Research Article An Accurate Block Hybrid Collocation Method for Third Order Ordinary Differential Equations

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1 Applied Mathematics, Article ID , 9 pages Research Article An Accurate Bloc Hybrid Collocation Method for Third Order Ordinary Differential Equations Lee Ken Yap, 1, Fudziah Ismail, and Noraza Senu 1 Department of Mathematical and Actuarial Sciences, Universiti Tunu Abdul Rahman, Setapa, 533 Kuala Lumpur, Malaysia Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 434 Serdang, Selangor, Malaysia Correspondence should be addressed to Lee Ken Yap; lyap@utar.edu.my Received 4 January 14; Revised 4 April 14; Accepted 7 May 14; Published 7 May 14 Academic Editor: Kai Diethelm Copyright 14 Lee Ken Yap et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. The bloc hybrid collocation method with two off-step points is proposed for the direct solution of general third order ordinary differential equations. Both the main and additional methods are derived via interpolation and collocation of the basic polynomial. These methods are applied in bloc form to provide the approximation at five points concurrently. The stability properties of the bloc method are investigated. Some numerical examples are tested to illustrate the efficiency of the method. The bloc hybrid collocation method is also implemented to solve the nonlinear Genesio equation and the problem in thin film flow. 1. Introduction Consider the general third order ordinary differential equations (ODEs): y =f(x,y,y,y ), (1) with the initial conditions y (a) =y, y (a) =y, y (a) =y, x [a, b. () In particular, the third order differential equations arise in many physical problems such as electromagnetic waves, thin film flow, and gravity-driven flows (see [1 6). Therefore, third order ODEs have attracted considerable attention. Many theoretical and numerical studies dealing with such equations have appeared in the literature. The popular approach for solving third order ODEs is by converting the problems to a system of first order ODEs and solving it using the method available in the literature. Awoyemi and Idowu [7, Jator [8, Mehranoon [9, and Bhrawy and Abd-Elhameed [1 remared the drawbac of this approach whereby it required complicated computational wor and lengthy execution time. The studies on direct approach to higher order ODEs demonstrated the advantages in speed and accuracy. Some attentions [8, have been focused on direct solution of second order ODEs. Fatunla [1 suggested the zero-stable -point bloc method to solve special second order ODEs. On the other hand, Omar et al. [13 andmajid and Suleiman [14 studied parallel implementation of the direct bloc methods. Jator [8, 11proposedaclassofhybrid collocation methods and emphasized the accuracy advantage on self-starting method. The only necessary starting value for evaluation at the next bloc is the last value from the previous bloc. Since the loss of accuracy does not affect the subsequent points, the order of the method is maintained. Some attempts have been made to solve third order ODEs directly using collocation method. Awoyemi [15 considered the P-stable linear multistep collocation method. Meanwhile, Awoyemi and Idowu [7 proposed the hybrid collocation method with an off-step point, x n+3/. Both schemes are implemented in predictor-corrector mode to obtain the approximation at x n+3. The Taylor series expansion is employed for the computation of initial values. Olabode and Yusuph [16 applied the interpolation and collocation technique on power series to derive 3-step bloc method, and it was implemented as simultaneous integrator to special third order ODEs. Bhrawy and Abd-Elhameed [1 developed the shifted Jacobi-Gauss collocation spectral method for general nonlinear third order differential equations.

