- #1

umby

- 50

- 8

- TL;DR Summary
- Numerical solution of a partial differential equation containing the derivative of the unknown at a point

What is the best way to solve numerically the following equation using Comsol 5.3.

##\frac{\partial T}{\partial t}=\frac{\partial ^2T}{\partial x^2}+\text{St}\left[1+\left(\frac{\partial T}{\partial x}\right)_{x=0}\right]\frac{\partial T}{\partial x}##

##T(0,t)=1##

##T(\infty ,t)=0##

##T(x,0)=\exp \left(-\frac{x^2}{\pi }\right)-x \text{erfc}\left(\frac{x}{\sqrt{\pi }}\right)##

where ##\text{St}## is a parameter which can varies from 0.01 to 100.

I am particularly interested in following the evolution of ##\text{St}\left[1+\left(\frac{\partial T}{\partial x}\right)_{x=0}\right]## with time.

Thanks in advance.

##\frac{\partial T}{\partial t}=\frac{\partial ^2T}{\partial x^2}+\text{St}\left[1+\left(\frac{\partial T}{\partial x}\right)_{x=0}\right]\frac{\partial T}{\partial x}##

##T(0,t)=1##

##T(\infty ,t)=0##

##T(x,0)=\exp \left(-\frac{x^2}{\pi }\right)-x \text{erfc}\left(\frac{x}{\sqrt{\pi }}\right)##

where ##\text{St}## is a parameter which can varies from 0.01 to 100.

I am particularly interested in following the evolution of ##\text{St}\left[1+\left(\frac{\partial T}{\partial x}\right)_{x=0}\right]## with time.

Thanks in advance.