Lapacian module

Created on Thu Aug 29 09:36:54 2024

@author: ppranjiv

class laplace_matrix.galerkin

Bases: object

Computes Lapalician matrix for (d2dx Pn^{lpha, eta}, Pm^{lpha, eta})_w

where,

n, m > 0, Pn-th order polynomail weight, w : coressponds to Jacobi polynomail with alpha , beta of the TEST function

Note

Here we do NOT use integration by parts, rather take the second derivative of trail function, then compute the inner product with respect to weights associated with test functions.

legendre()

Computes Mass matrix Legendre Polynomials in [-1,1] of dim = (k) x (k), where k-1 is highest degree

Parameters

degreeint

degree of the polynomial

Returns

Matrix Legendre polynomials within the interval [-1,1]

M: numpy array

Matrix of dim (k) X (k), where k-1 is the highest degree of Legengre Polynomials

homogeneous_legendre()

Computes Lapalce matrix Homogeneus basis using Legendre Polynomials in [-1,1] of dim = (k) x (k), where k-1 is highest degree

Parameters

degreeint

degree of the polynomial

Returns

Matrix Homogeneus basis using Legendre polynomials within the interval [-1,1]

M: numpy array

Matrix of dim (k) X (k), where k-1 is the highest degree of Legengre Polynomials

Notes

phi_n = l_{n+2} - l_{n},

where, l_{n} is an nth order Legendre Polynomial and, phi_n is a homogeneous basis function

shift_legendre()

Computes Mass matrix Shifted Legendre Lapalcian in [0,1] of dim = (k) x (k), where k-1 is highest degree

Parameters

degreeint

degree of the polynomial

Returns

Matrix Legendre Lapalcian within the interval [0,1]

M: numpy array

Matrix of dim (k) X (k), where k-1 is the highest degree of Shifted Legengre Lapalcian

jacobi(beta, degree)

Computes Mass matrix Jacobi Lapalcian in [-1,1] of dim = (k) x (k), where k-1 is highest degree

Parameters

alphafloat

paramter assocated with Jacobi Lapalcian, alpha > -1

betafloat

paramter assocated with Jacobi Lapalcian, beta > -1

degreeint

degree of the polynomial

Returns

Matrix Jacobi Lapalcian within the interval [-1,1]

M: numpy array

Matrix of dim (k) X (k), where k-1 is the highest degree of Jacobi Lapalcian

chebyshev_first_kind()

Computes Laplacpe chebyshev_first_kind s in [-1,1] of dim = (k) x (k), where k-1 is highest degree

Parameters

degreeint

degree of the polynomial

Returns

Laplace martix chebyshev_first_kind within the interval [-1,1]

M: numpy array

Matrix of dim (k) X (k), where k-1 is the highest degree

shift_jacobi(beta, degree)

Computes Mass matrix Shifted Jacobi Lapalcian in [0,1] of dim = (k) x (k), where k-1 is highest degree

Parameters

alphafloat

paramter assocated with Jacobi Lapalcian, alpha > -1

betafloat

paramter assocated with Jacobi Lapalcian, beta > -1

degreeint

degree of the polynomial

Returns

Matrix Jacobi Lapalcian within the interval [0,1]

M: numpy array

Matrix of dim (k) X (k), where k-1 is the highest degree of Shifted Jacobi Lapalcian

Q_poly(beta, degree)

Computes Mass matrix Q Lapalcian in [0,1] of dim = (k) x (k), where k-1 is highest degree

Parameters

alphafloat

paramter assocated with Q Lapalcian, alpha > -1

betafloat

paramter assocated with Q Lapalcian, beta > -1

degreeint

degree of the polynomial

Returns

Matrix Q Lapalcian within the interval [0,1]

M: numpy array

Matrix of dim (k) X (k), where k-1 is the highest degree of Q Lapalcian

Notes

Q (x) := P_n (1-2x), where P_n is a Jacobi polynomial with paramter, alpha and beta.

shift_arbitary_jacobi(b, alpha, beta, degree)

Computes Mass matrix Shifted (arbitary) Jacobi Lapalcian in [a,b] of dim = (k) x (k), where k-1 is highest degree

Parameters

afloat

min of interval [a, b]

bfloat

max of interval [a, b]

alphafloat

paramter assocated with Jacobi Lapalcian, alpha > -1

betafloat

paramter assocated with Jacobi Lapalcian, beta > -1

degreeint

degree of the polynomial

Returns

Matrix Jacobi Lapalcian within the interval [0,1]

M: numpy array

Matrix of dim (k) X (k), where k-1 is the highest degree of Shifted Jacobi Lapalcian

class laplace_matrix.petrov_galerkin

Bases: object

Computes Lapalician (Petrov-Galerkin) matrix for (d2dx Pn, Pm^{lpha, eta})_w

where,

n, m > 0, Pn-th order polynomail weight, w : coressponds to Jacobi polynomail with alpha , beta of the TEST function

Note

• Here we do NOT use integration by parts, rather take the second derivative of trail function,

then compute the inner product with respect to weights associated with test functions.

• Computes Lapalce matrix Patrov-Galkerin implies the test and trial functions are different.

homogeneous_legendre()

Computes Lapalce matrix Patrov-Galkerin

Trial function : Homogeneous Legendre Test function : Legendre

Parameters

degreeint

degree of the polynomial

Returns

M: numpy array

Matrix of dim (k) X (k), where k-1 is the highest degree of Legengre Polynomials Lapaciam Matrix within the interval [-1,1]

Notes

phi_n = l_{n+2} - l_{n},

where, l_{n} is an nth order Legendre Polynomial and, phi_n is a Homogeneous Legendre basis function

homogeneous_shift_legendre()

Computes Lapalce matrix Patrov-Galkerin

Trial function : Homogeneous Shifted Legendre Test function : Shifted Legendre

Parameters

degreeint

degree of the polynomial

Returns

M: numpy array

Matrix of dim (k) X (k), where k-1 is the highest degree of Shifted Legengre Polynomials Lapaciam Matrix within the interval [0,1]

Notes

phi_n = l_{n+2} - l_{n},

where, l_{n} is an nth order Shifted Legendre Polynomial and, phi_n is a Homogeneous Shifted Legendre basis function