An Accurate Five Off-Step Points Implicit Block Method for Direct Solution of Fourth Order Differential Equations

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**Subject Areas:** **Ordinary Differential Equation**

1. Introduction

In sciences and engineering, mathematical models are developed to understand as well as to interpret physical phenomena, many of such phenomena, when modeled, often result into higher order ordinary differential equations of the form:

(1)

An old conventional way to solve (1) is the method of first reducing (1) to system of first order differential equation of the form:

(2)

and to solve the resulting system of equations by any of the existing methods of solving first order ordinary differential equations. Literatures abounded in this old conventional method of solving problems of type (1) numerically are [1] - [3] . The drawbacks of this method include computational cumbersomeness and longer computer tine and space. In addition, [4] observes that these methods do not utilize additional information associated with a specific ordinary differential equation, such as oscillatory nature of the solution. To circumvent these drawbacks, many researchers have solved (1) directly; amongst these are [5] - [7] who develop blocked methods for numerical solution of fourth order ordinary differential equations. [8] develops linear multistep method for solution of fourth order ordinary differential equations whose implementation is Predictor-Corrector mode. Consequently, my motivation in this work is the success story of the adoption of single step method with five off-step points for direct numerical solution of fourth order ordinary differential equations which eliminate the use of predictors by providing sufficiently accurate simultaneous difference equations from a single continuous formula and its derivatives.

2. Derivation of the Method

We take our basis function to be a power series of the form:

(3)

The fourth derivative of (3) gives

(4)

By putting (4) into (1) we have the differential system:

(5)

where’s are the parameters to be determining while denotes number of collation and interpolation points. By collocating (5) at the mesh points and interpolating (3) at yields a system of equations:

(6)

(7)

By putting these system of equations in the matrix form and then solved to obtain values of parameters’s, which when substituted in (3), yields, after some manipulation, a hybrid linear method with continuous coefficients of the form:

(8)

The coefficient of and are

, ,

,

(9)

where.

We evaluate (9) at to obtain the discrete one step formula

(10a)

(10b)

(10c)

The first derivative of and in (9) gives:

(11)

Similarly, the second derivative of and in (9) gives

(12)

The third derivative of and in (9) gives

(13)

It is noted that the general fourth order odes involve the first, second and third derivatives. The derivatives can be obtained by imposing that:

(14)

By using (14) and evaluating (11), (12) and (13) at we obtain the first, second and the third derivative scheme as follows:

(15)

By combining the schemes (10), the first, second, third derivatives schemes (15) together and write them in block form, using the definition of implicit block method in [9] to obtain the block formula describe as follows:

(16)

is the power of the derivative of the continuous method and p is the order of the problem to solved:.

This equation is solved and we obtained values for and as follows:

(17)

(18)

(19)

(20)

3. Analysis of the Properties of the Block

In this section, we carry out the analysis of the basic properties of the new method.

3.1. Order of the Method

3.1.1. Order of the Block (17)

The linear operator of the block (17) is defined as:

(21)

By expanding and in Taylor series, (21) becomes:

(22)

The block (17) and associated linear operator are said to have order p if

See [10] .

The term is called the error constant and implies that the local truncation error is given by: (23)

Hence the block (17) has order 7 with error constant:

3.1.2. Order and Error Constant of the Main Method (10c)

By rewriting the main method (10c) in the form:

(24)

Expanding (24) in Taylor series in the form:

(25)

Since but see [10] ; then the main scheme is of order 7 and the error constant is:

.

3.2. Zero Stability of the Block

The block (17) is said to be Zero stable if the roots of the characteristic polynomial, satisfies and the root has multiplicity not exceeding the order of the differential equation. Moreover as

,

where is the order of the differential equation, for the block (19),

Hence our method is Zero stable.

3.3. Consistency of the Main Method (10c)

From main method (10c), the first and second characteristics polynomials of the method are given by:

and

the method (10c) is consistent since it satisfies the following conditions:

1. The order of the method is which is obvious.

2. For the method, , , and, thus .

3. .

4. it follows from here that showing that the condition (3) is satisfied as well.

5. Note that:

.

