Differential equations for generalized Jacobi polynomials

Differential Equations for Symmetric Generalized Ultraspherical Polynomials

We look for differential equations satisfied by the generalized Jacobi polynomials { P n (x) }∞ n=0 which are orthogonal on the interval [−1, 1] with respect to the weight function Γ(α+ β + 2) 2α+β+1Γ(α+ 1)Γ(β + 1) (1− x)(1 + x) +Mδ(x+ 1) +Nδ(x− 1), where α > −1, β > −1, M ≥ 0 and N ≥ 0. In the special case that β = α and N = M we find all differential equations of the form ∞ ∑ i=0 ci(x)y (x) =...

Solving the fractional integro-differential equations using fractional order Jacobi polynomials

In this paper, we are intend to present a numerical algorithm for computing approximate solution of linear and nonlinear Fredholm, Volterra and Fredholm-Volterra  integro-differential equations. The approximated solution is written in terms of fractional Jacobi polynomials. In this way, firstly we define Riemann-Liouville fractional operational matrix of fractional order Jacobi polynomials, the...

Fractional differential equations with Legendre polynomials

An Integral Formula for Generalized Gegenbauer Polynomials and Jacobi Polynomials

Generalized Jacobi functions and their applications to fractional differential equations

In this paper, we consider spectral approximation of fractional differential equations (FDEs). A main ingredient of our approach is to define a new class of generalized Jacobi functions (GJFs), which is intrinsically related to fractional calculus and can serve as natural basis functions for properly designed spectral methods for FDEs. We establish spectral approximation results for these GJFs ...