One Step Continuous Hybrid Block Method for the Solution of

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1 ISSN 4-86 (Paper) ISSN -9 (Online) Vol.4 No. 4 One Step Continuous Hybrid Block Method for the Solution of y = f ( x y y y ). K. M. Fasasi * A. O. Adesanya S. O. Adee Department of Mathematics Modibbo Adama University of echnology PMB 76 Yola Nigeria * of the corresponding author: kolawolefasasi@yahoo.com Abstract In this paper we present a block method for the direct solution of third order initial value problems of ordinary differential equations. Collocation and interpolation approach was adopted to generate a continuous linear multistep method which was then solved for the independent solution to give a continuous block method. We evaluated the result at selected grid points to give a discrete block which eventually gave simultaneous solutions at both grid and off grid points. he one-step block method is consistent and A -stable with good region of absolute stability. Experimental results confirmed the superiority of the new scheme over an existing method. Keywords: consistent convergent collocation hybrid points independent solution interpolation zero stable. Introduction We consider the solution to general third order initial value problem of the form k y = f ( x y y y ) y ( x ) = y k = yε R () k x is the initial point and f is continuous within the interval of integration and satisfies the existence and uniqueness condition. Many real life problems in sciences engineering biology and social sciences are model of third order ordinary differential equations. Some of these models do not always have theoretical solutions thus numerical methods are often employed to solve them. Researchers in most cases always use method of reduction of higher order ODEs into system of first order ODEs to solve (). his technique though quite good is bedeviled with many problems such tediousness complexity of the method waste of time and the need for large computer storage memories because of too many auxiliary functions etc. his approach has been extensively discussed in the literature (see Awoyemi Awoyemi & Kayode ). o address the challenges associated with method of reduction to system of first order scholars developed direct methods for higher order ODEs among whom are Awoyemi et al. () Kayode & Obaruah () Adesanya et al. () Jator (7) and Anake et al. (). When the problem to be solved is stiff and oscillatory Yakubu et al. () proposed hybrid methods. A threestep p-stable collocation method for direct third order ODEs was also developed by Yahaya (7). Some of the schemes can only solve special third order differential equations of the form y = f (x) y = f (y) and the level of accuracy is not high enough. In order to avoid the limitation of the methods mentioned above scholars adopted predictor -corrector approach which has also been reported to be inefficient because the cost of developing separate predictors human and computer time involved in the execution of the method are too costly (James et al. Adesanya et al. ). Scholars later developed block methods in which approximations are simultaneously generated at different grid points in the interval of integration and is less expensive in terms of the number of function evaluations compared to linear multistep methods. his assertion has been reported by Jator () Adesanya et al. () Anake et al. () and Awoyemi et al. (). In this paper we developed a one step with four off-grid points implemented in block method. he grid points were carefully selected to ensure zero stability of the new method.. Derivation of the Method We define the general power series approximate solution in the form: y( x) r+ s = a x () = r and s are the numbers of interpolation and collocation points respectively. he third derivative of () gives

2 ISSN 4-86 (Paper) ISSN -9 (Online) Vol.4 No. 4 Substituting () into () gives r + s ( )( ) () y ( x) = a x = Interpolating () at equation x n r r r+ s (4) = ( ) = ( )( ) f x y y y a x + = and collocating () at [ ] A = a a a a a a a a a AX U y y y f f f f f f + and = n 4 n n n + n n n n x n s s ( ) + = results into a system of nonlinear = U () xn xn xn xn xn xn xn x n x x x x x x x x n+ n+ n+ n+ n+ n+ n+ n x x x x x x x x n+ n+ n+ n+ n+ n+ n+ n xn 6xn xn xn 6x n 4 X = 6 4x 6x x x 6x n+ n+ n+ n+ n x 6x x x 6x n+ n+ n+ n+ n x 6x x x 6x n+ n+ n+ n+ n x 4 6x 4 x 4 x 4 6x n+ n+ n+ n+ n xn+ 6xn+ xn+ xn+ 6xn+ Solving () for the unknown constant gives a continuous hybrid linear multistep method in the form a using Gaussian elimination method and substituting back into () 's 4 y( x) = α y α y + α y + h β fn βv fn v v n+ n = = (6) α = t t α = t t + α = t t 6

