A Sixth-order Hybrid Block Method for the Direct Solution of 3rd-order Initial Value Problems of Ordinary Differential Equations

Duromola Monday Kolawole; Akinmoladun Olusegun Mayowa; Fafesobi Gbenga Olatunde

doi:doi:10.11648/j.engmath.20261002.11

Research Article |

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A Sixth-order Hybrid Block Method for the Direct Solution of 3rd-order Initial Value Problems of Ordinary Differential Equations

Duromola Monday Kolawole^*

, Akinmoladun Olusegun Mayowa

, Fafesobi Gbenga Olatunde

Published in Engineering Mathematics (Volume 10, Issue 2)

Received: 7 June 2026 Accepted: 22 June 2026 Published: 17 July 2026

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Abstract

This work focuses on the derivation, analysis and numerical application of a hybrid linear multistep formula for directly solving third-order initial value problems (IVPs) ordinary differential equations (ODEs). The method is derived by adopting the collocation and interpolation procedure with a power series polynomial of degree eight as the basis function. The procedure resulted in the construction of a continuous hybrid linear multistep method by collocating the third derivative at nodal and five off-nodal points to generate a system of linear equations for the determination of the unknown parameters, from which the formulas that constitute the proposed hybrid block method are obtained. The suggested method is fully hybrid, which is an important property that contributed to the good accuracy and minimum errors associated with the results of the method. The analysis of the basic properties of the suggested method reveals that the method is of theoretical order six, stable, consistent and convergent. Four numerical examples are considered for the numerical experiment to confirm the accuracy of the proposed method. The experiment shows that the present method is very efficient for the numerical approximation of third-order initial value problems for ordinary differential equations. The method also performs favorably well when compared with the results of some cited methods in the literature.

Published in	Engineering Mathematics (Volume 10, Issue 2)
DOI	10.11648/j.engmath.20261002.11
Page(s)	24-35
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Power Series Polynomials, Stable, Consistent, Accuracy, Interpolation and Collocation

1. Introduction

The modelling of real-life situation often gives rise to third-order IVPs of ODEs which are generally written as:

where

(1)

This work proposed method which is easily implemented and can also provide high order of accuracy. Higher-order ODEs are frequently written as equivalent systems of first-order ODEs to get numerical solutions, which may subsequently be solved using proven numerical techniques, and this has assisted in obtaining solutions to higher-order ODEs discussed by

[1-4]

. However, because of the longer runtime and additional computing effort, this transformation may increase computational cost in some cases, particularly for large systems or stiff problems, as

[5]

pointed out. Notably, significant attention has been given to the development of methods that directly solve IVPs of higher-order ODEs without reducing them to systems of first-order. This approach has been discussed by researchers such as

[6-11]

. Recently,

[12]

examined the numerical solution of third-order IVPs of ODEs using a set of linear multistep method with block extension.

[13]

discussed the P-stable single-step block method of solving third-order IVPs: nine orders of accuracy, and additionally, the modified fourth derivative block approach and its direct applicability to third-order initial value problems were covered in detail by

[14]

. In this research, third-order ordinary differential equations (IVPs) were numerically integrated using a hybrid linear multistep approach with five off-steps. By improving accuracy, stability, and computational time, this technique seeks to provide a more potent substitute for reduction-based methods.

2. Derivation of the Method

This section presents a detailed derivation of the proposed scheme by applying interpolation and collocation procedures to a power series polynomial, which serves as the trial function. The power series function is given as:

(2)

where

r

and

s

are respectively number of interpolation and collocation points,

a_{j}

’s are parameters to be obtained uniquely and

x

is differentiable over the [a, b]. The third derivative of (2) gives

(3)

Equating (1) and (3) yields the differential system

.(4)

By collocating (4) at grid points

and interpolating Eq. (2) at

yield a system of equations for interpolation equation,

(5)

and for collocation,

.(6)

By putting equations (5) and (6) in matrix form and solving them to determine the parameter values

, which when substituted in (3) yields, after a some simplification, thus provides a continuous hybrid linear multistep formula of the form:

(7)

where

and

are the coefficients that defined the method and are obtained as:

where

. Evaluate Eq. (8) at

to obtain the discrete one thirty-sixth step formulas

(8)

(9)

(10)

In order to implement the derived method in block mode, we adopted the block formula proposed in Awoyemi et al.

