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?