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Comparison of Adams-Bashforth-Moulton Method and Milne-Simpson Method on Second Order Ordinary Differential Equation

Adekoya Odunayo M.*, Z.O. Ogunwobi

Department of Mathematics Olabisi Onabanjo University, Ogun State, Nigeria

Abstract: This work considers the use of Adam Bashforth-Moulton method and Milne Simpson method to solve second order ordinary differential equation with initial value problem and to compare solution with the exact solution, to solve that we first convert the equation to two set of first order differential equation by order reduction method, then we use a single step method for approximation of initially orders which are required to start the linear multistep method. The result show that the numerical solutions are in good agreement with the exact solution. The result show that Adam Bashforth-Moulton method is better than Milne Simpson method in solving differential equation of second order.

Keywords: second order differential equation, Adam Bashforth Moulton Method, Milne Simpson method, convergence

Received November 23, 2020; Revised December 26, 2020; Accepted January 08, 2021

Citation: Adekoya Odunayo M., Z.O. Ogunwobi, Comparison of Adams-Bashforth-Moulton Method and Milne-Simpson Method on Second Order Ordinary Differential Equation. Turkish Journal of Analysis and Number Theory, 9 (1). 1-8. January 2021. doi:10.12691/tjant-9-1-1

1. Introduction

Differential equation is a mathematical equation that relates some functions with its derivative. In applications, the functions usually represent the rates of change. The equation defines the relationship between the two, because such equations are extremely common. Differential equation plays a prominent role in many disciplines including engineering, physics, economics, biology, and so on.

Most differential equations in science and technology are solved by numerical method because analytical solution are not possible or useful [1]. There are many methods for finding direct approximate solution to differential solution. These methods are referred to by a variety of different names including numerical methods, numerical integration or approximate solutions. The numerical method for solving ordinary differential equation are method of integrating a system of first order differential equation since higher ordinary differential equation can be reduced to a set of first order differential equation.

There are several existing algorithms designed for the integration of

[picture] | (1)

such as Runge Kutta and Euler method, Taylor series method as discussed by Lambert [1], hybrid method by Ademiluyi [2], numerical integration by Awoyemi [3,4,5] and one step method of integration by Ademiluyi and Kayode [6], non symmetric collation method by Awoyemi et al. [7] and a class of linear multi step method (LLM) for special ODE’s by Udo and Ademiluyi [8].

Linear multi-step methods are method used for the numerical solution of ordinary differential equation, conceptually a numerical method starts from an initial point and then takes a step forward in time to find the next solution point, the process continues with subsequent steps to map out the solution. Single step methods refer to only one previous point and its derivative to determine the current value. Methods such as Runge Kutta take some intermediate steps (for example, a half step) to obtain a higher order method, but then discard all previous information before taking a second step. Multistep methods attempt to gain efficiency by keeping and using the information from previous step rather than discarding it.

Linear multi-step method is one of the ways of getting approximate solution to a certain differential equation when the exact or analytical can be derived or not. Linear multi-step method also increases efficiency Haire and Wannal [9] by using the information from several previous solution approximations. Examples of linear multi-step method are Adams-Bashforth method, Adams-Moulton method, Nytrom method and Milne-Simpson method. This can be use as Predictor Corrector method.

1.1. Adams Methods

Adams methods are based on the idea of approximating the integrand with a polynomial with the interval using a kth order polynomial result in a k+1 order method. There are two types of Adam method, the explicit and the implicit types. The explicit type is called the Adams-Bashforth (AB) method which was designed by John Couch Adams to solve a differential equation modelling capillary action due to Francis Bashforth which was published by Bashforth in 1883 while the implicit type is called the Adams-Moulton (AM) method also due by John Couch Adams like the Adams Bashforth method, the name of Ray Moulton became associated with these method because he realise that they could be used in tandem with the Adams-Bashforth method as a Predictor Corrector method.

