# The Preparation of Fourier Series Matlab

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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.