3 ISSN 4-86 (Paper) ISSN -9 (Online) Vol.4 No. 4 β = (78t 7t + 747t 787 t + 479t 68t t 86 t) β = (96t 7t 6 t 69t t 96t 8 t) β = (96t 6t 8t 87t t 74t 8 t) β = (96t t + 47t 6t + t 46t + 48 t) β 4 = (96t 7t + 797t 67t + 6t 98t + 8 t) 8 β = (6t t 6t t 84t 9t 4 t) Solving (6) for the independent solution at the selected grid points give a continuous block method of the form m ( m) ( kh) m 4 yn+ k = y h σ fn+ σ v fn+ v v ( ) m= m! + = = (7) σ = (6 t t + 9 t 6 t + 86 t 44 t ) σ ( ) = t t + t t + t σ ( ) = t t + t t + t 4 ( t t t t t ) σ = σ ( 4 84 ) 4 = t t + t t + t σ = (6 t t + 4 t 4 t + 6 t ) 864 Evaluating ( 7 ) at t = ( ) gives a discrete block method of the form ( i) ( i) Ym = y y y y y ( ) ( ) ( ) ( ) 4 F( Ym ) = f f f f f 4 ( i) ( i) ( i) m i n i n i m = [ ( ) ( )] A Y = h e y + h d f y + b F Y (8) 7

4 ISSN 4-86 (Paper) ISSN -9 (Online) Vol.4 No. 4 ( i) ( i ) ( ) ( ) ( ) ( ) i i i i yn = y y y y 4 y n n n n n A is a X identity matrix. i = When e = e = 4 e = 8 9 d = b = When i = e = e = 4 d 6 7 = b = When i = e = d = b =

5 ISSN 4-86 (Paper) ISSN -9 (Online) Vol.4 No. 4. Implementation of the Method Writing equation (8) in the generalized form m n+ n+... n+ k µ µ Y = Ey + h DF( y ) + h BF( Y ) (9) m n n m Y = y y y µ is the order of the differential equation k is the step length E D and B are matrices. We then propose a prediction equation in the form λ =. x ( λ F ) ( y ) f ( x y y ) n λ y n () µ + λ λ m n n λ= Y = Ey + h F ( y ) () Substituting (9) into (8) gives Y = Ey + h DF y + h BF y () µ µ () m n ( n) ( m ) Equation () is our non-self-starting block method since the prediction equation is not obtained directly from the block formula (Awoyemi et al. ). 4. Basic Properties of the Developed Method 4. Order of the block L y( x) : h on (7) be Let the linear operator { } ( i) i ( i) i { ( ) : } = m i n [ ( n) + ( m )] L y x h A y h e y h df y bf y () Expanding y and f in aylor series and comparing the coefficients of h gives p p p+ p+ p+ p+ { } i= L y( x) : h = C y( x) + C y ( x) C h y ( x) + C h y ( x) + C h y ( x) +... p p+ p+ Definition: he linear operator L and associated block method are said to be of order p if C = C =... = C p = C p+ = C p+. p t C h y x O h C + is called the error constant and implies that the truncation error is given by ( ) p + p + p n k p ( ) + + = + +. Comparing the coefficients of h the order of the method is five with error constant [ ] c7 = Consistency A method is said to be consistent if it has order greater than one. From the above analysis it is obvious that our method is consistent. 4. Zero stability A block method is said to be zero stable as h if the roots of the first characteristics polynomial ρ ( r) = satisfy k A R and those roots with R = must be simple. For our method 9

6 ISSN 4-86 (Paper) ISSN -9 (Online) Vol.4 No. 4 ρ( r) = r = r 4 ( r ) = r = showing that our method is zero stable. 4.4 Convergence Definition: he necessary and sufficient conditions for a linear multistep method to be convergent are that it must be consistent and zero stable. Hence our method is convergent. 4. Stability region he method (6) is said to be absolutely stable if for a given h all roots zs of the characteristics polynomial f π ( z h) = ρ( z) + h σ ( z) = satisfies z < s =... n h = λ h and λ =. he s boundary locus method is adopted to determine the region of absolute stability. Substituting the test equation y = λ y y = and y y λ y = λ into the polynomial gives the stability region as shown in fig. y 8 Stability region of the block method 6 4 Imag(z) Numerical Experiments. Numerical Examples Problem. Solve y = cos x such that y() = y () = y () = x h =.. y( x) = x + x sin x +. Exact Solution: Source: Agam & Irhebbhude () (see the result is shown in able ). Problem. Solve y y + = such that y() = y () = y () = x [ ] h =.. Exact Solution: y( x) = ( cos x) + sin x. Source: Agam & Irhebbhude (). he result is shown in able 6

