A Class Of Collocation Methods For Second Order Ordinary Differential Equation

Second order ordinary differential equation has been applied widely in mathematics, physics and engineering areas, the relevant research on its numerical solution is still flourishing. During the research, Mathematicians have obtained a series of important results at home and abroad. In 2009,Gonzalez-Pinto and his parters have proposed a new method on the first-order stiff equation includes a free parameter ,which has many good properties compared with existing methods,for example it's strongly A-stabilty . In this paper we research a special type of second-order differential equations without first-order derivative.In the first three chapters of this paper ,we use their type of method to solve second order ordinary differential equation,and construct the direct and indirect collocation methods based on this method.Then we give their order results and stability properties.Due to stability region of the direct collocation method is finite, we then construct a new class of improved method ,research the order and stability of these methods in the fourth chapter, we find that the improved method does extend a bigger stability region ,so it has certain advantages on solving stiff problems .In the last Chapter ,we have done two numerical experiments,and find that the numerical results are very consistent with the theoretical analysis mentioned above.