Hybrid Block Method for Direct Integration of First, Second and Third Order IVPs

e-ISSN: 2564-7954 CUSJE 18(1): 001-008 (2021) Research Article Çankaya University Journal of Science and Engineering https://dergipark.org.tr/cankujse Hybrid Block Method for Direct Integration of First, Second and Third Order IVPs Emmanuel Oluseye Adeyefa1 1 , Adeyemi Sunday Olagunju2* Department of Mathematics, Federal University Oye-Ekiti, Ekiti State, Nigeria 2 Department of Mathematics, Federal University of Lafia, Nigeria Keywords Abstract Consistency, Convergence, Zero-Stability. Construction of numerical methods for the solution of initial value problems (IVPs) in ordinary differential equations (ODEs) has been considered overwhelmingly in literature. However, the use of a single numerical method for the integration of ODEs of more than one order has not been commonly reported. In this paper, we focus on the development of a numerical method capable of obtaining the numerical solution of first, second and third-order IVPs. The method is formulated from continuous schemes obtained via collocation and interpolation techniques and applied in a block-by-block manner as a numerical integrator for first, second and third-order ODEs. The convergence properties of the method are discussed via zero-stability and consistency. Numerical examples are included and comparisons are made with existing methods in the literature. 1. Introduction We propose in this paper, a block method of the form k k  k   2  ( t ) y = h  ( t ) f +  ( t ) f + h i (t ) g n +i +  j (t ) g n + j        i n +i i n +1 j n+ j  i =0  i =0   i =0   k + h 3   (t ) w +  (t ) w   i n +i j n + j    i =0  (1) (1) for direct solution of  y ' ( x) = f ( x, y ( x)), y ( x0 ) = y 0  '' ' ' '  y ( x) = f ( x, y ( x), y ( x)), y ( x0 ) = y 0 , y ( x0 ) = y 0  ' '' ' '' ' ' '' ''  y ( x) = f ( x, y ( x), y ( x), y ( x)), y ( x0 ) = y 0 , y ( x0 ) = y 0 , y ( x0 ) = y 0 (2) (2) where either of  0 (t ) and  0 (t ) do not varnish,  k (t ) = 1 ,  k (t )  0 and k = 1. The solution of either y' ' ' ( x) = f ( x, y( x), y' ( x), y' ' ( x)) or y' ' ( x) = f ( x, y( x), y' ( x)) has been extensively discussed in the literature using different approaches. Lambert [1, 2], Awoyemi [3] and Brugnano and Trigiante [4], among others, reduced higher-order initial value problems (IVPs) to a system of first-order equations. Resulting from this is the increase in the dimension of the problem which leads to more computation. Awoyemi ([5], [6]) and Kayode ([7]) successfully applied numerical algorithms as integrators of fourth-order initial value problems directly. However, the implementation in predictor-corrector mode has been reported to be more costly since the subroutines for incorporating the starting values lead to lengthy computational time, see * Corresponding Author: Received: May 29, 2020, Accepted: March 14, 2021 1 Adeyefa and Olagunju CUSJE 18(1): 001-008 (2021) (Jator [8]). This setback was addressed by Vigo-Aguiar and Ramos [9], Jator [10], Mohammed [11], Kayode et al. [12], Awoyemi et al. [13], Yap and Ismail [14], Hussain et al. [15], Ismail et al. [16], Ramos et al. [17], Adeyefa [18], among others who independently proposed block methods for solving higher order ordinary differential equation which do not require the development of separate predictors, but simultaneously generate approximations at different grid points within the interval of integration. Of recent, Ogunniran et al. developed Optimized three-step hybrid block method for stiff to integrate stiff IVPs [19] and later considered linear stability analysis of Runge-Kutta Methods for Singular Lane-Emden Equations [20]). Singh et al. developed an efficient optimized adaptive step-size hybrid block method for integrating differential systems [21]. These methods, though efficient in integrating the targeted ODEs but not capable to solve more than one order. The formulation of block method to integrate IVPs of order one or higher order has been widely reported in the literature but to use a formulated block method for integration of two or more order IVPs, say first, second and third-order ODEs, has not been commonly reported. Thus, the focus of this paper is to formulate a selfstarting method for the numerical integration of first, second and third-order IVPs. In what immediately follows in the next section, we consider the formulation of the proposed block method. 2. Materials and Methods In this section, we set i = 0,1, j = 3 and formulate a new one-step hybrid method capable of solving first, second 4 and third-order ODEs, employing Chebyshev polynomials as our basis function. Thus, we introduce k +8 y ( x) =  a j T j ( x) (3) j =0 3 4 Equation (3) is interpolated at x = x n , its first and second derivatives are collocated at x = x n +v , v = 0, ,1 while 3 4 its third derivative is collocated at x = x n +c , c = 0, . As a result, we have   j =0  k +8  ja j Tnj+−m1 = f n + m   j =1   k +8 j −2  j ( j − 1)a j Tn + m = g n + m   j =2  k +8  j −3 j ( j − 1)( j − 2)a j Tn + c = wn + c   j =3  k +8  a T (x ) = y j j n n (4) Solving equation (4) using the Gaussian elimination approach to get the unknown variables a' s that are substituted into equation (3). This yields a continuous implicit scheme of the form: (  1   1  y ( x) = h   i (t ) f n +i +  3 (t ) f n + 3  + h 2   i (t ) g n +i +  3 (t ) g n + 3  + h 3  0 (t ) w0 +  3 (t ) wn+ 3 4 4 4 4 4 4  i =0   i =0  2 x − 2 xn − h where t = . h 2 ) (5) Adeyefa and Olagunju CUSJE 18(1): 001-008 (2021) 3 1 i.e. t = 1, respectively, yields 4 2  fn   gn   wn         y n + 3  1 2 3  4  =   y n + hD f n + 3  + h E  g n + 3  + h F  wn + 3   y n +1  1  f 4 g 4 w 4   n +1   n +1   n +1  Equation (5), when evaluated at x = xn +c j , c j = 1, (6) where the values of D, E, and F are 181521   88593 1203  747 −    286720 , E =  20480 D =  286720 1120 19   2626 10496  59 −    8505 35   8505  1620 153 4480 128 2835 2187   963   57344 , F =  573440 1   19   28   11340 99  0 5120  8 0  405  Equation (6) is our proposed first, second and third-order IVPs solver. 3. Basic Properties of the Method We shall consider in this section, the analysis of the basic properties of this method such as order, error constant, zero stability and consistency is investigated. 3.1. Order and Error Constant Equation (6) derived is a discrete scheme belonging to the class of LMMs of the form k k k k j =0 j =0 j =0 j =0   j y n + j = h w j f n + j + h 2   j g n + j + h 3   j Gn + j (7) Following the work of Fatunla [26] and Lambert [2], we define the local truncation error associated with equation (7) by the difference operator k L[ y ( x) : h] =  [ j y ( xn + jh ) − hw j f ( xn + jh ) − h 2  j g ( xn + jh ) − h 3 j G ( xn + jh )] (8) j =0 where y(x) is an arbitrary function, continuously differentiable on [ a, (...truncated)