Implicit Three-Point Block Numerical Algorithm for Solving Third Order Initial Value Problem Directly with Applications

Allogmany, Reem; Ismail, Fudziah

doi:10.3390/math8101771

Cited by 11 publications

(20 citation statements)

References 33 publications

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“…This problem has recently appeared [27] and [28]. The solution of numerical test problem 1 is considered within [0, 1] for h = 0.1.…”

Section: Numerical Test Problemmentioning

confidence: 99%

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Modified Fourth Derivative Block Method and its direct applications to third-order initial value problems

Momoh,

DUROMOLA,

KUSORO

2024

J. Math. Anal. Model.

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A theoretical order eight Modified Fourth Derivative four-step block method (MFDFBM) has been derived, analysed and numerically applied to solve multiple problems originating from Fluid Dynamics, engineering and other sciences. The MFDFBM was derived by applying collocation and interpolation techniques to a power series approximation. Further introducing fourth derivative terms at each of the collocating points yields a block method with an improved order of accuracy. It was observed that the order of the block method increases with the number of fourth derivative terms introduced into the integration interval. Numerical experiments are presented to test MFDFBM on numerical examples, including non-linear homogeneous thin film flow (NHTFF) problems and two non-linear initial value problems(IVPs). The experiments confirm the good impact of adding the fourth derivative terms, which help improve the order of accuracy of the derived MFDFBM, thereby minimising error and agreeing with analytical solution up to at least seven decimal places.

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“…This problem has recently appeared [27] and [28]. The solution of numerical test problem 1 is considered within [0, 1] for h = 0.1.…”

Section: Numerical Test Problemmentioning

confidence: 99%

“…[27] and ITBDM in Ref. [28] are shown in Figure 2. MFDFBM showed good accuracy over the other two methods when applied to numerical test 1.…”

Section: Numerical Test Problemmentioning

confidence: 99%

Modified Fourth Derivative Block Method and its direct applications to third-order initial value problems

Momoh,

DUROMOLA,

KUSORO

2024

J. Math. Anal. Model.

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show abstract

“…Ukpebor [14]. 7 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 ( 1) 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0…”

Section: Zero Stabilitymentioning

confidence: 99%

“…Lambert [1][2] discussed extensively the approach of reducing higher order ODEs to system of lower order, specifically first order equations and then applying various methods available for solving the resulting system of first order IVPs. The direct solution of higher order numerically without reducing to a system of first order initial value problems have been studied by various authors such as Mohammed and [3][4][5][6][7]. Kuboye and Omar [8] proposed seven-step block method for solving third order ODEs.…”

Section: Introductionmentioning

confidence: 99%

Three-step Four-point Optimized Hybrid Block Method for Direct Solution of General Third Order Differential Equations

Obarhua

2023

ARJOM

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This research work considers derivation of three step four point optimized hybrid block method for solving general third order differential equations (odes) without reduction to systems of lower order odes. A combination of power series and exponential function is used as an approximate solution to the general third order ode problems. Continuous linear multistep method is developed by interpolating the basis function at both grid and off-grid points and collocating the differential function at only grid points. The unknown parameters in the system of linear equations arising from the collocation and interpolation functions were determined and the values substituted in the approximate solution to the problem. The required continuous method is obtained after necessary simplification. The derived method is tested and found to be consistent, symmetric and of low error constant. The results obtained showed a better performance than the existing methods in literature under review.

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“…Third-order ordinary differential equations (ODEs) are widely applied in the fields of engineering and physic, such as thermohaline convection, incompressible Newtonian fluid, turbulent transport of viscoelastic fluids in penetrable channels, jerk mechanical system with jerk curves, thin film flow systems and other disciplines (Herrera, 2019;Allogmany & Ismail, 2020).…”

Section: Introductionmentioning

confidence: 99%

Efficient Frequency-Dependent Coefficients of Explicit Improved Two-Derivative Runge-Kutta Type Methods for Solving Third-Order IVPs

Chien¹,

Senu²,

Ahmadian³

et al. 2023

JST

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This study aims to propose sixth-order two-derivative improved Runge-Kutta type methods adopted with exponentially-fitting and trigonometrically-fitting techniques for integrating a special type of third-order ordinary differential equation in the form u^''' (t)=f(t,u(t),u^' (t)). The procedure of constructing order conditions comprised of a few previous steps, k-i for third-order two-derivative Runge-Kutta-type methods, has been outlined. These methods are developed through the idea of integrating initial value problems exactly with a numerical solution in the form of linear composition of the set functions e^ѡt and e^(-ѡt)for exponentially fitted and e^iѡt and e^(-iѡt) for trigonometrically-fitted with ѡ ϵ R. Parameters of two-derivative Runge-Kutta type method are adapted into principle frequency of exponential and oscillatory problems to construct the proposed methods. Error analysis of proposed methods is analysed, and the computational efficiency of proposed methods is demonstrated in numerical experiments compared to other existing numerical methods for integrating third-order ordinary differential equations with an exponential and periodic solution.

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Implicit Three-Point Block Numerical Algorithm for Solving Third Order Initial Value Problem Directly with Applications

Cited by 11 publications

References 33 publications

Modified Fourth Derivative Block Method and its direct applications to third-order initial value problems

Modified Fourth Derivative Block Method and its direct applications to third-order initial value problems

Three-step Four-point Optimized Hybrid Block Method for Direct Solution of General Third Order Differential Equations

Efficient Frequency-Dependent Coefficients of Explicit Improved Two-Derivative Runge-Kutta Type Methods for Solving Third-Order IVPs

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