2 Applied Mathematics Adesanya et al. [17 proposed a self-starting bloc predictorcorrector method whereby the derivation involved interpolation and collocation of power series at x n+j,forj = 1(1)3 and j = (1)4,respectively. Several direct variable step methods have also been proposed in literature to solve general third order ODEs. For instance, Mehranoon [9 implemented the direct threepoint bloc multistep method of Adams type formulas in PECE mode with variable step size and Gauss Seidel iteration. Majid et al. [18 presented the -point 4-step implicit bloc method with the application of the simple form in Adams- Moultonmethodusingvariablestepsize. Here, we are going to derive the bloc hybrid collocation method for the direct solution of general third order ODEs. The method is along the lines proposed by Jator [11 and Awoyemi and Idowu [7. The derivation involves interpolation and collocation of the basic polynomial. The collocation method approximates the solution of y with basic polynomial which satisfies the initial conditions and differential equations at all points. This approach generates the main and the additional methods which can be combined and used as bloc method. In m-point bloc method, the interval is divided into series of blocs with each bloc containing m-points. The application of m-point bloc method generates a bloc of new solution concurrently.. Derivation of Bloc Hybrid Collocation Methods The hybrid collocation method that produces approximations y n+, y n+,andy n+ to the general third order ODEs is given as follows: α j y n+j + α j y n+j =h 3 ( β j f n+j + β j f n+j ). In order to obtain (3), we approximate the solution by the interpolating function Y(x) of the form where Y (x) = r+s 1 (3) φ j x j, (4) (i) x [a,b, (ii) φ j are unnown coefficients to be determined, (iii) r is the number of interpolations for 1 r,and (iv) s isthenumberofdistinctcollocationpointswiths>. The continuous approximation is constructed by imposing the following conditions: Y(x n+j )=y n+j,,1,,...,r 1, (5) Y (x n+μ )=f n+μ, μ = {j, 1, },,1,,...,, (6) where 1 and are not integers. Interpolating (5) atthe points x n, x n+1, x n+ and collocating (6) at the points x n, x n+1/, x n+1, x n+3/, x n+,andx n+3 lead to a system of nine equations, which can be solved by Mathematica software to obtain the coefficient φ j.thevaluesofφ j are substituted into (4) to obtain the continuous multistep method of the form Y (x) = α j y n+j +h 3 ( β j f n+j + β j f n+j ), (7) where α j, β j,andβ j are constant coefficients. Hence, the bloc hybrid collocation method can be derived as follows. Main Method. Consider y n+3 =3y n+ 3y n+1 +y n + 9 (f n + 36f n f n+3/ + 36f n+ +f n+3 ). Additional Method. Consider y n+3/ = 3 8 y n y n y n ( 19f n 648f n+1/ 979f n+1 14f n+3/ 9f n+ f n+3 ), y n+1/ = 1 8 y n y n y n (9f n + 4f n+1/ + 369f n f n+3/ + 39f n+ f n+3 ). It is noted that the general third order ODEs involve the first and second derivatives. These derivatives can be obtained by imposing that Y (x) = 1 h ( α j y n+j +h 3 ( Y (x) = 1 h ( α j y n+j +h 3 ( β The values of φ j are substituted into Y (x) = r+s 1 jφ j x j 1, Y (x) = β j f n+j + β β j f n+j )) j f n+j + j f n+j )). r+s 1 j= j(j 1)φ j x j (8) (9) (1) (11)

3 Applied Mathematics 3 to generate the formula for the first and second derivatives of the method. Thus, we obtain hy n = 1 y n+ +y n+1 3 y n (118f n + 196f n+1/ + 819f n+1 hy n+1/ =y n+1 y n f n+3/ 18f n+ +f n+3 ), h ( 449f n 931f n+1/ 99f n+1 hy n+1 = 1 y n+ 1 y n + 3f n+3/ + 15f n+ 5f n+3 ), 5 ( f n 8f n+1/ 58f n+1 hy n+3/ =y n+ y n+1 + 8f n+3/ f n+ ), h (5f n 549f n+1 899f n+3/ 549f n+ +5f n+3 ), hy n+ = 3 y n+ y n y n (f n + 4f n+1/ + 99f n f n+3/ + 138f n+ f n+3 ), hy n+3 = 5 y n+ 4y n y n (343f n 196f n+1/ f n+1 f n+3/ f n+ + 46f n+3 ), h y n =y n+ y n+1 +y n + 5 ( 398f n 1584f n+1/ 1f n+1 4f n+3/ + 66f n+ 3f n+3 ), h y n+1/ =y n+ y n+1 +y n + 43 (345f n 5744f n+1/ 13443f n+1 91f n+3/ 41f n+ + 15f n+3 ), h y n+1 =y n+ y n+1 +y n ( 11f n + 64f n+1/ + 9f n+1 75f n+3/ + 51f n+ f n+3 ), h y n+3/ =y n+ y n+1 +y n + 43 (11f n + 187f n+1/ + 193f n f n+3/ 645f n+ + 15f n+3 ), h y n+ =y