scholarly journals Emmy Noether and Linear Evolution Equations

10.14311/1811 ◽  
2013 ◽  
Vol 53 (3) ◽  
Author(s):  
P. G. L. Leach
Keyword(s):  
Variational Principle ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Evolution Equations ◽  
First Integrals ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Linear Evolution ◽  
Action Integral ◽  
Linear Evolution Equations

Noether’s Theorem relates the Action Integral of a Lagrangian with symmetries which leave it invariant and the first integrals consequent upon the variational principle and the existence of the symmetries. These each have an equivalent in the Schrödinger Equation corresponding to the Lagrangian and by extension to linear evolution equations in general. The implications of these connections are investigated.

scholarly journals Linear evolution equations on the half-line with dynamic boundary conditions

2021 ◽  
pp. 1-33
Author(s):  
D. A. SMITH ◽  
W. Y. TOH
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Evolution Equations ◽  
Half Line ◽  
Robin Problem ◽  
Dynamic Boundary Conditions ◽  
Transform Method ◽  
Linear Evolution ◽  
Robin Condition ◽  
Linear Evolution Equations

The classical half-line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $$bq(0,t) + {q_x}(0,t) = 0$$ is replaced with a dynamic Robin condition; $$b = b(t)$$ is allowed to vary in time. Applications include convective heating by a corrosive liquid. We present a solution representation and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half-line, with arbitrary linear dynamic boundary conditions.

A fractal modification of Chen–Lee–Liu equation and its fractal variational principle

2021 ◽  
pp. 2150214
Author(s):  
Ji-Huan He ◽  
Chun-Hui He ◽  
Tareq Saeed
Keyword(s):  
Variational Principle ◽  
Solitary Wave ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Inverse Method ◽  
Ultrashort Pulses

The Chen–Lee–Liu equation is a modified Schrödinger equation to describe a solitary wave of ultrashort pulses in optics, which lead to a discontinuous time, so a fractal modification is suggested and a fractal variational principle is established by the semi-inverse method.

scholarly journals Lumpability of linear evolution Equations in Banach spaces

2017 ◽  
Vol 6 (1) ◽  
pp. 15-34 ◽  
Author(s):  
Fatihcan M. Atay ◽  
◽  
Lavinia Roncoroni ◽  
Keyword(s):  
Banach Spaces ◽  
Evolution Equations ◽  
Linear Evolution ◽  
Linear Evolution Equations
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