Continuous block backward differentiation formula for solving stiff ordinary differential equations

Akinfenwa, O. A.; Jator, S. N.; Yao, Nianmin

doi:10.1016/j.camwa.2012.03.111

Cited by 44 publications

(39 citation statements)

References 10 publications

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“…This problem has appeared in the work of Akinfenwa et al and Amodio and Mazzia among others. The integration interval was [0,1].…”

Section: Numerical Experimentsmentioning

confidence: 98%

Development of a new Runge‐Kutta method and its economical implementation

Ramos

2019

Comp and Math Methods

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A two-step hybrid block method is developed for numerically solving first-order initial-value problems. The formulas will be obtained from a continuous approximation derived via interpolation and collocation at different points where the two intermediate points are chosen through the optimization of the local truncation errors. The main characteristics of the method are discussed after considering its formulation in vector form. As with the Runge-Kutta (RK) methods, there is no need to provide starting values by using other approaches; thus, it is a self-starting method. The proposed method may be reformulated as a RK one, but in this way, its implementation requires more computational cost. Nevertheless, a straightforward reformulation of the method reducing the number of occurrences of the source term f results in the most efficient formulation. This strategy might be applied to the numerous existing block methods. Some numerical tests considering different problems that appeared in the literature show the performance of the presented schemes, confirming the best behavior of the proposed economical reformulation. KEYWORDSA-stability, efficient reformulation, first-order initial-value problem, hybrid block method, Runge-Kutta methodComp and Math Methods. 2019;1:e1016.wileyonlinelibrary.com/journal/cmm4

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“…This problem has appeared in the work of Akinfenwa et al and Amodio and Mazzia among others. The integration interval was [0,1].…”

Section: Numerical Experimentsmentioning

confidence: 98%

Development of a new Runge‐Kutta method and its economical implementation

Ramos

2019

Comp and Math Methods

View full text Add to dashboard Cite

show abstract

“…Some attention by some eminent scholars has been given to solve the problem directly, such as [1]- [4]. However, Awoyemi [5], Akinfenwa and Jator [6]developed methods of higher order ODEs directly. The techniques for solving the problem directly are (i) using interpolation and collocation technique, which is the most common, see [7]- [10], and (ii) using interpolation and integration technique.…”

Section: Introductionmentioning

confidence: 99%

Implicit Two-point Block Method with Third and Fourth Derivatives for Solving General Second Order ODEs

Allogmany¹,

Ismail²,

İbrahim³

2019

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In this paper we present an implicit two-point block method for solving directly the general second order ordinary differential equations (ODEs). The method incorporates the first and second derivatives of f(x, y, y ), which are the third and fourth derivatives of the solution. The method is derived using Hermite Interpolating Polynomial as the basic function. A performance comparison of the two-point block method is compared in term of accuracy to several existing methods, which have order almost equal or higher than that of the new method. Numerical results interpret the accuracy and efficacy of the new m ethod. Application of the new method is discussed.

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“…where h is step size, is a deterrmine quantity unambiguously set implying construction of k order (Akinfenwa et al, 2013;Oghonyon et al, 2015).…”

Section: Introductionmentioning

confidence: 99%

“…Section 3 programming codes for implementing block Milnes' device. Section 4 Conclusion as cited (Akinfenwa et al, 2013;Oghonyon et al, 2016).…”

Section: Introductionmentioning

confidence: 99%

Programming codes of block-Milne's device for solving fourth-order ODEs

Oghonyon¹,

Bishop²,

Eke³

2018

Int. j. adv. appl. sci.

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Block-Milne's device is an extension of block-predictor-corrector method and specifically developed to design a worthy step size, resolve the convergence criteria and maximize error. In this study, programming codes of blockMilne's device (P-CB-MD) for solving fourth order ODEs are considered. Collocation and interpolation with power series as the basic solution are used to devise P-CB-MD. Analysing the P-CB-MD will give rise to the principal local truncation error (PLTE) after determining the order. The P-CB-MD for solving fourth order ODEs is written using Mathematica which can be utilized to evaluate and produce the mathematical results. The P-CB-MD is very useful to demonstrate speed, efficiency and accuracy compare to manual computation applied. Some selected problems were solved and compared with existing methods. This was made realizable with the support of the named computational benefits.

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Continuous block backward differentiation formula for solving stiff ordinary differential equations

Cited by 44 publications

References 10 publications

Development of a new Runge‐Kutta method and its economical implementation

Development of a new Runge‐Kutta method and its economical implementation

Implicit Two-point Block Method with Third and Fourth Derivatives for Solving General Second Order ODEs

Programming codes of block-Milne's device for solving fourth-order ODEs

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