The Preparation of Fourier Series Matlab

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Task:

Using Fourier series methods, solve the heat equation following the inhomogeneous initial boundary value problem using Matlab:

Solution:

First we need to set the constant:

The solution to this problem can be written as follows:

Where the coefficient a_n (t) satisfies the initial value problem:

The values c1, c2, c3 … are the Fourier sine coefficients of the constant function 10-7:

The coefficient a_n (t) is given by the following formula:

Next we can define the partial Fourier series of the solution:

S=symsum(a*sin(n*pi*x/100),n,1,10);

Fourier series is required for a qualitatively correct solution.

The wiggles in the graph suggest that 10 terms is not enough for a correct graph.

Therefore, we will try again with 20 terms:

Now let’s try with 40 terms:

It appears that 20 terms is enough for a qualitatively correct graph at t=600.

 

 

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