A class of single-step hybrid block methods with equally spaced points for general third-order ordinary di ff erential equations

This study presents a class of single-step, self-starting hybrid block methods for directly solving general third-order ordinary di ff erential equations (ODEs) without reduction to first order equations. The methods are developed through interpolation and collocation at systematically selected evenly spaced nodes with the aim of boosting the accuracy of the methods. The zero stability, consistency and convergence of the algorithms are established. Scalar and systems of linear and nonlinear ODEs are approximated to test the e ff ectiveness of the schemes, and the results obtained are compared against other methods from the literature. Significantly, the study shows that an increase in the number of intra-step points improves the accuracy of the solutions obtained using the proposed methods.


Introduction
Many problems in the fields of mathematics and engineering are formulated using initial and boundary value problems (IVP and BVP) for third-order ODEs.Most physical problems, including the deflection of a curved beam with a constant or varying cross-section, a three-layer beam, the motion of a rocket, thin film flow, electromagnetic waves, gravity-driven flows, physical oceanography, and the context of a variational inequality involve the use of third-order ODEs of the form y ′′′ (x) = f (x, y, y ′ , y ′′ ), (1) subject to initial conditions y(x 0 ) = y 0 , y ′ (x 0 ) = δ 0 , y ′′ (x 0 ) = ω 0 , boundary conditions y(x 0 ) = y 0 , y ′ (x 0 ) = δ 0 , y ′ (x N ) = ω 0 , or with mixed boundary conditions y(x 0 ) = y 0 , y ′ (x 0 ) = δ 0 , y(x N ) − y ′ (x N ) = ω 0 , where x 0 , x N , y 0 , δ 0 , ω 0 ∈ ℜ, and f is a continuous function and satisfies a Lipschitz condition as given Henrici [1].
Researchers have published a considerable amount of literature on Linear Multistep Methods (LMM) for solving Equation 1(1).Through Taylor expansions, some 3 and 5-step schemes for the special third-order ODEs were presented by Rajabi et al. [2].A one-step block method with four equidistant generalized hybrid points was presented by Adeyeye and Omar [3].Via numerical integration, a two-point four-step direct implicit block method implemented at two points simultaneously in a block using four backward steps was proposed by Majid et al. [4].Also, a fourth and fifth derivative, three-point implicit block method was presented by Allogmany and Ismail [5].Using interpolation and collocation techniques, implicit continuous LMM for solving Equation (1) was presented by Jator [6].Linear multistep methods have also been used to solve other types of differential equations in Refs.[7,8].
Proposed to overcome the Dahlquist Barrier theorem, hybrid methods were introduced and have continued to generate interest among numerical analysts as given in Ref. [9,10].In this paper, we derive a class of hybrid block methods via interpolation and collocation technique, that directly solves general third-order ODEs.We reiterate that the most typical methods for attempting to solve (1) typically entail reducing the problem to a system of first-order differential equations and then solving the system using one of the available methods, which has been shown to have some significant computational drawbacks like requiring more time and labour from the user as as presented by Jator et al. [11].Within the decade researchers have clearly shown an unequivocal dependence of the accuracy of block hybrid methods on the number and type of grid points incorporated in the derivation process as presented in Refs.[12][13][14][15].
In this paper, a self-starting class of single-step hybrid block methods are presented for numerically integrating general thirdorder ODEs.We formulate certain members of the single-step hybrid block methods, prove the convergence and perform numerical tests to check the precision of the suggested methods by contrasting them with other schemes from the literature.

Derivation of the Methods
In this section, we describe the formulation of a class of onestep block hybrid methods via the interpolation and collocation approach as proposed by Onumanyi et al. [16], which will be utilized to generate a number of discrete solutions for solving (1).
We start by deriving a block hybrid method of the form Y(x) = α 0 (x)y n + α 1 2 (x)y n+ 1 2 + α 1 (x)y n+1 + h 3 1 j=0 β j (x)y n+ j where p v ∈ (0, k) are a countable number m ∈ O of equally spaced, non-integer off-grid points with 1  2 as midpoint derived by Equation ( 1) contains the first and second derivative, hence, the derivatives of Equation ( 2) are given as The following conditions are imposed on Equations ( 4) and ( 5)