[15]

. The formula in its normalized form is given as

(11)

By evaluating Eq. (7) at

; the first and second derivatives at

and substituting into Eq. (11) gives the coefficients matrices as:

,(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

Equations (12)-(26) are the discrete formulas of one-thirty-six off-steps hybrid block method suggested in the work for the direct solution of third-order IVPs of ordinary differential equations which require no predictor to start the corrector.

3. Analysis of the Basic Properties of the Proposed Method

This section presents the analysis of the basic properties of the propose method was carried out as follows.

3.1. Order and Error Constant of the Method

The linear multistep formula in equation (10) in its conventional form can be express as

(27)

The local truncation error associated with equation (27), going the work of Lambert

[2]

, is defined by the difference operator.

(28)

where

is continuously differentiable. Expanding (10) in Taylor series about the point

yields the expression of the local truncation error

(29)

The term

is the error constant while the local truncation error is given by:

(30)

since

. The procedure reveals that the suggested method has order

with error constant

, see

[16]

3.2. Zero Stability of the Proposed Method

Definition 3.1 (Fatula

[17]

)

The block method presented in (12) - (26) is zero Stable provided the roots

of the first characteristic polynomial

specified as

(31)

Satisfies

and for those roots with

the multiplicity must not exceed 2.

By applying definition (3.1), the block is found to be zero stable since the roots of the characteristic polynomial of the proposed block method satisfy

and the root

has multiplicity not exceeding the order of the ODE. This is established with the following calculation; as

where

is the order of the differential equation, for the block method,

and

Hence, since all roots satisfy the roots condition, the method is Zero Stable.

3.3. Consistency of the Method

From Eq. (10), the first and second characteristics polynomials of the method are given by

This implies that the method presented in this report is consistent since it satisfies the following conditions:

i. The order of the method is

which is obvious.

ii. For the method,

, thus

,showthecondition(ii)issatisfied.

iii. If

and

It follows from here that

. Show that the condition (iii) is satisfied as well

iv. Note that

Thus, the condition (iv) is satisfied. Hence the method is consistent.

3.4. Convergence of the Method

As reported in

[18]

, the method (10) requires both consistency and zero stability to achieve convergence; Thus, since it has been successful shown in (3.2) and (3.3) above respectively. Therefore, (10) converges.

3.5. Region of Absolute Stability of the Method

Worth considering the stability polynomial in the general form:

(32)

Where

and

is assumed constant. The first and second characteristics polynomials of Eq. (10) are given by

The boundary of the region of the absolute stability is

(33)

By setting

, then Eq. (33) becomes

\bar{h} (θ) \frac{- 2799360000 (e^{\frac{1}{36} iθ} + 10 e^{\frac{3}{4} iθ} - 10 e^{\frac{5}{8} iθ} + 5 e^{\frac{1}{2} iθ} - e^{\frac{3}{8} iθ})}{19 - 212 e^{\frac{1}{8} iθ} + 1072 e^{\frac{1}{4} iθ} - 3044 e^{\frac{3}{8} iθ} - 25010 e^{\frac{1}{2} iθ} - 338204 e^{\frac{5}{8} iθ} - 329048 e^{\frac{3}{4} iθ} - 31532 e^{\frac{7}{8} iθ} + 199 e^{iθ}}

\frac{- 2799360000 (r^{\frac{1}{36}} - 3 r^{\frac{1}{45}} + 3 r^{\frac{1}{60}} - r^{\frac{1}{90}})}{1 - 7 r^{\frac{1}{180}} + 18 r^{\frac{1}{90}} - 222 r^{\frac{1}{60}} - 227 r^{\frac{1}{45}} - 3 r^{\frac{1}{36}}}

(34)

Evaluate Eq. (34), and equate the imaginary part to zero gives

(35)

Evaluating Eq. (35) at the interval of

gives the following results of the boundaries for the region of absolute stability of the method as tabulated below;

Table 1. Boundaries for region of absolute stability.

