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.

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.

In this paper, we compute the approximate numerical solution for the Volterra-Fredholm integral equation (VFIE) by using the shifted Jacobi collocation (SJC) method which depends on the operational matrices. Some properties of the shifted Jacobi polynomials are introduced. These properties allow us to transform the VolterraFredholm integral equation into a system of algebraic equations in a nice form with the expansion coefficients of the solution. Also, the convergence and error analysis are studied extensively. Finally, some examples which verify the efficiency of the given method are supplied and compared with other methods.

In this paper, a Laguerre collocation method is presented in order to obtain numer-ical solutions for linear and nonlinear Lane-Emden type equations and their initial conditions. The basis of the present method is operational matrices with respect to modi, ed generalized Laguerre polynomials(MGLPs) that transforms the solu-tion of main equation and its initial conditions to the solution of a matrix equation corresponding to the system of algebraic equations with the unknown Laguerre co-e, cients. By solving this system, coe, cients of approximate solution of the main problem will be determined. Implementation of the method is easy and has more accurate results in comparison with results of other methods.

The nonlinear space time dynamics have been discussed in terms of a hyper-bolic equation known as a sine-Gordon equation. The proposed equation has been discretized using the Bessel collocation method with Bessel poly-nomials as base functions. The proposed hyperbolic equation has been transformed into a system of parabolic equations using a continuously dif-ferentiable function. The system of equations involves one linear and the other nonlinear diffusion equation. The convergence of the present tech-nique has been discussed through absolute error, L2-norm, and L∞-norm.The numerical values obtained from the Bessel collocation method have been compared with the values already given in the literature. The present technique has been applied to different problems to check its applicability. Numerical values obtained from the Bessel collocation method have been presented in tabular as well as in graphical form.

In this article we use discrete collocation method for solving Fredholm–Volterra integro–differential equations, because these kinds of integral equations are used in applied sciences and engineering such as models of epidemic diffusion, population dynamics, and reaction–diffusion in small cells. Also the above integral equations with convolution kernel will be solved by discrete collocation method. In this method we approximate solution of problem by no smooth piecewise polynomial. Numerical results show a high accuracy and validity discrete collocation method.