Interpolation Lagrange method

Orthogonal Collocation The orthogonal collocation method has found widespread application in chemical engineering, particularly for chemical reaction engineering. In the collocation method, the dependent variable is expanded in a series of orthogonal polynomials. See "Interpolation and Finite Differences Lagrange Interpolation Formulas. ... 

The higher order ODEs are reduced to systems of first-order equations and solved by the Runge-Kutta method. The missing condition at the initial point is estimated until the condition at the other end is satisfied. After two trials, linear interpolation is applied after three or more, Lagrange interpolation is applied. 

Method of Lines. The method of lines is used to solve partial differential equations (12) and was already used by Cooper (I3.) and Tsuruoka (l4) in the derivation of state space models for the dynamics of particulate processes. In the method, the size-axis is discretized and the partial differential a[G(L,t)n(L,t)]/3L is approximated by a finite difference. Several choices are possible for the accuracy of the finite difference. The method will be demonstrated for a fourth-order central difference and an equidistant grid. For non-equidistant grids, the Lagrange interpolation formulaes as described in (15 ) are to be used. 

On solving these equations for the coefficients A j, the solution of minimum norm is the interpolated gradient vector P, such that pi = P 2 0, at the interpolated coordinate vector Q. The Lagrange multiplier /x in this method provides an estimate of the residual error. 

This expression is used as a trial-function expansion for T in much the same way as the Lagrange interpolation polynomial is in the polynomial collocation method of Villadsen and Stewart (10,1 1 There are four unknown constants associated with each node, giving a total of 4(n+l)(nH-1) unknowns in the expansion. 

The Lagrange interpolation polynomial was used to develop the spatial finite difference formulas used for the distance method of lines calculation. For example, the two point polynomial for the solids flux variable F(t,z) can be expressed by... 

In order to integrate we use Simpson s formula. We obtain U and a as we would like to see it, with the help of a four-point interpolation following Lagrange s method. 

Choosing the generating function UNg as the polynomial, UNf (q) = (q - qx)(q - 2) " (q qi) (q q ) leads to the well-known Lagrange interpolation formula. Figure 4 shows the expansion function g (q) which is based on the zeros of the Cheby-chev orthogonal polynomial of order Ng. Another choice appropriate for evenly distributed sampling points is based on the global function NNg(q) = sin(2 Tr /A ). It is closely related to the Fourier method described in the next section. 

In this section, we will develop two interpolation methods for unequally spaced data the Lagrange polynomials and spline interpolation. 

Method of Solution The Lagrange interpolation is done based on Eqs. (3.132) and (3.136). The order of interpolation is an input to the function. The cubic spline interpolation is done based on Eq. (3.143). The values of the second derivatives at base points, assuming a natural spline, are calculated from Eq. (3.147). 

Discussion of Results Order of the Lagrange interpolation is chosen to be three for comparison of the results with that of the cubic spline, which is also third-order interpolation. Fig. E3.2 shows the results of calculations. There is no essential difference between the two methods. The cubic spline, however, passes smoothly through the base points, as expected. Because the Lagrange interpolation is performed in the subsets of four base points with no restriction related to their neighboring base points, it can be. seen that the slope of the resulting curve is not continuous through most of the base points. 

Gregory-Newton forward interpolation method. Lagrange polynomial interpolation method. Cubic splines interpolation method. 

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