Solving the heat equation using FFCT (Finite Fourier Cosine Trans)

[Aows]

Homework Statement

Solve the following heat Eq. using FFCT:
A metal bar of length L is at constant temperature of Uo, at t=0 the end x=L is suddenly given the constant temperature U1, and the end x=0 is insulated. Assuming that the surface of the bar is insulated, find the temperature at any point x of the bar at any time t>0, assume thermal diffusivity coefficient (k) =1

Homework Equations

heat equation: dˆ2U/dxˆ2=(1/k) dU/dt

FFCT equation of derivative:
F (dˆ2U/dxˆ2)= -( n*pi/b)ˆ2 *F(n,t)+(-1)ˆn * (ux(b,t)-ux(0,t)

The Attempt at a Solution

my attempt has many mistake at the start of transforming.

 

[RUber]

@Aows,
The FFCT assumes that
$C(x,t) = \sum_{n = 0}^{N} a_n \cos \frac{n\pi x}{L}$
Where
$a_0 (t)= \frac{1}{L} \int_0^L f(x,t) dx \\ a_n(t) = \frac{2}{L} \int_0^L \cos \frac{n \pi x}{L} f(x,t) dx.$

Apply the transform to the PDE, as you have done:
$\frac{\partial^2 f}{\partial x^2} = \frac{\partial f}{\partial t} \\ \frac{\partial^2 }{\partial x^2}\left(\sum_{n = 0}^{N} a_n \cos \frac{n\pi x}{L}\right) = \sum_{n = 0}^{N} \frac{\partial }{\partial t}a_n(t) \cos \frac{n\pi x}{L}$
For each n, you get the equation you provided for the 2nd derivative w.r.t. x. This is inconvenient, since your boundary conditions don't seem to provide
$u_x(L,0) \text{ or } u_x(0,0).$
Does other information in your problem tell you what they should be?

 

[Aows]

@Aows,
The FFCT assumes that
$C(x,t) = \sum_{n = 0}^{N} a_n \cos \frac{n\pi x}{L}$
Where
$a_0 (t)= \frac{1}{L} \int_0^L f(x,t) dx \\ a_n(t) = \frac{2}{L} \int_0^L \cos \frac{n \pi x}{L} f(x,t) dx.$

Apply the transform to the PDE, as you have done:
$\frac{\partial^2 f}{\partial x^2} = \frac{\partial f}{\partial t} \\ \frac{\partial^2 }{\partial x^2}\left(\sum_{n = 0}^{N} a_n \cos \frac{n\pi x}{L}\right) = \sum_{n = 0}^{N} \frac{\partial }{\partial t}a_n(t) \cos \frac{n\pi x}{L}$
For each n, you get the equation you provided for the 2nd derivative w.r.t. x. This is inconvenient, since your boundary conditions don't seem to provide
$u_x(L,0) \text{ or } u_x(0,0).$
Does other information in your problem tell you what they should be?

Hello Dr. Ruber,
here is the problem (the solution provided in this picture is by using Laplace) and am required to solve it using FFCT:
https://i.imgur.com/F5LlyM0.jpg

please, excuse me for attaching the image

 

[Aows]

Dr. RUber @RUber those are the formulas used:
https://i.imgur.com/z0lc9rk.jpg