Thread: understanding PDE's

Hi im new to Pde's , a few key concepts I am trying to get a better understanding of, : integrals, characteristic curves and a envelopes for a pde. I understand they are key components in constructing a general solution. If we stick to 3 dimensions, the solution is a surface(?), then is there a visual interpretation for those other parts aswell? many thanks in advance

Originally Posted by Richyie

Hi im new to Pde's , a few key concepts I am trying to get a better understanding of, : integrals, characteristic curves and a envelopes for a pde. I understand they are key components in constructing a general solution. If we stick to 3 dimensions, the solution is a surface(?), then is there a visual interpretation for those other parts aswell? many thanks in advance

That an integral curve is a curve that fits all the directions in a direction field..ergo being a solution to the partial differntial equation

in not sure what you mean by a direction field ?
say we had a quasilinear pde a*du/dx+b*du/dy = c . Lecture notes say: charactaristics satisfy dx/a = dy/b = du/c
from the above we can deduct integrals I, J such that I=constant and J = constant.. fair enough
and when these integrals intersect we get a charactaristic curve.. . ..
or am i understanding this incorrectly, just trying to picture it, its confusing what does it look like geometrically ? ?

Originally Posted by Richyie

in not sure what you mean by a direction field ?
say we had a quasilinear pde a*du/dx+b*du/dy = c . Lecture notes say: charactaristics satisfy dx/a = dy/b = du/c
from the above we can deduct integrals I, J such that I=constant and J = constant.. fair enough
and when these integrals intersect we get a charactaristic curve.. . ..
or am i understanding this incorrectly, just trying to picture it, its confusing what does it look like geometrically ? ?

but if you assigned the two derivatives a value c and d...then you assign them numbers and graphed them you would get a three dimensional graph which shows the slope at each point

hmmm i think im going to try this on matlab , thnx anyway