What is the Outcome of Solving a Partial Differential Equation?

[Paolo]

Partial Differential Equations

Can someone tell me what do we get when we solve a Partial Differential Equation? Do we get a Partial Solution or the whole thing, Thanks a lot

 

[Zurtex]

Say you have z=f(x,y) and you work out:

$$\frac{\partial z}{\partial x}$$

You have worked the rate of change of z with respect to x and nothing else, I think, I've only just started it lol.

 

[Galileo]

The solution of any differential equation is the set containing all solutions. (The solution set).
There are many ways to tackle PDE's. Some will give the general answer, some will give a subset of the solution set. It all depends on the particular problem.

 

[dextercioby]

It's worth saying that the ones quasilinear and nonlinear are not integrable exactly,meaning that u cannot do anything to get the set of solutions...


Daniel.

 

[fourier jr]

the first thing i learned was separation of variables. you assume your solution u(x,y) has the form u(x,t) = f(x)g(t). then, say, for the

1-dimensional heat equation: $$\frac{\partial u}{\partial t} = k\frac{\partial^2 u}{\partial x^2}$$. rewrite it using the form above

$$\frac{\partial u}{\partial t} - k\frac{\partial^2 u}{\partial x^2} = f(x)g'(t) - kf"(x)g(t) = 0$$

fiddle with that last bit to get this:

$$\frac{f(x)}{kf"(x)} = \frac{g(t)}{g'(t)} = -\lambda$$

from which you get 2 ORDINARY differential equations:

$$\frac{d^2 f}{dx^2} + \lambda f = 0$$

$$\frac{dg}{dt} - \lambda g = 0$$

& you get the f & g from this system

 

[saltydog]

Partial Differential Equations

Can someone tell me what do we get when we solve a Partial Differential Equation? Do we get a Partial Solution or the whole thing, Thanks a lot

You get a solution, a function of several variables, such as f(x,y). It's a surface (for PDEs of 2 indep. variables) which if you back-plug the values of the function and the values of the derivatives (the partial ones) at any point in the domain, they will satisfy the PDE. PDEs: the crown-prince of Mathematics!

Salty

 

[fourier jr]

PDEs: the crown-prince of Mathematics!

Salty

i think sophus lie said that PDEs was the most important area of math. i don't know why he said that though. maybe it's PDE's proximity to physics & the real world?

 

[Sirus]

Partial Differential Equations

Can someone tell me what do we get when we solve a Partial Differential Equation? Do we get a Partial Solution or the whole thing, Thanks a lot

Welcome Paolo.

I think Zurtex hit on the explanation you're looking for. Partial derivatives give rates of change, just like regular ones, except we are dealing with multi-variable functions. Given a function $z=f(x,y)$, taking the partial derivative with respect to x,

$$\frac{\partial{z}}{\partial{x}}$$

gives the rate of change of the function z as we change x and hold y constant. Similarly, taking the partial derivative with respect to y,

$$\frac{\partial{z}}{\partial{y}}$$

gives the rate of change of z as we vary y, holding x constant.