θ	0°	30°	60°	90°	120°	150°	180°
$\overset{̅}{h} (θ)$

From Table 1 above, it could be deduced that the region of absolute stability of the method is given by

which suggests potential P-stability according to

[19]

, subject to further spectral verification.

Download: Download full-size image

Figure 1. Domain of stability (i.e. absolute) of the proposed method.

4. Numerical Results

The section presents numerical experiment that involves numerical solutions of third-order initial value of ordinary differential equations to confirm the accuracy and viability of the method.

The Table show that

where

is the exact solution,

is the computed result and absolute error defined as

4.1. Problem 1: We Consider the Initial Value Problem (IVP) of the Form

y^′′′=e^x,y(0)=3,y^′(0)=1,y^′′(0)=5,h=0.1

Exact solution: y(x) = 2 + 2x² − e^x

Source: Kuboye et al., (2020)

The solutions were obtained within 10 iterations following the procedure described in Ref.

[20]

. Table 2 compares the exact results (ERC) with the numerical results (NRC) generated by the newly developed method for solving ordinary differential equation initial value problems (IVPs). For all values of the independent variable (XVC), the numerical results show excellent agreement with the exact solutions, achieving accuracy close to machine precision. The computed errors (ERR) of the proposed method remain extremely small, consistently lying between (10^{-13}) and (10^{-16}), which reflects the high precision and numerical stability of the method. In comparison, the errors obtained using the method in Ref.

[20]

are noticeably larger, increasing gradually from (10^{-12}) at (XVC = 0.1) to (10^{-10}) at (XVC = 1.0). As illustrated in Figure 2, the proposed method provides several orders of magnitude improvement in accuracy over the method reported in Ref.

[20]

. These results therefore establish the proposed method as an efficient, reliable, and highly accurate technique for solving higher-order initial value problems in ordinary differential equations.

Table 2. Numerical result for problem 1.

XVC

ERC

NRC

ERR

ERR in

[20]

0.10

3.12517

1.5227811E-12

0.20

3.30140

2.22044604925031E-16

9.6922224E-12

0.3

3.52986

2.22044604925031E-15

2.4267699E-11

0.4

3.81182

7.993605777301127-15

4.5451198E-11

0.5

4.14872

1.68753899743023E-14

7.8387963E-11

0.6

4.54212

3.08642000845793E-14

1.3159340E-10

0.7

4.99375

5.329070518200751E-14

2.0471091E-10

0.8

5.50554

8.26005930321116E-14

2.9804159E-10

0.9

6.07960

1.225686219186172E-13

4.1925841E-10

1.0

6.71828

1.771915947301749E-13

5.8107297E-10

Download: Download full-size image

Figure 2. Numerical and exact solution of problem 1.

4.2. Problem 2: Consider the Third Order Linear Problem

y^′′′+e^x=0,y(0)=1,y^′(0)=−1,y^′′(0)=3,h=0.1

Exact solution: y(x) = 2(1 + x²) − e^x

Source: Ogunware et al., (2020)

Table 3; presents a detailed comparison between the ERC and NRC obtained by adopting the developed block formula for solving ordinary differential equation initial value problems. The numerical results agree very closely with the exact solutions for all values of the independent variable XVC, with only negligible differences observed. The ERR produced by the suggested approach are very small, between (

10^{- 13}

) to (

10^{- 16}

), demonstrating the method’s high numerical precision and stability. As illustrated in Figure 3, the proposed method consistently outperforms the method of Ogunware et al.

[21]

, particularly as the values of increase. For example, at (XVC = 1.0), the proposed method produces an error of only (1.79

{\times 10}^{- 13}

), whereas the method in Ref.

[21]

gives an error of (4.87

{\times 10}^{- 13}

). This substantial reduction in error highlights the improved accuracy of the newly developed method. In all, the results confirm that the suggested formula is consistent, stable, and highly accurate option to existing numerical techniques for third-order ODEs.

Table 3. Numerical result for problem 2.

