scholarly journals Lie Symmetries, Painlev\’{e} analysis and global dynamics for the temporal equation of radiating stars

2021 ◽  
Author(s):  
Genly Leon ◽  
Megandhren Govender ◽  
Paliathanasis Andronikos
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Master Equation ◽  
Similarity Solution ◽  
Analytic Solution ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Global Dynamics ◽  
Radiating Stars ◽  
The Given

We study the temporal equation of radiating stars by using three powerful methods for the analysis of nonlinear differential equations. Specifically, we investigate the global dynamics for the given master ordinary differential equation to understand the evolution of solutions for various initial conditions as also to investigate the existence of asymptotic solutions. Moreover, with the application of Lie’s theory, we can reduce the order of the master differential equation, while an exact similarity solution is determined. Finally, the master equation possesses the Painlev\’{e} property, which means that the analytic solution can be expressed in terms of a Laurent expansion.

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

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.

Resonance and non-resonance in terms of average values. II

2001 ◽  
Vol 131 (5) ◽  
pp. 1217-1235
Author(s):  
M. N. Nkashama ◽  
S. B. Robinson
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Nonlinear Term ◽  
Spectral Gap ◽  
Elliptic Boundary ◽  
Sufficient Conditions ◽  
Existence Results ◽  
Elliptic Boundary Value ◽  
Semilinear Elliptic ◽  
The Given

We prove existence results for semilinear elliptic boundary-value problems in both the resonance and non-resonance cases. What sets our results apart is that we impose sufficient conditions for solvability in terms of the (asymptotic) average values of the nonlinearities, thus allowing the nonlinear term to have significant oscillations outside the given spectral gap as long as it remains within the interval on the average in some sense. This work generalizes the results of a previous paper, which dealt exclusively with the ordinary differential equation (ODE) case and relied on ODE techniques.

On the Approximate Solutions of Heat-Conduction Problems

1963 ◽  
Vol 85 (3) ◽  
pp. 203-207 ◽  
Author(s):  
Fazil Erdogan
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Approximate Solutions ◽  
Integral Transforms ◽  
Partial Differential ◽  
The Given

Integral transforms are used in the application of the weighted residual methods to the solution of problems in heat conduction. The procedure followed consists in reducing the given partial differential equation to an ordinary differential equation by successive applications of appropriate integral transforms, and finding its solution by using the weighted-residual methods. The undetermined coefficients contained in this solution are functions of transform variables. By inverting these functions the coefficients are obtained as functions of the actual variables.

scholarly journals Scalar-Torsion Theories in Four Dimensional Static Spacetimes

2021 ◽  
Vol 32 (2) ◽  
pp. 12-15
Author(s):  
Mulyanto . ◽  
Fiki Taufik Akbar ◽  
Bobby Eka Gunara
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Exact Solution ◽  
Master Equation ◽  
Equations Of Motion ◽  
Scalar Potential ◽  
Torsion Theory ◽  
Linear Ordinary Differential Equation ◽  
Four Dimensions ◽  
Torsion Theories

In this paper, we consider a class of static spacetimes scalar-torsion theories in four dimensioanal static spacetimes with the scalar potential turned on. We discover that the 2-dimensional submanifold must admit constant triplet structures, one of which is the torsion scalar. This indicates that these equations of motion can be reduced to a single highly non-linear ordinary differential equation known as the master equation. Then, we show that there are no exact solution of the scalar-torsion theory in four dimensions considering the Sinh-Gordon potential.

