Runge Kutta Collocation Method for the Solution of First Order Ordinary Differential Equations

Transcription

1 Nonlinear Analysis and Differential Equations, Vol. 4, 2016, no. 1, HIKARI Ltd, Runge Kutta Collocation Method for the Solution of First Order Ordinary Differential Equations A. O. Adesanya 1 Department of Mathematics Modibbo Adama University of Technology Yola, Adamawa State, Nigeria A. U. Fotta Department of Mathematics Adamawa State Polytechnic Yola, Adamawa State, Nigeria R. O. Onsachi Department of Mathematics Modibbo Adama University of Technology Yola, Nigeria, Adamawa State, Nigeria Copyright c 2015 A. O. Adesanya, A. U. Fotta and R. O. Onsachi. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We introduce one step continuous Runge Kutta collocation method with three free parameters for the solution of stiff first order ordinary differential equations. We adopt interpolation and collocation of the approximate solution at some selected grid points to give system of non linear equations. Using Crammer s rule to solve for the unknown parameters and substituting into the approximate solution gives the continuous method. To determine how best to fix the free parameters, we consider 1 Corresponding author

2 18 A. O. Adesanya, A. U. Fotta and R. O. Onsachi three cases; the Guass type, the equal interval, and the Radau type. Numerical results show that the farer the free parameters from the center, the better the results. Keywords: Runge Kutta collocation, interpolation, collocation, approximate solution, grid points, continuous method, stiff problems This paper considers solution to 1. Introduction y = f (x, y), y (x 0 ) = y 0 (1) where f is continuous and satisfies Lipschitz s condition, x 0 is the initial point and y 0 is the solution at x 0. Adoption of collocation and interpolation of power series approximate solution for the solution of initial value problems have been studied by [1 11], most of these methods fails when the problem is stiff or stiff oscillatory. Linear multistep method has been reported to be efficient and easier to implement for the solution of ordinary differential equations [12, 13]. [12 15] studied development of hybrid methods, they reported that hybrid methods are difficult to develop, but gives methods with good stability properties due to the reduction in the step length. [14 16] adopted approximate solution: n 1 y (x) = a j x j + a n e nx (2) j=0 where n is the numbers of interpolation and collocation points. They discovered that methods developed using (2) posses good stability properties which is good for stiff problems. In this paper, we introduce a new continuous Runge-Kutta collocation method with three free parameters using new approximate solution. The introduction of the free parameters and the approximate solution make this work different from the existing methods. Our interest also include investigation into how best to fix the free parameters, we consider three cases; Guass type, Radau type and equal interval methods. 2. Methods We consider an approximate solution N M y(x) = a n x n + a n e nx (3) n=0 N+1

3 Runge Kutta collocation method for the solution of first order ODE 19 The first derivative of (3) is y (x) = substituting (4)into (1)gives f (x, y) = N na n x n n=1 N na n x n n=1 M na n e nx (4) N+1 M na n e nx (5) we then construct a continuous approximation by imposing the following conditions N+1 Y (x n+j ) = y n+j, j = 0, 1,..., s (6) Y (x n+j ) = f n+j, j = 0, 1,..., r where s and r are the numbers of interpolation and collocation points respectively. we sought the solution of (1) on the partition Π N : x 0 < x 1 < x 2 <... < x N. over a constant stepsize h = x n+1 x n Interpolating (3) at x n+s, s = 0 and collocating (4) at points x n+r, gives a system of non linear equation XA = U (7) where A = [ a 0 a 1 a 2 a 3 a 4 a 5 a k ] T U = [ ] T y n f n f n+u f n+v f n+w f n+1 f n+k 1 x n x N n e (N+1)xn e Mxn 1 x n+1... x N n+1 e (N+1)x n+1 e Mx n x X = n+n... x N n+n e (N+1)x n+n e Mx n+n Nxn N 1 (N + 1) e (N+1)xn Me Mxn Nx N 1 n+n (N + 1) e (N+1)x n+1 Me Mx n Nx N 1 n+m (N + 1) e (N+1)x n+m Me Mx n+m Solving (7) for the a ks, gives an s-stage Runge-Kutta method s y n+1 = y n + h β i ϖ i i=0 ϖ i = f(x n + c i h, Y i ) s Y i = y n + h a ij ϖ j (8) j=0

