Block Method for Numerical Integration of Initial Value Problems in Ordinary Differential Equations
Chapter One
OBJECTIVES OF THE STUDY
The overall aim of the study is to modify polynomial basis function
The specific objectives are as follows:
- To derive an order ten continuous hybrid block method.
- To investigate the stability properties, characteristics, consistency and the nature of convergence of the integrator constructed.\
- To implement the integrators and compare it with existing methods.
- Apply MAPLE programming for the implementation of the integrator and compare their performance with some established results.
CHAPTER TWO
LITERATURE REVIEW
In this section, we shall look precisely into various numerical methods, which involves the idea of discretization, in which the continuous interval of x is replaced by discrete point defined by (2.1)
The parameter h is called the step length.
If donate an approximation to the solution of of the Initial Value Problems at the points then
Numerical method is a difference equation involving a number of consecutive approximation yn+y for which it could be possible to compute sequentially the sequence such that. Numerical methods are usually needed for solving differential equations of the form such that
Hence, we shall examine various methods to use in solving the problem.
One Step Methods
The conservational one step method for the IVP is described as follows:
Where (2.3)
is the movement function and is the step size adopted in sub-interval
The following definitions highlight and describe one step method.
Definitions Associated with one step Methods
- The one step method is said to be consistent provided the incremental function satisfies the relation then consistency of the one step method is to enable that the method is at least of order one
- The Local Truncation Error (LTE) of one step method is given as (2.4)
Definition Fatunla (1988)
The one step method is said to be of order p, if p is the positive integer such that (2.5a)
A global error of the one step method is the difference between the theoretical solution and numerical solution.
CHAPTER THREE
RESEARCH METHODOLOGY
We consider the polynomial basis function of the form
Using collocating and interpolating procedure. and t be the number of interpolation point. i.e t = 1.
Let s be the number of collocation point. i.e, s = 10
Therefore i.e 10+1-1=10. (3.1b)
Given the general form of the block method
We use the interpolant (3.1).
Differentiating (3.1) with respect to x yield
CHAPTER FOUR
STABILITY ANALYSIS OF THE METHOD
PREAMBLE
THE BASIC PROPERTIES OF THE METHOD
Order and Error Constant
The nine difference schemes (3.15) are discrete schemes belonging to the class of linear multistep method (LMM) of the form
CHAPTER FIVE
CONCLUSION AND SUMMARY
This chapter deals with the implementation of the methods in solving Initial Value Problems of Ordinary Differential Equations.
Algorithm
MAPLE software was used to code the schemes derived and they were tested on some numerical problems.
Numerical Experiment and Results
The methods are tested on some numerical problems to test the accuracy of the proposed method that were implemented in Block method and are also compared with results obtained using existing methods.
The following problems are taken as test problems.
Problem 1:
Consider the non-stiff initial value problem
FINDINGS
- The method are convergent consistent and zero stable, it is also is A-stable
- The table of the results show that the performance level of our new methods competes favorably with the existing methods
RECOMMENDATION AND CONTRIBUTION TO KNOWLEDGE
- A self-starting block method of solution of Initial Value Problems in Ordinary Differential Equations was implemented in block form
- The results obtained were comparable and even better than some existing results
CONCLUSION
A collocation and interpolation approach which produced an order ten scheme has been proposed for the numerical integration of Initial Value Problems in Ordinary Differential Equations. The errors arising from the problems in Table 5.3c using the proposed method is compared with those obtained by skwame eltal (2012), Abubakar eltal (2014), Areo and Adeniyi, (2009), and serisenal eltal (2004) respectively who earlier solved the same problem while the error arising from Table 5.3b were compared with Abubakar M. Bakoji Ali M. Bukar, Muktar I. Bello (2014).
The comparative analysis of our literature in the comparative analysis of our table revealed that we have successfully derived a higher order with better accuracy when compared with existing methods in our literature
REFERENCES
- Abubakar M. Bakoji Ali M. Bukar, Muktar I. Bello (2014) Vol. 5. No 03, Formulation of Predictor-Corrector, Methods from 2 step Hybrid Adams methods for the Solution of Initial value Problems of Ordinary differential equations.
- Areo E. A. Adenuyi, R. B. (2009), One step embedded Butcher type two-step block hybrid method for IVPs in ODEs. Advances in Maths. Vol. 1 proceeding of a numerical conference in honour of late Professor C. O. A. Sowunmi, University of Ibadan, Nigeria 120 – 128.
- Areo E. A. and R B. Adenuyi (2013), Sixth order hybrid block method for the numerical of first order IVP, journal of mathematical theory and modeling, vol. 3, No 8, pp 113 -120
- Areo E. A. and R. B. Adenuyi (2004), Block implicit one-step method for numerical integration of IVP in ordinary differential equation. International journal of mathematics and statistics. Vol. 2, vol. 3, pp 4 – 13.
- Awoyemi D. O., Kayode S. J., Adoghe L. O. (2015); A six step continuous multi-step method for the solution of general fourth order initial value problems of ODEs. Journal of Natural Science research. Vol. 5, No. 5. www.iiste.org
- B. Samuyi and D. J. Evans, the numerical oscillatory problems. Intex j computer, math 31 (1989), 237 – 255.
- Butcher J.C., A Modified Multistep Method for the Numerical Integration of Ordinary Differential Equations, J, Assoc. Comput. Mach., 1965, vol. 12, pp. 124 – 135.
- Chu M. T. and Hamiton (1987) Parallel solution of ODE by multi block methods. SIAM journal of scientific and statistical computation. Vol. 8, pp 342 – 553