scholarly journals Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain

2021 ◽  
pp. 1-27
Author(s):  
Jane Allwright
Keyword(s):  
Diffusion Equation ◽  
Linear Growth ◽  
Airy Function ◽  
Higher Dimensions ◽  
Time Dependent ◽  
Exact Form ◽  
Logarithmic Term ◽  
Time Dependent Domain ◽  
Dependent Domain ◽  
Growth Diffusion

Abstract A linear growth-diffusion equation is studied in a time-dependent interval whose location and length both vary. We prove conditions on the boundary motion for which the solution can be found in exact form and derive the explicit expression in each case. Next, we prove the precise behaviour near the boundary in a ‘critical’ case: when the endpoints of the interval move in such a way that near the boundary there is neither exponential growth nor decay, but the solution behaves like a power law with respect to time. The proof uses a subsolution based on the Airy function with argument depending on both space and time. Interesting links are observed between this result and Bramson's logarithmic term in the nonlinear FKPP equation on the real line. Each of the main theorems is extended to higher dimensions, with a corresponding result on a ball with a time-dependent radius.

scholarly journals Conditionally Integrable Nonlinear Diffusion with Diffusivity 1/u

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 804 ◽  
Author(s):  
Philip Broadbridge ◽  
Joanna M. Goard
Keyword(s):  
Diffusion Equation ◽  
Nonlinear Diffusion ◽  
Meromorphic Functions ◽  
Nonlinear Diffusion Equation ◽  
Direct Application ◽  
Time Dependent ◽  
Flux Boundary Conditions ◽  
General Complex ◽  
Laser Heated ◽  
Special Time

An explicit mapping is given from the space of general complex meromorphic functions to a space of special time-dependent solutions of the 1 + 2-dimensional nonlinear diffusion equation with diffusivity depending on concentration as D = 1 / u. These solutions have constant-flux boundary conditions. Some simple examples are constructed, including that of a line source enclosed by a cylindrical barrier. This has direct application to electron diffusion in a laser-heated plasma.

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