# Solving Du/dt=A*d^2u/dx^2 w/ Boundary Conditions

• timsea81
In summary, the given problem is a heat equation with constant A and boundary conditions u(x,0)=f(x), u(0,t)=0, and u(L,t)=V. To find a solution, the transformation u=u(x,t) - (V/L)*x is used, and the resulting differential equation is solved using the same method as the heat equation on Wikipedia. The relation u(x,t) = w(x/t) + (V/L)*x is then used to find the final solution.
timsea81
du/dt = A*d^2u/dx^2
Where u=u(x,t) and A is a constant.
The boundary conditions are:

u=f(x) when t=0
u=0 when x=0, independent of t
u=V when x=L, independent of t, where V and L are both non-zero constants.

Is there a general solution to this, or do I need to solve it numerically?

I tried solving it by separation of variables similar to what was done here:
http://en.wikipedia.org/wiki/Heat_equation#Solving_the_heat_equation_using_Fourier_series

But in the link the last boundary condition is u=0 when x=L, which allows us to conclude something about lambda that I can't with a non-zero boundary condition.

If it is separable, we have u(x,t)=X(x)T(t)=V when x=L independent of t. Following the method used in the link, to the point where we apply the different boundary conditions and my problem becomes different than what is in the link:

u(x,t) = [B sin(rootlambda x)] * [A e^-(lambda alpha t)]

Since u(L,t) is a non-zero constant independent of t, I think that I need to conclude A=0 leading to a trivial solution. I got this far with it before and concluded that the solution was not separable and therefore can only be evaluated numerically. Do you agree, or am I missing something?

Here is a link to another forum where I am looking for the same answer:

The transformation ##u \to u-\frac{V}{L}x## will not influence your differential equation, but give u(L)=0. Solve the equation for the modified u, and add that linear part afterwards?

mfb said:
The transformation ##u \to u-\frac{V}{L}x## will not influence your differential equation, but give u(L)=0. Solve the equation for the modified u, and add that linear part afterwards?

Interesting. Let's develop that a little:

u=u(x,t)
du/dt = A*(d^2u/dx^2)
u(x,0) = f(x)
u(0,t) = 0
u(L,t) = V

Let w(x,t) = u(x,t) - (V/L)*x

Now,
w(x,0) = u(x,t) - (V/L)*x = f(x)-[(V/L)*x]
We can just call this g(x) so that's fine

w(0,t) = u(0,t) - (V/L)*0 = 0-0 = 0
w(L,t) = u(L,t) - (V/L)*L = V-V = 0

dw/dt = du/dt - 0 = du/dt
dw/dx = du/dx - (V/L)
d^2w/dt^2 = d^2u/dt^2 - 0 = d^2u/dt^2

du/dt = A*(d^2u/dx^2) is given in the problem, and from above, this means that

dw/dt = A*(d^2w/dx^2)

So I solve dw/dt = A*(d^2w/dx^2) the same way they do for the heat equation on wikipedia, then use the relation

u(x,t) = w(x/t) + (V/L)*x

To find u(x,t)

I think that will work! Thanks!

## What is the meaning of the equation "Du/dt=A*d^2u/dx^2"?

The equation describes the change in a function u over time (t) and space (x), where the rate of change is proportional to the second derivative of u with respect to x, with a constant of proportionality A.

## What is the significance of the boundary conditions in this equation?

The boundary conditions specify the values or behavior of the function u at the edges of the spatial domain. They are necessary for uniquely solving the equation and determining the behavior of u at all points in the domain.

## What type of problem does this equation model?

This equation models a diffusion problem, where the function u represents the concentration or diffusion of a substance over time and space.

## What are some common methods for solving this type of equation?

Some common methods for solving this type of equation include finite difference methods, finite element methods, and spectral methods. The appropriate method to use depends on the specific problem and the desired level of accuracy.

## Can this equation be applied to other areas of science besides diffusion?

Yes, this equation can be applied to other areas of science, such as heat transfer, population dynamics, and fluid mechanics. It is a general equation for describing the change of a quantity over time and space, and can be applied to a wide range of phenomena.

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