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I don't understand what is meant by there being a "solution" to an equation. For example, what does a solution to the wave or heat equation represent, and what are we solving for?

Of course, we can use such equations to perform computations, but what can the "solutions" tell us, and how can we use them? (Please excuse my knowing almost nothing about differential equations, but I am curious)

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It would help if you gave a specific example for your question. But in general, a solution to an equation is like finding a result. If 2 lines cross and you know the equations for the lines, you can determine the location where the lines cross by solving the equation. –  LDC3 Jun 9 at 2:35
    
@LDC3 For example, the heat equation is $\frac{\partial u}{\partial t} - \alpha \nabla^2 u=0$, but what would a solution to this be? –  yoyo311 Jun 9 at 2:41
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For example, the heat equation is ∂u/∂t−α∇2u=0, but what would a solution to this be?

A solution to this equation is a function $u(\vec{x},t)$ that describes the temperature of the material in question at different locations, and how it evolves over time.

We say this is a solution to the equation, because out of all possible functions $u(\vec{x},t)$, we have to find the one that satisfies the equation given (the heat equation).

The wiki article talks about solving the equation with the Fourier series. So essentially, it's not always possible to solve differential equations in the same way we can solve simple linear or quadratic equations, so instead we have to approximate them through various methods?

The issue is more that there are an infinite number of possible solutions. For example, in the one-dimensional version of the problem, any function of the form

$u(x,t)=A\exp(-\lambda\alpha{}t)\left(\sin(\sqrt{\lambda}x) + B \cos(\sqrt{\lambda}x)\right)$

will be found to satisfy the heat equation itself.

So we need some additional information to narrow down the solution to one that fits our physical situation. These are the boundary conditions and the initial conditions.

For example, we might have a situation where the left end of a bar (of length L) is connected to a ideal heatsink at 100 C, and the right end of the bar is connected to an ideal heatsink at 0 C. Then our boundary conditions are

$u(0, t) = 100$

$u(L, t) = 0$

And we might have an initial condition from the fact that before the bar was connected to those heat sinks it was heated to a uniform temperature of 25 C.

$u(x, 0) = 25$.

We use Fourier analysis to find the combination of all the possible solutions that satisfy these conditions.

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On Wikipedia, the graph on the right shows the solution for a bar of metal at various times where one end was heated and is now distributing the heat along the bar. en.wikipedia.org/wiki/Heat_equation#General_description –  LDC3 Jun 9 at 3:08
    
I think I'm starting to understand. The wiki article talks about solving the equation with the Fourier series. So essentially, it's not always possible to solve differential equations in the same way we can solve simple linear or quadratic equations, so instead we have to approximate them through various methods? –  yoyo311 Jun 9 at 3:20
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@yoyo311 for algebraic equations it's also not always possible to solve them in a simple way. See e.g. Abel-Ruffini theorem. So, in many cases you'll have to use numerical methods to find even roots of polynomials, and these problems are not even the most complicated ones. Differential equations are not much different in this sense. –  Ruslan Jun 9 at 11:06
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As I understand it, a solution to an equation in a physical sense means that, for a given set of conditions of a system, we find a formula or equation that expresses the "state" of the system at some future or past time. By "state" I mean a variety of things; for instance, in classical mechanics problems we use the kinematics equations and Newton's laws to find the position, velocity, and acceleration of an object (or objects) over a given time. In quantum mechanical problems we use Schroedinger's equation to find the wave function(s) of particle(s). A solution to the heat equation for a given system would describe how the temperature of an object changes over time.

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