n+ y n+1 +y n + 5 ( 6f n + 8f n+1/ + 471f n f n+3/ + 458f n+ 3f n+3 ), h y n+3 =y n+ y n+1 +y n + 5 (95f n 1584f n+1/ + 49f n+1 443f n+3/ f n+ + 69f n+3 ). (1) 3. Order and Stability Properties Extending the idea of Henrici [19 and Jator[8, the linear difference operator L associated with (3) is defined by L[y(x) ;h= + [α j y(x+jh) h 3 β j y (x + jh) [α j y(x+ j h) h 3 β j y (x + j h), (13) where y(x) is an arbitrary function that is sufficiently differentiable. We expand the test function y(x+jh)and its third derivative y (x+jh)about x and collect the terms to obtain L[y(x) ;h=c y (x) +C 1 hy (x) + +C q h q y (q) (x) + (14) whose coefficients C q for q =, 1,... are constants and given as C = C 1 =. C q = 1 q! [ [ α j + jα j + α j j α j j q α j + q j α j 1 (q 3)! [ j q 3 β j + [ q 3 j β j. (15)

4 4 Applied Mathematics The associated linear multistep method and the linear difference operator are said to be of order p, ifc =C 1 = = C p+ =and C p+3 =.Themainmethodandthetwo additional methods have order p=6with the error constants; C 9 are 19/196, 41/619315, and41/619315, respectively.theblochybridcollocationmethodisconsistentsince it has order p>1. To analyze the zero stability, (8) (1) arenormalized. Zero stability can be described by matrix finite difference equation as follows: with I 5 Y m+1 =A (1) Y m +h 3 (B () F m+1 +B (1) F m ) +h C (1) Y m +hd(1) Y m Y m+1 =[y n+1/,y n+1,y n+3/,y n+,y n+3 T Y m =[y n,y n 3/,y n 1,y n 1/,y n T F m+1 =[f n+1/,f n+1,f n+3/,f n+,f n+3 T and constant matrices F m =[f n,f n 3/,f n 1,f n 1/,f n T, Y m =[y n,y n 3/,y n 1,y n 1/,y n T Y m =[y n,y n 3/,y n 1,y n 1/,y n T I 5 =5 5identity matrix; A (1) 3 9 = [ 3 9 [ B () = [ 3 [ ; ; (16) (17) B (1) = [ 1 1 [ ; C (1) = 3 9 ; D (1) 3 9 = 1. [ 1 [ [ [ (18) The first characteristic polynomial is defined as ρ (z) = Det [za () A (1) =z 4 (z 1). (19) Since the roots of the first characteristic polynomial are modulus less than or equal to one, the method is zero stable. This property, together with the consistency of the method, implies the convergence (see Henrici [19) of the bloc method. 4. Numerical Examples and Discussion To illustrate the effectiveness of the bloc hybrid collocation method, the test Problems 1 3 are solved numerically. The bloc hybrid collocation method is implemented and leads to a system of fifteen equations. The code is written and executed in Mathematica 8. to obtain the numerical solutions. A built-in Mathematica pacage, namely, LinearSolve[m, b, is applied in the algorithm to solve the linear system for numerical results. In general, this pacage solves the matrix equation m x=band returns a vector x. The bloc hybrid collocation method is then compared with the existing methods [7, 15 17, for direct solution of general third order ODEs. The bloc hybrid collocation method is also compared with the Adams Bashforth-Adams Moulton method. The well-nown fourth order Runge Kutta methodisappliedtoobtainthestartingvaluesforadams method, whereby the third order ODEs are reduced to system of first order ODEs. Problem 1. Consider y y 3y + 1y = 34xe x 16e x 1x +6x+34 y () =3, y () =y () =, x [, b. Exact Solution: y(x) = x e x x +3. Source: Majid et al. [18. ()

5 Applied Mathematics 5 Table 1: Numerical results for Problem 1. b h Method Step Maxe Adams Awoyemi() Adams Awoyemi() Problem. Consider Adams Awoyemi(1) Adams Awoyemi(1) y +y = y () =, y () =1, y () =, x [, b. Exact Solution: y(x) = (1 cos x) + sin x. Source: Majid et al. [18. Problem 3. Consider y =3sin x (1) y () =1, y () =, y () =, x [, b. () Exact Solution: y(x) = 3 cos x+x /. Source: Adesanya et al. [17. The performance comparison between bloc hybrid collocation method with the existing results [7, 9, 15 17, and the Adams Bashforth-Adams Moulton method is presented in Tables 1 4. The followingnotationsare used in thetables. h:stepsize. : bloc hybrid collocation method. Adams: Adams Bashforth-Adams Moulton method. Awoyemi (1): P-stable multistep method in Awoyemi [15. Awoyemi (): hybrid collocation method in Awoyemi and Idowu [7. Awoyemi (3): nonsymmetric collocation method in Awoyemi et al. [. Adesanya: bloc predictor-corrector