A tenth order<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si131.gif" overflow="scroll"><mml:mi mathvariant="script">A</mml:mi></mml:math>-stable two-step hybrid block method for solving initial value problems of ODEs

Solving stiff, singular, and singularly perturbed initial value problems (IVPs) has always been challenging for researchers working in different fields of science and engineering. In this research work, an attempt is made to devise a family of nonlinear methods among which second‐ to fourth‐order methods are not only 𝒜− stable but 𝒜− acceptable as well under order stars' conditions. These features make them more suitable for solving stiff and singular systems in ordinary differential equations. Methods with remaining orders are either zero‐ or conditionally stable. The theoretical analysis contains local truncation error, consistency, and order of accuracy of the proposed nonlinear methods. Furthermore, both fixed and variable stepsize approaches are introduced wherein the latter improves the performance of the devised methods. The applicability of the methods for solving the system of IVPs is also described. When used to solve problems from physical and real‐life applications, including nonlinear logistic growth and stiff model for flame propagation, the proposed methods are found to have good results.

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“…Problem Consider the following nonlinear stiff system: 56

\begin{align} y_{1}^{'} false(t false) = prefix- 200 y_{2}^{2} false(t false), y_{1} false(0 false) = 1, \\ y_{2}^{'} false(t false) = prefix- 100 y_{2} false(t false) y_{2} false(0 false) = 1, \end{align}

with exact solution

y_{1} (t) = \exp (- 200 t), y_{2} (t) = \exp (- 100 t)

and

0 \leq t \leq 1.0

.…”

Section: Numerical Dynamics With Results and Discussionmentioning

confidence: 99%

A new family of 𝒜− acceptable nonlinear methods with fixed and variable stepsize approach

Qureshi

Soomro

Hınçal³

2021

Comp and Math Methods

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“…Block methods contain two parts viz: main and complementary methods [16]. Some of the advantages of block methods include but not limited to permission of easy change of step length [17], selfstarting and thus avoiding the use of other method(s) to get the starting solution, overcoming the overlapping of pieces of solution and obtaining numerical solution at more than one point at a time [18].…”

Section: Original Research Articlementioning

confidence: 99%

Trigonometrically Fitted Block Backward Differentiation Methods for First Order Initial Value Problems with Periodic Solution

Abdulganiy

2018

JAMCS

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A family of Trigonometrically Fitted Backward Differentiation Formula (TBDF) whose coefficients depend on the frequency and step size for periodic initial value problems is presented. The method is constructed based on collocation techniques. The primary method of TBDF is obtained from its continuous version while the additional methods are derived from the derivative of the continuous version which are combined and applied in block form as simultaneous numerical integrators. The stability properties of the method are discussed and numerical experiments show that the TBDF is an accurate numerical integrator.

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“…Block methods have the advantage over Runge-Kutta methods that they are less expensive in terms of number of function evaluations for a given order. In recent years, work on block methods (Singh et al 2019;Ramos and Singh 2017;Singla et al 2021) has drawn attention of many researchers.…”

Section: Introductionmentioning

confidence: 99%

Using a cubic B-spline method in conjunction with a one-step optimized hybrid block approach to solve nonlinear partial differential equations

2021

Self Cite

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In this paper, we develop an optimized hybrid block method which is combined with a modified cubic B-spline method, for solving non-linear partial differential equations. In particular, it will be applied for solving three well-known problems, namely, the Burgers equation, Buckmaster equation and FitzHugh–Nagumo equation. Most of the developed methods in the literature for non-linear partial differential equations have not focused on optimizing the time step-size and a very small value must be considered to get accurate approximations. The motivation behind the development of this work is to overcome this trade-off up to much extent using a larger time step-size without compromising accuracy. The optimized hybrid block method considered is proved to be A-stable and convergent. Furthermore, the obtained numerical approximations have been compared with exact and numerical solutions available in the literature and found to be adequate. In particular, without using quasilinearization or filtering techniques, the results for small viscosity coefficient for Burgers equation are found to be accurate. We have found that the combination of the two considered methods is computationally efficient for solving non-linear PDEs.

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A tenth orderA-stable two-step hybrid block method for solving initial value problems of ODEs

Cited by 21 publications

References 9 publications

A new family of 𝒜− acceptable nonlinear methods with fixed and variable stepsize approach

A new family of 𝒜− acceptable nonlinear methods with fixed and variable stepsize approach

Trigonometrically Fitted Block Backward Differentiation Methods for First Order Initial Value Problems with Periodic Solution

Using a cubic B-spline method in conjunction with a one-step optimized hybrid block approach to solve nonlinear partial differential equations

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