Abstract
A family of tensor-based methods called Proper Generalized Decomposition (PGD) methods have been recently introduced for the a priori construction of the solution of several partial differential equations. This strategy was tested with success to demonstrate the capability of representing the solution with a significant reduction of the calculation and storage cost. In this paper, we suggest to test the efficiency of a such approach in solving general nonseparable fractional derivative elliptic problem. We will illustrate by several numerical experiments the efficiency of PGD, especially when the mesh or the coefficients vary with high contrast ratio. Although the PGD scheme considered in this paper is based on spectral method, it is extendable to other methods such as finite element method.
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References
R.A. Adams, J.J.F. Fournier, Sobolev Spaces, vol. 140 (Academic, New York, 2003)
D.A. Benson, R. Schumer, M.M. Meerschaert, S.W. Wheatcraft, Fractional dispersion, Lévy motion, and the MADE tracer tests. Transp. Porous Media 42(1), 211–240 (2001)
F. Chinesta, R. Keunings, A. Leygue, The Proper Generalized Decomposition for Advanced Numerical Simulations: A Primer (Springer Science & Business Media, New York, 2013)
K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type (Springer, Berlin, 2010)
R. Gorenflo, F. Mainardi, Fractional Calculus and Continuous-Time Finance III: The Diffusion Limit (Birkh\(\ddot{a}\) user, Basel, 2001)
X. Li, C. Xu, A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47(3), 2108–2131 (2009)
X. Li, C. Xu, Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation. Commun. Comput. Phys. 8(5), 1016 (2010)
Z. Mao, J. Shen, Efficient spectral–Galerkin methods for fractional partial differential equations with variable coefficients. J. Comput. Phys. 307, 243–261 (2016)
A. Oustaloup, P. Coiffet, Systèmes asservis linéaires d’ordre fractionnaire: théorie et pratique. Masson (1983)
Y.A. Rossikhin, M.V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50(1), 15–67 (1997)
Acknowledgements
This work was supported by IDEX-Bordeaux research programm and CAMPUS FRANCE PHC CAI YUANPEI Research program.
Chuanju Xu is supported by NSF of China (Grant numbers 11471274, 11421110001, 51661135011, and 91630204.
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Lin, S., Azaiez, M., Xu, C. (2017). Using PGD to Solve Nonseparable Fractional Derivative Elliptic Problems. In: Bittencourt, M., Dumont, N., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016. Lecture Notes in Computational Science and Engineering, vol 119. Springer, Cham. https://doi.org/10.1007/978-3-319-65870-4_13
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DOI: https://doi.org/10.1007/978-3-319-65870-4_13
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