Numerical Solution of Second Order Initial Value Problems of Bratu-type Equations using Sixth Order Runge-Kutta Seven Stages Method

In this paper, second order initial value problem of Bratu-type ordinary differential equations is solved numerically using sixth order Runge-Kutta seven stages method. The stability of the method is checked and verified. In order to justify the validity and effectiveness of the method, two model examples are solved and the numerical solutions are compared to the corresponding exact solutions. Furthermore, the results obtained using the current method are compared with the numerical results obtained by other researchers. The numerical results in terms of point-wise absolute errors presented in tables and plotted graphs show that the present method approximates the exact solutions very well.


Numerical Solution of Second Order Initial Value Problems of Bratu-type Equations using Sixth Order Runge-Kutta Seven Stages Method
Hibist Bazezew Fenta and Getachew Adamu Derese Abstract-In this paper, second order initial value problem of Bratu-type ordinary differential equations is solved numerically using sixth order Runge-Kutta seven stages method. The stability of the method is checked and verified. In order to justify the validity and effectiveness of the method, two model examples are solved and the numerical solutions are compared to the corresponding exact solutions. Furthermore, the results obtained using the current method are compared with the numerical results obtained by other researchers. The numerical results in terms of point-wise absolute errors presented in tables and plotted graphs show that the present method approximates the exact solutions very well.

I. INTRODUCTION
P HYSICAL situations concerned with the rate of change of one quantity with respect to another give rise to a differential equation. Differential equations are absolutely fundamental to modern science and engineering. Almost all of the known laws of physics and chemistry are actually differential equations.
Many authors have attempted to solve initial value problems (IVP) to obtain high accuracy rapidly by using numerous methods such as Taylor's method and Runge-Kutta methods. Runge-Kutta methods are more efficient than many numerical methods in terms of accuracy. The methods were developed by two German Mathematicians Carl Runge (1856-1927) and Wilhelm Kutta (1867Kutta ( -1944. The general Runge-Kutta methods are widely used for approximating initial value problems. In particular when the computation of higher derivatives is complicated. They are distinguished by their orders in the sense that they agree with Taylor series solution up to terms of h r where r is the order of the method.
A sixth order Runge-Kutta method was derived by [1] depending on the fifth order Runge-Kutta method of David Goeken and Olin Johnson which needs only five function evaluations. The standard Bratu problem is used in a large variety of applications, such as the fuel ignition model of the theory of thermal combustion, the thermal reaction process model, the Chandrasekhar model of the expansion of the universe, radiated heat transfer, nanotechnology and theory of chemical reaction [2]. Manuscript  According to [3], in 1850, Liouville solved the second order PDE u xv + λ e u = 0, which is associated with the Liouville model in quantum theory. In 1914, Bratu solved almost similar second order ODE u xx + λ e u = 0 which is a basic onedimensional model in combustion theory.
Because of the mathematical and physical properties, the Bratu initial value problems have been studied extensively by many researchers, for example [4] studied a numerical solution of Bratu-type equations by using the variational iteration method; [5] considered Bratu's problems by means of modified homotopy perturbation method; [6] applied Adomian decomposition method to study the Bratu-type equations; [7] used successive differentiation method to solve Bratu equation and Bratu-type equations; [8] developed an algorithm using Runge-Kutta methods of orders four and five for first order systems of ordinary differential equations. The numerical solution of second order initial value problems of Bratu-type equations was studied by [9] using fifth order Runge-Kutta method.
The objective of the present study is to investigate numerical solutions of second order initial value problems of Bratu-type equations using sixth order Runge-Kutta seven stages method. The exact solutions will be compared with numerical solutions obtained using graphs and tables of point-wise absolute errors.

II. FORMULATION OF THE PROBLEM
Consider the second order initial value problem of Bratu-Type equation of the form: subject to the initial conditions: where λ , α, β are given constant numbers and y(x) is unknown functions.

III. STABILITY ANALYSIS OF THE METHOD
Here, the second order initial value problems of Bratu-type equation of the form of equation (1) is reduced into first-order system of equations of the form (4). The second equation in equation (4) is: The nonlinear equation (9) can be linearized by expanding the function G in Taylor series about the point (x 0 , y 0 , z 0 ). Truncating it after the first term gives: By differentiation rules of function of several variables, equation (10) can be written as: where C = g 0 − λ e y 0 + (x − x 0 )(g 0 − λ y 0 e y 0 ) + λ (y 0 − y)e y 0 . Then z = C which is linear in the function of z. Consider the test equation: Then, the solution of the test equation (12) at the n-th discrete point is: where y 0 is constant. Now, by applying equation (7) on equation (12), we have: By substituting the values of k 1 and k 3 − k 7 into y n+1 = y n + 1 120 (11k 1 + 81k 3 + 81k 4 − 32k 5 − 32k 6 + 11k 7 ) we obtain: where E(λ h) = 1 + λ h + 1 2 (λ h) 2 + 1 6 (λ h) 3 + 1 24 (λ h) 4 + 1 120 (λ h) 5 + 1 720 (λ h) 6 + 1 2160 (λ h) 7 . The error in numerical computation does not grow if the propagation error tends to zero or if at least bounded [10]. Now, from equation (13), it is easily observed that the exact value of y(x i ) increases for the constant λ > 0 and decreases for λ < 0 with the factor of e λ h . Meanwhile, from equation (14), the approximate value of y(x n ) increases or decreases with the factor of E(λ h). If λ h ≥ 0, then e λ h ≥ 1. So, the sixth order Runge-Kutta method with seven stage is relatively stable if λ h ≤ 0 (i.e., λ ≤ 0). Then, the sixth order Runge-Kutta method with seven stages is absolutely stable in the interval −3.17 ≤ λ h ≤ 0.

V. DISCUSSIONS AND CONCLUSIONS
In this paper, higher order Runge-Kutta (RK6) method with seven stages is formulated to approximate the solutions of second order initial value problems of Bratu-type equation. The stability of the method is also discussed. To justify the applicability of the proposed method, tables of pointwise absolute errors and graphs have been plotted for three model examples to compare the exact solution and numerical solution at different mesh size h. From Tables II-VII, it is evident that all the absolute errors decrease rapidly as the mesh size h decreases which in turn show that smaller mesh size provides the better approximate value. Figures 1-6 show that the present method approximates the exact solution in an excellent manner.     2.3677e −11 0.2 5.0353e −9 2.2004e −10 0.3 7.9863e −9 6.3819e −10 0.4 1.1616e −8 1.3888e −9 0.5 1.6395e −8 2.6999e −9 0.6 2.3102e −8 5.0521e −9 0.7 3.3168e −8 9.5241e −9 0.8 4.9480e −8 1.8764e −8 0.9 7.8591e −8 4.0066e −8 1.0 1.3772e −7 9.6681e −8