American Journal of Computational and Applied Mathematics

p-ISSN: 2165-8935    e-ISSN: 2165-8943

2017;  7(6): 143-148

doi:10.5923/j.ajcam.20170706.01

 

A Numerical Integration for Solving First Order Differential Equations Using Gompertz Function Approach

Ogunrinde R. B., Ayinde S. O.

Department of Mathematics, Faculty of Science, Ekiti State University, Ado Ekiti, Nigeria

Correspondence to: Ayinde S. O., Department of Mathematics, Faculty of Science, Ekiti State University, Ado Ekiti, Nigeria.

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Abstract

In this paper, we present a new numerical integration of a derived interpolating function using the Gompertz Function approach for solving first order differential equations. The new numerical integration obtained was used to solve some oscillatory and exponential problems. The effectiveness of the new Integrator was verified and the results obtained show that the Integrator is computational reliable and stable.

Keywords: Numerical integration, Oscillatory and exponential problems, Gompertz Function, Interpolating Function

Cite this paper: Ogunrinde R. B., Ayinde S. O., A Numerical Integration for Solving First Order Differential Equations Using Gompertz Function Approach, American Journal of Computational and Applied Mathematics , Vol. 7 No. 6, 2017, pp. 143-148. doi: 10.5923/j.ajcam.20170706.01.

1. Introduction

Differential equations are used to model problems in science and engineering that involve the change of some variables with respect to another. Most of these problems required the solution of an initial value problem. Many scholars [1, 3, 5] have approached the problems in various ways by formulating an interpolating function through which an integration scheme which is particularly well suited were developed and used to solve the problems. Similarly, Mathematician scholars have approached the growth problems by formulating functions and distributions which play an important role in modeling survival times, human mortality and actuarial data. [2, 4]
Gompertz proposed and showed that if the average exhaustions of a man’s power to avoid death were such that at the end of equal infinitely small intervals of time, he lost equal portions of his remaining power to oppose destruction. The number of survivors was given by the equation
(1)
Winsor in 1932 for more convenient was written in the form
(2)
in which k and b are essentially positive quantities.
The Gompertz curve or function which has been useful for the empirical representation of growth phenomena. These functions have for long been applied to Actuarial and Demographic problems such as: growth of organisms, psychological growth and population growth.
In this research, we consider an interpolating function in similitude to Gompertz function with additional terms added and derived a numerical integration that can compete favorably in solving some physical problems of growth phenomena, the two parameters (Scale and shape) were considered in the formulation.
Therefore, we formulate or derived new computational numerical integration scheme from the method based on the theoretical solution to the initial value problem of the form:
(3)
in the interval by considering the interpolating function
(4)
Where are real undetermined coefficients, are the shape and scale parameters, K represent the saturation level using Gompertz approach. The intervals defined are and .
Some conditions will be imposed on (4) to guarantee the derivation of the method. The numerical experiments will be carried out on some initial value problems and the test results show the reliability of the method.

2. Derivation of the Integrator

We can assume that the theoretical solution to the initial value problem (3) can be locally represented in the interval by the non-polynomial interpolating function (4) above, that is
We shall assume is a numerical estimate to the theoretical solution and
We define mesh points as follows: and
(5)
We can impose the following constraints on the interpolating function (4) in order to get the undetermined coefficients.

2.1. Constraints

a. The interpolating function must coincide with the theoretical solution at . Hence we required that
(6)
and
(7)
b. Secondly, the derivatives of the interpolating function are required to coincide with the differential equation as well as its first, second, and third derivatives with respect to
We denote the i-th total derivatives of with respect to x with such that
(8)

2.2. Derivation of the Numerical Integrator

Following from (8), we differentiate (4) to obtain
(9)
(10)
(11)
We formed a system of equation to solve for from (9) to (11),
Hence we have,
(12)
Taking this system of equation such as , using Cramer rule it gives
(13)
(14)
(15)
(16)
Therefore, we can deduce that
(17)
(18)
(19)
Since and
Implies that and
Then
(20)
and therefore we shall have from putting (6) and (7) into (20),
(21)
Simplifying (21) gives
(22)
(23)
Therefore, by expansion
(24)
Hence, (24) can be written compactly as
Where
(25)
Substituting for from (17, 18, 19) in (25), we have
(26)
where
is the new numerical schemes for the solution of the first order differential equation.

3. The Implementation of the Integrator

In this paper, however, we limit the numerical integration (26) to first test on some oscillatory and exponential problems of the first order differential equations so as to show the reliability and stability of the integration before applying it to solve the physical problems of growth phenomena.
Example 1
Using the Integrator (26) to solve the initial value problem
Table 1
     
Figure 1. The graph of the Numerical and Exact solution of
Figure 2. The graph of the Absolute Error of Numerical and Exact solution of
Example 2
Using the Integrator (26) to solve the initial value problem
Table 2
     
Figure 3. The graph of the Numerical and Exact solution of
Figure 4. The graph of the Absolute Error of Numerical and Exact solution of

4. Conclusions

In conclusion, we have presented a new numerical integration of a derived interpolating function using the Gompertz Function approach for solving first order differential equations. The interpolating function in comparison with Gompertz function with little modification was used, and the scale and the shape parameters were considered. The reliability of the new Integrator was verified and the results obtained show that the Integrator is computational reliable and stable, when the new numerical integration obtained was used to solve some oscillatory and exponential problems.
However the Integrator will be used to solve growth and population problems in the next paper where Gompertz Function and the equation has been widely used.

References

[1]  Ayinde, S. O. et al. (2015). “A New Numerical Method for solving First Order Differential Equations” American Journal of Applied Mathematics and Statistics. Vol. 3, No. 4, 156-160.
[2]  Charles, P. W. (1932). “The Gompertz Curve as a Growth Curve”. Proceedings of the National Academy of Sciences. Vol. 18, No 1. 1-8.
[3]  Fatunla, S. O. (1976). “A New Algorithm for Numerical Solution of ODEs”. Computer and Mathematics with Applications. Vol. 2, pp 247 – 253.
[4]  Gompertz, Benjamin (1825). “On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Method of Determining the value of Life Contingencies”. Phil. Transactions Royal Society. 513 – 585.
[5]  Ibijola E.A. et al (2010). “On a New Numerical Scheme for the Solution of Initial Value Problems in ODE”. Australian Journal of Basic and Applied Sciences. 4(10): 5277-5282.