Numerical Integration Formulas for Solving the



Numerical Integration Formulas for Solving the

Initial Value Problem of Ordinary Differential Equations

Maitree Podisuk and Wannaporn Sanprasert

Department of Mathematics and Computer Science

King Mongkut’s Institute of Technology Chaokhuntaharn Ladkrabang

Ladkrabang Bangkok 10520

THAILAND

Abstract: In this paper, we will use four numerical integration formulas to solve the initial value problem of the ordinary differential equations by solving the integral equation instead of the ordinary differential equation. We will use these four formulas to find the numerical solutions of some examples and compare these results with the known Runge-Kutta formulas, Euler’s formula, Goeken-Johnson formula and Wu’s formula.

Key-Words: Runge-Kutta Midpoint Trapezoidal Simpson Goeken Johnson Wu

1. Introduction

The initial value problem of the ordinary differential equation is of the form

1) [pic]

with the initial condition

2) [pic].

If we divide the close interval [pic] into n subintervals at the points [pic], where [pic] for [pic] then some known numerical formulas for finding the numerical solution of the above equations at these points are

(3) [pic]

which is Euler formula which we shall denote by RK1,

4) [pic]

where [pic]

[pic]

which is two points Runge-Kutta formula which we shall denote by RK2,

5) [pic]

where [pic]

[pic]

[pic]

which is three points Runge-Kutta formula which we shall denote by RK3.

In 1999 Goeken and Johnson[1] introduced the following four points Runge-Kutta method for finding the numerical solutions of the autonomous initial value problem of the ordinary differential equation and it is of the form

6) [pic]

[pic]

where [pic]

[pic][pic]

[pic]

[pic]

[pic] which we shall denote by GJ.

In 2003 Wu[2] introduced the two-step formula Runge-Kutta method for finding the numerical solution of the autonomous initial value problem of the ordinary differential equations and it is of the form

7) [pic][pic]

[pic] [pic]

[pic]

[pic]

[pic]

which we shall denote by WU.

2 Problem Formulation

We know that the above equations (1)-(2) are equivalent to the integral equation

8) [pic].

By the method of successive approximation, the sequence [pic] converges to the solution of the above differential equation where

9) [pic]

So we will use the fact that the sequence [pic] converges to the solution of the above differential equation to approximate the numerical solution at the point [pic] by using the numerical approximation of

(10) [pic].

The four numerical integration formulas that we will use in this paper are;

11) [pic]

which is Midpoint formula,

(12) [pic]

which is Trapezoidal formula,

(13) [pic]

[pic] which is the Modified Trapezoidal formula and

(14)[pic]

[pic]

which is Simpson’s formula.

By equation (11), (12), (13) and (14), we obtain four formulas which are

(15)[pic]

where [pic] which we shall denote by PS1,

16) [pic]

where [pic]

[pic] which we shall denote by PS2 ,

17) [pic]

[pic]

where [pic]

[pic]

[pic][pic]which we shall denote by PS3 and

18) [pic]

[pic]

[pic]

which we shall denote by PS4.

3 Examples

There will be five examples in this section. We will use the above nine formulas to find the numerical solutions of these five examples.

3.1 Example 1

Find the numerical solution of the equation

(19) [pic]

with the initial condition

20) [pic]

The analytical solution of the above equation is

[pic].

The numerical results are in following table 1, 2 and 3.

| |[pic] | |

| |[pic] | |

| | [pic]y | [pic] |

|RK1 |[pic] |[pic] |

|RK2 |[pic] |[pic] |

|RK3 |[pic] |[pic] |

|PS1 |[pic] |[pic] |

|PS2@ |[pic] |[pic] |

|PS3 |[pic] |[pic] |

|PS4 |[pic] |[pic] |

Table 1

| |[pic] | |

| |[pic][pic] | |

| |[pic] | |

| | [pic] | [pic] |

|RK1 |[pic] |[pic] |

|RK2 |[pic] |[pic] |

|RK3 |[pic] |[pic] |

|PS1 |[pic] |[pic] |

|PS2 |[pic] |[pic] |

|PS3 |[pic] |[pic] |

|PS4 |[pic] |[pic] |

Table 2

| |[pic] | |

| |[pic][pic] | |

| |[pic] | |

| | [pic] | [pic] |

|RK1 |[pic] |[pic] |

|RK2 |[pic] |[pic] |

|RK3 |[pic] |[pic] |

|PS1 |[pic] |[pic] |

|PS2 |[pic] |[pic] |

|PS3 |[pic] |[pic] |

|PS4 |[pic] |[pic] |

Table 3

3.2 Example 2

Find the numerical solution of the equation

(21) [pic]

with the initial condition

22) [pic]

The analytical solution of the above equation is [pic].

The numerical results are in following table 4, 5 and 6.

| |[pic],[pic] | |

| |[pic] | |

| | [pic]y | [pic] |

|RK1 |[pic] |[pic] |

|RK2 |[pic] |[pic] |

|RK3 |[pic] |[pic] |

|PS1 |[pic] |[pic] |

|PS2 |[pic] |[pic] |

|PS3 |[pic] |[pic] |

|PS4 |[pic] |[pic] |

Table 4

| |[pic] | |

| |[pic][pic] | |

| |[pic] | |

| | [pic] | [pic] |

|RK1 |[pic] |[pic] |

|RK2 |[pic] |[pic] |

|RK3 |[pic] |[pic] |

|PS1 |[pic] |[pic] |

|PS2 |[pic] |[pic] |

|PS3 |[pic] |[pic] |

|PS4 |[pic] |[pic] |

Table 5

| |[pic] | |

| |[pic] | |

| |[pic] | |

| | [pic]y | [pic] |

|RK1 |[pic] |[pic] |

|RK2 |[pic] |[pic] |

|RK3 |[pic] |[pic] |

|PS1 |[pic] |[pic] |

|PS2 |[pic] |[pic] |

|PS3 |[pic] |[pic] |

|PS4 |[pic] |[pic] |

Table 6

3. Example 3

Find the numerical solution of the equation

(23) [pic]

with the initial condition

(24) [pic]

The analytical solution of the above equation is [pic].

