Numerical Method Solutions ()

[Lucky mkhonza]

I have been given the following problem as assignment:
Find a numerical solution for the 1-D heat conduction (using the Explicit Method):

$$\left\{\begin{array}U_{xx} = U_{t},\\ U(x,0) = \sin \pi x, \\ U(0,t) = U(1,t) = 0$$

Use h = 1, k = 0.005125 and M = 200.

Can anyone help by giving me a hint of this problem.

Thank you in advance...

 

[H_man]

Is this any help? ... it seems to be roughly the same problem.

http://en.wikipedia.org/wiki/Finite_difference

 

[ravenprp]

Sure, I would be happy to give you some guidance on this problem. The first thing to note is that this is a problem in 1-D heat conduction, which means we are looking for a solution u(x,t) that represents the temperature at a given point (x) and time (t). The equation given is the heat equation, which relates the change in temperature over time (U_{t}) to the change in temperature over space (U_{xx}). The initial condition (U(x,0) = sin(pi*x)) represents the temperature distribution at time t = 0, and the boundary conditions (U(0,t) = U(1,t) = 0) represent the temperature at the boundaries of the domain (x = 0 and x = 1) at any given time t.

To solve this problem numerically, we can use the Explicit Method, which is a finite difference method that approximates the solution at a given point (x_i,t_n) using the values at neighboring points (x_{i-1},t_{n-1}) and (x_{i+1},t_{n-1}). The step sizes h and k represent the spatial and temporal discretization, respectively, and M is the number of spatial points we will use to discretize the domain (in this case, we are given M = 200).

To get started, let's first discretize the domain into M points, with a step size of h = 1. This means that our domain will be divided into 200 equally spaced points, with x_0 = 0, x_1 = 0.005, x_2 = 0.01, and so on, up to x_{200} = 1. Next, we need to choose a time step k, which is given as k = 0.005125. This means that our time domain will be divided into time steps of 0.005125, with t_0 = 0, t_1 = 0.005125, t_2 = 0.01025, and so on.

Now, let's use the Explicit Method to approximate the solution at each point in our domain. We can start by using the initial condition (U(x,0) = sin(pi*x)) to determine the temperature at each point in the first time step (t_1 = 0.005125). This can be done