scholarly journals Second-Order Conditional Lie-Bäcklund Symmetry and Differential Constraint of Radially Symmetric Diffusion System

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Jianping Wang ◽  
Huijing Ba ◽  
Yaru Liu ◽  
Longqi He ◽  
Lina Ji
Keyword(s):  
Dynamical Systems ◽  
Exact Solutions ◽  
Surface Condition ◽  
Original System ◽  
Second Order ◽  
Diffusion System ◽  
Differential Constraint ◽  
Invariant Surface ◽  
Symmetric Diffusion ◽  
Symmetry Method

The classifications and reductions of radially symmetric diffusion system are studied due to the conditional Lie-Bäcklund symmetry method. We obtain the invariant condition, which is the so-called determining system and under which the radially symmetric diffusion system admits second-order conditional Lie-Bäcklund symmetries. The governing systems and the admitted second-order conditional Lie-Bäcklund symmetries are identified by solving the nonlinear determining system. Exact solutions of the resulting systems are constructed due to the compatibility of the original system and the admitted differential constraint corresponding to the invariant surface condition. For most of the cases, they are reduced to solving four-dimensional dynamical systems.

scholarly journals Conditional Lie-Bäcklund Symmetries and Differential Constraints of Radially Symmetric Nonlinear Convection-Diffusion Equations with Source

Entropy ◽  
2020 ◽  
Vol 22 (8) ◽  
pp. 873
Author(s):  
Lina Ji ◽  
Rui Wang
Keyword(s):  
Dynamical Systems ◽  
Diffusion Equations ◽  
Two Dimensional ◽  
Nonlinear Convection ◽  
Differential Constraint ◽  
Invariant Surface ◽  
Differential Constraints ◽  
Convection Diffusion ◽  
Constraint Method ◽  
Symmetry Method

A conditional Lie-Bäcklund symmetry method and differential constraint method are developed to study the radially symmetric nonlinear convection-diffusion equations with source. The equations and the admitted conditional Lie-Bäcklund symmetries (differential constraints) are identified. As a consequence, symmetry reductions to two-dimensional dynamical systems of the resulting equations are derived due to the compatibility of the original equation and the additional differential constraint corresponding to the invariant surface equation of the admitted conditional Lie-Bäcklund symmetry.

scholarly journals Functional Separation of Variables in Nonlinear PDEs: General Approach, New Solutions of Diffusion-Type Equations

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 90 ◽  
Author(s):  
Andrei D. Polyanin
Keyword(s):  
Exact Solutions ◽  
Applied Mathematics ◽  
Separation Of Variables ◽  
Surface Condition ◽  
Traveling Wave Solutions ◽  
Mathematical Physics ◽  
Variable Coefficients ◽  
Nonlinear Pdes ◽  
Diffusion Type ◽  
Differential Constraint

The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied mathematics and mathematical physics, based on a special transformation with an integral term and the generalized splitting principle. The effectiveness of this approach is illustrated by nonlinear diffusion-type equations that contain reaction and convective terms with variable coefficients. The focus is on equations of a fairly general form that depend on one, two or three arbitrary functions (such nonlinear PDEs are most difficult to analyze and find exact solutions). A lot of new functional separable solutions and generalized traveling wave solutions are described (more than 30 exact solutions have been presented in total). It is shown that the method of functional separation of variables can, in certain cases, be more effective than (i) the nonclassical method of symmetry reductions based on an invariant surface condition, and (ii) the method of differential constraints based on a single differential constraint. The exact solutions obtained can be used to test various numerical and approximate analytical methods of mathematical physics and mechanics.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

A Class of Bounded and Partially Bounded Nonlinear Controllers for First and Second Order Dynamical Systems

2021 ◽  
pp. 1-1
Author(s):  
Eddie Clemente ◽  
M. C. Rodriguez-Linan ◽  
Marlen Meza-Sanchez ◽  
Luis Monay-Arredondo ◽  
Leonardo Herrera
Keyword(s):  
Dynamical Systems ◽  
Second Order ◽  
Nonlinear Controllers

scholarly journals Second-order dynamical systems associated to variational inequalities

2016 ◽  
Vol 96 (5) ◽  
pp. 799-809 ◽  
Author(s):  
Radu Ioan Boţ ◽  
Ernö Robert Csetnek
Keyword(s):  
Dynamical Systems ◽  
Variational Inequalities ◽  
Second Order

scholarly journals Analysis of IoT-Based Load Altering Attacks Against Power Grids Using the Theory of Second-Order Dynamical Systems

2021 ◽  
pp. 1-1
Author(s):  
Subhash Lakshminarayana ◽  
Sondipon Adhikari ◽  
Carsten Maple
Keyword(s):  
Dynamical Systems ◽  
Second Order ◽  
Power Grids

Multidimensional exact solutions to the reaction-diffusion system with power-law nonlinear terms

2017 ◽  
Vol 58 (4) ◽  
pp. 619-632 ◽  
Author(s):  
A. A. Kosov ◽  
E. I. Semenov
Keyword(s):  
Exact Solutions ◽  
Power Law ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System
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