Abstract
AbstractThe nonlocal nature of the fractional derivative makes the numerical treatment of fractional differential equations expensive in terms of computational accuracy in large domains. This paper presents a new multiple-step adaptive pseudospectral method for solving nonlinear multi-order fractional initial value problems (FIVPs), based on piecewise Legendre–Gauss interpolation. The fractional derivatives are described in the Caputo sense. We derive an adaptive pseudospectral scheme for approximating the fractional derivatives at the shifted Legendre–Gauss collocation points. By choosing a step-size, the original FIVP is replaced with a sequence of FIVPs in subintervals. Then the obtained FIVPs are consecutively reduced to systems of algebraic equations using collocation. Some error estimates are investigated. It is shown that in the present multiple-step pseudospectral method the accuracy of the solution can be improved either by decreasing the step-size or by increasing the number of collocation points within subintervals. The main advantage of the present method is its superior accuracy and suitability for large-domain calculations. Numerical examples are given to demonstrate the validity and high accuracy of the proposed technique.
Highlights
Fractional calculus has a long history because, starting from a letter from Leibniz to L’Hospital in 1695, it has been developing up to now
We propose an efficient multiplestep adaptive pseudospectral method based on shifted Legendre polynomials for the numerical solution of the linear and nonlinear multi-order fractional initial value problems (FIVPs), which involve Caputo fractional derivatives
An efficient multiple-step pseudospectral method based on the shifted Legendre–Gauss (ShLG) collocation points has been proposed for numerically solving the multi-order FIVPs
Summary
Fractional calculus has a long history because, starting from a letter from Leibniz to L’Hospital in 1695, it has been developing up to now. Global pseudospectral methods use global polynomials together with Gaussian quadrature collocation points which is known to provide accurate approximations that converge exponentially for problems whose solutions are smooth over the whole domain of interest [35]. In global pseudospectral methods it is not convenient to resolve the corresponding discrete system with very large number of collocation points To remove this deficiency, adaptive version of pseudospectral method, which is based on the domain decomposition procedure, is considered [38,39,40]. We propose an efficient multiplestep adaptive pseudospectral method based on shifted Legendre polynomials for the numerical solution of the linear and nonlinear multi-order fractional initial value problems (FIVPs), which involve Caputo fractional derivatives.
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