scholarly journals Regularity of solutions of the anisotropic hyperbolic heat equation with nonregular heat sources and homogeneous boundary conditions

2017 ◽  
Vol 41 ◽  
pp. 461-482 ◽  
Author(s):  
Juan Antonio LÓPEZ MOLINA ◽  
Macarena TRUJILLO
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Regularity Of Solutions ◽  
Heat Sources ◽  
Hyperbolic Heat Equation

scholarly journals Calculation of the power of thermal energy consumed for heating the reactor of a bioenergy plant

2021 ◽  
Vol 264 ◽  
pp. 04069
Author(s):  
Nodira Imomova ◽  
Ochil Komilov ◽  
Jurabek Majitov ◽  
Jukhriddin Ergashov ◽  
Kamol Usmonov
Keyword(s):  
Power Consumption ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Thermal Energy ◽  
Temperature Difference ◽  
Heat Sources ◽  
Additional Source ◽  
Additional Heat ◽  
Bioenergy Plant ◽  
Biogas Reactor

The issues of calculating the power of thermal energy consumed for heating biomass in the reactor of a bioenergy plant are considered. Based on the Fourier heat equation, a solution for the axisymmetric cylindrical problem under boundary conditions of the first kind is obtained, and the power of additional heat sources in a cylindrical biogas reactor is calculated. The influence of the height of the bioreactor and the temperature difference of the biomass on the power consumption of an additional source of thermal energy is analyzed

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.

scholarly journals An Application Of The Theory Of Scale Of Banach Spaces

2015 ◽  
Vol 29 (1) ◽  
pp. 51-59
Author(s):  
Łukasz Dawidowski
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Banach Spaces ◽  
Abstract Cauchy Problem ◽  
Initial Boundary ◽  
Scale Of Banach Spaces ◽  
The Cauchy Problem ◽  
Selection Of

AbstractThe abstract Cauchy problem on scales of Banach space was considered by many authors. The goal of this paper is to show that the choice of the space on scale is significant. We prove a theorem that the selection of the spaces in which the Cauchy problem ut − Δu = u|u|s with initial–boundary conditions is considered has an influence on the selection of index s. For the Cauchy problem connected with the heat equation we will study how the change of the base space influents the regularity of the solutions.

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