Shifted Legendre polynomials

There are two polynomial forms that are often used to solve engineering problems. These are the shifted Legendre poljmomials and the Jacobi polynomials. The shifted Legendre polynomials are used for problems without symmetry and the Jacobi polynomials for problems with symmetry. 

We consider the spatial domain x e [0,1] and that the problem has no special symmetry properties over that domain. In this case we choose a trial solution 

For the collocation method, we will force the differential equation to be satisfied at the collocation points. In order to do so, we need to 

The set of collocation points N in number) and the boundary points (2 in number) can be expressed in terms of an unknown vector of coefficients d as 

The first derivative of the t rial function at a collocation point is given 

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Orthogonal collocation on two finite elements is used in the radial direction, as in the steady-state model (1), with Jacobi and shifted Legendre polynomials as the approximating functions on the inner and outer elements, respectively. Exponential collocation is used in the infinite time domain (4, 5). The approximating functions in time have the form... 

Chang, R.Y. and Wang, M.L. (1984) Modelling the batch crystallization process via shifted Legendre polynomials. Industrial and Engineering Chemistry Process Design... 

Due to the variable gas velocity and the nonlinear rate law the model equations represent a set of coupled nonlinear algebraic and differential equations of boundary value type which must be solved numerically. For this purpose the nonlinear equations are entirely linearized using the cjuasilinearization technique (12) and the linearized differential equations are solved using the orthogonal collocation method based on shifted Legendre polynomials (13). 

Table 8.6 Roots ( Kj) for Shifted Legendre Polynomials (Source Finlayson, 1972)...

Since we have not used the symmetry boundary condition that dTfdy = 0 at y = 1/2, we will solve the problem by performing an orthogonal collocation in the spatial domain using the shifted Legendre polynomials as the basis function. This will reduce the problem to a set of initial-value ordinary differential equations that can be solved using IMSL ordinary differential equation routines. As discussed by Cooper et al. (1986), seven internal collocation points accurately describe the solution to the partial differential equations. Therefore, we use equation (8.12.14) to approximate the second spatial derivative. This reduces the original partial differential equation of (8.12.15) to... 

For this purpose, use the shifted Legendre polynomials whose roots and matrices are provided below. 

These partial differential equations can be converted to ordinary differential equations nsing orthogonal collocation. For this purpose, we will use the shifted Legendre polynomials whose roots and matrices are provided below. 

The orientational order parameter is the second Legendre polynomial of cos9 and, where the chemical shift dispersion is reduced by rotational averaging, is given by... 

The second-order quadrupole interaction depends on the Wigner rotation matrices, which become time-dependent when the sample is rotated about an angle 6 with respect to B. The average second-order quadrupolar shift then depends on the Legendre polynomials... 

For first-order interactions (dipole-dipole coupling, chemical shift anisotropy, first-order quadrupolar interaction) the expansion in Eq. (23) contains only the zero- and second-rank terms. As shown in Fig. 3, the Legendre polynomial of rank 2 is zero at the magic angle (P2(cosXm)=0 for Xm=54.74°). Therefore, under MAS, and provided that the spinning rate is larger than the anisotropic linewidth, the anisotropic terms of the first-order interactions are averaged to zero and isotropic spectra are obtained. 

Notice that the coefficient + 1) in the original equation becomes — 1) + 2), ( — 2)( + 3), and ( — 3) ( + 4) for the successive species being the same as Eqs. (52-56), while the eigenvalues hf are also shifted depending on the coefficients involved in the second derivatives of fA(Xi)- These changes are reminiscent of the familiar ones for the ordinary and associated Legendre polynomials, and their connections with the actions of ladder operafors. We are exploring the possibilities for the Lame functions themselves and their connections with Section 4.2.2. 

The scattering amplitudes f 6) can be expanded in terms of Legendre polynomials and the derivation of the phase from that of gives the phase shifts rji for each / as... 

We can solve Eq. (7.2) for cos 6 and insert the result in the explicit formula for the second Legendre polynomial. This expresses the angular distribution in terms of the shift, called the detuning, between the nominal absorption frequency vq... 

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