1.2. Milne Method

Another Predictor Corrector scheme is known as the Milne or Milne-Simpson method (Milne [10]), its Predictor is based on integration of the slope function f(t, y(t)) over the interval and then applying the Simpson rule. The Predictor uses the Lagrange polynomial approximation for f(t,y(t)) based on four mesh points . It integrated over the interval the formula was discovered by William Edmund Milne in 1953; Milne method has second order accuracy and is dahlquist stable.

2. Methodology

This section deals with the derivation of the methods

2.1. Description of Adams-Bashforth Method

Integrating

[picture] | (2)

with the interval

[picture] | (3)

gives

[picture] | (4)

Equation (4) is equivalently written as

[picture] | (5)

The following are the explicit method through the k data values

The Newton backward difference interpolating polynomial of degree k-1is given as

[picture] | (6)

setting

then equation (6) gives

[picture] | (7)

noting that

replacing by in the equation (5) yields

[picture] | (8)

Integrating equation (8) gives

[picture] | (9)

Applying the backward difference formula into equation (9), the result is then introduced back into equation (9)

[picture] | (10)

The error term is then given as

[picture] | (11)

2.2. Description of Adams-Moulton Method

The following are the implicit method through the k data values

The Newton backward difference interpolating polynomial of degree k is given as

[picture] | (12)

setting

then equation (12) gives

[picture] | (13)

noting that

replacing by in equation (13) yields

[picture] | (14)

integrating equation (14), gives

[picture] | (15)

applying backward difference formula into equation (14), the result is then introduced back into equation (14)

[picture] | (16)

the error term is then given as

[picture] | (17)

2.3. Description of Milne-Simpson Predictor Method

Integrating

[picture] | (18)

with the interval

[picture] | (19)

gives

[picture] | (20)

equation (20) is equivalently written as

[picture] | (21)

the following are explicit method through the k data values

the Newton backward difference interpolating polynomial of degree k-1is given as

[picture] | (22)

setting

then equation (22) gives

[picture] | (23)

replacing the integrand on the right hand side by the same backward difference polynomial

[picture] | (24)

integrating equation (24), gives

[picture] | (25)

applying backward difference formula into equation (26), the result is then introduced back into equation (26)

[picture] | (27)

2.4. Description of Milne-Simpson Corrector Method

Integrating

[picture] | (28)

in the interval

[picture] | (29)

gives

[picture] | (30)

or

[picture] | (31)

these are implicit method through the k data values

the Newton backward difference interpolating polynomial of degree k gives

[picture] | (32)

setting

then equation (32) gives

[picture] | (33)

replacing the integrand in the right hand side by the same backward difference polynomial

[picture] | (34)

integrating equation (34), the resulting equation is

[picture] | (35)

applying backward difference formula into equation (36), the result is then introduced back into equation (36)

[picture] | (37)

3. Numerical Experiments

Example 1

Consider second order non homogeneous differential equation problem

[picture] | (38)

(https://www.chegg.com/homework-help/ questions and answer/ )

The exact solutions are

Reducing the second order equation to get two set of first order equation

Example 2

Consider second order non homogeneous differential equation problem

[picture] | (39)

(Schaum’s outlines, Advance calculus second edition by Robert C. wrede)

The exact solution are

Reducing the second order equation to get two set of first order equation

4. Results

This chapter presents computational results for the examples solved using Adams-Bashforth-Moulton and Milne-Simpson methods. The results obtained by comparing our exact solution with the approximated solutions using two step size (0.1 and 0.05) to obtain the absolute errors were presented in tables. The results were seen to be accurate and converge.

5. Discussion of Result

From the numerical examples solved, the result was shown in Table 1 - Table 8, step size 0.1 and 0.05 was used and maximum errors were calculated. In the result, it was observed that the linear multistep method produce close results to the exact solution, it clearly seen from the table that both method (Adams-Bashforth-Moulton method and Milne-Simpson method) yield better approximation that converges slowly to the exact solution.

In Table 1 and Table 2, for numerical example 1, the result for Adams-Bashforth-Moulton converge first to the exact solution than that of Milne-Simpson from the step size 0.1 and in Table 3 and Table 4, Milne-Simpson converges first using step size 0.05.

