# The 1D heat equation

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Solve the heat equation

u_t = ku_xx

for 0<x<L

With boundary conditions u(0,t)=u(L,t)=0

Solve for the initial value conditions:

a. u(x,0) = sin(5*Pi*x/L)

b. u(x,0) = x

c. For part b, plot the solution at t=0, 0.1, 1

https://brainmass.com/math/numerical-analysis/1d-heat-equation-241523

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The heat equation is:

(1.1)

The boundary conditions are:

(1.2)

And the initial condition is:

(1.3)

We start with setting:

(1.4)

That is, and are single-variable independent functions. Thus:

(1.5)

And:

(1.6)

Plugging (1.5) and (1.6) into (1.1) we get:

(1.7)

Dividing both sides by and we get a separated equation:

(1.8)

Now, the left hand side is independent of t while the right hand side is independent of x. This can be true if and only if the two sides of the equation are equal to the same constant.

Thus:

...

#### Solution Summary

The 9 pages solution includes full derivations explanations of the method of separation of variables and Fourier series expansion.