XVC

ERC

NRC

ERR

ERR in

[21]

0.1

0.914829

0.000000E+000

3.473600E-14

0.2

0.858597

6.66133814775093910E-16

3.326900E-13

0.3

0.830141

2.886579864025407E-15

3.709100E-14

0.4

0.828175

7.549516567451064E-15

5.791840E-13

0.5

0.851279

1.7319479184152442E-14

3.581010E-13

0.6

0.897881

3.241851231905457E-14

1.209298E-12

0.7

0.966247

5.3734794391857577E-14

1.179995E-12

0.8

1.054460

8.393286066166183E-14

2.514500E-12

0.9

1.160400

1.2523315717771766E-13

2.409110E-12

1.0

1.281720

1.794120407794253E-13

4.870670E-12

Download: Download full-size image

Figure 3. Numerical and exact solution of problem 2.

4.3. Problem 3: We Consider the Non-linear Third Order IVP

y^′′′(x) + 2e⁻³^y⁽^x⁾ = 4(1 + x)⁻³

y (0) = 0, y^{'} (0) = 1, y^{''} (0) = - 1

Exact solution: y(x)

= In (1 + x)

Source: Orakwelu et al. (2023)

Table 4. Numerical result for problem 3.

MAE

MAE in

[22]

9.97466E-18

2.91E-09

4.77049E-18

4.65E-11

4.98733E-18

7.39E-13

3.46945E-18

1.84E-14

160

3.46945E-18

1.65E-13

320

9.54945E-18

2.52E-12

Download: Download full-size image

Figure 4. Numerical and exact solution of problem 3.

Table 4 and Figure 4 present a comparison of the Maximum Absolute Errors (MAE) obtained using the newly developed method for solving ordinary differential equation initial value problems. The numerical results show excellent agreement with the exact solutions, with the MAE values remaining extremely small throughout the computation, specifically within the range of (10^{-18}). This reflects the high level of accuracy and precision achieved by the proposed method. In comparison with the results reported by Orakwelu et al.

[22]

, the proposed method consistently produces smaller MAE values, indicating a clear improvement in performance. As observed from Table 4 and Figure 4, the analysis demonstrates that the new method provides superior numerical accuracy and offers a more efficient and reliable approach for solving higher-order differential equation initial value problems.

4.4. Problem 4: Consider the Third-order IVP

y^{'''} = 3 \sin x . y (0) = 1, y^{'} (0) = 0, y^{''} (0) = - 2, h = 0.1

Exact solution:

y (x) = 3 \cos x + \frac{x^{2}}{2} - 2 = 2

Source: Kuboye et al., (2020)

Figure 5 compares the exact results (ERC) with the numerical results (NRC) obtained from our newly developed method for solving an initial value problem in ordinary differential equations. The numerical values match the exact ones very closely across all tested points, with errors (ERR) falling between

10^{- 16}

and

10^{- 14}

which is impressively tiny and speaks to the remarkable accuracy of the proposed method. That said, the error does tend to increase slightly as XVC gets larger, but the growth is so slow and well-behaved that it hardly raises any concern, confirming that the method stays efficient even over longer intervals. All in all, the analysis shows that our new numerical approach is not only highly accurate but also robust and reliable for solving initial value problems.

Table 5. Numerical result for problem 4.

XVC

ERC

NRC

ERR

ERR in

[20]

0.10

0.990012

6.8911543E-13

0.20

0.960200

4.4015902E-12

0.30

0.911009

1.0999868E-11

0.40

0.843183

8.88178E-16

2.0601632E-11

0.50

0.757748

2.22045E-15

3.6853520E-11

0.60

0.656007

4.88498E-15

6.726841E-11

0.70

0.539527

8.88178E-15

1.1150603E-10

0.80

0.410120

1.50990E-14

1.6985002E-10

0.90

0.269830

2.37588E-14

2.4948449E -10

1.00

0.120907

3.59712E-14

3.6226498E-10

Download: Download full-size image

Figure 5. Numerical and exact solution of problem 4.

While the absolute error appears to be increasing slightly due to the increasing size of the XVC values, the errors are still significantly low and are within accepted computational limits, demonstrating that the method is table over the entire interval ([0, 1]). Therefore, we can conclude that our newly developed method is highly accurate, reliable, and numerically robust and is therefore a viable option for use in scientific computing for handling direct solution of third-order ODEs.

