Finite Dimensional Dynamics and Exact Solutions of Burgers – Huxley Equation

2019 ◽  
Author(s):  
Alexei G. Kushner ◽  
Ruslan I. Matviychuk
Keyword(s):  
Exact Solutions ◽  
Finite Dimensional ◽  
Huxley Equation

scholarly journals Exact solutions of infinite dimensional total-variation regularized problems

2018 ◽  
Vol 8 (3) ◽  
pp. 407-443 ◽  
Author(s):  
Axel Flinth ◽  
Pierre Weiss
Keyword(s):  
Inverse Problems ◽  
Exact Solutions ◽  
Total Variation ◽  
Linear Mapping ◽  
Scattered Data Approximation ◽  
Convex Programs ◽  
Linear Measurements ◽  
Data Approximation ◽  
Infinite Dimensional ◽  
Finite Dimensional

Abstract We study the solutions of infinite dimensional inverse problems over Banach spaces. The regularizer is defined as the total variation of a linear mapping of the function to recover, while the data fitting term is a near arbitrary function. The first contribution describes the solution’s structure: we show that under mild assumptions, there always exists an $m$-sparse solution, where $m$ is the number of linear measurements of the signal. Our second contribution is about the computation of the solution. While most existing works first discretize the problem, we show that exact solutions of the infinite dimensional problem can be obtained by solving one or two consecutive finite dimensional convex programs depending on the measurement functions structures. We finish by showing an application on scattered data approximation. These results extend recent advances in the understanding of total-variation regularized inverse problems.

Exact solutions of the Burgers-Huxley equation

2004 ◽  
Vol 68 (3) ◽  
pp. 413-420 ◽  
Author(s):  
O.Yu. Yefimova ◽  
N.A. Kudryashov
Keyword(s):  
Exact Solutions ◽  
Huxley Equation

Exact solutions of the Rosenau–Hyman equation, coupled KdV system and Burgers–Huxley equation using modified transformed rational function method

2018 ◽  
Vol 32 (24) ◽  
pp. 1850282 ◽  
Author(s):  
Yong-Li Sun ◽  
Wen-Xiu Ma ◽  
Jian-Ping Yu ◽  
Chaudry Masood Khalique
Keyword(s):  
Exact Solutions ◽  
Rational Function ◽  
Function Method ◽  
Bernoulli Equation ◽  
Huxley Equation ◽  
Transformed Rational Function Method ◽  
The Given

In this research, we study the exact solutions of the Rosenau–Hyman equation, the coupled KdV system and the Burgers–Huxley equation using modified transformed rational function method. In this paper, the simplest equation is the Bernoulli equation. We are not only obtain the exact solutions of the aforementioned equations and system but also give some geometric descriptions of obtained solutions. All can be illustrated vividly by the given graphs.

scholarly journals Recursion Operator and Local and Nonlocal Symmetries of a New Modified KdV Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Qian Suping ◽  
Li Xin
Keyword(s):  
Exact Solutions ◽  
Kdv Equation ◽  
Symmetry Algebra ◽  
Original Model ◽  
Identity Transformation ◽  
Recursion Operator ◽  
Nonlocal Symmetries ◽  
Symmetry Reductions ◽  
Finite Dimensional ◽  
Modified Kdv Equation

The recursion operator of a new modified KdV equation and its inverse are explicitly given. Acting the recursion operator and its inverse on the trivial symmetry 0 related to the identity transformation, the infinitely many local and nonlocal symmetries are obtained. Using a closed finite dimensional symmetry algebra with both local and nonlocal symmetries of the original model, some symmetry reductions and exact solutions are found.

scholarly journals Dynamics and exact solutions of the generalized Harry Dym equation

2020 ◽  
Vol 12 (4) ◽  
pp. 50-59
Author(s):  
Ruslan Matviichuk
Keyword(s):  
Exact Solutions ◽  
Soliton Solutions ◽  
Third Order ◽  
Finite Dimensional ◽  
Scattering Transform ◽  
Painlevé Property ◽  
All Solutions ◽  
Petviashvili Equation ◽  
Dym Equation ◽  
Harry Dym Equation

The Harry Dym equation is the third-order evolutionary partial differential equation. It describes a system in which dispersion and nonlinearity are coupled together. It is a completely integrable nonlinear evolution equation that may be solved by means of the inverse scattering transform. It has an infinite number of conservation laws and does not have the Painleve property. The Harry Dym equation has strong links to the Korteweg – de Vries equation and it also has many properties of soliton solutions. A connection was established between this equation and the hierarchies of the Kadomtsev – Petviashvili equation. The Harry Dym equation has applications in acoustics: with its help, finite-gap densities of the acoustic operator are constructed. The paper considers a generalization of the Harry Dym equation, for the study of which the methods of the theory of finite-dimensional dynamics are applied. The theory of finite-dimensional dynamics is a natural development of the theory of dynamical systems. Dynamics make it possible to find families that depends on a finite number of parameters among all solutions of evolutionary differential equations. In our case, this approach allows us to obtain some classes of exact solutions of the generalized equation, and also indicates a method for numerically constructing solutions.

scholarly journals A New Numerical Scheme for the Generalized Huxley Equation

2016 ◽  
Vol 16 ◽  
pp. 105-111
Author(s):  
Bilge Inan
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Exact Solutions ◽  
Difference Method ◽  
Present Method ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
Alternative Method ◽  
Huxley Equation

In this paper, an implicit exponential finite difference method is applied to compute the numerical solutions of the nonlinear generalized Huxley equation. The numerical solutions obtained by the present method are compared with the exact solutions and obtained by other methods to show the efficiency of the method. The comparisons showed that proposed scheme is reliable, precise and convenient alternative method for solution of the generalized Huxley equation.

scholarly journals Conditional Lie-Bäcklund Symmetry Reductions and Exact Solutions of a Class of Reaction-Diffusion Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Xinyang Wang ◽  
Junquan Song
Keyword(s):  
Dynamical Systems ◽  
Exact Solutions ◽  
Separation Of Variables ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Financial Mathematics ◽  
Symmetry Reductions ◽  
Finite Dimensional ◽  
Wide Range

The method of conditional Lie-Bäcklund symmetry is applied to solve a class of reaction-diffusion equations ut+uxx+Qxux2+Pxu+Rx=0, which have wide range of applications in physics, engineering, chemistry, biology, and financial mathematics theory. The resulting equations are either solved exactly or reduced to some finite-dimensional dynamical systems. The exact solutions obtained in concrete examples possess the extended forms of the separation of variables.

scholarly journals Dynamics of media with stationary parameters and telegraph equations

2021 ◽  
Vol 2091 (1) ◽  
pp. 012069
Author(s):  
A G Kushner ◽  
E N Kushner
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Natural Extension ◽  
Mathematical Physics ◽  
Telegraph Equation ◽  
Partial Differential ◽  
Finite Dimensional ◽  
Telegraph Equations ◽  
Equations Of Mathematical Physics

Abstract The paper proposes an approach for constructing exact solutions of differential equations of mathematical physics, in particular, the telegraph equation. The method is based on the theory of finite-dimensional dynamics of systems of evolutionary differential equations. This theory is a natural extension of the theory of dynamical systems to partial differential equations. It allows one to construct exact solutions of partial differential equations even in the case when equations do not have symmetry algebras sufficient for integration.

Exact solutions of the Burgers–Huxley equation via dynamics

2020 ◽  
Vol 151 ◽  
pp. 103615 ◽  
Author(s):  
Alexei G. Kushner ◽  
Ruslan I. Matviichuk
Keyword(s):  
Exact Solutions ◽  
Huxley Equation
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