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

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1 American Journal of Computational and Applied Mathematics 2017, 7(6): DOI: /j.ajcam 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 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 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 LL xx = kkgg ee xx. (1) Winsor in 1932 for more convenient was written in the form yy = kkee ee aa bbbb (2) in which k and b are essentially positive quantities. * Corresponding author: biskay2003@yahoo.com (Ayinde S. O.) Published online at Copyright 2017 Scientific & Academic Publishing. All Rights Reserved 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 yy(xx) to the initial value problem of the form: yy = ff(xx, yy), yy(xx 0 ) = yy 0 (3) in the interval [xx nn, xx nn+1 ] by considering the interpolating function FF(xx,yy) KK = αα 1 ee ββββ + αα 2 BB xx + αα 3 cccccccc (4) Where αα 1, αα 2, αα 3 are real undetermined coefficients, ββ aaaaaa BB are the shape and scale parameters, K represent the saturation level using Gompertz approach. The intervals defined are xx [0,1] and kk (0,1]. 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.

2 144 Ogunrinde R. B. et al.: A Numerical Integration for Solving First Order Differential Equations Using Gompertz Function Approach 2. Derivation of the Integrator We can assume that the theoretical solution yy(xx) to the initial value problem (3) can be locally represented in the interval [xx nn, xx nn+1 ], nn 0 by the non-polynomial interpolating function (4) above, that is FF(xx, yy) = αα KK 1 ee ββββ + αα 2 BB xx + αα 3 cccccccc We shall assume yy nn is a numerical estimate to the theoretical solution yy(xx) and ff nn = ff(xx nn, yy nn ). We define mesh points as follows: xx nn = aa + nnh, and xx nn+1 = aa + (nn + 1)h, nn = 0, 1, 2, (5) We can impose the following constraints on the interpolating function (4) in order to get the undetermined coefficients Constraints and a. The interpolating function must coincide with the theoretical solution at xx = xx nn aaaaaa xx = xx nn+1. Hence we required that FF(xx nn, yy nn ) = KK(αα 1 ee ββxx nn + αα 2 BB xx nn + αα 3 ccccccxx nn ) (6) FF(xx nn+1, yy nn+1 ) = KK(αα 1 ee ββxx nn +1 + αα 2 BB xx nn +1 + αα 3 ccccccxx nn+1 ) (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 xx aaaa xx = xx nn We denote the i-th total derivatives of ff(xx, yy) with respect to xx with ff (ii) such that 2.2. Derivation of the Numerical Integrator Following from (8), we differentiate (4) to obtain FF 1 (xx nn ) = ff nn, FF 2 (xx nn ) = ff nn 1, FF 3 (xx nn ) = ff nn 2 (8) ff nn = kkαα 1 ββee ββxx nn + kkαα 2 BB xx nn llllll BB kkαα 3 ssssssxx nn (9) ff nn 1 = kkαα 1 ββ 2 ee ββxx nn + kkαα 2 BB xx nn (llllll BB) 2 kkαα 3 ccccccxx nn (10) ff nn 2 = kkαα 1 ββ 3 ee ββxx nn + kkαα 2 BB xx nn (llllll BB) 3 + kkαα 3 ssssssxx nn (11) We formed a system of equation to solve for αα 1, αα 2, aaaaaa αα 3 from (9) to (11), Hence we have, KKKKee ββxx nn KKBB xx nn llllllll KKKKKKKKxx nn αα 1 KKββ 2 ee ββxx nn KKBB xx nn (llllllll) 2 KKKKKKKKxx nn αα 2 = KKββ 3 ee ββxx nn KKBB xx nn (llllllll) 3 KKKKKKKKxx αα 3 nn Taking this system of equation such as AAAA = BB, using Cramer rule it gives ff nn 1 ff nn ff nn 2 (12) AA = KKKKeeββxx nn ((llllllll) 2 ssssssxx nn + (llllllll) 3 ccccccxx nn ) KKKKKKKKKKee ββxx nn (ββ 2 ssssssxx nn + ββ 3 ccccccxx nn ) KKKKKKKKxx nn ee ββxx nn (ββ 2 (llllllll) 3 ββ 3 (llllllll) 2 )) (13) XX 1 = ff nn(llllll BB) 2 ssssssxx nn + (llllll BB) 3 ccccccxx nn (llllllll)(ff nn 1 ssssssxx nn + ff nn 2 ccccccxx nn ) ssssssxx nn (ff nn 1 (llllllll) 3 ff nn 2 (llllllll) 2 ) (14) Therefore, we can deduce that XX 2 = ββ(ff nn 1 ssssssxx nn + ff nn 2 ccccccxx nn ) ff nn (ββ 2 ssssssxx nn + ββ 3 ccccccxx nn ) ssssssxx nn (ββ 2 ff nn 2 ββ 3 ff nn 1 ) (15) XX 3 = ββ(llllllbb 2 ff nn 2 llllllbb 3 ff nn 1 ) llllllll (ββ 2 ff nn 2 ββ 3 ff nn 1 ) + ff nn (ββ 2 llllllbb 3 ββ 3 llllllbb 2 ) (16) αα 1 = ff nn (llllll BB) 2 ssssss xx nn +(llllll BB) 3 cccccc xx nn (llllllll ) ff nn 1 ssssss xx nn +ff nn 2 cccccc xx nn ssssss xx nn (ff nn 1 (llllllll ) 3 ff nn 2 (llllllll ) 2 ) KKKK ee ββ xx nn ((llllllll ) 2 ssssss xx nn +(llllllll ) 3 cccccc xx nn ) KKKKKKKKKK ee ββ xx nn (ββ 2 ssssss xx nn +ββ 3 cccccc xx nn ) KKKKKKKK xx nn ee ββ xx nn (ββ 2 (llllllll ) 3 ββ 3 (llllllll ) 2 )) (17)

