Partial Diff Equations

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In summary: So when you solve a partial differential equation, you are finding a function that satisfies the given equation and takes into account the rates of change with respect to all the variables involved. In summary, solving a Partial Differential Equation results in a solution, often a function of multiple variables, that satisfies the given equation and takes into account the rates of change with respect to all the variables involved. This solution can be a partial solution or the entire solution set, depending on the specific problem being solved.
  • #1
Paolo
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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
:uhh:
 
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  • #2
Say you have z=f(x,y) and you work out:

[tex]\frac{\partial z}{\partial x}[/tex]

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.
 
  • #3
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.
 
  • #4
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... :yuck:


Daniel.
 
  • #5
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: [tex]\frac{\partial u}{\partial t} = k\frac{\partial^2 u}{\partial x^2}[/tex]. rewrite it using the form above

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

fiddle with that last bit to get this:

[tex] \frac{f(x)}{kf"(x)} = \frac{g(t)}{g'(t)} = -\lambda[/tex]

from which you get 2 ORDINARY differential equations:

[tex]\frac{d^2 f}{dx^2} + \lambda f = 0[/tex]

[tex]\frac{dg}{dt} - \lambda g = 0[/tex]

& you get the f & g from this system
 
  • #6
Paolo said:
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
:uhh:

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
 
  • #7
saltydog said:
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?
 
  • #8
Paolo said:
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
:uhh:

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 [itex]z=f(x,y)[/itex], taking the partial derivative with respect to x,

[tex]\frac{\partial{z}}{\partial{x}}[/tex]

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,

[tex]\frac{\partial{z}}{\partial{y}}[/tex]

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

1. What are partial differential equations (PDEs)?

Partial differential equations are mathematical equations that involve multiple independent variables and their partial derivatives. They are used to model physical phenomena in fields such as physics, engineering, and economics.

2. How are PDEs different from ordinary differential equations (ODEs)?

The main difference between PDEs and ODEs is that PDEs involve multiple independent variables and their partial derivatives, while ODEs only involve a single independent variable and its derivatives.

3. What are some applications of PDEs?

PDEs are used to model various phenomena in fields such as fluid dynamics, electromagnetism, heat transfer, and quantum mechanics. They are also used in economics and finance to model stock prices and interest rates.

4. What are the different types of PDEs?

PDEs can be classified into different types based on their order, linearity, and boundary conditions. Some common types include elliptic, parabolic, and hyperbolic PDEs.

5. How are PDEs solved?

The method for solving a PDE depends on its type and complexity. Some common methods include separation of variables, Fourier analysis, and numerical methods such as finite difference and finite element methods.

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