Parabola Method in Ordinary Differential Equation

Abstract

In this paper we used the method of parabola approximation to study some nonlinear differential equations. We derive exact, explicit solutions to the parabolic equations and use this analytical results in the numerical computations for the general equations. We then draw the comparison of between the solutions of original and approximated equations. Moreover, we apply such method to the population growth problem. The error of the difference between the solutions of the differential equations and the numerical results caused by the discrete approximations is reasonable.

Applications of Parameterized Nonlinear Ordinary Differential Equations and Dynamic Systems: An Example of the Taiwan Stock Index

“…As for applying dynamic system and differential equations to stock prices, Chiang-Lin et al [15] applied "parabola approximation" and "dynamic integration" proposed in [16] to model the Taiwan stock index, TAIEX, and evaluate its derivative by considering the index as a dynamic system. Li et al [17] also applied the same methods to model German DAX and tried to develop a risk detecting instrument of approaching financial catastrophe.…”

“…We apply forward difference method to numerical analysis since the results do not differ much from applying backward difference method [16]. For fitting with the dynamic models, the future information is included in calculations.…”

The Space-Jump Model of the Movement of Tumor Cells and Healthy Cells

“…According to above assumptions, ( , ) ≡ ( ) = ( ) + ]( ) and ( ) = ( )/ and ]( ) = V( )/ , ( ) can be restored to ( ( )/ ) + (V( )/ ), where is a constant. In that case, equation (20) can be transformed into the form…”

“…If it is possible, we hope the solutions of (10) could be obtained by using our methods that were analytically used in [11][12][13][14][15][16][17][18] or numerically used in [19,20].…”

Taiex Index Option Model by Using Nonlinear Differential Equation

“…Our goal is to construct a model more suitable for TXO. We use -Parabola Approximation‖ proposed by Li et al (2011) to solve the differential equations and try to find the model which fits our data the most. Empirical study shows the model used all produce accurate estimates of TXO prices.…”