3 American Journal of Computational and Applied Mathematics 2017, 7(6): αα 2 = ββ ff nn 1 ssssss xx nn +ff nn 2 cccccc xx nn ff nn (ββ 2 ssssss xx nn +ββ 3 cccccc xx nn ) ssssss xx nn (ββ 2 ff nn 2 ββ 3 ff nn 1 ) KKKK BB xx nn ((llllllll ) 2 ssssss xx nn +(llllllll ) 3 cccccc xx nn ) KKBB xx nn llllllll (ββ 2 ssssss xx nn +ββ 3 cccccc xx nn ) KKBB xx nn ssssss xx nn (ββ 2 (llllllll ) 3 ββ 3 (llllllll ) 2 )) αα 3 = ββ llllll BB2 ff nn 2 llllll BB 3 ff nn 1 llllllll ββ 2 ff nn 2 ββ 3 ff nn 1 +ff nn (ββ 2 llllll BB 3 ββ 3 llllll BB 2 ) KKKK BB xx nn ((llllllll ) 2 ssssss xx nn +(llllllll ) 3 cccccc xx nn ) KKBB xx nn llllllll (ββ 2 ssssss xx nn +ββ 3 cccccc xx nn ) KKBB xx nn ssssss xx nn (ββ 2 (llllllll ) 3 ββ 3 (llllllll ) 2 )) (18) (19) Since FF(xx nn+1 ) = yy(xx nn+1 ) and FF(xx nn ) = yy(xx nn ) Implies that yy(xx nn+1 ) = yy nn+1 and yy(xx nn ) = yy nn Then FF(xx nn+1 ) FF(xx nn ) = yy nn+1 yy nn (20) and therefore we shall have from putting (6) and (7) into (20), yy nn+1 yy nn = KK(αα 1 ee ββxx nn +1 + αα 2 BB xx nn +1 + αα 3 ccccccxx nn+1 ) KK(αα 1 ee ββxx nn + αα 2 BB xx nn + αα 3 ccccccxx nn ) (21) Simplifying (21) gives yy nn+1 yy nn = KKαα 1 ee ββxx nn +1 ee ββxx nn KKαα 2 [BB xx nn +1 BB xx nn ] + KKKK 3 [ccccccxx nn+1 ccccccxx nn ] (22) Recall that xx nn = aa + nnh, xx nn+1 = aa + (nn + 1)h with nn = 0,1,2 (23) Therefore, by expansion yy nn+1 yy nn = KKαα 1 ee ββxx nn ee ββh 1 KKαα 2 BB xx nn (BB h 1) + KKKK 3 [cccccc(xx nn + h) ccccccxx nn ] (24) Hence, (24) can be written compactly as Where yy nn+1 = yy nn + PP + QQ + RR PP = KKαα 1 ee ββxx nn (ee ββh 1) QQ = KKαα 2 BB xx nn (BB h 1) (25) RR = KKKK 3 (cccccc(xx nn + h) ccccccxx nn ) Substituting for αα 1, αα 2, and αα 3 from (17, 18, 19) in (25), we have where yy nn+1 = yy nn + PP + QQ + RR (26) PP = ee ββ h 1 ff nn llllllll ff nn 1 ff nn 1 (llllllll ) 2 +ff nn 2 llllllll ssssss xx nn + ff nn (llllllll ) 2 ff nn 2 cccccc xx nn [(llllllll ββ 2 ββ 2 (llllllll ) 2 +ββ 3 llllllll )ssssss xx nn +(ββ (llllllll ) 2 ββ 3 )cccccc xx nn ] QQ = 1 BB h [ ff nn 1 ββff nn ββff nn 2 +ββ 2 ff nn 1 ssssss xx nn + ff nn 2 ββ 2 ff nn cccccc xx nn ] [(llllllll ) 2 ββββββββββ ββ (llllllll ) 3 +ββ 2 (llllllll ) 2 ]ssssss xx nn +[(llllllll ) 3 ββ 2 llllllll ]cccccc xx nn RR = (cccccc (xx nn +h) cccccc xx nn ) (ββff nn ff nn 1 )(llllllll ) 2 +(ff nn 2 ββ 2 ff nn )llllllll + ββ 2 ff nn ββββ nn 2 (llllllll ββ ββ(llllllll ) 2 )ssssss xx nn +((llllllll ) 2 ββ 2 )cccccc xx nn +ββ 2 llllllll 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 yy = yy, yy(0) = 1, iiii tthee iiiiiiiiiiiiiiii 0 xx 1, The analytical solution yy(xx) = ee xx, h = 0.1, BB = , ββ = 1.162

