World Applied Sciences Journal (Special Issue of Applied Math): 8-86, 3 ISSN 88-495; IDOSI Publications, 3 DOI:.589/idosi.wasj.3..am.43 Solving Delay Differential Equations (DDEs) using Naashima s Stages 4 th Order Pseudo-Runge-Kutta Method Nur Ain Ayunni Sabri and Mustafa bin Mamat Department of Business and Finance, Faculty of Entrepreneurship and Business, Universiti Malaysia Kelantan (UMK), 6 Kota Bharu, Kelantan Department of Mathematics, Faculty of Science and Technology, Universiti Malaysia Terengganu, 3 K. Terengganu, Malaysia Abstract: This paper is investigate numerically the problem on solving Delay Differential Equations (DDEs) using Naashima s Stages 4th Order Pseudo-Runge-Kutta Method and also developed the algorithm of Naashima s Stages 4th Order Pseudo-Runge-Kutta Method incorporates with Hermite Interpolation. Numerical result is applied to a real life problem which is Food-Limited Model. Mathematica 7 software and Microsoft Excel are used to develop the calculation. Key words: Delay Differential Equations (DDEs) Pseudo-Runge Kutta method Hermite interpolation INTRODUCTION Delay differential equations (DDEs) are used to describe many phenomena in physics, biology and chemistry [, 7]. Pseudo-Runge-Kutta (PRK) Method is also one of the Runge-Kutta (RK) Methods family. In this article, Stages 4th Order Pseudo-Runge-Kutta Method, which is a type of Pseudo Runge-Kutta (PRK) Methods will be discussed. According to F.Constabile (99), Pseudo-Runge Kutta (PRK) Methods require fewer evaluations than Runge-Kutta (RK) Method of the same order. Fourth Order Runge-Kutta (RK4) Method will be used to solve Initial Value Problems (IVPs) in order to get the first term. Many real life problem phenomena in physics, engineering, biology, medicine, economics etc can be modelled by an Initial Value Problem (IVP) or Cauchy problem for Ordinary Differential Equations (ODEs ) of the type ( ) y y' t gt,y t,t t yt (.) where the function y(t) called the state variable represents some physical quantity that evolves over the time []. ODEs are usually solved using the step-sie methods that. The method that commonly been used to solve ODEs are RK Method, Euler Method and any other step-sie methods. RK Method is the most In mathematics, DDEs are differential equations in which the derivative of the unnown function at certain time is given in terms of the values of the function at previous times. DDEs differ from ODEs in that the derivatives at any time depend on the solution (and in the case of neutral equations on the derivatives) at prior times. The simplest constant delay equation has the form ( ( τ) ( τ) ( τ) ) ' y t f t,y t,y t,yt,,yt (.) where the time delay (lags) τ j are positive constant. More generally, state dependent delays may depends on the solution, that is ττ i i( t,yt ). In more general models, the derivative y (t) may depend on y and y itself at some past value-τ. In this case, Equation (.) changes into the form ' ' φ y t f t,y t,y t τ,y t τ, t t yt t, t t popular method that been used to solve ODEs. Corresponding Author: Nur Ain Ayunni Sabri, Department of Business and Finance, Faculty of Entrepreneurship and Business, Universiti Malaysia Kelantan (UMK), 6 Kota Bharu, Kelantan 8 (.3) where the function φ(t) is assumed to be at least continuous. Equation (.3) is called a Delay Differential Equation of neutral type (NDDE). The methods that used to solve DDEs in this paper are RK Method and one of the step-sie methods, which is Hermite Interpolation. For the recent years, there are many papers have been written regarding on solving DDEs [5-5]. In this
World Appl. Sci. J., (Special Issue of Applied Math): 8-86, 3 research, we solve DDEs using Naashima.s Stages 4th Order Pseudo-Runge-Kutta Method and we used this method incorporates with Hermite Interpolation. PSEUDO-RUNGE-KUTTA AND HERMITE INTERPOLATION Pseudo-Runge-Kutta formulae that we are considering is Naashima s Stages 4 th Order Pseudo- Runge-Kutta [3]: 7 5 yn+ yn + h + + 74 74 74 ( ) f x,y n n f x,y n n x +.7h,.56y +.56y n n n f +.833h +.3h (.) Interpolation is to obtain an approximation solution for a specific value of x, it is possible to shorten the final step, if necessary, to complete the step exactly at the right place. It is usually more convenient to rely on step sie control mechanism that is independent of output requirements and to produce require output results by interpolation as the opportunity arises. The use of interpolation maes it also possible to produce output at multiple and arbitrary