4 20 A. O. Adesanya, A. U. Fotta and R. O. Onsachi where s is the internal stage, with the condition s c i = β i (9) To implement (8), [11] proposed a prediction equation in the form k Y i = y n + (c i ) i δ i f(x, y) δxi i=1 i=0 (xn,y n) Writing (5) compactly in a partitioned Butcher table gives c 1 a 11 a a 1s c [ ] 2 a 21 a a 2s.... c A b T = c s a s1 a s2... a ss b 1 b 2... b s (10) (11) [17] defined the stability function for the s-stage implicit R-K scheme as a rational function U 1 (z) given by U 1 (z) = 1 + zb T (I za) 1 e (12) where z = λh, I ia an identity matrix ans e is the s-vector e = [ ] T 2.1. Specification of the method. We consider X = 0 u < v < w < 1 A = [ a 0 a 1 a 2 a 3 a 4 a 5 ] T U = [ y n f n f n+u f n+v f n+w f n+1 ] T 1 x n e xn e 2xn e 3xn e 4xn 0 1 e xn 2e 2xn 3e 3xn 4e 4xn 0 1 e x n+u 2e 2x n+u 3e 3x n+u 4e 4x n+u 0 1 e x n+v 2e 2x n+v 3e 3x n+v 4e 4x n+v 0 1 e x n+w 2e 2x n+w 3e 3x n+w 4e 4x n+w 0 1 e x n+1 2e 2x n+1 3e 3x n+1 4e 4x n+1 Solving for the unknown constants, (8) gives y n+1 = y n + h(β 1 F 1 + β 2 F 2 + β 3 F 3 + β 4 F 4 + β 5 F 5 ) (13) where α 0 = 1 [ ] (5u + 5v + 5w 10uv 10uw 10vw + 30uvw 3) β 1 = 60uvw

5 Runge Kutta collocation method for the solution of first order ODE 21 (5v + 5w 10vw 3) β 2 = 60u (u v) (u w) (u 1) (5u + 5w 10uw 3) β 3 = 60v (u v) (v w) (v 1) (5u + 5v 10uv 3) β 4 = 60w (u w) (v w) (w 1) (15u + 15v + 15w 20uv 20uw 20vw + 30uvw 12) β 5 = 60 (u 1) (v 1) (w 1) The internal stages are: Y 1 = y n (14) Y 2 = y n + h(γ 1 F 1 + γ 2 F 2 + γ 3 F 3 + γ 4 F 4 + γ 5 F 5 ) (15) γ 1 = (10u2 v 5u 3 v + 10u 2 w 5u 3 w 5u 3 + 3u u 2 vw 30uvw) 60vw γ 2 = u (20uv 15u2 w 15u 2 v + 20uw 30vw 15u u uvw) 60 (u v) (u w) (u 1) γ 3 = u2 (10uw 5u 2 w 5u 2 + 3u 3 ) 60v (u v) (v w) (v 1) γ 4 = u2 (10uv 5u 2 v 5u 2 + 3u 3 ) 60w (u w) (v w) (w 1) γ 5 = u2 (3u 3 5u 2 w 5u 2 v + 10uvw) 60 (u 1) (v 1) (w 1) Y 3 = y n + h (µ 1 F 1 + µ 2 F 2 + µ 3 F 3 + µ 4 F 4 + µ 5 F 5 ) (16) µ 1 = (10uv2 5uv v 2 w 5v 3 w 5v 3 + 3v uv 2 w 30uvw) 60uw µ 2 = v2 (10vw 5v 2 w 5v 2 + 3v 3 ) 60u (u v) (u w) (u 1) µ 3 = v (20uv 15v2 w 15uv 2 30uw + 20vw 15v v uvw) 60 (u v) (v w) (v 1) µ 4 = v2 (10uv 5uv 2 5v 2 + 3v 3 ) 60w (u w) (v w) (w 1) µ 5 = v2 (3v 3 5v 2 w 5uv uvw) 60 (u 1) (v 1) (w 1) Y 4 = y n + h(η 1 F 1 + η 2 F 2 + η 3 F 3 + η 4 F 4 + η 5 F 5 ) (17) η 1 = (10uw2 5uw vw 2 5vw 3 5w 3 + 3w uvw 2 30uvw) 60uv η 2 = w2 (10vw 5vw 2 5w 2 + 3w 3 ) 60u (u v) (u w) (u 1)

