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Hello, I am taking an introductory class on non relativistic classical field theory and right now we are doing the more mathematical aspect of things right now. The types of differential equations in the function ##f(\vec{r},t)## that are considered in this course are linear in the following components.
##f##
##\nabla^{2}f##
##\frac{\partial f}{\partial t}##
##\frac{\partial^{2} f}{\partial t^{2}}##
And some function ##g(\vec{r},t)##
Resulting in the heat and wave type of equations.
The boundary conditions were defined as the function ##f(\vec{R},t)## where ##\vec{R}## is bounded to the boundary of the volume that we consider the equation in at any time.
After some vague argument I don't quite understand we arrive at the following general relationship for the boundary function:
##a f(\vec{R},t) + b \vec{n}.\nabla{f}+ h(\vec{R},t) = 0 ##
Where ##\vec{n}## is the normal vector to the boundary sufrace, ##h(\vec{R},t)## is some function which we don't know the characteristics of yet and ##a## and ##b## are just constants.
Can anyone explain me intuitively how that relationship is deduced? I understand there's probably some difficult formal math behind this but the argument made in class was an intuitive one. It was based on the fact that if you know the values at the boundary you can deduce values at any point in the volume using some Taylor series type of reasoning. Anyway I'm hoping if someone could provide a similar argument. Thanks
Bonus: How do initial value conditions ##f(\vec{r},0)## and it's derivative come into play?
##f##
##\nabla^{2}f##
##\frac{\partial f}{\partial t}##
##\frac{\partial^{2} f}{\partial t^{2}}##
And some function ##g(\vec{r},t)##
Resulting in the heat and wave type of equations.
The boundary conditions were defined as the function ##f(\vec{R},t)## where ##\vec{R}## is bounded to the boundary of the volume that we consider the equation in at any time.
After some vague argument I don't quite understand we arrive at the following general relationship for the boundary function:
##a f(\vec{R},t) + b \vec{n}.\nabla{f}+ h(\vec{R},t) = 0 ##
Where ##\vec{n}## is the normal vector to the boundary sufrace, ##h(\vec{R},t)## is some function which we don't know the characteristics of yet and ##a## and ##b## are just constants.
Can anyone explain me intuitively how that relationship is deduced? I understand there's probably some difficult formal math behind this but the argument made in class was an intuitive one. It was based on the fact that if you know the values at the boundary you can deduce values at any point in the volume using some Taylor series type of reasoning. Anyway I'm hoping if someone could provide a similar argument. Thanks
Bonus: How do initial value conditions ##f(\vec{r},0)## and it's derivative come into play?