Spectral Collocation Method for Fractional Differential/Integral Equations with Generalized Fractional Operator

Abstract

Generalized fractional operators are generalization of the Riemann-Liouville and Caputo fractional derivatives, which include Erdélyi-Kober and Hadamard operators as their special cases. Due to the complicated form of the kernel and weight function in the convolution, it is even harder to design high order numerical methods for differential equations with generalized fractional operators. In this paper, we first derive analytical formulas for − ℎ ( > 0) order fractional derivative of Jacobi polynomials. Spectral approximation method is proposed for generalized fractional operators through a variable transform technique. Then, operational matrices for generalized fractional operators are derived and spectral collocation methods are proposed for differential and integral equations with different fractional operators. At last, the method is applied to generalized fractional ordinary differential equation and Hadamard-type integral equations, and exponential convergence of the method is confirmed. Further, based on the proposed method, a kind of generalized grey Brownian motion is simulated and properties of the model are analyzed.

A New Optimal Homotopy Asymptotic Method for Fractional Optimal Control Problems

“…. , n, converges where ψ k (t, C j ) is governed by equation (22) under the definitions equations (19) and (20), it achieves the solution of equations ( 12) and (13).…”

“…Fractional calculus applications are but not limited to liquid-containing gas bubbles [6], neurodynamics system [7], the FOCPs with a general derivative operator [8], a new adaptive synchronization and hyperchaos control of a biological snap oscillator [9], a new mathematical model for Zika virus transmission [10], the fractional features of a harmonic oscillator with position-dependent mass [11], electrohydrodynamic flow in a cylindrical conduit [12], fractional order of HIV infection model [13], and novel solution methods for IBVPs of arbitrary-order with conformable differentiation [14]. Many areas of FDEs have been researched by many authors: the multistage ADM for solving NLP problems over a nonlinear fractional dynamical system [15], the existence of the solution to FOCPs [16], stable analytical techniques for solving FDEs [17], Caputo-Fabrizio operator sense for brucellosis model [18], and spectral collocation method of generalized fractional operator sense for fractional calculus equations [19]. Currently, to obtain exact solutions for FOCPs is still difficult and remain a challenge.…”