A Laguerre Approach for the Solutions of Singular Perturbated Differential Equations

In this paper, a Laguerre method is presented to solve singularly perturbated two-point boundary value problems. By means of the matrix relations of the Laguerre polynomials and their derivatives, original problem is transformed into a matrix equation. Later, we use collocation points in the matrix equation and thus the considered problem is reduced to a system of linear algebraic equations. The solution of this system gives the coefficients of the desired approximate solution. Also, an error estimation based on the residual function is introduced for the method. The Laguerre polynomial solution is improved by using this error estimation. Finally, error estimation and residual improvement are illustrated by examples and comparisons are given with other methods.

Method of Taylor Expansion Moment Incorporating Fractal Theories for Brownian Coagulation of Fine Particles

Fine particles aggregating into larger units or flocculation body is a random combination process. Increasing the size and density of flocculation body is the main approach to rapid particle removal or sedimentation in water. Aiming at the Brownian coagulation of fine particles, a new method of Taylor expansion moment construction of fractal flocs has been developed in this paper, incorporating the Taylor expansion approach based on the moment method and the fractal dimension of the floc structure originated from fractal theories. This method successfully overcomes the limit of previous moment methods that require pre-assumed particle size distribution. Results of the zero and second order moments of Brownian flocs from the proposed method are compared with those from the Laguerre method, integral moment method and finite element method. It is found that the higher accuracy and efficiency of computation have been achieved by the new method, compared to the previous ones. Effects of the fractal dimension on the zero and second order moments, geometric average volume and standard deviation are also analyzed using this method. The self-conservation characteristics of particle distribution is observed without presumption of initial distributions.