Modified Parker-Sochacki method for solving nonlinear oscillators

Abdelrazik, Ismail M.; Elkaranshawy, Hesham A.; Abdelrazek, Amr M.

doi:10.1080/15397734.2016.1201425

Cited by 5 publications

(10 citation statements)

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“…It is suitable for approximating a divergent series function. e approximant is derived by expanding the function as a ratio of two power series and determining both the numerator and denominator coefficients [28][29][30][31][32][33][34]. If we have a function f(u) that can be represented in a power series form as…”

Section: Laplace-padé Resummationmentioning

confidence: 99%

“…Power series solutions are first obtained for the system of ODE, and then, the Laplace-Padé resummation method (PSLP) is used to obtain the entailed approximate analytical solution. e technique used depends on the methodology developed in [27][28][29][30][31][32][33][34] for the general system of ODE. e solution is used for the analysis of the standard viral dynamic model of HCV after any type of DAA treatment initiation.…”

Section: Introductionmentioning

confidence: 99%

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Innovative Approximate Analytical Solution for Standard Model of Viral Dynamics: Hepatitis C with Direct‐Acting Agents as an Implemented Case

Elkaranshawy

Ezzat

Abouelseoud

et al. 2019

Mathematical Problems in Engineering

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In this article, a novel approximate analytical solution is presented for solving the standard viral dynamic model. Basically, the standard model is used to study viral dynamics in patients for a wide range of viruses like HIV, HPV, HBV, and HCV. In this research work, the standard model for hepatitis C virus (HCV) is considered in detail; however, the analysis and results can be applicable for all other viruses. This standard model is used to study viral dynamics in patients treated with direct-acting antiviral agents (DAAs). Power series solution combined with Laplace–Padé resummation method (PSLP) is used to obtain the approximate analytical solutions for the model. To test the capability as well as the validity of the proposed method, results are compared with available viral load data published in the literature and with published simulation results. Good fits are obtained in the comparison for all cases considered. Given that medical specialists and physicians are more interested in solutions that yield direct and simple predictions, it is expected that the proposed approximate analytical solution would be attractive to them and help them to obtain a straightforward and a proper estimation regarding the viral load due to variations in treatment and/or patient’s parameters.

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Section: Laplace-padé Resummationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Innovative Approximate Analytical Solution for Standard Model of Viral Dynamics: Hepatitis C with Direct‐Acting Agents as an Implemented Case

Elkaranshawy

Ezzat

Abouelseoud

et al. 2019

Mathematical Problems in Engineering

Self Cite

View full text Add to dashboard Cite

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“…When N tends to infinity, an exact solution can be obtained. Abdelrazik et al 2 actually searched for an approximate solution in the form…”

Section: Periodic Solution Versus Chaotic Solutionmentioning

confidence: 99%

“…Nonlinear vibrations arise everywhere in engineering, and it is of great importance to accurately predict its frequency. There are many analytical methods for this purpose, for example, the Parker–Sochacki method, 1,2 the homotopy perturbation method, 3–7 the variational iteration method, 8–13 the variational principle 14–17 and the fixed point theory, 18–21 just to name a few, among which the homotopy perturbation method is the most powerful tool to nonlinear oscillators, while the Parker–Sochacki method always ends in a wrong result. In this paper, we will study a nonlinear oscillator with high nonlinearity and point out some misunderstandings in the analytical approach to nonlinear oscillators.…”

Section: Introductionmentioning

confidence: 99%

Periodic solution to nonlinear oscillators without artificial noises by the homotopy perturbation method

2018

Journal of Low Frequency Noise, Vibration and Active Control

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“…For example, PSM was used as the primary solver in [38] for gathering simulation data in testing the times of firing events in neurons and scheduling algorithms used for synaptic event delivery. See [31,30,1,2,4,5,6,11,13,17,20,21,23,26,27,32,33,35,37,39] for a sample of recent studies that report the advantages (and some disadvantages) of PSM. PSM and work from the Automatic Differentiation (AD) community have developed nearly simultaneously [9,16,25], and there has been a number of benefits in collaborations between these two communities, including strategies in this work.…”

mentioning

confidence: 99%

An Adaptive, Highly Accurate and Efficient, Parker-Sochacki Algorithm for Numerical Solutions to Initial Value Ordinary Differential Equation Systems

Guenther¹

2019

SIURO

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The Parker-Sochacki Method (PSM) allows the numerical approximation of solutions to a polynomial initial value ordinary differential equation or system (IVODE) using an algebraic power series method. PSM is equivalent to a modified Picard iteration and provides an efficient, recursive computation of the coefficients of the Taylor polynomial at each step. To date, PSM has largely concentrated on fixed step methods. We develop and test an adaptive stepping scheme that, for many IVODEs, enhances the accuracy and efficiency of PSM. PSM Adaptive (PSMA) is compared to its fixed step counterpart and to standard Runge-Kutta (RK) foundation algorithms using three example IVODEs. In comparison, PSMA is shown to be competitive, often outperforming these methods in terms of accuracy, number of steps, and execution time. A library of functions is also presented that allows access to PSM techniques for many non-polynomial IVODEs without having to first rewrite these in the necessary polynomial form, making PSM a more practical tool.

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Modified Parker-Sochacki method for solving nonlinear oscillators

Cited by 5 publications

References 38 publications

Innovative Approximate Analytical Solution for Standard Model of Viral Dynamics: Hepatitis C with Direct‐Acting Agents as an Implemented Case

Innovative Approximate Analytical Solution for Standard Model of Viral Dynamics: Hepatitis C with Direct‐Acting Agents as an Implemented Case

Periodic solution to nonlinear oscillators without artificial noises by the homotopy perturbation method

An Adaptive, Highly Accurate and Efficient, Parker-Sochacki Algorithm for Numerical Solutions to Initial Value Ordinary Differential Equation Systems

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