7 ISSN 4-86 (Paper) ISSN -9 (Online) Vol.4 No. 4 Error = Exact result Computed result AI = Error in Agam & Irhebbhude () YB = Error in Yahaya and Badmus (8) NM= Error in New Method X able : Comparison of absolute errors for Problem I Error in NM Error in AI Error in YB..( 6) 6.4( 8).4( 6)..( ).6( 7).6( 6)..997( ).( 7).( ) ( ).( 7).( )..799( 4).7( 7).( ) X able : Comparison of absolute errors for Problem Error in NM Error in AI Error in YB..6( 4).( 7).4( 9)..9( ).7( 7) 9.84( 9). 8.94( ).47( 7).68( 8).4.964( ) 4.86( 7) 4.7( 8)..7( ) 7.97( 7).98( 8) b Note a( b) : = a. 6. Discussion of Result wo numerical examples were used to test the efficiency of our developed scheme. Agam & Irhebbhude () solved Problems and they developed modified Runge-Kutta methods with step size h =.. Our block method competes favourably with this method and with less computational cost. he accuracy of the method is demonstrated in ables and. 6. Conclusion An improved block method for direct solution of third order ordinary differential equations has been developed and implemented in this paper. he good convergent and stability properties of our method make it more attractive for numerical integrator of linear initial value problems of third order ordinary differential equations. Its accuracy and effectiveness are shown clearly in tables and. Our method proves to be a good estimate of the exact solution for the test examples. References Adesanya A.O. Odekunle M. R. & Udoh M.O (). Four steps continuous method for the solution of y = f ( x y y ) American Journal of Computational Mathematics Adesanya A.O. Udoh A. M. & Aileye A. M. (). A new hybrid block method for the solution of general third order initial value problems of ordinary differential equations International Journal of Pure and Applied Mathematics 86() 7-48 Adesanya A. O. Udoh M.O. & Alkali A.M. (). A new block predictor-corrector algorithm for the solution y = f x y y y American Journal of Computational Mathematics 4-44 of ( ) Agam S.A. & Irhebbhude M. E. (). A modification of the fourth order Runge- Kutta method for third order ordinary differential equations ODEs. ABACUS Journal of the Mathematical Association of Nigeria 6

8 ISSN 4-86 (Paper) ISSN -9 (Online) Vol.4 No. 4 8() 87-9 Anake. A. Awoyemi D. O. & Adesanya A.O. (). A one step method for the solution of general second order ordinary differential equations. International Journal of Science and echnology (4)9-6 Anake. A. Awoyemi D. O. & Adesanya A. O. (). One step implicit hybrid block method for the direct solution of general second order ordinary differential equations. IAENG International Journal of Applied Mathematics 4(4) 4-44 Awoyemi D. O. (). A New Sixth-order algorithm for general second order ordinary differential equation International Journal of Computer Mathematics Awoyemi D.O. Adebile E.A. Adesanya A. O. & Anake. A. (). Modified block method for the direct solution of second order ordinary differential equation. International Journal of Applied Mathematics and Computation () 8-88 Awoyemi D. O. & Kayode S. J. (). A maximal order collocation method for direct solution of initial value problems of general second order ordinary differential equation Proceeding of the Conference Organized by the National Mathematical Centre Abua Nigeria James A. A. Adesanya A. O. & Joshua S. (). Continuous block method for the solution of second order initial value problems of ordinary differential equations International Journal of Pure and Applied Mathematics 8() 4-46 Jator S. N. (7). A sixth order linear multistep method for the direct solution of International Journal of Pure and Applied Mathematics 4() Jator S. N. (). On a class of hybrid method for y f ( x y y ) 9(4) 8-9 = Intern. J. of Pure and Appl. Math. Obarhua F. O. & Kayode S. J. (). Continuous y-function hybrid methods for direct solution of differential equations International Journal of Differential equations and Applications () 6-7 Yahaya Y. A. & Badmus A. M. (8). A -step hybrid collocation method for special third order initial value problems of ODE International Journal of Numerical Mathematics 6-4 Yahaya Y. A. Badmus A. M. & Mishelia D. (7). Construction of a step p-stable collocation method for direct special third order ODEs. Journal of Numerical Mathematics 6-4 Yakubu D. G. Madaki A. G. & Kwani A. M. (). Stable two step Runge -Kutta collocation methods for oscillatory system of initial value problems in ODEs American Journal of Computational and Applied Mathematics () 9-6

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