It was also seen in Table 5 and Table 6 that Adams-Bashforth-Moulton converges first also before Milne Simspon but using step size 0.05 Milne-Simpson converges first in Table 7 and Table 8, so a numerical solution converge to the exact solution if the step size leads to decreased errors such that in the limit when the step size tends to zero the errors go to zero.

We see that the Adam Bashfort-Moulton and Milne Simspon method using step size 0.1 and 0.05 converges rapidly to exact solution. This shows that the small step size provides better approximation.

Table 1. Numerical approximation for example 1(Y) (Adams-Bashforth - Moulton and Milne-Simpson method) and the error estimation for step size h = 0.1

Table 2. Numerical approximation for example 1(Z) (Adams-Bashforth - Moulton and Milne-Simpson method) and the error estimation for step size h = 0.1

Table 3. Numerical approximation for example 1(Y) (Adams-Bashforth - Moulton and Milne-Simpson method) and the error estimation for step size h = 0.05

Table 4. Numerical approximation for example 1(Z) (Adams-Bashforth - Moulton and Milne-Simpson method) and the error estimation for step size h = 0.05

Table 5. Numerical approximation for example 2(Y) (Adams-Bashforth - Moulton and Milne-Simpson method) and the error estimation for step size h = 0.1

Table 6. Numerical approximation for example 2(Z) (Adams-Bashforth - Moulton and Milne-Simpson method) and the error estimation for step size h = 0.1

Table 7. Numerical approximation for example 2(Y) (Adams-Bashforth - Moulton and Milne-Simpson method) and the error estimation for step size h = 0.05

Table 8. Numerical approximation for example 2(Z) (Adams-Bashforth - Moulton and Milne-Simpson method) and the error estimation for step size h = 0.05

6. Conclusion

Linear multistep methods are used for solving Ordinary Differential Equation (ODE) with initial value problem. Two examples of second ordinary differential equation have been considered to verify the convergence using two different step sizes (0.1 and 0.05). The numerical solutions obtained by the two methods are in good agreement with exact solutions.

Comparing the result of the two methods, we observed that Adams-Bashforth-Moulton was found to be more accurate and also the approximate solution converged faster to the exact solution when compared to Milne-Simpson method. It can be concluded that Adams-Bashforth-Moulton method is powerful and more efficient in finding numerical solution of initial value problem (IVP).

References

[1]   | Lambert J.O. “Computational method in ordinary differential equation”. John Wiley and Sons New York, 1973.
[2]   | Ademiluyi R.A “New hybrid method for solving system of stiff ordinary differential equation”. Ph.D Thesis University of Benin, Benin City, 1987.
[3]   | Awoyemi D.O.”A class of continuous methods for the solution of general second order ordinary differential equations”.International Journal of Computational Mathematics, 72, pp 29-32, 1999.
[4]   | Awoyemi D.O. “A p- stable liner multistep method for solving general third order initial value problems”. International Journal of Computational Mathematics 80(8) pp 985-991, 2003.
[5]   | Awoyemi D.O. “A four point fully implicit method for the numerical integration of third order ordinary differential equation”. International Journal of Physics Science, 9 pp. 7-12, ISSN 1992-1950, 1996.
[6]   | Ademiluyi R.A and Kayode “A comparative study of a class of implicit multi-derivative for numerical solution of non stiff and stiff first order ordinary differential equation”. African Journal of Mathematics and Computer science. 4 (2), pp. 120-135, ISSN 2001-9731, 2001.
[7]   | Awoyemi D.O, Udo M.O. and Adesanya A.O. “Non-symmetric collocation method for direct solution of general third order initial value problems of ordinary differential equation” Journal of Nat and Applied Sciences, 7(1) pp 31-37, 2006.
[8]   | Udo M.O. and Awoyemi D.O. “An algorithm for solving initial value problem of first order ordinary differential equation”. Global Journal of Mathematical Sciences, 8(2), pp 123-130, 2007.
[9]   | Hairer Ernst and Wannal Gerhard “Linear multistep method” scholarpedia, 5(4), p 4591, 2010.
[10]   | Milne William “Numerical solution of differential equation” new york, 1953.