method in Adesanya et al. [17. Mehranoon: variable step three-point bloc multistep method in Mehranoon [9. Olabode: bloc method for special third order ODEs in Olabode and Yusuph [16. Step: total number of steps taen to obtain solution. Table : Numerical results for Problem. b h Method Step Maxe Adams Awoyemi(3) Adesanya Adams Awoyemi(1) Adams Awoyemi(1) Adams Awoyemi(1) Adams Awoyemi(1) Table 3: Numerical results for Problem 3. b h Method Step Maxe Adams Olabode Adesanya Table 4: Comparison of numerical results for nonlinear Genesio equation. b Mehranoon TOL Step y h Step y Time: execution time taen in microseconds. Maxe: magnitude of the maximum error of the computed solution. The maximum error is defined as Maxe = max ( 1 i Step y(x i) y i ). (3) Tables 1 3 show that the bloc hybrid collocation method has better performance in terms of accuracy and number of steps to obtain the solution compared to Adams Bashforth- Adams Moulton method. In Table 1, a direct comparison is made between Awoyemi methods [7, 15 and our bloc hybrid collocation method. For b = 1, has better approximation compared to Awoyemi hybrid collocation method [7 forstepsizesh =.15 and h =.65. Even in the larger interval(b = 4), has higher order of accuracy when compared to Awoyemi P-stable multistep method [15usingh =.1 and

6 6 Applied Mathematics Log 1 Maxe Adams Awoyemi() Log 1 step size Figure 1: Performance comparison for Problem 1 when b=1. h =.5. It is obvious that requires less steps to obtain the solutions compared to Awoyemi methods [7, 15. Table shows that manages to achieve better accuracy and less total steps compared to Awoyemi P-stable multistep method [15 for constant step size.5 when b= 5, 1, 15, and. Furthermore, gain better accuracy compared to Awoyemi nonsymmetric collocation method [ and Adesanya bloc predictor-corrector method [17 when h =.1 and b = 1. Table 3 shows the superiority of in terms of accuracy over Adesanya bloc predictorcorrector method [17 and Olabode bloc method [16 for special third order ODEs. Figures 1,, and 3 depict the performance comparison between and the existing methods: Awoyemi hybrid collocationmethod[7, Awoyemi P-stable multistep method [15, and Adams Bashforth-Adams Moulton method, respectively. It shows that is more efficient than the existing methodsintermsofaccuracy.alsorequiresless computational time compared to the existing methods. This is expected since calculates the approximation at three main points concurrently. 5. Application to Solve Nonlinear Genesio Equation Here we consider the nonlinear chaotic system from Genesio and Tesi [1 with x +Ax +Bx f(x (t)) = (4) f (x (t)) = Cx(t) +x (t) (5) where A = 1., B =.9,andC=6are the positive constants that satisfied AB < C for the existence of the solution. is implemented together with a built-in Mathematica pacage, namely, FindRoot, for the solution of nonlinear system based on Newton s method. In order to apply and Newton s method at the next bloc, the last value from previous bloc is used as starting value and initial guess, respectively. The numerical solutions are compared with the numerical solutions obtained in Mehranoon [9, Bataineh et al. [, and the Mathematica built-in pacage NDSolve. Table 4 shows the comparison in the numerical approximation of y at the end points b = 1 and b = 4. achieves similar approximation as Mehranoon variable step three-point bloc multistep method [9. Figure 4 depicts the numerical solutions for the nonlinear Genesio equation [1 intheinterval[, 4.5. Batainehetal. [ stated that NHAM is more stable than the numerical solution obtained by classical HAM. In fact, the solutions obtained by are in agreement with the observation of Bataineh et al. [ using NHAM and Mathematica built-in pacage NDSolve. 