An Implicit Method for Numerical Solution of Singular and Stiff Initial Value Problems

An implicit method has been presented for solving singular initial value problems. The method is simple and gives more accurate solution than the implicit Euler method as well as the second order implicit Runge-Kutta (RK2) (i.e., implicit midpoint rule) method for some particular singular problems. Diagonally implicit Runge-Kutta (DIRK) method is suitable for solving stiff problems. But, the derivation as well as utilization of this method is laborious. Sometimes it gives almost similar solution to the two-stage third order diagonally implicit Runge-Kutta (DIRK3) method and the five-stage fifth order diagonally implicit Runge-Kutta (DIRK5) method. The advantage of the present method is that it is used with less computational effort. 1. Introduction Mathematical models of numerous applications from physics, chemistry, and mechanics take the form of systems of time-dependent partial differential equations subject to initial or boundary conditions. For the investigation of stationary solutions, many of these models can be reduced to singular systems of ordinary differential equations, especially when symmetries problem in the geometry and polar, cylindrical, or spherical coordinates can be used. The leading-edge model describing the avalanches dynamics [1] has the form of a singular initial value problem for a scalar ordinary differential equation. Koch et al. [2, 3] applied implicit Euler method (backward) to evaluate the approximate solutions of this type of singular initial value problem and finally used an acceleration technique known as the Iterated Defect Correction (IDeC) to improve the approximations. The second order implicit Runge-Kutta (RK2) method (i.e., implicit midpoint rule) is a higher-order solver than the implicit Euler method for solving singular initial value problems. For some singular initial value problems, the present method gives more accurate solutions than those of implicit Euler method as well as second order implicit Runge-Kutta (RK2) method. Implicit methods are more suitable than explicit methods for solving stiff problems because of their higher-order accuracy. Stiff differential equations arise in a variety of physical applications, such as network analysis and chemical or nuclear kinetics [4]. Stiff differential equations also occur in many kinds of studies, such as biochemistry, biomedical system, weather prediction, mathematical biology, electronics, fluids and heat transfer [5]. For example, the stiffness in heat transfer originates physically in one of two ways: sharp changes in the thermal environment or large