6 22 A. O. Adesanya, A. U. Fotta and R. O. Onsachi η 3 = w2 (10uw 5uw 2 5w 2 + 3w 3 ) 60v (u v) (v w) (v 1) η 4 = w (20uw 15vw2 30uv 15uw vw 15w w uvw) 60 (u w) (v w) (w 1) η 5 = w2 (3w 3 5vw 2 5uw uvw) 60 (u 1) (v 1) (w 1) Y 5 = y n + h(β 1 F 1 + β 2 F 2 + β 3 F 3 + β 4 F 4 + β 5 F 5 ) F 1 = f(x n, Y 1 ); F 2 = f(x n + uh, Y 2 ); F 3 = f(x n + vh, Y 3 ); F 4 = f(x n + wh, Y 4 ); F 5 = f(x n + h, Y 5 ) Eqn (11) reduces to 0 u v w γ 1 γ 2 γ 3 γ 4 γ 5 µ 1 µ 2 µ 3 µ 4 µ 5 η 1 η 2 η 3 η 4 η 5 β 1 β 2 β 3 β 4 β 5 β 1 β 2 β 3 β 4 β Cases. Three cases are considered: (i): the equal interval method: u = 1, v = 1, w = (ii): the Guass type: 3 3, v = 1, w = (iii): the Radau Type: u = 6 6, v = 1, w = Analysis of the basic properties of the method (18) 3.1. Order and truncation error of the method. We associate with (13) the linear operator N (y (x) ; h) specified by N (y (x) ; h) = y (x + h) β 1 F 1 β 2 F 2 β 3 F 3 β 4 F 4 β 5 F 5 (19) The method (19) is said to be of order p if N (y (x) ; h) = O (h p+1 ) and the local truncation error T n+1 at x n+1 is given by N (y (x n ) ; h) where y (x n ) is now the theoretical solution y n = y (x n ) = T n+1 = y(x n+1 ) y n+1 Expanding (19) in Taylor series about x n,equating the coefficient of h, the truncation error gives ( ) T n+1 = h6 3u + 3v + 3w 5uv + O ( h uw 5vw + 10uvw 2 7) (20) 3.2. Consistency. Method (13) is said to be consistent if y n+1 y n lim h 0 h It can be shown that (13) is consistent = y n

7 Runge Kutta collocation method for the solution of first order ODE Zero stability. Method (13) is said to be zero stable if lim h 0 y n+1 = y n It can be shown that (13) is zero stable 3.4. Convergence. (i): Method (13) is said to be convergent if lim y (x) y (x n) 0 h 0 where y (x) is the numerical result, y (x n ) is the exact solution which is y n. It can be shown that (13) is convergence (ii): Method (13) is said to be convergent iff it is consisitent and zero stable. It can be shown that our method is consistence and sero stable, hence convergent 3.5. Stability of the method. The stability matrix becomes, where N (x) = D (x) = u 1 (z) = N (x) D (x) 36h 2 z 2 + 8h 3 z 3 + h 4 z hz 18h 2 uz 2 18h 2 vz 2 6h 3 uz 3 18h 2 wz 2 6h 3 vz 3 h 4 uz 4 6h 3 wz 3 h 4 vz 4 h 4 wz 4 24huz 24hvz 24hwz + 6h 2 uvz 2 + 6h 2 uwz 2 +4h 3 uvz 3 + 6h 2 vwz 2 + 4h 3 uwz 3 + h 4 uvz 4 + 4h 3 vwz 3 +h 4 uwz 4 + h 4 vwz 4 2h 3 uvwz 3 h 4 uvwz Convergence implies: 24hz + 6h 2 uz 2 + 6h 2 vz 2 + 6h 2 wz 2 24huz 24hvz 24hwz + 6h 2 uvz 2 + 6h 2 uwz 2 2h 3 uvz 3 + 6h 2 vwz 2 2h 3 uwz 3 2h 3 vwz 3 2h 3 uvwz 3 + h 4 uvwz (w 1) (v 1) (u 1) lim R (z) h uvw which is true for all values of u, v, w. It implies that the method is bounded s = { R (z) < 1} which shows that the method is A-stable 1

8 24 A. O. Adesanya, A. U. Fotta and R. O. Onsachi 4. Numerical Examples The following are use in the tables Error = y (x) y (x n ) where y (x) is the exact result and y n (x) is the computed result Error 1 = Error in case (i) Error 2 = Error in case (ii) Error 3 = rror in case (iii) Problem 1: We consider the highly stiff ordinary differential equation: y = 10(y 1) 2, y(0) = 2, h = 0.01 Exact solution: 1 y(x) = x Source: [16] Table 1: Results of problem I x Exact Error 1 Error 2 Error ( 10) ( 10) ( 10) ( 10) ( 10) ( 11) ( 11) ( 11) ( 11) ( 11) ( 11) ( 11) ( 11) ( 11) ( 11) ( 11) ( 11) ( 11) ( 12) ( 12) ( 12) Problem II: We consider the highly stiff ordinary differential equation: y = xy, y(0) = 1, h = 0.01 Exact solution: 1 y(x) = x Source: [16] Table 2: Results of problem II x Exact Error 1 Error 2 Error ( 16) ( 16) ( 15) ( 15) ( 15) ( 16) ( 14) ( 14) ( 14) ( 14) ( 14) ( 14) ( 13) ( 13) ( 14) ( 13) ( 13) ( 13) ( 13) ( 13) ( 13) ( 13) ( 13) ( 13) ( 12) ( 12) ( 13) ( 12) ( 12) ( 12) Problem III: We consider the highly stiff ordinary differential equation:

