In this paper we present a new method to find simple or multiple roots of functions in a finite interval. In this method using bisection method we can find an interval such that this function is one to one on it, thus we can transform problem of finding roots in this interval into an ordinary differential equation with boundary conditions.By solving this equation using collocation method we can find a root for given function in the special interval. We also present convergence analysis of the new method. Finally some examples are given to show efficiency of the presented method.

The spline collocation method is used to approximate solutions of boundary value problems. The convergence analysis is given and the method is shown to have second-order convergence. A numerical illustration is given to show the pertinent features of the technique.

In the pharmaceutical industry, a great number of products are in the form of particulate solids. Since the mid-1980s, a new method of powder generation has appeared involving crystallisation with supercritical fluids (SCF). The rapid expansion of supercritical solutions (RESS) is a promising new process for the production of small and uniform particles.Several variables can influence the RESS process: the nozzle temperature, preexpansion pressure, the nozzle diameter, and geometry. In this work, a two-dimensional Lagrange interpolation method has been proposed to describe the size of nanoparticle forming through the rapid expansion of supercritical solutions, as a function of preexpansion pressure and nozzle temperature.

A meshless approach, collocation discrete least square (CDLS) method, is extended in this paper, for solving elasticity problems. In the present CDLS method, the problem domain is discretized by distributed field nodes. The field nodes are used to construct the trial functions. The moving least-squares interpolant is employed to construct the trial functions. Some collocation points that are independent of the field nodes are used to form the total residuals of the problem. The least-squares technique is used to obtain the solution of the problem by minimizing the summation of the residuals for the collocation points. The final stiffness matrix is symmetric and therefore can be solved via efficient solvers. The boundary conditions are easily enforced by the penalty method. The present method does not require any mesh so it is a truly meshless method. Numerical examples are studied in detail, which show that the present method is stable and possesses good accuracy, high convergence rate and high efficiency.