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A new computational method based on fractional Lagrange functions to solve multi-term fractional differential equations. (English) Zbl 1501.65079

Summary: 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.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
58C40 Spectral theory; eigenvalue problems on manifolds
35S10 Initial value problems for PDEs with pseudodifferential operators
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations

Software:

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