Please see the attached file for the fully formatted problems.
Solve the heat equation
u_t = ku_xx
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
Please see the attached file for the complete solution.
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The heat equation is:
The boundary conditions are:
And the initial condition is:
We start with setting:
That is, and are single-variable independent functions. Thus:
Plugging (1.5) and (1.6) into (1.1) we get:
Dividing both sides by and we get a separated equation:
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.
The 9 pages solution includes full derivations explanations of the method of separation of variables and Fourier series expansion.