Zhang, "Numerical solution of the fractional partial differential equations by the two-dimensional fractional-order Legendre functions
," Abstract and Applied Analysis, vol.
We define the fractional-order Legendre functions
(FLFs) by introducing the change of variable t = [x.sup.[alpha]] and [alpha] > 0 on shifted Legendre polynomials.
A new operational matrix of integration for fractional-order Legendre functions
(FLFs) is first derived.
The general solution of this equation is suitable written as a linear combination of associated the third kind Legendre functions
[R.sup.[+ or -]1/2.sub.v] .
The most important special functions are known as: Bessel functions, Hermite functions, Legendre functions
, Laguerre functions, Chebyshev functions etc., .
Here [P.sup.D.sub.m] denotes the associated Legendre functions
, defined by
S (Approximations of Special Functions) has new routines for polygamma functions, zeros of Bessel functions, jacobian functions, elliptic integrals and associations Legendre functions
It has more than 800 built-in functions or objects, from simple trig relations to complex Legendre functions
In , the authors applied fractional order Legendre functions
method depending on the choices of two parameters to solve the fractional diffusion-wave equations.
Lu, "Couple of the variational iteration method and fractional-order legendre functions
method for fractional differential equations," The Scientific World Journal, vol.