Similarity solutions and viscous gravity current adjustment times

2019 ◽  
Vol 874 ◽  
pp. 285-298 ◽  
Author(s):  
Thomasina V. Ball ◽  
Herbert E. Huppert
Keyword(s):  
Differential Equation ◽  
Similarity Solution ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Gravity Current ◽  
Similarity Solutions ◽  
Partial Differential ◽  
Initial Shape ◽  
Wide Range

A wide range of initial-value problems in fluid mechanics in particular, and in the physical sciences in general, are described by nonlinear partial differential equations. Recourse must often be made to numerical solutions, but a powerful, well-established technique is to solve the problem in terms of similarity variables. A disadvantage of the similarity solution is that it is almost always independent of any specific initial conditions, with the solution to the full differential equation approaching the similarity solution for times $t\gg t_{\ast }$, for some $t_{\ast }$. But what is $t_{\ast }$? In this paper we consider the situation of viscous gravity currents and obtain useful formulae for the time of approach, $\unicode[STIX]{x1D70F}(p)$, for a number of different initial shapes, where $p$ is the percentage disagreement between the radius of the current as determined by the full numerical solution of the governing partial differential equation and the similarity solution normalised by the similarity solution. We show that for any initial shape of volume $V,\unicode[STIX]{x1D70F}\propto 1/(\unicode[STIX]{x1D6FD}V^{1/3}\unicode[STIX]{x1D6FE}_{0}^{8/3}p)$ (as $p\downarrow 0$), where $\unicode[STIX]{x1D6FD}=g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/(3\unicode[STIX]{x1D707})$, with $g$ representing the acceleration due to gravity, $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ the density difference between the gravity current and the ambient, $\unicode[STIX]{x1D707}$ the dynamic viscosity of the fluid that makes up the gravity current and $\unicode[STIX]{x1D6FE}_{0}$ the initial aspect ratio. This framework can used in many other situations, including where it is not an initial condition (in time) that is studied but one valid for specified values at a special spatial coordinate.

scholarly journals The Novel Integral Homotopy Expansive Method

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1204
Author(s):  
Uriel Filobello-Nino ◽  
Hector Vazquez-Leal ◽  
Jesus Huerta-Chua ◽  
Jaime Ramirez-Angulo ◽  
Darwin Mayorga-Cruz ◽  
...  
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
The Novel ◽  
Practical Applications ◽  
Linear And Nonlinear ◽  
Scientific Phenomena ◽  
Analytical Expressions

This work proposes the Integral Homotopy Expansive Method (IHEM) in order to find both analytical approximate and exact solutions for linear and nonlinear differential equations. The proposal consists of providing a versatile method able to provide analytical expressions that adequately describe the scientific phenomena considered. In this analysis, it is observed that the proposed solutions are compact and easy to evaluate, which is ideal for practical applications. The method expresses a differential equation as an integral equation and expresses the integrand of the equation in terms of a homotopy. As a matter of fact, IHEM will take advantage of the homotopy flexibility in order to introduce adjusting parameters and convenient functions with the purpose of acquiring better results. In a sequence, another advantage of IHEM is the chance to distribute one or more of the initial conditions in the different iterations of the proposed method. This scheme is employed in order to introduce some additional adjusting parameters with the purpose of acquiring accurate analytical approximate solutions.

On the existence and uniqueness of a positive solution to a boundary value problem for a nonlinear ordinary differential equation of even order

2021 ◽  
pp. 341-347
Author(s):  
Gusen E. Abduragimov ◽  
Patimat E. Abduragimova ◽  
Madina M. Kuramagomedova
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Ordinary Differential Equation ◽  
Even Order

In the article, we consider a boundary value problem for a nonlinear ordinary differential equation of even order which, obviously, has a trivial solution. Sufficient conditions for the existence and uniqueness of a positive solution to this problem are obtained. With the help of linear transformations of T. Y. Na [T. Y. Na, Computational Methods in Engineering Boundary Value Problems, Acad. Press, NY, 1979, ch. 7], the boundary value problem is reduced to the Cauchy problem, the initial conditions of which make it possible to uniquely determine the transformation parameter. It is shown that the transformations of T. Y. Na uniquely determine the solution of the original problem. In addition, based on the proof of the uniqueness of a positive solution to the boundary value problem, a sufficiently effective non–iterative numerical algorithm for constructing such a solution is obtained. A corresponding example is given.

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