One-Step Four Point Hybrid Block Scheme Designed for the Solution of General Third Order Ordinary Differential Equations

@ 2025 | PUBLISHED BY GJR PUBLICATION, INDIA 20 Global Journal of Research in Engineering & Computer Sciences ISSN: 2583-2727 (Online) Volume 05| Issue 05 | Sept.-Oct. | 2025 Journal homepage: https://gjrpublication.com/gjrecs/ Research Article One-Step Four Point Hybrid Block Scheme Designed for the Solution of General Third Order Ordinary Differential Equations *Joshua, S 1, Raymond, D 2, Sabo, Z 2, Ebenezer, O. S 3 1 Department of Mathematical Sciences, Taraba State University, Jalingo 2 Department of Mathematics, Federal University Wukari 3 Department of Mathematics, Federal College of Education Obudu, Cross River State Nigeria. *Corresponding author: Joshua, S Department of Mathematical Sciences, Taraba State University, Jalingo 1. INTRODUCTION Ordinary Differential Equations (ODEs) are commonly used in mathematically formulated models designed to describe physical phenomena across science and engineering disciplines. They play a crucial role and find wide-ranging applications, not only in the physical sciences but also in various other areas such as medical sciences, thermodynamics, chemical engineering, control theory, operations research, and behavioural sciences. Let us examine the general third-order ordinary differential equations (ODEs) expressed using a sixth-order power series of the form: ( ) j k j jxbxy  = = 0 (1) Which is recommend as general third order derivative solution of initial value problems of the form ( ) ( ) ( ) ( ) ( ) ''0'','0',,'',',,''' 000 yyyyyxyyyyxfxy ==== (2) The resolution of (1) has been explored by numerous scholars, such as: Abdulazeez, Kayode Jimoh [1] proposed twostep hybrid block method for the numerical solution third order differential equations. He adopted the used of approximate power series as an interpolation equation and its derivatives as a collocation equation that is used in the development of the method. Ishaq et al. [2] introduced a novel three-step block method designed to directly tackle thirdorder initial value problems through the method of collocation. Their method was zero-stable, convergent and the region of stability is absolutely stable. Dalatu et al. [3] developed a hybrid block method for solving third-order derivative with initial value problems of ordinary differential equations. Their method was derived by collocating and interpolating the approximate solution using power series. Abdelrahim and Omar [4] developed one-step blocks method for the direct solution of third-order initial value problems of ordinary differential equations using the power series as the basis function. Their method was developed to solve third-order initial value problems. Atabo et al. [5] developed a selected Abstract In this paper, one-step four point hybrid block technique is formulated and employed to solve general third-order ordinary differential equations directly. The construction of the method utilized interpolation and collocation, with power series serving as the basis function. An analysis of its properties such as order, convergence, consistency, zero stability and stability region was examined. When tested on third order ODEs problems, the results indicated that the method produced superior performance compared to those in the literature. Keywords: One-step, off-grid point, basis function, Hybrid block formula, interpolation, collocation, Local Truncation Error. Global J Res Eng Comput Sci. 2025; 5(5), 20-33 @ 2025 | PUBLISHED BY GJR PUBLICATION, INDIA 21 single step hybrid block formula for solving third-order ordinary differential equations with application in thin film flow. Their method has advantage of selecting only odd off-grid points within a single-step interval and collocated at all points. Modebei et al. [6] proposed a three-step fourth derivatives method for numerical integration of third order ordinary differential equations. They used a three-step hybrid block method with three mid-step grid points based on linear multistep method to presented in their work for direct approximation of solution of third-order initial and boundary value problems. Muhammed and Adeniyi [7] developed three-step implicit hybrid linear method for solution of third-order ordinary differential equations. Adeyeye and Omar [8] developed third-order ordinary differential equations using onestep block method with four equidistant generalized hybrid points. The equation for the generalized linear block method takes a similar form as the conventional linear multistep method, however the form produces the needed family of scheme required simultaneously evaluate the solution of the third-order ordinary differential equations at individual grid points in a self-starting mode. Joshua, S. [9] a hybrid block technique with two-step optimization for handling general third order ordinary differential equations. 