The Modified Differential Quadratures and Their Applications

In this paper, a number of modifications are instituted in implementing the quadrature method for solving chemical engineering problems with semi-infinite domains and/or steep gradients. This improvement in the curve-fitting ability of differential quadratures is achieved by adopting trial functions of forms other than the polynomials. Formal criteria are first developed (and proved) for the selection of proper function forms. If the trial functions are restricted to the products of polynomials and some auxiliary functions, explicit formulae are derived to facilitate the calculation of the corresponding modified quadrature coefficients. If, in addition, the grid points are chosen to be the zeros of an orthogonal polynomial, e.g. Jacobi, Laguerre and Hermite, further simplifications can be realized to promote the efficiency and accuracy of the computation procedure. The modified differential quadratures have been applied to various example problems. From the data we have collected so far, it can be concluded that the proposed approach yields more accurate results in regions where most of the variations in the dependent variables occur and tends to lose its edge at locations where negligible changes can be detected in the numerical solutions.