The numerical results are in following table 7, 8 and 9.

| |[pic],[pic] | |

| |[pic] | |

| | [pic]y | [pic] |

|RK1 |[pic] |[pic] |

|RK2 |[pic] |[pic] |

|RK3 |[pic] |[pic] |

|PS1 |[pic] |[pic] |

|PS2 |[pic] |[pic] |

|PS3 |[pic] |[pic] |

|PS4 |[pic] |[pic] |

Table 7

| |[pic] | |

| |[pic][pic] | |

| |[pic] | |

| | [pic] | [pic] |

|RK1 |[pic] |[pic] |

|RK2 |[pic] |[pic] |

|RK3 |[pic] |[pic] |

|PS1 |[pic] |[pic] |

|PS2 |[pic] |[pic] |

|PS3 |[pic] |[pic] |

|PS4 |[pic] |[pic] |

Table 8

| |[pic] | |

| |[pic] | |

| |[pic] | |

| | [pic]y | [pic] |

|RK1 |[pic] |[pic] |

|RK2 |[pic] |[pic] |

|RK3 |[pic] |[pic] |

|PS1 |[pic] |[pic] |

|PS2 |[pic] |[pic] |

|PS3 |[pic] |[pic] |

|PS4 |[pic] |[pic] |

Table 9

4. Example 4

Find the numerical solution of the equation

(25) [pic]

with the initial condition

(26) [pic]

The analytical solution of the above equation is [pic].

The numerical results are in following table 10, 11 and 12.

| |[pic],[pic][pic] | |

| | [pic]y | [pic] |

|RK1 |[pic] |[pic] |

|RK2 |[pic] |[pic] |

|RK3 |[pic] |[pic] |

|PS1 |[pic] |[pic] |

|PS2 |[pic] |[pic] |

|PS3 |[pic] |[pic] |

|PS4 |[pic] |[pic] |

Table 10

| |[pic] | |

| |[pic][pic] | |

| |[pic] | |

| | [pic] | [pic] |

|RK1 |[pic] |[pic] |

|RK2 |[pic] |[pic] |

|RK3 |[pic] |[pic] |

|PS1 |[pic] |[pic] |

|PS2 |[pic] |[pic] |

|PS3 |[pic] |[pic] |

|PS4 |[pic] |[pic] |

Table 11

| |[pic] | |

| |[pic] | |

| |[pic] | |

| | [pic]y | [pic] |

|RK1 |[pic] |[pic] |

|RK2 |[pic] |[pic] |

|RK3 |[pic] |[pic] |

|PS1 |[pic] |[pic] |

|PS2 |[pic] |[pic] |

|PS3 |[pic] |[pic] |

|PS4 |[pic] |[pic] |

Table 12

5. Example 5

Find the numerical solution of the equation

(27) [pic]

with the initial condition

(28) [pic][pic]

The analytical solution of the above equation is [pic].

The numerical results are in following table 13, 14 and 15.

| |[pic][pic] | |

| | [pic] | [pic] |

|RK1 |[pic] |[pic] |

|RK2 |[pic] |[pic] |

|RK3 |[pic] |[pic] |

|GJ |[pic] |[pic] |

|WU* |[pic] |[pic] |

|PS1 |[pic] |[pic] |

|PS2 |[pic] |[pic] |

|PS3 |[pic] |[pic] |

|PS4 |[pic] |[pic] |

Table 13

| |[pic] | |

| |[pic] | |

| |[pic] | |

| | [pic] | [pic] |

|RK1 |[pic] |[pic] |

|RK2 |[pic] |[pic] |

|RK3 |[pic] |[pic] |

|GJ |[pic] |[pic] |

|WU* |[pic] |[pic] |

|PS1 |[pic] |[pic] |

|PS2 |[pic] |[pic] |

|PS3 |[pic] |[pic] |

|PS4 |[pic] |[pic] |

Table 14

| |[pic] | |

| |[pic] | |

| |[pic] | |

| | [pic] | [pic] |

|RK1 |[pic] |[pic] |

|RK2 |[pic] |[pic] |

|RK3 |[pic] |[pic] |

|GJ |[pic] |[pic] |

|WU* |[pic] |[pic] |

|PS1 |[pic] |[pic] |

|PS2 |[pic] |[pic] |

|PS3 |[pic] |[pic] |

|PS4 |[pic] |[pic] |

Table 15

* WU formula used [pic] and [pic] as the initial two points.

Conclusion

All four new formulas are good numerical formulas for finding the numerical solutions of the initial value problem of the ordinary differential equations. These four new formulas will give us more freedom to find the way of finding the numerical solutions of the initial value problem of the ordinary differential equations. However we have to keep in mind that the above numerical solutions are just for only these five equations and the Wu formula is the 2-step formula. We strongly recommend the formula PS1 and the formula PS4.

References:

[1] David Goeken and Olin Johnson, Fifth-Order Runge-Kutta with Higher Order Derivative Approximations, Electronic Journal of Differential Equations, Vol.2, November 1999, pp.1-9.

[2] Xinyuan Wu, A Class of Runge-Kutta of Order Three and Four with Reduced Evaluations of Function, Applied Mathematics and Computation, Vol.146, December 2003, pp.417-432.

-----------------------

[pic]

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download