Development of a Nonlinear Hybrid Numerical Method

Aliya, Tasneem; Shaikh, Asif Ali; Qureshi, Sania

doi:10.17654/de019030275

Cited by 17 publications

(14 citation statements)

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“…The logarithm of absolute errors for the solutions obtained is compared with other methods discussed in [12] as given in Figure 4.…”

Section: Problemmentioning

confidence: 99%

“…Since rational functions are more appropriate for the representation of functions close to singularities than polynomials, the limitation is overcome by a local representation of the theoretical solution with a rational expression. Interestingly, this approach appears to be promising as several methods are now being constructed in this direction [6] [12] showed that solution around singularity point are well approximated by this approach. In this work, an explicit single-step nonlinear method involving higher derivatives of the state function for solving (1) is presented.…”

Section: Introductionmentioning

confidence: 99%

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An Explicit Single-Step Nonlinear Numerical Method for First Order Initial Value Problems (IVPs)

Bakre¹,

Wusu²,

Akanbi³

2020

JAMP

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Interest in the construction of efficient methods for solving initial value problems that have some peculiar properties with it or its solution is recently gaining wide popularity. Based on the assumption that the solution is representable by nonlinear trigonometric expressions, this work presents an explicit single-step nonlinear method for solving first order initial value problems whose solution possesses singularity. The stability and convergence properties of the constructed scheme are also presented. Implementation of the new method on some standard test problems compared with those discussed in the literature proved its accuracy and efficiency.

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“…The logarithm of absolute errors for the solutions obtained is compared with other methods discussed in [12] as given in Figure 4.…”

Section: Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An Explicit Single-Step Nonlinear Numerical Method for First Order Initial Value Problems (IVPs)

Bakre¹,

Wusu²,

Akanbi³

2020

JAMP

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show abstract

“…It is challenging to find the exact solution of differential equation‐based models in various situations, mainly when the problem is either nonlinear, stiff, singular, or singularly perturbed. In these conditions, we move toward the nonlinear, trigonometrically‐fitted, Runge–Kutta collocation, and BDF‐type Chebyshev numerical methods 17‐24 . After being motivated with research works recently carried out for devising or modifying nonlinear numerical methods that are suitable for stiff and singular IVPs, we attempt to derive a new family of nonlinear methods with second to the fourth‐order of accuracy, and

𝒜 -

stability feature of the methods is also established.…”

Section: Introductionmentioning

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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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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“…In certain cases, however, analytical methods are not capable of solving such complicated or complex differential equations. Numerical methods are used to achieve the solution to the complicated differential equations [2,3]. With the help of computer programming, numerical methods are very valuable tools for solving complex problems very quickly.…”

Section: Introductionmentioning

confidence: 99%

New Modification on Heun’s Method Based on Contraharmonic Mean for Solving Initial Value Problems with High Efficiency

Workie

2020

Journal of Mathematics

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In this paper, small modification on Improved Euler’s method (Heun’s method) is proposed to improve the efficiency so as to solve ordinary differential equations with initial condition by assuming the tangent slope as an average of the arithmetic mean and contra-harmonic mean. In order to validate the conclusion, the stability, consistency, and accuracy of the system were evaluated and numerical results were presented, and it was recognized that the proposed method is more stable, consistent, and accurate with high performance.

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Development of a Nonlinear Hybrid Numerical Method

Cited by 17 publications

References 0 publications

An Explicit Single-Step Nonlinear Numerical Method for First Order Initial Value Problems (IVPs)

An Explicit Single-Step Nonlinear Numerical Method for First Order Initial Value Problems (IVPs)

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

New Modification on Heun’s Method Based on Contraharmonic Mean for Solving Initial Value Problems with High Efficiency

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