Differential invariant method for seeking nonlocally related systems and nonlocal symmetries. I: General theory and examples

2021 ◽  
Vol 477 (2250) ◽  
pp. 20200908
Author(s):  
George W. Bluman ◽  
Rafael de la Rosa ◽  
María Santos Bruzón ◽  
María Luz Gandarias
Keyword(s):  
Differential Equation ◽  
Differential Invariant ◽  
Nonlocal Symmetries ◽  
Independent Variables ◽  
New Exact Solutions ◽  
Pde Systems ◽  
Nonlocal Conservation Laws ◽  
Two Independent Variables ◽  
Nonlocally Related Systems ◽  
Ode Systems

Nonlocally related systems, obtained through conservation law and symmetry-based methods, have proved to be useful for determining nonlocal symmetries, nonlocal conservation laws, non-invertible mappings and new exact solutions of a given partial differential equation (PDE) system. In this paper, it is shown that the symmetry-based method is a differential invariant-based method. It is shown that this allows one to naturally extend the symmetry-based method to ordinary differential equation (ODE) systems and to PDE systems with at least three independent variables. In particular, we present the situations for ODE systems, PDE systems with two independent variables and PDE systems with three or more independent variables, separately, and show that these three situations are directly connected. Examples are exhibited for each of the three situations.

scholarly journals Differential invariant method for seeking nonlocally related systems and nonlocal symmetries. II: Connections with the conservation law method

2021 ◽  
Vol 477 (2250) ◽  
pp. 20200909
Author(s):  
George W. Bluman ◽  
Rafael de la Rosa ◽  
María Santos Bruzón ◽  
María Luz Gandarias
Keyword(s):  
Differential Equation ◽  
Conservation Law ◽  
Differential Invariant ◽  
Related System ◽  
Nonlocal Symmetries ◽  
Independent Variables ◽  
Linear Pde ◽  
Special Case ◽  
Two Independent Variables ◽  
Nonlocally Related Systems

In this paper, we show direct connections between the conservation law (CL)-based method and the differential invariant (DI)-based method for obtaining nonlocally related systems and nonlocal symmetries for a given partial differential equation (PDE) system. For a PDE system with two independent variables, we show that the CL method is a special case for the DI method. For a PDE system with at least three independent variables, we show that the CL method, for a curl-type CL, is a special case for the DI method. We also consider the situation for a self-adjoint, i.e. variational, linear PDE system. Here, a solution of the linear PDE system yields a nonlocally related system for both approaches. In particular, the resulting nonlocally related systems need not be invertibly equivalent. Through an example, we show that three distinct nonlocally related systems can be obtained from an admitted point symmetry.

Conservation Laws and Nonlocally Related Systems of Two-Dimensional Boundary Layer Models

2017 ◽  
Vol 72 (11) ◽  
pp. 1031-1051
Author(s):  
R. Naz ◽  
A.F. Cheviakov
Keyword(s):  
Boundary Layer ◽  
Conservation Laws ◽  
Direct Method ◽  
Local Conservation ◽  
Solution Sets ◽  
Nonlocal Symmetries ◽  
Pde Systems ◽  
Nonlocal Conservation Laws ◽  
Constitutive Parameters ◽  
Nonlocally Related Systems

AbstractLocal conservation laws, potential systems, and nonlocal conservation laws are systematically computed for three-equilibrium two-component boundary layer models that describe different physical situations: a plate flow, a flow parallel to the axis of a circular cylinder, and a radial jet striking a planar wall. First, local conservation laws of each model are computed using the direct method. For each of the three boundary layer models, two local conservation laws are found. The corresponding potential variables are introduced, and nonlocally related potential systems and subsystems are formed. Then nonlocal conservation laws are sought, arising as local conservation laws of nonlocally related systems. For each of the three physical models, similar nonlocal conservation laws arise. Further nonlocal variables that lead to further potential systems are considered. Trees of nonlocally related systems are constructed; their structure coincides for all three models. The three boundary layer models considered in this work provide rich and interesting examples of the construction of trees of nonlocally related systems. In particular, the trees involve spectral potential systems depending on a parameter; these spectral potential systems lead to nonlocal conservation laws. Moreover, potential variables that are not locally related on solution sets of some potential systems become local functions of each other on solution sets of other systems. The point symmetry analysis shows that the plate and radial jet flow models possess infinite-dimensional Lie algebras of point symmetries, whereas the Lie algebra of point symmetries for the cylinder flow model is three-dimensional. The computation of nonlocal symmetries reveals none that arise for the original model equations, which is common for partial differential equations (PDE) systems without constitutive parameters or functions, but does reveal nonlocal symmetries for some nonlocally related PDE systems.

scholarly journals Deflection of Plates by Using Principle of Quasi Work

2016 ◽  
Vol 6 (2) ◽  
pp. 21-24
Author(s):  
A. Mahavir Singh ◽  
B. I.K. Pandita ◽  
C. S.K. Kheer
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Fourth Order ◽  
Present Method ◽  
Elementary Mathematics ◽  
Order Partial Differential Equation ◽  
Simply Supported ◽  
Independent Variables ◽  
Reference Equation ◽  
Two Independent Variables

Abstract A new methodology based on Principle of Quasi Work is used for calculating the deflections in plates. The basis of this methodology is concept of topologically similar systems. Present method uses a priory known solution for deflection of a simply supported plate for arriving at the deflection of any other topologically similar plate with different loading and boundary conditions. This priory known solution is herein referred to as reference equation. Present methodology is easy as deflections are obtained mostly by elementary mathematics for point loads and for other loads by integration that’s integrant is reference equation multiplied by the equation of load. In the present methodology solution of fourth order partial differential equation in two independent variables as used in lengthy and not so easy conventional method is bypassed.

scholarly journals Memoir on the integration of partial differential equations of the second order in three independent variables, when an intermediary integral does not exist in general

1898 ◽  
Vol 62 (379-387) ◽  
pp. 283-285
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
General Feature ◽  
Partial Differential ◽  
First Order ◽  
Independent Variables ◽  
Dependent Variables ◽  
Two Independent Variables

The general feature of most of the methods of integration of any partial differential equation is the construction of an appropriate subsidiary system and the establishment of the proper relations between integrals of this system and the solution of the original equation. Methods, which in this sense may be called complete, are possessed for partial differential equations of the first order in one dependent variable and any number of independent variables; for certain classes of equations of the first order in two independent variables and a number of dependent variables; and for equations of the second (and higher) orders in one dependent and two independent variables.

scholarly journals Different forms of Euler’s theorem forhomogeneous functions to solve partial differential equation

2021 ◽  
Vol 12 (2) ◽  
pp. 2810-2813
Author(s):  
Amit Kumar, Et. al.
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential ◽  
Independent Variables ◽  
Euler’S Theorem ◽  
Different Order ◽  
Two Independent Variables

In this paper we will discuss Euler’s theorem for homogenous functions to solve different order partial differential equations. We will see that how we can predict the solution of partial differential Equation using different approaches of this theorem. In fact we also consider the case when more than two independent variables will be involved in the partial differential equation whenever dependent functions will be homogenous functions. We will throw a light on one method called Ajayous rules to predict the solution of homogenous partial differential equation.

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