and Karniadakis, G. E., Fractional spectral collocation method, SIAM J. Sci. Comput., 36 (2014), pp. 40–62.CrossRefGoogle Scholar