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A new computational method based on fractional Lagrange functions to solve multi-term fractional differential equations

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Abstract

This paper explores a new method, called fractional pseudospectral method (FPM), which solves the linear and nonlinear fractional ordinary/partial differential equations (FODEs/FPDEs). After the required basic definitions are explained, we define a new class of interpolants, called fractional Lagrange functions (FLFs), so that they satisfy in the Kronecker delta function at collocation points. These functions can use as a new basis for the pseudospectral methods and can apply for developing a framework or theory in these methods. The Caputo fractional differentiation matrices are obtained for the FLFs; it has been shown that calculating these matrices is very simple and they are the generalization of differentiation matrices in the classical Lagrange functions. Furthermore, the matrices for combining the Ritz method and the fractional pseudospectral method are calculated, and Chebyshev’s theorem and the error estimate for interpolations are extended and proven on FLFs. To demonstrate the efficiency and convergence of FPM, three critical classes of well-known linear and nonlinear FODEs/FPDEs in engineering, physics, and applied sciences based on the five classes of the collocation points are examined.

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Acknowledgments

The authors are very grateful to reviewers and editor for carefully reading the paper and for their comments and suggestions which have improved the paper.

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  1. Department of Computer Sciences, Shahid Beheshti University, G.C., Tehran, Iran

    Mehdi Delkhosh & Kourosh Parand

  2. Department of Cognitive Modelling, Institute for Cognitive and Brain Sciences, Shahid Beheshti University, G.C., Tehran, Iran

    Kourosh Parand

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  1. Mehdi Delkhosh
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Correspondence to Mehdi Delkhosh.

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Delkhosh, M., Parand, K. A new computational method based on fractional Lagrange functions to solve multi-term fractional differential equations. Numer Algor 88, 729–766 (2021). https://doi.org/10.1007/s11075-020-01055-9

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