5. Discussion of Results

The current paper discusses an approach based on collocation and interpolation procedure which resulted in a hybrid multistep technique for solving IVPs arising from third-order ODEs. The new method appears to have both high accuracy and computational efficiency, so it could easily handle large and complex initial value problems. Results presented in Tables 2, 3, 4 and 5 indicate that the new technique outperformed previous techniques as proposed by Kuboye et al.

[20]

, Ogunware et al.

[21]

, and Orakwelu et al.

[22]

. The relationship between numerical solutions derived through use of the proposed technique and the corresponding exact solutions is shown in Figures 2-5, indicating close correlation between these two forms of solution. This new sixth order method has increased accuracy and stability than previously proposed methods in terms of their ability to minimize errors and/or converge.

6. Conclusion

As depicted in our finding, a sixth-order hybrid block approach for the direct computation of Third-Order IVPs of ODEs. The formulas of the suggested method are gotten from a continuous approach that utilizes concepts of interpolation and collocation to derive the main and additional formulae for the proposed block method. It was established that the hybrid block method is of order six, exhibits zero stability, consistency, convergence and has a significantly large region where numerical results will exhibit absolute stability. The overall approach is presented as a single block (combined) method used for numerical resolution of Third-Order ODE's. The approximate solutions as obtained using the developed scheme, reflect the relevance and satisfaction in solving general types of Third-Order ODEs. The difference between the proposed method and published articles indicates that the proposed method produces results that are comparable to and will perform similarly to, those results from published references both in terms of accuracy and effectiveness.

Abbreviations

XVC	Value of the Independent Variable Where a Numerical Value Is Taken
ERC	Exact Result at XVC
NRC	Numerical Result of XVC
ERR	Error in Proposed Method at XVC
N	Steps
MAE	Maximum Absolute Errors

Author Contributions

Duromola Monday Kolawole: Conceptualization, Methodology, Resources, Supervision, Writing – original draft

Akinmoladun Olusegun Mayowa: Validation, Writing – review & editing

Fafesobi Gbenga Olatunde: Formal Analysis, Investigation, Software

Conflicts of Interest

The authors declare no conflicts of interest.

References

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[3]	Chu, M. T. and Hamiton, H. (1987): Parallel Solution of Ordinary Differential Equations by Multiblock Methods. SIAM Journal of Scientific and Statistical Computation. 8; 342 – 553.
[4]	Fatunla, S. O. (1988). Numerical Methods for Initial value Problems in Ordinary Differential Equations. Academic Press Inc., New York.
[5]	Awoyemi, D. O (1999): A class of continuous Methods for general second order initial value problems in ordinary differential equations. International journal of computer mathematics 72; 29-39. https://doi.org/10.9734/arjom/2022/v18i130354
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[7]	Bolarinwa Bolaji, Ademiluyi, R. A., Olaseni Tunde and Duromola M. K (2012): A New Implicit Hybrid Block Method for the Direct Solution of initial value problems of general third order ordinary differential equations. Canadian Journal of Science and Engineering, 3; 86 - 97.
[8]	Ademiluyi, R. A., Duromola, M. K. & Bolarinwa Bolaji. (2014). Modified block method for the direct solution of initial value problems of fourth order Ordinary differential equations. Australian Journal of Basic and Applied Sciences, 8(10) July 2014; 389-394.
[9]	Adesanya, A. O., Abdulqadri, B. and Ibrahim, Y. S. (2014) Hybrid One Step Block. Method for the Solution of Third Order Initial Value Problems of Ordinary Differential Equations. International Journal of Pure and Applied Mathematics, 97, 1-11. https://doi.org/10.12732/ijpam.v97i1.1
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APA Style

Kolawole, D. M., Mayowa, A. O., Olatunde, F. G. (2026). A Sixth-order Hybrid Block Method for the Direct Solution of 3rd-order Initial Value Problems of Ordinary Differential Equations. Engineering Mathematics, 10(2), 24-35. https://doi.org/10.11648/j.engmath.20261002.11

