The solution needs to be found using a Fourier Series expansion as in Post #2.

Not necessarily.

Any method of solving, correctly carried out, leads to the same result.

In the present case, the solution of the PDE with the third and fourth conditions leads to a unique result : the function u(x,t) in attachment.

Any other method will leads to this function and not to a different function.

But the patern of the function can be different. For example the function can be presented on the form of a Fourier series.

So the method which consists to solve the PDE thanks to Fourier series leads to the same result, but on the form of a Fourier series, which is the Fourier series corresponding to u(x,t) in attachment.

We know that the Fourier series correctly express the function on a limited range, beteween the borders, but not exactly on the border in many cases (for example the square wave function, the sawtooth wave function etc.). That is what append with the PDE considered here.

Of course, the Fourier series which is obtained satisfy the border conditions u(1,t)=0 and du/dx(0,t)=0. But it is an "artificial" agreement because the true fuction u(x,t) is deformed on the border in order to fit the patern of a Fourier series.