# pde – solution for \$ u_ {t} = alpha ^ {2} u_ {xx} \$ (problem with Fourier series of the initial state)

Solve the partial differential equation $$u_ {t} = alpha ^ {2} u_ {xx}$$ with conditions:
$$begin {cases} u (x, 0) = f (x) \ u (0, t) = u_ {x} (L, t) = 0 end {cases}$$
from where
$$f (x) = begin {cases} frac {2} {L} x, & x in left[0frac{L}{2}right)\1xinleft[Frac{L}{2}Lright)end{cases}[0frac{L}{2}right)1xinleft[Frac{L}{2}Lright)end{cases}[0frac{L}{2}right)1&xinleft[frac{L}{2}Lright)end{cases}[0frac{L}{2}right)&xinleft[frac{L}{2}Lright)end{cases}$$
They must justify the convergence of the series and, if necessary, justify aspects of odd and even extensions.

My attempt

Step 1. Separation of variables:

Write $$u (x, t) = F (x) G (t)$$, Thus, $$F (x) G # (t) = alpha ^ {2} F & # 39; ((X) G (t)$$i.e.
$$frac {F & # 39; & # 39; (x)} {F (x)} = y = frac {1} { alpha ^ {2}} frac {G # (t)} {G (t)}$$
where the parameter $$y$$ no depends on $$x$$ and $$t$$, We have two ODEs:
$$F & # 39; & # 39; (x) – yF (x) = 0 tag {1}$$
and
$$G # (t) – y alpha ^ {2} G (t) = 0 tag {2}$$

Step 2: Bound conditions.

we have

• If $$y = 0$$, $$F equiv 0$$,
• If $$y> 0$$, $$F equiv 0$$,
• If $$y <0$$, to take $$y = – lambda ^ {2}$$ then $$F_ {n} (x) = c_ {2} sin left ( frac {n pi x} {2L} right)$$

where the solution for $$y> 0$$ is $$F (x) = c_ {1} cos ( lambda x) + c_ {2} sin ( lambda x)$$, The solution for $$G # (t) – y alpha ^ {2} G (t) = 0$$ is $$G (t) = ce ^ {- y alpha ^ {2} t}$$, So for $$n = pm 1, pm 2, dots$$, we have
$$u_ {n} (x, t) = Ke ^ {- frac {n pi alpha} {2L} t} sin left ( frac {n pi x} {2L} right).$$

Step 3. Find the Fourier series from $$f$$,

I tried to get an odd or even extension of it $$f$$ and so to find the extension's Fourier series (maybe it's not possible to get one $$sin$$ or $$cos$$ The Fourier series). But I could not find a periodic extension of $$f$$, Actually, I want to get a Fourier series of the form
$$sum_ {1} ^ { infty} c_ {n} sin left ( frac {n pi x} {2L} right).$$
With this,
$$u (x, t) = K sum_ {1} ^ { infty} c_ {n} e ^ {- frac {n pi alpha} {2L} t} sin left ( frac { n pi x} {2L} right)$$
is a solution.

Can someone help me?