EXPLICIT EXACT SOLUTIONS TO THE (2+1) DIMENSIONAL SINH-POISSON EQUATION

2010 ◽  
Vol 24 (17) ◽  
pp. 3395-3409
Author(s):  
YUANXI XIE ◽  
SHIYU PENG
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Exact Solutions ◽  
Poisson Equation ◽  
Variable Separation ◽  
Simple Manner

By introducing an auxiliary ordinary differential equation and solving it by the method of variable separation, many explicit exact solutions of the (2+1) dimensional sinh-Poisson equation are presented in a simple manner.

EXPLICIT AND EXACT SOLUTIONS TO THE mKdV-SINE-GORDON EQUATION

2008 ◽  
Vol 22 (15) ◽  
pp. 1471-1485 ◽  
Author(s):  
YUANXI XIE
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Exact Solutions ◽  
Gordon Equation ◽  
Variable Separation ◽  
Sine Gordon Equation ◽  
Simple Manner

By introducing an auxiliary ordinary differential equation and solving it by the method of variable separation, rich types of explicit and exact solutions of the mKdV-sine-Gordon equation are presented in a simple manner.

scholarly journals The Cauchy Problem for the Generalized Hyperbolic Novikov–Veselov Equation via the Moutard Symmetries

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2113
Author(s):  
Alla A. Yurova ◽  
Artyom V. Yurov ◽  
Valerian A. Yurov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cauchy Problem ◽  
Exact Solutions ◽  
Airy Function ◽  
The Real ◽  
The Cauchy Problem

We begin by introducing a new procedure for construction of the exact solutions to Cauchy problem of the real-valued (hyperbolic) Novikov–Veselov equation which is based on the Moutard symmetry. The procedure shown therein utilizes the well-known Airy function Ai(ξ) which in turn serves as a solution to the ordinary differential equation d2zdξ2=ξz. In the second part of the article we show that the aforementioned procedure can also work for the n-th order generalizations of the Novikov–Veselov equation, provided that one replaces the Airy function with the appropriate solution of the ordinary differential equation dn−1zdξn−1=ξz.

The Kerr metric and stationary axisymmetric gravitational fields

1978 ◽  
Vol 358 (1695) ◽  
pp. 405-420 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Exact Solutions ◽  
Event Horizon ◽  
Kerr Metric ◽  
Symmetric Pair ◽  
General Solutions ◽  
Gravitational Fields ◽  
First Order ◽  
Metric Functions

A treatment of Einstein’s equations governing vacuum gravitational fields which are stationary and axisymmetric is shown to divide itself into three parts: a part essentially concerned with a choice of gauge (which can be chosen to ensure the occurrence of an event horizon exactly as in the Kerr metric); a part concerned with two of the basic metric functions which in two combinations satisfy a complex equation (Ernst’s equation) and in one combination satisfies a symmetric pair of real equations; and a third part which completes the solution in terms of a single ordinary differential equation of the first order. The treatment along these lines reveals many of the inner relations which characterize the general solutions, provides a derivation of the Kerr metric which is direct and verifiable at all stages, and opens an avenue towards the generation of explicit classes of exact solutions (an example of which is given).

Analysis of a Nonlinear Model of Heat Transfer Through a Rectangular Fin: Exact Solutions and Their Multiplicity

2016 ◽  
Vol 12 (2) ◽  
Author(s):  
Mohammad Danish ◽  
Shashi Kumar ◽  
Surendra Kumar
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Thermal Conductivity ◽  
Ordinary Differential Equation ◽  
Exact Solutions ◽  
Reaction Diffusion ◽  
Substitution Method ◽  
Porous Catalyst ◽  
Exact Analytical Solutions ◽  
The Heat Transfer Coefficient