points [4]. Hermite Interpolation is a method of interpolating data points as a polynomial function, in the field of numerical analysis. The generated Hermite Polynomial is closely related to the Newton Polynomial in that both are derived from the calculation of Divided Differences. Hermite Interpolation matches data points in both value and first derivative. This means that n derivatives values ( x,y ),,( x,y ) must be given in addition n n to n data points ( x,y ),,( x,y ). n n NAKASHIMA S STAGES 4TH ORDER PSEUDO-RUNGE-KUTTA METHOD INCORPORATE WITH HERMITE INTERPOLATION AND THE ALGORITHM y n y + + n ( 3 4) forj,,,,m 6 + + + (3.) where hf t,y n n h hf t n+,yn + h 3 hf t n +,yn + hf t + h,y + 4 n n 3. From the Classical RK4 Method which is Equation (3.), the equations becomes y y + + 6 + + + n n 3 4 ( ) f t,y,y t τ n n n h h h f t n +,yn +,y tn + τ h h h f t,y,y t 3 + + + τ n n n ( ) f t + h,y + h,yt + h τ 4 n n 3 n (3.) where y is the delay argument which depends on the question and τ given in the question. 3. From the first step, we will get the delay argument that is not fix point. Therefore, here we need do some approximation using Hermite Interpolation. The mesh point,,, n that we going to choose must: a) Depends on the delay argument that we want to approximate and the value of h. For each point that we choose, that point is multiple becomes two points. b) Contain all the argument in the interval of mesh points. 4. Next, substitute the value of mesh point that already choose to find new argument of From the meaning of pseudo and RK Method, PRK Method is the method that is not actually RK Method but having the appearance of RK Method. It has been proof that PRK Method is one of the family members of RK Method. Algorithm. Runge-Kutta Method use the formula t + t +h and 8 [ ] y ( ) [ ] y ( ) y f f f y f n 5. Then, using divided difference to find the approximation. The finite difference formula that need to find are: n
World Appl. Sci. J., (Special Issue of Applied Math): 8-86, 3 [ ] y ( ) f[ ] f [ ] f, f[ ] f,,f, y 3 f, n n f f, y n n n n n n n 6. Continue the calculation for, f,, f,, 3 f,, n n n f, f, f, f, 3 3 f, f, n n n n n n Next, Until f,,, 3 f,,, 3 4 f,,, n 3 n n n [ ] f,,,, n f,, f,, 3 3 f,, f,, 3 4 3 4 f,, f,, n n n n 3 n n n n 3 [ ] [ ] f,,, f,,,, n n n 7. Finally, substitute into Hermite Interpolation formulae. ( ) [ ] [ ]( n ) [ ]( n )( n ) [ 3]( n )( n )( n ) [ ] f t,y,y t τ n n n f + f, t τ + f,, t τ t τ + f,,, t τ t τ t τ + + f,,,, t τ t τ t τ t τ n n n n n n (3.3) h h h f t n +,yn +,y tn + τ h h h f[ ] + f [,] tn + τ + f,, [ ] tn + τ tn + τ h h h + f,,, [ 3] tn + τ tn + τ tn + τ + + h h h h + f,,, [,n] tn + τ tn + τ tn + τ tn + τ n h h h 3 f t n +,yn +,y tn + τ h h h f[ ] + f [,] tn + τ + f,, [ ] tn + τ tn + τ h h h + f,,, [ 3] tn + τ tn + τ tn + τ + + h h h h + f,,, [,n] tn + τ tn + τ tn + τ tn + τ n ( ) [ ] [ ]( n ) [ ]( n )( n ) f,,, [ 3]( tn h )( tn h )( tn h ) [ ] f t + h,y + h,yt + h τ 4 n n 3 n f + f, t + h τ + f,, t + h τ t + h τ + + τ + τ + τ + + + f,,,, t + h τ t + h τ t + h τ t + h τ n n n n n n (3.4) (3.5) (3.6) 8. Substitute all the value of,, 3 and 4 into equation 83
World Appl. Sci. J., (Special Issue of Applied Math): 8-86, 3 h y n y + + n ( 3 4) + + + to get y. 9. To calculate y, y 3,, we will use Naashima s Stages 4 th Order Pseudo-Runge-Kutta Method incorporate with Hermite Interpolation. From the formula of Naashima s Stages 4 th Order Pseudo-Runge-Kutta Method and Equation (.) becomes, 7 5 yn+ yn + h + + 74 74 74 (3.7) ( ( )) ( ) ( ) f t,y,yt τ n n n f t,y,y t τ n n n f t +.7h,.56y +.56y +.833h +.3h,y t +.7h τ n n n n. From this step, we will get the delay argument that is not fix point. Therefore, here we need do some approximation using Hermite Interpolation. The mesh point,,, n that we going to choose must : a) pends on the delay argument that we want to approximate and the value of h. For each point that we choose, that point is multiple becomes two points. b) Contain all the argument in the interval of mesh point. Repeat steps 4, 5 and 6 for calculations of finite difference.. Next, substitute the values that we get previously in Hermite Interpolation formula. ( ( )) [ ] [ ]( n ) [ ]( n )( n ) f,,, [ 3]( tn )( tn )( tn ) f,,,, [, ]( t )( t )( t ) ( t ) f t,y,yt τ n n n f + f, t τ + f,, t τ t τ + τ τ τ + + + τ τ τ τ 3 n n n n n n ( ) [ ] [ ]( n ) [ ]( n )( n ) f,,, [ 3]( tn )( tn )( tn ) [ ] f t,y,y t τ n n n f + f, t τ + f,, t τ t τ + τ τ τ + + + f,,,, 3,n tn τ tn τ tn τ tn τ n ( ) [ ] [ ]( n ) [ ]( n )( n ) f,,, [ 