Fractional Derivative using Convolution Orthogonality over Homogeneous Basis#

Created on Mon Sep 2 08:51:44 2024

@author: ppranjiv

frac_differ_conv_homogeneous.gm(x)#

Convolution Spectral methods for fractional derivative,

exmaple Power function : x^k1 - x^k2. Since the function is homogenous, we use Sifted Homogeneuos Legendre Basis, where,

Trial function : phi_n = kappa * (L_n+2 - L_n ), Test function : (L_n+2 - L_n );

where L_n is nth order legendre polynomial and Kappa is a sonine kernel, assocaited with integral - in this case fractional

Note : Outer convolution orthogonality is used, instead of inner product.

To fix : Jacobi Quadture weights and quad points.

frac_differ_conv_homogeneous.exp(x, a1, degree, alpha)#

Reconsturct the function f (x) = SUM_n a_n P_n,

where,: P_n is shifted Legendre polynomails over the interval [0,1] and, a_n are the coeffcients.

Parameters#

frac_differ_conv_homogeneous.func1(x, k1, k2)#

Test function

k1 : int

k2 : int

frac_differ_conv_homogeneous.rhs_conv(degree, k1, k2, alpha)#

Computes the outer-convolution from [0,1] of

rhs = g * P_n,

where,: P_n is shifted homogenehous Legendre polynomial and f is the function

k1 : int

k2 : int

alpha : float