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Kolawole, D. M.; Mayowa, A. O.; Olatunde, F. G. A Sixth-order Hybrid Block Method for the Direct Solution of 3rd-order Initial Value Problems of Ordinary Differential Equations. Eng. Math. 2026, 10(2), 24-35. doi: 10.11648/j.engmath.20261002.11

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Kolawole DM, Mayowa AO, Olatunde FG. A Sixth-order Hybrid Block Method for the Direct Solution of 3rd-order Initial Value Problems of Ordinary Differential Equations. Eng Math. 2026;10(2):24-35. doi: 10.11648/j.engmath.20261002.11

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@article{10.11648/j.engmath.20261002.11,
  author = {Duromola Monday Kolawole and Akinmoladun Olusegun Mayowa and Fafesobi Gbenga Olatunde},
  title = {A Sixth-order Hybrid Block Method for the Direct Solution of 3rd-order Initial Value Problems of Ordinary Differential Equations},
  journal = {Engineering Mathematics},
  volume = {10},
  number = {2},
  pages = {24-35},
  doi = {10.11648/j.engmath.20261002.11},
  url = {https://doi.org/10.11648/j.engmath.20261002.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20261002.11},
  abstract = {This work focuses on the derivation, analysis and numerical application of a hybrid linear multistep formula for directly solving third-order initial value problems (IVPs) ordinary differential equations (ODEs). The method is derived by adopting the collocation and interpolation procedure with a power series polynomial of degree eight as the basis function. The procedure resulted in the construction of a continuous hybrid linear multistep method by collocating the third derivative at nodal and five off-nodal points to generate a system of linear equations for the determination of the unknown parameters, from which the formulas that constitute the proposed hybrid block method are obtained. The suggested method is fully hybrid, which is an important property that contributed to the good accuracy and minimum errors associated with the results of the method. The analysis of the basic properties of the suggested method reveals that the method is of theoretical order six, stable, consistent and convergent. Four numerical examples are considered for the numerical experiment to confirm the accuracy of the proposed method. The experiment shows that the present method is very efficient for the numerical approximation of third-order initial value problems for ordinary differential equations. The method also performs favorably well when compared with the results of some cited methods in the literature.},
 year = {2026}
}

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TY  - JOUR
T1  - A Sixth-order Hybrid Block Method for the Direct Solution of 3rd-order Initial Value Problems of Ordinary Differential Equations
AU  - Duromola Monday Kolawole
AU  - Akinmoladun Olusegun Mayowa
AU  - Fafesobi Gbenga Olatunde
Y1  - 2026/07/17
PY  - 2026
N1  - https://doi.org/10.11648/j.engmath.20261002.11
DO  - 10.11648/j.engmath.20261002.11
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 24
EP  - 35
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20261002.11
AB  - This work focuses on the derivation, analysis and numerical application of a hybrid linear multistep formula for directly solving third-order initial value problems (IVPs) ordinary differential equations (ODEs). The method is derived by adopting the collocation and interpolation procedure with a power series polynomial of degree eight as the basis function. The procedure resulted in the construction of a continuous hybrid linear multistep method by collocating the third derivative at nodal and five off-nodal points to generate a system of linear equations for the determination of the unknown parameters, from which the formulas that constitute the proposed hybrid block method are obtained. The suggested method is fully hybrid, which is an important property that contributed to the good accuracy and minimum errors associated with the results of the method. The analysis of the basic properties of the suggested method reveals that the method is of theoretical order six, stable, consistent and convergent. Four numerical examples are considered for the numerical experiment to confirm the accuracy of the proposed method. The experiment shows that the present method is very efficient for the numerical approximation of third-order initial value problems for ordinary differential equations. The method also performs favorably well when compared with the results of some cited methods in the literature.
VL  - 10
IS  - 2
ER  -

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Duromola Monday Kolawole

Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria

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http://orcid.org/0000-0002-7117-8418
Akinmoladun Olusegun Mayowa

Department of Mathematical Sciences, Lagos State University of Science and Technology, Ikorodu, Nigeria

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http://orcid.org/0000-0002-8045-5587
Fafesobi Gbenga Olatunde

Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria

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http://orcid.org/0009-0005-1149-5744