4 146 Ogunrinde R. B. et al.: A Numerical Integration for Solving First Order Differential Equations Using Gompertz Function Approach Table 1 xx nn Numerical Exact Absolute Error Yn NUMERICAL SOLUTION EXACT SOLUTION Xn Figure 1. The graph of the Numerical and Exact solution of yy = yy Figure 2. The graph of the Absolute Error of Numerical and Exact solution of yy = yy

5 American Journal of Computational and Applied Mathematics 2017, 7(6): Example 2 Using the Integrator (26) to solve the initial value problem yy = sec(xx), yy(0) = 1, iiii tthee iiiiiiiiiiiiiiii 0 xx 1, The analytical solution yy(xx) = 1 + ln sec(xx) + tan (xx) h = 0.025, BB = , ββ = Table 2 Numerical Exact Absolute xx nn Error Yn NUMERICAL SOLUTION EXACT SOLUTION Xn Figure 3. The graph of the Numerical and Exact solution of yy = sec(xx)

6 148 Ogunrinde R. B. et al.: A Numerical Integration for Solving First Order Differential Equations Using Gompertz Function Approach Figure 4. The graph of the Absolute Error of Numerical and Exact solution of yy = sec(xx) 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, [2] Charles, P. W. (1932). The Gompertz Curve as a Growth Curve. Proceedings of the National Academy of Sciences. Vol. 18, No [3] Fatunla, S. O. (1976). A New Algorithm for Numerical of ODEs. Computer and Mathematics with Applications. Vol. 2, pp [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 [5] Ibijola E.A. et al (2010). On a New Numerical Scheme for the of Initial Value Problems in ODE. Australian Journal of Basic and Applied Sciences. 4(10):