Hybrid methods for direct integration of special third order ordinary differential equations

Jikantoro, Yusuf Dauda; Ismail, Fudziah; Senu, Norazak; İbrahim, Zarina Bibi

doi:10.1016/j.amc.2017.10.003

Cited by 10 publications

(8 citation statements)

References 11 publications

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“…To see if our proposed method agrees with some of the available numerical results, we firstly solve the problem in (23) using FDTHBS by fixing h ini = 10 −1 , x ∈ [0, 1] and varying the tolerances to have the same number of steps with the methods in [8,17]. The numerical results in Figure 9 confirm that the proposed method agrees well with the ones presented in [8,17].…”

Section: Numerical Test Problemsupporting

confidence: 65%

A variable step-size fourth-derivative hybrid block strategy for integrating third-order IVPs, with applications

Rufai

Ramos

2021

International Journal of Computer Mathematics

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In this paper, an efficient fourth-derivative two-step hybrid block strategy (FDTHBS) to get the approximate solution of a third-order IVP with applications to problems in Fluid Dynamics and Engineering is constructed. The mathematical derivation of the proposed FDTHBS is based on the interpolation and collocation of the exact solution and its derivatives at the selected equidistant grid and off-grid points. The theoretical characteristics of the proposed method are analysed. An embedding-like procedure is considered and executed in variable step-size mode to get better performance of the newly developed strategy. Some test problems, including the wellknown Blasius equation and three different types of non-linear thin-film flow problems, are integrated numerically to ascertain the superior impact of our developed error estimation and control strategy. It is worth concluding that the proposed technique is not only efficient in term of CPU time, but also minimizes errors and support the analytical results.

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Section: Numerical Test Problemsupporting

confidence: 65%

A variable step-size fourth-derivative hybrid block strategy for integrating third-order IVPs, with applications

Rufai

Ramos

2021

International Journal of Computer Mathematics

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“…Replace, f t, y, y 0 , y 00 ð Þ in (6), (7) and (8) by Hermite interpolating polynomial in (2), which is defined as P 2 t ð Þ, where…”

Section: Two-point Additional Derivative Block Methodsmentioning

confidence: 99%

“…Jator et al 6 constructed a new class of implicit continuous linear multistep methods (LMMs) which were used to solve general third order initial and boundary value problems. Jikantoro et al 7 developed hybrid method for direct integration of special third order ODEs.…”

Section: Introductionmentioning

confidence: 99%

Direct integrator of block type methods with additional derivative for general third order initial value problems

Turki

Ismail

Senu

et al. 2014

Advances in Mechanical Engineering

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This paper presents the construction of the two-point and three-point block methods with additional derivatives for directly solving [Formula: see text]. The proposed block methods are formulated using Hermite Interpolating Polynomial and approximate the solution of the problem at two or three-point concurrently. The block methods obtain the numerical solutions directly without reducing the equation into the first order system of initial value problems (IVPs). The order and zero-stability of the proposed methods are also investigated. Numerical results are presented and comparisons with other existing block methods are made. The performance shows that the proposed methods are very efficient in solving the general third order IVPs.

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“…However, there is a dearth of research in methods that solve higher-order stiff ODEs, especially third-order stiff ODEs. Most past research focused on the special and non-stiff third order ODEs as discussed in [10][11][12][13]. We, therefore, aim to develop an efficient direct method to provide numerical solutions for third-order ODEs.…”

Section: Introductionmentioning

confidence: 99%

High Order Block Method for Third Order ODEs

Asnor¹,

Yatim²,

İbrahim³

et al. 2021

Computers, Materials &Amp; Continua

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Many initial value problems are difficult to be solved using ordinary, explicit step-by-step methods because most of these problems are considered stiff. Certain implicit methods, however, are capable of solving stiff ordinary differential equations (ODEs) usually found in most applied problems. This study aims to develop a new numerical method, namely the high order variable step variable order block backward differentiation formula (VSVO-HOBBDF) for the main purpose of approximating the solutions of third order ODEs. The computational work of the VSVO-HOBBDF method was carried out using the strategy of varying the step size and order in a single code. The order of the proposed method was then discussed in detail. The advancement of this strategy is intended to enhance the efficiency of the proposed method to approximate solutions effectively. In order to confirm the efficiency of the VSVO-HOBBDF method over the two ODE solvers in MATLAB, particularly ode15s and ode23s, a numerical experiment was conducted on a set of stiff problems. The numerical results prove that for this particular set of problem, the use of the proposed method is more efficient than the comparable methods. VSVO-HOBBDF method is thus recommended as a reliable alternative solver for the third order ODEs.

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Hybrid methods for direct integration of special third order ordinary differential equations

Cited by 10 publications

References 11 publications

A variable step-size fourth-derivative hybrid block strategy for integrating third-order IVPs, with applications

A variable step-size fourth-derivative hybrid block strategy for integrating third-order IVPs, with applications

Direct integrator of block type methods with additional derivative for general third order initial value problems

High Order Block Method for Third Order ODEs

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