6. Application to Solve Problem in Thin Film Flow The proposed method is also applied to solve the well-nown physical problem regarding the thin film flow of a liquid. In fluid dynamics, Tuc and Schwartz [3 investigated some third order ODEs that are relevant to draining and coating flows. They discussed the motion of the fluid on a plane surface in which the flow is in the direction of motion along the plane. This fluid dynamics problem was formulated in an autonomous third order ODEs where f(y)= 1+y, y =f(y), (7) f(y)= 1+(1+δ+δ )y (δ+δ )y 3, f(y)= y y 3, f(y)= y. (8) In the literature, some numerical methods for solving special third order ODEs have been extended to solve the problem in thin film flow. Some numerical investigation was presented in Momoniat and Mahomed [4and Mechee et al. [5 concerning the special third order ODEs y =y (9) that is subject to the following initial conditions: x () =., x () =.3, x () =.1, t [, b, (6) with the initial conditions y() = y () = y () = 1 for the cases =and =3. Instead of using the conventional approach of reduction to system of first order ODEs, Momoniat and Mahomed [4 applied the successive reduction by writing the third order

7 Applied Mathematics Log 1 Maxe 1 15 Log 1 time (μs) Log 1 step size Log 1 step size Adams Awoyemi(1) Awoyemi() Adams Awoyemi(1) Awoyemi() (a) Maxe versus step size (b) Time versus step size Figure : Performance comparison for Problem when b=4. Log 1 Maxe Adams Log 1 step size Figure 3: Performance comparison for Problem 3 when b=1. ODEs (9) in terms of the differential invariants and solved it by fourth order Runge Kutta method. Mechee et al. [5 appliedthethree-stagefifthorderrungekuttamethodto solve the third order physical problems (9)directly. Here we will apply the proposed bloc hybrid collocation method () to solve the third order ODEs (9). The numerical solutions are compared with Momoniat and Mahomed [4 and Mechee et al.[5 in the literature. The results are displayed in Tables 5 8. Table 5 demonstrates that performs slightly better compared to Mechee et al. [5 withh =.1 for the case =. In Table 6, we observe that Momoniat and Mahomed [4, Mechee et al. [5, and have similar order of accuracy. Tables 7 and 8 show the numerical results for the case =3with h =.1 and h =.1, respectively. In fact, the case =3cannot be solved analytically. Table 7 shows that manages to achieve the numerical result which agrees to six decimal places when compared to Mechee et al. [5for Table 5: Numerical results for problem in Thin Film Flow (9)with h =.1, =. x Error Mechee Table 6: Numerical results for problem in Thin Film Flow (9)with h =.1, =. x Error Momoniat Mechee Table 7: Numerical results for problem in Thin Film Flow (9)with h =.1, = 3. x Mechee h =.1. In Table 8,the numerical results for agree to nine decimal places when compared to Momoniat and

8 8 Applied Mathematics x(t)..1 x(t) t RK78 HAM NHAM (a) Solutions obtained by seven- and eight-order Runge Kutta method (RK78), homotopy analysis method (HAM), and new variant of HAM (NHAM) (adapted from Bataineh et al. [) t.3 NDSolve h =.1 (b) Solutions obtained by with h=.1 Figure 4: Plot of solutions for nonlinear Genesio equation Total steps Total steps Mechee Mechee Momoniat h =.1 h =.1 (a) Total steps when h=.1 (b) Total steps when h =.1 Figure 5: Total steps VS method. Table 8: Numerical results for problem in Thin Film Flow (9)with h =.1, = 3. x Momoniat Mechee Mohamed [4, while they agree to ten decimal places when compared to Mechee et al. [5forh =.1. Figure 5 shows the total number of steps taen to obtain the solutions versus method. It is obvious that require less steps to obtain the solution compared to Mechee et al. [5 and Momoniat and Mahomed [4. The results are expected since the calculates the values of y at 3 points simultaneouslywhiletherungekuttamethodinmecheeet al. [5 and Momoniat and Mahomed [4 calculatesonlyone value of y at a time. is clearly superior in solving the problem in thin film flow (9) since it involves less computational wor and yields highly accurate solutions. 7. Conclusion As a whole, the numerical results demonstrate the efficiency of the bloc hybrid collocation method. It is observed that the bloc hybrid collocation method requires less number of total steps compared to Awoyemi methods [7, 15,. It reduces the total number of steps to almost one-third to obtain the solution. These results are expected since the 3- point bloc methods calculate the values of y at three main points concurrently.

