Three Improved Euler Methods for a Class of Simplified Quasi-cubics Function

Abstract

The simplified quasi-cubics function is used quite common in the ordinary differential equation, which is used to describe the biological neuron model. As complexity of the biological neuron model, its analytical solution hardly can be solved in common situation. Numerical solution of the model is very important in reality. Three fast convergence methods are proposed for the simplified function by means of changing the step size of the classical Euler method in this manuscript. The results obtained are pretty good. First method is to correct the step size of two directions. Second method is to correct and encrypt two directions' step size. And the last method is to search unidirectional step size. On one hand, the feasibility of the three methods are proved based on the theoretical analysis. On the other hand, numerical results and comparison show that the advised three methods are effective to get the numerical solution with small population size, enhanced convergence speedup and improved precision.