9 Runge Kutta collocation method for the solution of first order ODE 25 y = 100xy 2, y(1) = 1 51, h = 0.25 Exact solution: 1 y(x) = x 2 Source: [16] Table 3: Results of problem III x Exact Error 1 Error 2 Error ( 08) ( 08) ( 08) ( 09) ( 09) ( 09) ( 10) ( 10) ( 10) ( 10) ( 10) ( 10) ( 10) ( 10) ( 10) ( 11) ( 11) ( 11) ( 11) ( 11) ( 11) 5. Discussion of Result We consider three numerical examples; case (iii) which is the Radau type gives the best results followed by Case (ii) which is the Gauss type. The implication is that the farer u and w from v, the better the results. 6. Conclusion We discuss development of Runge Kutta collocation method with five internal stages for the solution of first order initial value problems. Three free parameters are considered, further work can be done by assigning different values to the free parameters to substantiate our claims above. The comparison of our method with the existing methods is not shown but our method compete favourably with the existing methods. References [1] P. Onumanyi, D. O. Awoyemi, S. N. Jator, U. W. Sirisena, New linear multistep method with constant coefficients for first order initial value problems, J. of Nig. Maths. Soc., 13 (1994), [2] P. Onumayi, U. W. Sirisena, S. N. Jator, Continuous finite difference approximation for solving differential equations, Intern. J. of Comput. Maths., 72 (1999), [3] J. D. Lambert, Computational Methods in Ordinary Differential Equations, John Wiley, New York, [4] J. D. Lambert, Safe point methods for separable stiff systems of ordinary differential equations, UK, University of Dundee, Department of Mathematics report No NA/31, (1979). [5] A. A. James, A. O. Adesanya, A note on the construction of constant order predictor corrector algorithm for the solution of y = f (x, y), British Journal of Mathematics and Computer Science, 4 (2014), no. 6,

10 26 A. O. Adesanya, A. U. Fotta and R. O. Onsachi [6] U. W. Sirisena, An accurate implementation of the butcher hybrid formula for the initial value problems in ordinary differential equations, Nig. J. Maths. and Appls., 12 (1999), [7] S. N. Jator, A Sixth Order Linear Multistep Method for the Direct Solution of y = f(x, y, y ), IJPAM, 40 (2007), no. 4, [8] D. O. Awoyemi, A new sixth-order algorithm for general second order ordinary differential equations, Intern. J. Compu. Math., 77 (2001), [9] A. O. Adesanya, A. A. James, S. Joshua, Hybrid block predictor-hybrid block corrector for the solution of first order ordinary differential equations, Eng. Math. Lett., 2014 (2014), [10] A. O. Adesanya, M. O. Udoh, A. M. Alkali, A new block-predictor corrector algorithm for the solution of y = f(x, y, y, y ), American Journal of Computational Mathematics, 2 (2012), [11] A. O. Adesanya, M. R. Odekunle, M. O. Udoh, Four steps continuous method for the solution of y = f(x, y, y ), American Journal of Computational Mathematics, 3 (2013), [12] T. A. Anake, D. O. Awoyemi, A. O. Adesanya, One step implicit hybrid block method for the direct solution of general second order ordinary differential equations, Intern. J. Appl. Math., (2012). [13] M. K. Fasasi, A. O. Adesanya, S. O. Adee, Block numerical integrator for the solution of y = f(x, y, y, y ), International Journal of Pure and Apllied Mathematics, 92 (2014), [14] J. Sunday, M.R. Odekunle, A. O. Adesanya, A. A. James, Extended block integrator for first order stiff and oscillatory differential equations, American Journal of Computational and Applied Mathematics, 3 (2013), no. 6, [15] J. Sunday, A. O. Adesanya, M. R. Odekunle, A self-starting four step fifth order block integrator for stiff and oscillatory differential equations, J. Maths. and Compu. Sci., 4 (2014), no. 1, [16] A. A. Momoh, A. O. Adesanya, M. K. Fasasi, A. Tahir, A new numerical integrator for the solution of stiff first order ordinary differential equations, Eng. Math. Lett., 2014 (2014), [17] S. O. Fatunla, Applied Numerical Method for Initial Value Problems in Ordinary Differential Equations, Academic Press Cambridge, Received: August 18, 2015; Published: October 16, 2015