For the principal root r = 1: it is observed that the last condition above is satisfied, hence the main method is consistent.

3.4. Convergence

The necessary and sufficient condition for a numerical method to be convergent is for it to be consistent and Zero stable. Thus since it has been successfully shown from the above condition, it could be seen that our method is convergent.

3.5. Region of Absolute Stability of the Method

We consider the stability polynomial written in general form:

where and is assumed constant. The stability polynomial of the main method (10c) becomes:

(26)

Adopting the boundary locus method whose equation is given by:

(27)

By inserting the values of and into (27) and evaluate, we obtain the following results as displayed in the table below:

From here, it could be seen that the region of absolute stability of the method is given by which satisfies the condition for A-stability, similarly the interval of periodicity lies in interval.

4. Numerical Experiments

To test the accuracy, workability and suitability of the method, I adopted our method to solving some initial value problems of fourth order ordinary differential equations.

Test Problem 1

I consider special fourth order problem:

Whose exact solution is:

My method was used to solve the problem and result compared with [6] . The result is as shown in Table 1.

Test Problem 2

I consider a linear fourth order problem

Whose exact solution is given by:

My method was used to solve the problem and result compared with [8] . The result is as shown in Table 2.

Numerical Results

I make use of the following Notations in the table of results:

XVAL: Value of the independent variable where numerical value is taken.

ERC: Exact result at XVAL.

NRC: Our Numerical result at XVAL.

ERR: Error of our result at XVAL.

5. Discussion of Results

In this paper, I propose an accurate five off-step points modified implicit block algorithm for the numerical solution of initial value problems of fourth order ordinary differential equations. For better performance of the method, step size is chosen within the stability interval.

Table 1. Showing results for problem 1.

Table 2. Showing results for problem 2.

The order of my method is of order 7 higher than that of [6] of order 4, which collaborates the principle, that the higher the order of a method is, the more accurate it is. The absolute errors in [6] are more than those of the new methods; this also means that the new methods are accurate than [6] which is of order 4 and implemented in block mode.

The results of my new method when also compared with the block method proposed by [8] showed that my method is more accurate.

References

[1] Lambert, J.D. (1973) Computational Methods in ODEs. John Wiley & Sons, New York.

[2] Fatunla, S.O (1991) Block Method for Second Order IVPs. International Journal of Computer Mathematics, 41, 55-63.

http://dx.doi.org/10.1080/00207169108804026

[3] Brujnano, L. and Trigiante, D. (1998) Solving Differential Problems by Multistep Initial and Boundary Value Methods. Amsterdam, Gordon and Breach Science Publishers, Netherlands.

[4] Vigo-Aguiar, J. and Ramos, H. (2006) Variable Step Size Implementation of Multistep Method for . Journal of Computational and Applied Mathematics, 192, 114-131.

http://dx.doi.org/10.1016/j.cam.2005.04.043

[5] Omar, Z. (1999) Developing Parallel Block Method for Solving Higher Orders ODES Directly. PhD Thesis, University Putra, Malaysia.

[6] Mohammed, U. (2010) A Six Step Block Method for Solution of Fourth Order Ordinary Differential Equations. The Pacific Journal of Science and Technology, 11, 258-265

[7] Ademiluyi, R.A, Duromola, M.K. and Bolaji, B. (2014) Modified Block Method for the Direct Solution of Initial Value Problems of Fourth Order Ordinary Differential Equations. Australian Journal of Basic and Applied Sciences, 8, 389-394.

[8] Kayode, S.J. (2008) A Zero Stable Method for Direct Solution of Fourth Order Ordinary Differential Equation. American Journal of Applied Sciences, 5, 1461-1466.

http://dx.doi.org/10.3844/ajassp.2008.1461.1466

[9] Shampine, L.F. and Watts, H.A. (1969) Block Implicit One Step Methods. Mathematics of Computation, 23, 731-740.

http://dx.doi.org/10.1090/S0025-5718-1969-0264854-5

[10] Badmus, A.M. and Yahaya, Y.A. (2014) New Algorithm of Obtaining Order and Error Constants of Third Order Linear Multistep Method. Asian Journal of Fuzzy and Applied Mathematics, 2, 190-194.