Specification of the methods
In this section, Equation ( 2) is utilized in obtaining a particular one-step block hybrid method with equally spaced off-grid points by specifying m.For illustrative purposes, the results for m = 3 is provided.Evaluating Equation ( 2) at x = x n+ 1 4 , x n+ 1 2 , x n+ 3  4 , we generate the two main methods ( 8) and ( 9) given below as: The starters Equations ( 10) and ( 11) are obtained from Equation (7) and are given as: It is worth mentioning that the derivatives are generated by δ(x n+τ m ) = δ n+τ m and ω(x n+τ m ) = ω n+τ m as follows: For the one-step block hybrid method with m = 5 equally spaced off-grid points, we evaluate Equation (2) at We generate the four main methods (20), ( 21),( 22) and (23) given below as: The starters Equations ( 24) and (25) are obtained from Equation (7) and are given as: It is worth mentioning that the derivatives are generated by δ( For the one-step block hybrid method with m = 7 equally spaced off-grid points, we evaluate Equation (2) at We generate the six main methods (38), ( 39), ( 40), ( 41), ( 42) and ( 43) given below as: The starters Equations ( 44) and ( 45) are obtained from Equation (7) and are given as: It is worth mentioning that the derivatives are generated by δ( For brevity, we have omitted the schemes for m > 7.

Analysis of the methods
We report the findings from our analysis of the characteristics of our proposed method with evenly spaced off-grid collocation nodes in this section.We focus in particular on the zero stability analysis and truncation error.

Local Truncation Errors and Order
We express our main methods and starters for third order IVPs in terms of a linear operator L defined as where ζ = 1, 2, . . ., m.Assuming that y(x n ) is sufficiently differentiable, we can expand the terms y(x n + jh), y(x n + p ζ h), y ′′′ (x n + p ν h)and y ′′′ (x n + jh) as a Taylor series about the point x n to obtain the expression where Ĉ0 , Ĉ1 , . . .Ĉp are constant vectors.As stated in Henrici [1], we say that the method has order p if Ĉ0 , Ĉ1 , • • • , Ĉp , Ĉp+1 , and Ĉp+2 = 0 C p+3 0.
The vector Ĉp+3 is called the error constant and Ĉp+3 h p+3 y (p+3) (x n ) the principal part of the local truncation error at the end point x n .It is standard from our computations that our methods have order p > 1 and with comparatively small error constants.Consequently, for our method with m = 3, we have For our method with m = 5 and m = 7, we have 13h 10 y (10)

Zero Stability
The starters and primary methods are represented in matrixvector form as where m and D (0) m are square matrices.
For m = 3 we obtain we obtain the square matrices and .
The matrix system (67) is reduced to The zero stability is established from the nature of the zeros of the first characteristic polynomial defined as A method is said to be zero stable if the roots of ρ(R) satisfy | R j |≤ 1 and all the roots with | R j |= 1 have multiplicity that does not exceed 2. For our method with m = 3, we obtain the first characteristic polynomial given as which has principal root | R 0 |= 1 and spurious roots R j = 0, j = 1(1)3.For our method with m = 5, we obtain the first characteristic polynomial given as which has principal root | R 0 |= 1 and spurious roots R j = 0, j = 1(1)5.For our method with m = 7, we obtain the first characteristic polynomial given as which has principal root | R 0 |= 1 and spurious roots R j = 0, j = 1(1)7.In general, the first characteristic polynomial of our class of hybrid block methods alternates between and Following Jator [17], our class of hybrid block methods are zero stable, consistent and convergent.

Numerical Problems & Discussions
In this section, we have tested the performance of our methods on both scalar and system ODEs of the linear and nonlinear types.All results in this study were compared with the theoretical solutions, hence, the Absolute Errors (AE) and Maximum Absolute Errors (MAE) obtained.We compare our methods with the Generalized Linear Block Method (GLBM) of order 5 in Ref. [3], the Direct Two-Point Four-Step Variable Step (D2P4VS) methods of order 6 and 7 in Ref. [4], the Implicit Three-Point Block Method of order 9 (ITPBMO9) in Ref. [5], and the Boundary Value Methods (BVM5) of order 6 in Ref. [11] .All computations were implemented using Mathematica 13.
which was solved in Ref. [11], with the initial conditions, interval of integration and exact solution given as In this example, a comparison is made between our method m = 3 and the BVM5 in Ref. [11], both orders 5 and 6 respectively.
In Table 1, we compare the MAE of our method m = 3 with the BVM5 at different values of N. It is evident from the results that our method m = 3 of order 5 performs favourably well against the BVM5 of order 6.Our method m = 3 outperforms the BVM5 for all the values of N provided except at 320.Table 2 display the MAE for our method m = 3, 5 and 7.The results were computed using different values for N.

