Question

1. [5 marks] For the following function, sketch the Fourier series of $f(x)$ and determine the Fourier coefficients. $f(x) = egin{cases} 0 & x < 0 \ 1 & x > 0 end{cases}$ 2. [5 marks] For the following function, sketch the Fourier sine series of $f(x)$ and determine its Fourier coefficients. $f(x) = egin{cases} 1 & x < 1/6 \ 3 & 1/6 < x < 1/2 \ 0 & x > 1/2 end{cases}$ 3. [5 marks] Solve the PDE $u_t = u_{xx} + sinleft(frac{pi x}{l} ight)$, $0 < x < l$, $t > 0$, $u(0, t) = 0$ $u(l, t) = 0$, $t > 0$, $u(x, 0) = 1$, $0 < x < l$. 4. [5 marks] Solve the PDE $u_t = u_{xx}$, $0 < x < l$, $t > 0$, $u(0, t) = 0$ $u(l, t) = cos t$, $t > 0$, $u(x, 0) = x$, $0 < x < l$.

          1. [5 marks] For the following function, sketch the Fourier series of $f(x)$ and determine the Fourier coefficients.
$f(x) = egin{cases} 0 & x < 0 \ 1 & x > 0 end{cases}$
2. [5 marks] For the following function, sketch the Fourier sine series of $f(x)$ and determine its Fourier coefficients.
$f(x) = egin{cases} 1 & x < 1/6 \ 3 & 1/6 < x < 1/2 \ 0 & x > 1/2 end{cases}$
3. [5 marks] Solve the PDE
$u_t = u_{xx} + sinleft(frac{pi x}{l}
ight)$, $0 < x < l$, $t > 0$,
$u(0, t) = 0$  $u(l, t) = 0$, $t > 0$,
$u(x, 0) = 1$, $0 < x < l$.
4. [5 marks] Solve the PDE
$u_t = u_{xx}$, $0 < x < l$, $t > 0$,
$u(0, t) = 0$  $u(l, t) = cos t$, $t > 0$,
$u(x, 0) = x$, $0 < x < l$.

$1. [5 marks] For the following function, sketch the Fourier series of f(x) and determine the Fourier coefficients. f(x) = egincases 0 x < 0 1 x > 0 endcases 2. [5 marks] For the following function, sketch the Fourier sine series of f(x) and determine its Fourier coefficients. f(x) = egincases 1 x < 1/6 3 1/6 < x < 1/2 0 x > 1/2 endcases 3. [5 marks] Solve the PDE ut = uxx + sinleft(fracpi xl ight), 0 < x < l, t > 0, u(0, t) = 0 u(l, t) = 0, t > 0, u(x, 0) = 1, 0 < x < l. 4. [5 marks] Solve the PDE ut = uxx, 0 < x < l, t > 0, u(0, t) = 0 u(l, t) = cos t, t > 0, u(x, 0) = x, 0 < x < l.$

Added by Diego S.

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