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

In summary, the conversation discusses solving a heat equation using the FFCT method. The problem involves a metal bar with constant temperature at one end and insulated at the other, and the goal is to find the temperature at any point of the bar at any time. The FFCT method involves transforming the heat equation and applying boundary conditions, but additional information is needed to determine the values of ux(L,0) and ux(0,0). The formulas used in this method are also provided.
  • #1
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.
 
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  • #2
@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?
 
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Likes Aows
  • #3
RUber said:
@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
 
  • #4

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

1. What is FFCT?

FFCT stands for Finite Fourier Cosine Transform, which is a mathematical technique used to analyze and solve the heat equation. It transforms a function in the spatial domain into a function in the frequency domain, making it easier to solve the heat equation.

2. How does FFCT solve the heat equation?

FFCT works by breaking down the heat equation into smaller, simpler equations that can be solved using Fourier series or cosine transforms. These smaller equations are then combined to get the solution for the entire heat equation.

3. What are the advantages of using FFCT to solve the heat equation?

Some advantages of using FFCT include its ability to solve the heat equation for non-uniform and non-rectangular domains, as well as its efficiency in solving problems with variable coefficients. It also allows for easier implementation of boundary conditions.

4. Are there any limitations to using FFCT for solving the heat equation?

One limitation of FFCT is that it is only applicable to linear heat equations, meaning that it cannot be used for systems with nonlinear terms. It also requires a certain degree of mathematical knowledge and skill to properly implement and interpret the results.

5. How is FFCT used in real-world applications?

FFCT is commonly used in various fields such as engineering, physics, and mathematics to model and solve problems involving heat transfer. It has applications in areas such as heat conduction, heat convection, and heat radiation, and is used in the design and optimization of various systems and processes.

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