Approximate solution of some nonlinear problems of heat conduction theory

Abstract The paper deals with the application of a strong method called the modified Mickens iteration technique which is used for solving a strongly nonlinear system. The system describes the motion of a simple mathematical pendulum with a particle attached to it through a stretched wire. This model has great applications especially in the area of nonlinear vibrations and oscillation systems. The proposed method depends on determining the frequency and amplitude of the system through the modified Mickens iterative approach which is a modification of the regular Mickens approach. The preliminaries of the proposed technique are present and the application to the model is discussed. The method depends on the Mickens iteration approach which transforms the considered equation into a linear form and then is solving this equation result in the approximate solution. Some examples are given to validate and illustrate the effectiveness and convenience of the method. These results are compared with other relative techniques from the literature in terms of finding the frequency of the two examined models. The method produces more accurate results when compared to these methods and is considered a strong candidate for solving other nonlinear problems with applications in science and engineering.

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Universality in heat conduction theory: weakly nonlocal thermodynamics

Annalen der Physik ◽

10.1002/andp.201200042 ◽

2012 ◽

Vol 524 (8) ◽

pp. 470-478 ◽

Cited By ~ 56

Author(s):

P. Ván ◽

T. Fülöp

Keyword(s):

Heat Conduction ◽

Conduction Theory

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Approximate solution of heat-conduction and thermoelasticity problems taking account of the inhomogeneity of the medium

Soviet Applied Mechanics ◽

10.1007/bf00884165 ◽

1980 ◽

Vol 16 (5) ◽

pp. 376-381 ◽

Cited By ~ 1

Author(s):

B. M. Lisitsyn ◽

K. B. Bulyga

Keyword(s):

Approximate Solution ◽

Heat Conduction

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A New Approach for Solving the Undamped Helmholtz Oscillator for the Given Arbitrary Initial Conditions and Its Physical Applications

Mathematical Problems in Engineering ◽

10.1155/2020/7876413 ◽

2020 ◽

Vol 2020 ◽

pp. 1-7

Author(s):

Alvaro H. Salas S ◽

Jairo E. Castillo H ◽

Darin J. Mosquera P

Keyword(s):

Differential Equation ◽

Approximate Solution ◽

Analytical Solution ◽

Initial Conditions ◽

Negative Ions ◽

Nonlinear Problems ◽

New Approach ◽

Weierstrass Elliptic Function ◽

Electronegative Plasma ◽

The Given

In this paper, a new analytical solution to the undamped Helmholtz oscillator equation in terms of the Weierstrass elliptic function is reported. The solution is given for any arbitrary initial conditions. A comparison between our new solution and the numerical approximate solution using the Range Kutta approach is performed. We think that the methodology employed here may be useful in the study of several nonlinear problems described by a differential equation of the form z ″ = F z in the sense that z = z t . In this context, our solutions are applied to some physical applications such as the signal that can propagate in the LC series circuits. Also, these solutions were used to describe and investigate some oscillations in plasma physics such as oscillations in electronegative plasma with Maxwellian electrons and negative ions.

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Heat Transfer with Melting and Freezing : 1st Report, An Approximate Solution of Phase Front by Heat Conduction, when Cooling-Surface Temperature is a Function of Time

Transactions of the Japan Society of Mechanical Engineers ◽

10.1299/kikai1938.39.1881 ◽

1973 ◽

Vol 39 (322) ◽

pp. 1881-1889 ◽

Cited By ~ 3

Author(s):

Tamotsu IGARASHI ◽

Masaru HIRATA ◽

Niichi NISHIWAKI

Keyword(s):

Heat Transfer ◽

Approximate Solution ◽

Heat Conduction ◽

Surface Temperature ◽

Phase Front ◽

Cooling Surface

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An optimal iteration method with application to the Thomas-Fermi equation

Open Physics ◽

10.2478/s11534-010-0059-z ◽

2011 ◽

Vol 9 (3) ◽

Cited By ~ 6

Author(s):

Vasile Marinca ◽

Nicolae Herişanu

Keyword(s):

Approximate Solution ◽

Approximate Method ◽

Iteration Method ◽

Nonlinear Problems ◽

Analytical Approximate Solution ◽

Thomas Fermi ◽

Strongly Nonlinear ◽

Good Agreement

AbstractThe aim of this paper is to introduce a new approximate method, namely the Optimal Parametric Iteration Method (OPIM) to provide an analytical approximate solution to Thomas-Fermi equation. This new iteration approach provides us with a convenient way to optimally control the convergence of the approximate solution. A good agreement between the obtained solution and some well-known results has been demonstrated. The proposed technique can be easily applied to handle other strongly nonlinear problems.

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