A three point variable mesh finite difference method of third order, have been derived to solve the SINGULAR two-point BOUNDARY VALUE PROBLEMs. The method reduces to a method of order four for the uniform mesh case and may be considered as a modification of the well known Numerov"s method. The methods are self starting and are exact for y=1/x. The convergence of the fourth order method has also been discussed. The various measures of error for solution of two PROBLEMs, using the methods and the fourth order method in [1] are listed. The numerical results are also compared with five well known classical methods to show the self starting and accuracy of the present methods. Subject Classifications: AMS 64L10, CR: G1.7

Various fractional differential equations have been examined during the last decades. Among them, SINGULAR equations are more notable. In this article, by using control functions, the existence of a solution for a bi-SINGULAR fractional differential equation with MULTI-POINT initial VALUE conditions is considered. In the following, some examples elucidate our main result. In this paper by using control functions method, we prove the existence of the solution.

In this paper a mathematical simulation of a simplified internal combustion engine is presented. To contribute engine kinematics and its geometry, simple relations are derived for constrained motions. The equation of motion for the piston forms a SINGULAR BOUNDARY VALUE PROBLEM. The uniqueness of the solution was studied in the Banach space. For solving governing equations an iterative numerical algorithm was used and the numerical method has shown very fast convergency. With this simulation the thermodynamics of expansion process is coupled with piston dynamics. Simulating an engine working at constant torque and polytropic expansions has shown high frequency piston vibration under certain conditions.

Purpose: In this paper, we shall present an algorithm for solving more general SINGULAR second-order MULTI-POINT BOUNDARY VALUE PROBLEMs.Methods: The algorithm is based on the quasilinearization technique and the reproducing kernel method for linear MULTI-POINT BOUNDARY VALUE PROBLEMs.Results: Three numerical examples are given to demonstrate the efficiency of the present method.Conclusions: Obtained results show that the present method is quite efficient.

Variable mesh finite difference method of fifth order have been derived for solution of SINGULAR two point BOUNDARY VALUE PROBLEM, which reduces to the method of order six, for uniform mesh case. These methods are self starting and are exact for u=r-1. The convergence of the sixth order method is discussed. The absolute error, for solution of four PROBLEMs, two linear and two nonlinear are listed to show the accurate and effectiveness of the present method.

In this paper, the variational iteration method (VIM) is used to solve a class of SINGULAR BOUNDARY VALUE PROBLEMs. Three numerical examples are selected to illustrate the effectiveness and simplicity of the method. Numerical solutions obtained by the method are of high accuracy. Numerical results show that the method is a satisfactory method.

‎In this paper‎, ‎we consider a Sturm-Liouville equation with non-separated BOUNDARY conditions on a finite interval‎. ‎We discuss some properties of solutions of the Sturm-Liouville equation‎, ‎where the potential function has a SINGULARity in the finite interval‎. ‎We also calculate eigenVALUEs and prove the uniqueness of Borg's Theorem of this BOUNDARY VALUE PROBLEM‎.