Abstract
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.
Highlights
Mathematical models of numerous applications from physics, chemistry, and mechanics take the form of systems of timedependent 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
Summary
Mathematical models of numerous applications from physics, chemistry, and mechanics take the form of systems of timedependent partial differential equations subject to initial or boundary conditions. The leading-edge model describing the avalanches dynamics [1] has the form of a singular initial value problem for a scalar ordinary differential equation. The second order implicit Runge-Kutta (RK2) method (i.e., implicit midpoint rule) is a higherorder solver than the implicit Euler method for solving singular initial value problems. Alexander [6] solved stiff problems by using different stage and different order diagonally implicit Runge-Kutta (DIRK) methods. Ababneh and Ahmad [5] derived and applied a three-stage third order diagonally implicit Runge-Kutta (AM-DIRK3) method for solving stiff problems. The aim of this paper is to derive a simple implicit method specially for solving singular and stiff initial value problems
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