Sonin Kernel module#
Created on Mon Sep 2 13:56:04 2024
@author: ppranjiv
- class sonin_kernel.sonin#
Bases:
object- Sonine Kernels of power series type
k := h_{alpha} SUM a_n x^n and, kappa := h_{1-alpha} SUM b_n x^n, where, h_alpha = x^{alpha -1} / Gamma (alpha), Gamma is the Euler Gamma function
Reference#
Luchko, Y. (2021) General Fractional Integrals and Derivatives with the Sonine Kernels. Mathematics
- get_kappa_coeff(alpha)#
Computes the coeffcient for the sonin kernel, kappa given, the coeffcients of kernel, k
Parameters#
- aarray, float
coefficients of kernel, k
- alphafloat
fractional order, 0 < alpha < 1
Returns#
- barray, float
coefficients of kernel, kappa. same shape as k
Note#
- It uses the relationship between a and b as,
a[0] b[0] = 1, for n = 1
SUM_k^n Gamma(k+1-alpha) Gamma(alpha + n -k) a[n-k] b[k] = 0, for n > 1
Reference#
Luchko, Y. (2021) General Fractional Integrals and Derivatives with the Sonine Kernels. Mathematics
- verify_sonin(b1, k1, k2, alpha, degree)#
- Verifies the sonin contdion, for a kernel
k = a1 x^{k1+alpha-1} / Gamma (alpha), kappa = b1 x^{k2-alpha} / Gamma (1-alpha)
Parameters#
- a1float
coefficients of kernel, k
- b1float
coefficients of kernel, kappa
- k1int
parameter for power for kernel, k, which appears as x^{k1+alpha-1}
- k2int
parameter for power for kernel, kappa, which appears as x^{k2-alpha}
- alphafloat
fractional order, 0 < alpha < 1
- degreeint
Gauss Jacobi Quadrature degree
Returns#
- xarray, float
x in [0, 1], the points where sonin condition is verified
- sonine: array, float
the values of convolution of (k * kappa)(x), at x
- verify_condition(b, alpha, degree)#
- Verifies the sonin contion, for a kernel
k = SUM_k^n a_k x^{k+alpha-1} / Gamma (alpha), kappa = SUM_n^n b_k x^{k2-alpha} / Gamma (1-alpha)
Parameters#
- aarray, float
coefficients of kernel, k
- barray, float
coefficients of kernel, kappa
- alphafloat
fractional order, 0 < alpha < 1
- degreeint
Gauss Jacobi Quadrature degree
Returns#
- xarray, float
x in [0, 1], the points where sonin condition is verified
- sonine: array, float
the values of convolution of (k * kappa)(x), at x
Note#
uses sonin.verify_sonin for computations