Problem 2
Consider the singularly perturbed third order BVP which was solved in Refs.[6,11], with the boundary conditions, domain of integration and exact solution given as where The singularly perturbed third-order BVP has been solved with a variety of values ϵ, and it is shown that the results are still acceptable as ϵ → 0. In the Tables 3 and 4, the MAE of our method m = 3 of order 5 is compared with the BVM5 of order 6.From the results presented, our method m = 3 outperforms the BVM5 for ϵ = 0.01.Although, our methods perform better with a decreased number of steps, for example, our methods m = 3, 5 and 7 outperform those presented in Ref. [11] for smaller values of N, the BVM5 outperforms our methods at N = 320 and with ϵ = 0.1.For our methods, m = 3, 5 and 7, the numerical solutions were contrasted with the theoretical and the MAE presented in Table 5 and 6.

Problem 3
Consider the third order IVP with the initial conditions, domain of integration and exact solution given as Tables 7 and 8 display the AE at the endpoints x = 1 and x = 4 of the integration domain.The results were computed using different values for N. We compare our methods m = 3 of order 5, with the two-point four-step implicit block method (D2P4VS) of orders 6 and 7 presented in Ref. [4].
In the Tables 9 and 10, we provide the MAE within the integration domain x ∈ [0, b].From the results displayed in Table 9, our methods outperform the D2P4VS in a lesser number of steps with b = 1.In the Table 10, our method m = 3 outperforms the D2P4VS.

Problem 4
Consider the BVP with mixed boundary conditions which was solved in Ref. [11], with the mixed boundary conditions, domain of integration and exact solution given as In this problem, f (x) is deduced from the exact solution.All methods solve this problem accurately and are consequently exact to machine precision.Table 11 shows the computational comparison of the different methods for this problem.
In Table 12, we display the MAE within the integration domain of the problem for m = 3, 5 and 7.The results were computed using different values of N.

Problem 5
Consider the third order IVP which was solved in Ref. [3], with the initial conditions, domain of integration and exact solution given as Tables 13 and 14 show the AE at selected points in the integration domain for the problem solved with step size h =  and compared with the theoretical solution.In the Table 13, we compare the AE of our methods m = 3 against the GLBM in Ref. [3].It is instructive to note that both methods are of order 5.It is easy to see that our method m = 3 performs relatively well against the GLBM.From Table 13, both methods perform slightly better than the other at 5 selected points each within the domain of integration.The AE of our methods m = 5 and m = 7 is presented in Table 14.
which was solved in Ref. [3], with the initial conditions, domain of integration and exact solution given as Exact : y(x) = sin x.
In this example, a comparison is made between our method m = 3 and the GLBM in Ref. [3].In Table 15, we compare the AE of our method m = 3 with the GLBM at selected points within the integration domain of the problem solved with step size h = 1 10 .It is evident from the results that our method m = 3 performs relatively well against the GLBM.Our scheme m = 3 slightly outperforms the GLBM at the first five selected points in the Table 15, while the GLBM marginally outperforms our method m = 3 at the points x = 0.6, 0.7, 0.8, 0.9 and 1.0.The AE of our methods m = 5 and m = 7 is presented in Table 16.
which was solved in Ref. [11], with the initial conditions     Tables 19, 20 and 21 shows the AE at selected points in the integration domain for the problem solved with step size h =

Conclusion
We proposed a class of single-step hybrid block methods for solving third-order ODEs without first converting to an analogous first-order system.The convergence of the new methods was established, and the schemes are implemented without the use of starting values or predictors, avoiding the necessity for complex subroutines.To illustrate the effectiveness of the new methods, eight test problems of varied degrees of difficulty are considered.From the results, it is shown that the improvement in accuracy increases as the number of intra-step points m is increased.Tables 1-21 discuss the numerical outcomes in further depth.Our future research will focus on developing optimized single-step hybrid block schemes for solving third-order ODEs directly.

1 10
and compared with the theoretical solution.The numerical results for this problem are presented in Tables 19, 20 and 21.

Table 1 .
Comparison of the MAE of m = 3 and BVM5 (Problem 1)

Table 11 .
Comparison of the MAE methods m = 3 and BVM5 (Problem 4)

Table 13 .
Comparison of the AE methods m = 3 and GLBM (Problem 5)

Table 14 .
AE at selected points for method m = 5 and m = 7 (Problem 5)

Table 15 .
Comparison of the AE methods m = 3 and GLBM (Problem 6)

Table 16 .
AE at selected points for method m = 5 and m = 7 (Problem 6)