Abstract Exact analytical solutions for the temperature profile and the efficiency of a nonlinear rectangular fin model have been obtained in the forms of well-known algebraic/non-algebraic functions. In the considered nonlinear fin model, the thermal conductivity and the heat transfer coefficient have been assumed to vary as distinct power-law functions of temperature thereby yielding a nonlinear BVP in a 2nd order ODE (ordinary differential equation). These exact solutions have been obtained by employing the derivative substitution method which not only include the solutions of previously studied simplified cases of the same problem but also the solutions of a similar problem of reaction-diffusion process occurring in a porous catalyst slab. These exact solutions have been successfully validated against their numerical counterparts. Besides, effects of various parameters on the obtained solutions have been studied, and the conditions for their existence, uniqueness/multiplicity and stability/instability are analyzed and discussed in detail.

scholarly journals On finite series solutions of conformable time-fractional Cahn-Allen equation

2020 ◽  
Vol 9 (1) ◽  
pp. 194-200 ◽  
Author(s):  
Asim Zafar ◽  
Hadi Rezazadeh ◽  
Khalid K. Ali
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Exact Solutions ◽  
Fractional Order ◽  
Symbolic Computation ◽  
Nonlinear Ordinary Differential Equation ◽  
Hyperbolic Function ◽  
Series Solutions ◽  
Finite Series ◽  
New Exact Solutions

AbstractThe aim of this article is to derive new exact solutions of conformable time-fractional Cahn-Allen equation. We have achieved this aim by hyperbolic function and expa function methods with the aid of symbolic computation using Mathematica. This idea seems to be very easy to employ with reliable results. The time fractional Cahn-Allen equation is reduced to respective nonlinear ordinary differential equation of fractional order. Also, we have depicted graphically the constructed solutions.

Exact solutions to the space-time fraction Whitham–Broer–Kaup equation

2020 ◽  
Vol 34 (16) ◽  
pp. 2050178
Author(s):  
Damin Cao ◽  
Cheng Li ◽  
Fajiang He
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Space Time ◽  
Polynomial Discriminant

The objective work of this paper is to transform the nonlinear space-time fraction Whitham–Broer–Kaup equation into ordinary differential equation by using the conformal fractional derivative, and find the exact solutions through the complete polynomial discriminant system. At the same time, we build the appropriate solution for the identified parameters to show the existence of the solution. In addition, we provide the 3D and 2D graphics to show that the solutions are real and effective.

scholarly journals Symmetry properties and exact solutions of the time fractional Kolmogorov-Petrovskii-Piskunov equation

2019 ◽  
Vol 65 (5 Sept-Oct) ◽  
pp. 529 ◽  
Author(s):  
M. S. Hasheim ◽  
M. Inc ◽  
M. Bayram
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Lie Symmetry ◽  
Conformable Fractional Derivative ◽  
Symmetry Approach ◽  
Simplest Equation Method ◽  
Symmetry Properties ◽  
Lie Symmetry Approach

In this paper, the time fractional Kolmogorov-Petrovskii-Piskunov (FKP) equation is analyzed by means of Lie symmetry approach. The FKP is reduced to ordinary differential equation of fractional order via the attained point symmetries. Moreover, the simplest equation method is used in construct the exact solutions of underlying equation with recently introduced conformable fractional derivative.

Regarding Rational Exponential and Hyperbolic Functions Solutions of Conformable Time-Fractional Phi-four Equation

2018 ◽  
Author(s):  
Asim Zafar ◽  
Alper Korkmaz ◽  
Bushra Khalid ◽  
Hadi Rezazadeh
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Function Method ◽  
Wave Transformation ◽  
Hyperbolic Function ◽  
Hyperbolic Functions ◽  
Hyperbolic Function Method ◽  
Projected Methods

In this study, we actually want to explore the time-fractional Phi-four equation via two methods, the exp a function method and the hyperbolic function method. We transform a fractional order dierential equation into an ordinary differential equation using a wave transformation and the fractional derivative in conformable form. Then, the resulting equation has successfully been explored for new explicit exact solutions. The procured solutions are simply showed the effectiveness and plainness of the projected methods.

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