An embedded 3(2) pair of nonlinear methods for solving first order initial-value ordinary differential systems

Ramos, Higinio; Singh, Gurjinder; Kanwar, V.; Bhatia, Saurabh

doi:10.1007/s11075-016-0209-5

Cited by 16 publications

(12 citation statements)

References 25 publications

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“…To derive some high-order methods of this type, several techniques have been developed recently [11][12][13]. Here we show one general approach providing a simple way to obtain the Padé integration method of arbitrary order.…”

Section: Integration Methods Based On Padé Approximantsmentioning

confidence: 99%

“…The most promising application of these methods is the integration of problems with the singularity in the solution that can possess higher numerical efficiency in comparison to classical approaches [10][11][12]. Another attractive feature of these formulae is that they can achieve theoretical A-stability without implicit computations [13]. Although these methods deal well with singular nonlinear problems, no numerical evidence is found in the literature concerning their behavior in solving highly nonlinear ODEs, e.g., chaotic problems.…”

Section: Introductionmentioning

confidence: 99%

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The Effects of Padé Numerical Integration in Simulation of Conservative Chaotic Systems

Butusov

Каримов

Тутуева

et al. 2019

Entropy

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In this paper, we consider nonlinear integration techniques, based on direct Padé approximation of the differential equation solution, and their application to conservative chaotic initial value problems. The properties of discrete maps obtained by nonlinear integration are studied, including phase space volume dynamics, bifurcation diagrams, spectral entropy, and the Lyapunov spectrum. We also plot 2D dynamical maps to enlighten the features introduced by nonlinear integration techniques. The comparative study of classical integration methods and Padé approximation methods is given. It is shown that nonlinear integration techniques significantly change the behavior of discrete models of nonlinear systems, increasing the values of Lyapunov exponents and spectral entropy. This property reduces the applicability of numerical methods based on Padé approximation to the chaotic system simulation but it is still useful for construction of pseudo-random number generators that are resistive to chaos degradation or discrete maps with highly nonlinear properties.

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Section: Integration Methods Based On Padé Approximantsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Effects of Padé Numerical Integration in Simulation of Conservative Chaotic Systems

Butusov

Каримов

Тутуева

et al. 2019

Entropy

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“…The newly proposed scheme in ( 9) has been formulated with a constant stepsize h in the previous section. Nevertheless, as several authors such as [20] have pointed out, a numerical integrator must be more suitable for an adaptive step-size formulation that is based on an explicit formula. Now, the proposed scheme is discussed in adaptive step-size form by using a lower order approach to create a strategy to estimate the local error (LE n ) at the end-point on the interval.…”

Section: Adaptive Step-size Strategymentioning

confidence: 99%

A new nonlinear L-stable scheme with constant and adaptive step-size strategy

Arain¹,

Qureshi²,

Shaikh³

2021

J. Appl. Math. Comput. Mech.

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The present study proposes a new explicit nonlinear scheme that solves stiff and nonlinear initial value problems in ordinary differential equations. One of the promising features of this scheme is its fourth-order convergence with strong stability having an unbounded region. A modern approach for relative stability growth analysis is also presented under order stars conditions. The scheme is also good in dealing with singular and stiff type of models. Comparing numerical experiments using various errors, including maximum absolute global error over the integration interval, absolute error at the endpoint, average error, norm of errors, and the CPU times (seconds), shows better performance. An adaptive step-size approach seems to increase the performance of the proposed scheme. The numerical simulations assure us of L -stability, consistency, order, and rapid convergence.

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“…In this regard, many approximation methods have been studied for solving ODEs, for example using neural networks [20], embedded three and two pairs of nonlinear methods [21], electrical analogy [22], multi-general purpose graphical processing units [23], the differential transform method [24] and a Galerkin finite element method [25]. A numerical method based on the trapezoidal rule for the Cauchy-Smoluchowski problem was discussed by [26].…”

Section: Introductionmentioning

confidence: 99%

New Approximation Methods Based on Fuzzy Transform for Solving SODEs: I

ALKasasbeh

Perfilieva

Ahmad

et al. 2018

ASI

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In this paper, new approximation methods for solving systems of ordinary differential equations (SODEs) by fuzzy transform (FzT) are introduced and discussed. In particular, we propose two modified numerical schemes to solve SODEs where the technique of FzT is combined with one-stage and two-stage numerical methods. Moreover, the error analysis of the new approximation methods is discussed. Finally, numerical examples of the proposed approach are confirmed, and applications are presented.

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An embedded 3(2) pair of nonlinear methods for solving first order initial-value ordinary differential systems

Cited by 16 publications

References 25 publications

The Effects of Padé Numerical Integration in Simulation of Conservative Chaotic Systems

The Effects of Padé Numerical Integration in Simulation of Conservative Chaotic Systems

A new nonlinear L-stable scheme with constant and adaptive step-size strategy

New Approximation Methods Based on Fuzzy Transform for Solving SODEs: I

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