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APA Style

Kolawole, D. M., Mayowa, A. O., Olatunde, F. G. (2026). A Sixth-order Hybrid Block Method for the Direct Solution of 3rd-order Initial Value Problems of Ordinary Differential Equations. Engineering Mathematics, 10(2), 24-35. https://doi.org/10.11648/j.engmath.20261002.11

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ACS Style

Kolawole, D. M.; Mayowa, A. O.; Olatunde, F. G. A Sixth-order Hybrid Block Method for the Direct Solution of 3rd-order Initial Value Problems of Ordinary Differential Equations. Eng. Math. 2026, 10(2), 24-35. doi: 10.11648/j.engmath.20261002.11

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AMA Style

Kolawole DM, Mayowa AO, Olatunde FG. A Sixth-order Hybrid Block Method for the Direct Solution of 3rd-order Initial Value Problems of Ordinary Differential Equations. Eng Math. 2026;10(2):24-35. doi: 10.11648/j.engmath.20261002.11

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@article{10.11648/j.engmath.20261002.11,
  author = {Duromola Monday Kolawole and Akinmoladun Olusegun Mayowa and Fafesobi Gbenga Olatunde},
  title = {A Sixth-order Hybrid Block Method for the Direct Solution of 3rd-order Initial Value Problems of Ordinary Differential Equations},
  journal = {Engineering Mathematics},
  volume = {10},
  number = {2},
  pages = {24-35},
  doi = {10.11648/j.engmath.20261002.11},
  url = {https://doi.org/10.11648/j.engmath.20261002.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20261002.11},
  abstract = {This work focuses on the derivation, analysis and numerical application of a hybrid linear multistep formula for directly solving third-order initial value problems (IVPs) ordinary differential equations (ODEs). The method is derived by adopting the collocation and interpolation procedure with a power series polynomial of degree eight as the basis function. The procedure resulted in the construction of a continuous hybrid linear multistep method by collocating the third derivative at nodal and five off-nodal points to generate a system of linear equations for the determination of the unknown parameters, from which the formulas that constitute the proposed hybrid block method are obtained. The suggested method is fully hybrid, which is an important property that contributed to the good accuracy and minimum errors associated with the results of the method. The analysis of the basic properties of the suggested method reveals that the method is of theoretical order six, stable, consistent and convergent. Four numerical examples are considered for the numerical experiment to confirm the accuracy of the proposed method. The experiment shows that the present method is very efficient for the numerical approximation of third-order initial value problems for ordinary differential equations. The method also performs favorably well when compared with the results of some cited methods in the literature.},
 year = {2026}
}

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TY  - JOUR
T1  - A Sixth-order Hybrid Block Method for the Direct Solution of 3rd-order Initial Value Problems of Ordinary Differential Equations
AU  - Duromola Monday Kolawole
AU  - Akinmoladun Olusegun Mayowa
AU  - Fafesobi Gbenga Olatunde
Y1  - 2026/07/17
PY  - 2026
N1  - https://doi.org/10.11648/j.engmath.20261002.11
DO  - 10.11648/j.engmath.20261002.11
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 24
EP  - 35
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20261002.11
AB  - This work focuses on the derivation, analysis and numerical application of a hybrid linear multistep formula for directly solving third-order initial value problems (IVPs) ordinary differential equations (ODEs). The method is derived by adopting the collocation and interpolation procedure with a power series polynomial of degree eight as the basis function. The procedure resulted in the construction of a continuous hybrid linear multistep method by collocating the third derivative at nodal and five off-nodal points to generate a system of linear equations for the determination of the unknown parameters, from which the formulas that constitute the proposed hybrid block method are obtained. The suggested method is fully hybrid, which is an important property that contributed to the good accuracy and minimum errors associated with the results of the method. The analysis of the basic properties of the suggested method reveals that the method is of theoretical order six, stable, consistent and convergent. Four numerical examples are considered for the numerical experiment to confirm the accuracy of the proposed method. The experiment shows that the present method is very efficient for the numerical approximation of third-order initial value problems for ordinary differential equations. The method also performs favorably well when compared with the results of some cited methods in the literature.
VL  - 10
IS  - 2
ER  -

Copy | Download