9 Applied Mathematics 9 The numerical results also demonstrate the accuracy of the bloc hybrid collocation method. It gives precise approximation as the step size decreases. When compared with the existing methods [7, 15 17, andadams Bashforth-Adams Moulton method, the bloc hybrid collocation method meets better accuracy for various step sizes even in larger interval. As a conclusion, the 3-point bloc hybrid collocation method with two off-step points has been proposed and implemented as a self-starting method for third order ordinary differential equations. The results suggest a significant improvement in efficiency of the bloc hybrid collocation method in the direct solution of general third order ODEs. The bloc hybrid collocation method is applicable to solve the nonlinear Genesio equation [1 andthephysicalproblemin thin film flow [4, 5. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. References [1 R. Genesio and A. Tesi, Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems, Automatica, vol. 8, no. 3, pp , 199. [ A. S. Bataineh, M. S. M. Noorani, and I. Hashim, Direct solution of nth-order IVPs by homotopy analysis method, Differential Equations and Nonlinear Mechanics, vol. 9, Article ID 8494, 15 pages, 9. [3 E. O. Tuc and L. W. Schwartz, A numerical and asymptotic study of some third-order ordinary differential equations relevant to draining and coating flows, SIAM Review,vol.3,no.3, pp ,199. [4 E. Momoniat and F. M. Mahomed, Symmetry reduction and numerical solution of a third-order ODE from thin film flow, Mathematical and Computational Applications,vol.15,no.4,pp , 1. [5 M.Mechee,N.Senu,F.Ismail,B.Niouravan,andZ.Siri, A three-stage fifth-order Runge-Kutta method for directly solving special third-order differential equation with application to thin film flow problem, Mathematical Problems in Engineering,vol. 13, Article ID , 7 pages, 13. [6 Y.J. L. Guo, A third order equation arising in the falling film, Taiwanese Mathematics,vol.11,no.3,pp , 7. [7 D. O. Awoyemi and O. M. Idowu, A class of hybrid collocation methods for third-order ordinary differential equations, International Computer Mathematics, vol.8,no.1,pp , 5. [8 S. N. Jator, On a class of hybrid methods for y = f(x,y,y ), International Pure and Applied Mathematics, vol.59, no. 4, pp , 1. [9 S. Mehranoon, A direct variable step bloc multistep method for solving general third-order ODEs, Numerical Algorithms, vol. 57, no. 1, pp , 11. [1 A. H. Bhrawy and W. M. Abd-Elhameed, New algorithm for the numerical solutions of nonlinear third-order differential equations using Jacobi-Gauss collocation method, Mathematical Problems in Engineering, vol. 11, Article ID 83718, 14 pages, 11. [11 S. N. Jator, Solving second order initial value problems by a hybrid multistep method without predictors, Applied Mathematics and Computation,vol.17,no.8,pp ,1. [1 S. O. Fatunla, Bloc methods for second order ODEs, International Computer Mathematics, vol.41,no.1-,pp , [13 Z. Omar, M. Sulaiman, M. Y. Saman, and D. J. Evans, Parallel r-point explicit bloc method for solving second order ordinary differential equations directly, International Computer Mathematics, vol. 79, no. 3, pp ,. [14 Z. A. Majid and M. Suleiman, Parallel direct integration variable step bloc method for solving large system of higher order ordinary differential equations, World Academy of Science, Engineeringand Technology,vol.4,pp.71 75,8. [15 D. O. Awoyemi, A P-stable linear multistep method for solving general third order ordinary differential equations, International Computer Mathematics,vol.8,no.8,pp , 3. [16 B. T. Olabode and Y. Yusuph, A new bloc method for special third order ordinary differential equation, Mathematics and Statistics,vol.5,no.3,pp ,9. [17 A. O. Adesanya, M. O. Udo, and A. M. Alali, A new bloc-predictorcorrector algorithm for the solution of y = f(x, y, y ), The American Computational Mathematics,vol.,no.4,pp ,1. [18 Z. A. Majid, N. A. Azmi, M. Suleiman, and Z. B. Ibrahim, Solving directly general third order ordinary differential equations using two-point four step bloc method, Sains Malaysiana, vol. 41, no. 5, pp , 1. [19 P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley & Sons, New Yor, NY, USA, 196. [ D. O. Awoyemi, M. O. Udoh, and A. O. Adesanya, Nonsymmetric collocation method for direct solution of general third order initial value problems of ordinary differential equations, Natural and Applied Mathematics, vol.7, pp , 6.

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