scholarly journals The cost of approximate controllability of heat equation with general dynamical boundary conditions

2021 ◽  
Vol 78 (1) ◽  
pp. 65-99
Author(s):  
Idriss Boutaayamou ◽  
Salah-Eddine Chorfi ◽  
Lahcen Maniar ◽  
Omar Oukdach
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Approximate Controllability ◽  
Dynamical Boundary Conditions ◽  
The Cost

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.

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