A New Fifth-Order Weighted Runge-Kutta Algorithm Based on Heronian Mean for Initial Value Problems in Ordinary Differential Equations

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 IVP for the Logistic growth: 54

y^{'} (t) = \frac{y (t)}{4} (1 - \frac{y (t)}{20}), y (0) = 1, 0 \leq t \leq 2.0,

with exact solution

y (t) = \frac{20}{1 + 19 \exp (- \frac{t}{4})} .

…”

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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“…Having explained the entire process, we discuss the adaptive step-size approach in the following way. From (43), we obtain: .…”

Section: Adaptive Step-size Approachmentioning

confidence: 99%

A New Nonlinear Hybrid Technique with fixed and adaptive step-size approaches

Soomro¹

2022

Sigma J Eng Nat Sci - Sigma Müh Fen Bil Derg

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Linear and nonlinear numerical techniques are the most popular techniques for finding approximate solutions to initial value problems in numerous scientific fields. Due to the substantial importance of ordinary differential equations, an attempt has been made in the present research study to obtain a new nonlinear hybrid technique based upon contraharmonic and harmonic means having fourth-order accuracy. Theoretical analysis in terms of consistency, stability, asymptotic errors (local and global truncation errors), and convergence has also been carried out. The newly formulated technique is compared with some existing techniques having the same characteristics and observed to be much better because of errors, CPU time, and stability region. The adaptive step-size approach improves the performance of the proposed technique, and strategies to control the errors are developed. Some numerical experiments for scalar and vector initial value problems, including logistic growth, sinusoidal and industrial Robot Arm systems, are presented to show better performance of the proposed technique.

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“…The a nonlinear mid-point rule formula based on geometric means (GM) for the numerical solution of differential equations ( , ) y f x y ′ = was presented [11] with supporting numerical results. However, the New fifth order weighted Runge -kutta methods based on the Heronian mean for initial value problems in ordinary differential equations was developed and implemented [12]. In the paper Comparisons in terms of numerical accuracy and size of the stability region between new proposed Runge-Kutta(5,5) algorithm, Runge-Kutta (5,5) based on Harmonic Mean, Runge-Kutta (5,5) based on Contra Harmonic Mean and Runge-Kutta (5,5) based on Geometric Mean where also investigated.…”

Section: Introductionmentioning

confidence: 99%

Solution of an Initial Value Problemin Ordinary Differential Equations Using the Quadrature Algorithm Based on the Heronian Mean

Etin-Osa¹

2019

IJAMTP

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Over the years, the Quadrature Algorithm as a method of solving initial value problems in ordinary differential equations is known to be of low accuracy compared to other well known methods. However, It has been shown that the method perform well when applied to moderately stiff problems. In this present study, the nonlinear method based on the Heronian Mean (HeM), of the function value for the solution of initial value problems is developed. Stability investigation is in agreement with the known Trapezoidal method.

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A New Fifth-Order Weighted Runge-Kutta Algorithm Based on Heronian Mean for Initial Value Problems in Ordinary Differential Equations

Cited by 5 publications

References 13 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

A New Nonlinear Hybrid Technique with fixed and adaptive step-size approaches

Solution of an Initial Value Problemin Ordinary Differential Equations Using the Quadrature Algorithm Based on the Heronian Mean

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