3]( tn.7h )( tn.7h )( tn.7h ) f,,,, [, ]( t.7h )( t + τ )( + τ ) ( + τ ) f t +.7h,.56y +.56y +.833h +.3h,y t +.7h τ n n n n f + f, t +.7h τ + f,, t +.7h τ t +.7h τ + + τ + τ + τ + + + + τ 3 n n n.7h t.7h t.7h n n n (3.8) (3.9) (3.). Finally, substitute all the values of, and into to find the next y. 7 5 yn+ yn + h + + 74 74 74 SOLVING DDE S USING NAKASHIMA S STAGES 4TH ORDER PSEUDO-RUNGE-KUTTA METHOD We have chosen a model in order to apply Naashima s Stages 4 th Order Pseudo-Runge-Kutta Methods. The model that we choose was Food Limited Model. We also mae a comparison with 4 th Order Runge-Kutta Method or also nown as Classical Runge-Kutta Method in order to see the accuracy between these two methods. 84
World Appl. Sci. J., (Special Issue of Applied Math): 8-86, 3 Table 4.: Results for application food limited model using RK4 method and Naashima s stages 4 th order Pseudo-Runge- Kutta method incorporate with Hermite interpolation Naashima s stages 4 th order t RK4 method Pseudo-Runge-Kutta method..5.5.5.485379688443358.485379688443358..46976675987.4695387583994.5.4595639757739.4595545567333538..4584739853478.488457358795.5.448639567355387.448638884793688 3..65698798553979.4465576434997583 3.5.44636588638993.4465355789 4..4474747396998.44748486389347 4.5.449496555755893.449538988363966 5..4589876567476.4598457558546 5.5.4569678858444.456553548596 6..463945495.46345798799595 6.5.4649854774744433.46539687659848 7..4699395375368987.47455557885 7.5.4757348795893.475767654575 8..4833698798746.484494439373 8.5.4855646437836.48573836844457497 9..49853847634847.499798986678464 9.5.495964864596935.49799863474969..54386874468.579469578535 Food limited model: Food-limited model is a model for population growth which is proposed by Smith in 963. This model is an alternative to the logistic equation for food-limited population dx K xt r( x ),t dt K + rcn t (4.) Here N, r and K are the mass of the population, the rate of increase with unlimited food and value of N at saturation respectively. The constant /c is the rate replacement of mass in the population at saturation (this includes both the replacement of metabolic loss and of dead organism) [6]. The delay Food-Limited is where ( τ) dx K x t rx( t) dt K+ rcx t τ r.5,k.,c and τ 8 for t [,] and x t e for t<. 85 h.5. The initial function.5t We see that the solution from both methods is almost same although have small error between both method, this shows that Naashima.s Stages 4th Order Pseudo-Runnge-Kutta Mehod also can be applied to real life problem of DDEs. By comparing both methods, we can see the validity of the new method in solving the real life problem of DDEs. We conclude that Naashima.s Stages 4th Order Pseudo-Runge- Kutta Method can be used to solve DDEs. We used to compare both methods because we want to find other method to solve real life problems of DDEs and we found that Naashima s Stages 4 th Order Pseudo-Runge-Kutta Method is one of the method that we can use to solve DDEs. CONCLUSIONS In this study, the method that been used to solve Delay Differential Equations (DDEs) is Naashima s Stages 4 th Order Pseudo-Runge-Kutta method incorporate with Hermite Interpolation. From numerical results, we have observed that PRK method perform a good result in solving DDEs of single delay because the answer is accurate as Runge- Kutta method. PRK method is accurate as RK method and it is also one step method. From here we can mae a conclusion that Naashima s Stages 4 th Order Pseudo-Runge-Kutta method is another method to solve DDEs problem. We also conclude that, the solution using Naashima s Stages 4 th Order Pseudo-Runge- Kutta method need less calculation than RK4 method. This is because, using Naashima s Stages 4 th Order Pseudo-Runge-Kutta method, we only need to calculate until, but using RK4 method we need to calculate until 4. Therefore, we note that Naashima s Stages 4 th Order Pseudo-Runge-Kutta method is also can be used to solve DDEs and this is another method to solve DDEs. REFERENCES. Sharma, J.N., 4. Numerical Methods for Engineers and Scientist. Deemed University: Alpha Science International Ltd. Pangbourne England.. Bellen, A. and M. Zennaro, 3. Numerical Methods for Delay Differential Equations. United States: Oxford Science Publications. 3. Masaharu Naashima, 98. On Pseudo-Runge- Kutta Methods with and 3 Stages. RIMS Publications, 8 (3): 895-99. 4. Butcher, J.C., 8. Numerical Methods for Ordinary Differential Equation. nd Edn. John Willey and Sons Ltd.
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