2. Derivation of the Method The derivation of the one-step third derivative method is based on a finite power series function expressed as: ( ) ( ) j cl j jxbxy −+ = =1 0 Which is propose as general third order derivative solution of initial value problems of the form ( ) ( ) ( ) ( ) ( ) ''0'','0',,'',',,''' 000 yyyyyxyyyyxfxy ==== Where i and c denote points of interpolation and collocation respectively, so that the third derivative is ( ) ( ) ( ) 3 1 0 21)(''' − −+ = −−= j j cl j xbjjjxy (3) Interpolating (1) at 3 1 , 9 1 ,0, = +ix in and collocating at 1, 9 7 , 9 5 , 3 1 , 9 1 ,0, = +rx rn we obtained a system of nonlinear equations of the form UAX = (4)                                       +++++ +++++ +++++ ++++ + +++++ ++++++++ ++++++++ 2 1 2 1 2 1 2 124 2 9 7 2 9 7 2 9 7 2 9 7 3 56 2 9 5 2 9 5 2 9 5 2 9 5 3 40 2 3 1 2 3 1 2 3 1 2 3 18 5 9 1 4 9 1 3 9 1 2 9 1 3 8 5432 8 3 1 7 3 1 6 3 1 5 3 1 4 3 1 3 3 1 2 3 1 3 1 8 9 1 7 9 1 6 9 1 5 9 1 4 9 1 3 9 1 2 9 1 9 1 8765432 33621012060246000 33621012060246000 33621012060246000 33621012060246000 33621012060246000 33621012060246000 1 1 1 nnnnn nnnnn nnnnn nnnn n nnnnn nnnnn nnnnnnnn nnnnnnnn nnnnnnnn xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxxxxx xxxxxxxx xxxxxxxx   T n nnnn n nn n TffffffyyyUbbbbbbbbX       == + ++++++ 1 9 7 9 5 3 1 9 1 3 1 9 187654310 ,,,,,,,,,,,,,,,, Whose unknowns sb' are solved for using Gaussian elimination technique and results are substituted in to equation (1) to give a continuous linear multistep method of the form Global J Res Eng Comput Sci. 2025; 5(5), 20-33 @ 2025 | PUBLISHED BY GJR PUBLICATION, INDIA 22 ( )      ++++++++= + ++++++ 11 9 7 9 7 9 5 9 5 3 1 3 1 9 1 9 10 3 3 1 3 1 9 1 9 10 n nnnn n nn nffffffhyyyxy  (5) The expression for the s'  then written in terms of the parameters s j'  and s j'  as the following functions of t. The parameters ( ) ( ) tt jj  , are evaluated at 1, 9 7 , 9 5 , 3 1 , 9 1 ,0=t . The values obtained are then substituted in to (5) to obtain the implicit hybrid block method as follows Global J Res Eng Comput Sci. 2025; 5(5), 20-33 @ 2025 | PUBLISHED BY GJR PUBLICATION, INDIA 23 Differentiating (5) once, we have ( )      ++++++++= + ++++++ 1 ' 1 9 7 ' 9 7 9 5 ' 9 5 3 1 ' 3 1 9 1 ' 9 1 ' 0 3 3 1 ' 3 1 9 1 ' 9 1 ' 0 'n nnnn n nn nffffffhyyyxhy  (9) The derivatives of the parameters s j'  and s j'  are written as the following function of t. Global J Res Eng Comput Sci. 2025; 5(5), 20-33 @ 2025 | PUBLISHED BY GJR PUBLICATION, INDIA 24 The derivatives ss jj ','  at 1, 9 7 , 9 5 , 3 1 , 9 1 ,0=t the values obtained are then substituted in to (9) to obtain the following implicit hybrid block scheme Global J Res Eng Comput Sci. 2025; 5(5), 20-33 @ 2025 | PUBLISHED BY GJR PUBLICATION, INDIA 25 Differentiating (5) twice, we have ( )      ++++++++= + ++++++ 1 '' 1 9 7 '' 9 7 9 5 '' 9 5 3 1 '' 3 1 9 1 '' 9 1 '' 0 3 3 1 '' 3 1 9 1 '' 9 1 '' 0 2'' n nnnn n nn nffffffhyyyxyh  (16) The second derivatives of the parameter ss jj ','  are written as the following functions of t. The second derivatives ss jj ','  at 1, 9 5 , 9 5 , 3 1 , 9 1 ,0=t the values obtained are then substituted in to (16) to obtain the following implicit hybrid block scheme. Global J Res Eng Comput Sci. 2025; 5(5), 20-33 @ 2025 | PUBLISHED BY GJR PUBLICATION, INDIA 26 Equations (6) - (8), equations (10) - (14) and equations (16) - (22) are then put in matrix form to produce: QQQQ UFRFSTST ++= −− 11 (23) Where, T n nnnn n nnnn n nnnn QyyyyyyyyyyyyyyyT       =+ ++++ + ++++ + ++++ '' 1 '' 9 7 '' 9 5 '' 3 1 '' 9 1 ' 1 ' 9 7 ' 9 5 ' 3 1 ' 9 11 9 7 9 5 3 1 9 1,,,,,,,,,,,,,,   T nnnQ yyyT ''' 1,,= − ,   T nQ fF = −1 T n nnnn QfffffF       =+ ++++ 1 9 7 9 5 3 1 9 1,,,, The block matrices in equation (24) is then resolved by multiplying by 1− S to gives the following discrete scheme Global J Res Eng Comput Sci. 2025; 5(5), 20-33 @ 2025 | PUBLISHED BY GJR PUBLICATION, INDIA 27 Global J Res Eng Comput Sci. 2025; 5(5), 20-33 @ 2025 | PUBLISHED BY GJR PUBLICATION, INDIA 28 2.1 Analysis of the Method In this paper, the main properties of the one-step four-point hybrid block method for solving third order initial value problems are presented. The properties include the order and error constant, zero stability, linear stability, stability polynomial, consistency and convergence of the method. Consider the linear operator L associated with the implicit hybrid block method (25) – (39) defined as