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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 an (t) satisfies the initial value problem:
The values c1, c2, c3 … are the Fourier sine coefficients of the constant function 10-7:
The coefficient an (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.