Fourier transform of $frac{U_{j}^{n+1} - U_{j}^{n}}{Delta t} $
if $U_{j}^{n},V_{j}^{n}$ are approximations for $u_{j}^{n} = u(x_j,t_n)$ and $v_{j}^{n} = v(x_j,t_n)$
How does applying the fourier transform defined by:
$U_{j}^{n} = frac{1}{2pi}int_{-frac{pi}{Delta x}}^{frac{pi}{Delta x}} hat{U}^{n}(k)e^{ikjDelta x}dk$
to
$frac{U_{j}^{n+1} - U_{j}^{n}}{Delta t} = frac{1}{Delta x^2}(U_{j+1}^{n} - 2U_{j}^{n} + U_{j-1}^{n})$
give
$hat{U}^{n+1}(k) - hat{U}^{n}(k) = frac{Delta t}{Delta x^2}(-4sin^2(frac{kDelta X}{2}))hat{U}^n(k)$
I know that applying the fourier transform we get
$frac{1}{2pi} int_{-frac{pi}{Delta x}}^{frac{pi}{Delta x}} frac{hat{U}^{n+1}(k) - hat{U}^{n}(k)}{Delta t} e^{ikj Delta x} dk =frac{1}{2pi} int_{-frac{pi}{Delta x}}^{frac{pi}{Delta x}} hat{U}^n(k)e^{ikj Delta x} frac{e^{ikDelta x} -2 + e^{-ikDelta x}}{Delta x^2} dk $
but im stuck after here
differential-equations fourier-transform
asked Dec 26 '18 at 18:41
pablo_mathscobar
836
add a comment |
if $U_{j}^{n},V_{j}^{n}$ are approximations for $u_{j}^{n} = u(x_j,t_n)$ and $v_{j}^{n} = v(x_j,t_n)$
How does applying the fourier transform defined by:
$U_{j}^{n} = frac{1}{2pi}int_{-frac{pi}{Delta x}}^{frac{pi}{Delta x}} hat{U}^{n}(k)e^{ikjDelta x}dk$
to
$frac{U_{j}^{n+1} - U_{j}^{n}}{Delta t} = frac{1}{Delta x^2}(U_{j+1}^{n} - 2U_{j}^{n} + U_{j-1}^{n})$
give
$hat{U}^{n+1}(k) - hat{U}^{n}(k) = frac{Delta t}{Delta x^2}(-4sin^2(frac{kDelta X}{2}))hat{U}^n(k)$
I know that applying the fourier transform we get
$frac{1}{2pi} int_{-frac{pi}{Delta x}}^{frac{pi}{Delta x}} frac{hat{U}^{n+1}(k) - hat{U}^{n}(k)}{Delta t} e^{ikj Delta x} dk =frac{1}{2pi} int_{-frac{pi}{Delta x}}^{frac{pi}{Delta x}} hat{U}^n(k)e^{ikj Delta x} frac{e^{ikDelta x} -2 + e^{-ikDelta x}}{Delta x^2} dk $
but im stuck after here
differential-equations fourier-transform
asked Dec 26 '18 at 18:41
pablo_mathscobar
836
add a comment |
if $U_{j}^{n},V_{j}^{n}$ are approximations for $u_{j}^{n} = u(x_j,t_n)$ and $v_{j}^{n} = v(x_j,t_n)$
How does applying the fourier transform defined by:
$U_{j}^{n} = frac{1}{2pi}int_{-frac{pi}{Delta x}}^{frac{pi}{Delta x}} hat{U}^{n}(k)e^{ikjDelta x}dk$
to
$frac{U_{j}^{n+1} - U_{j}^{n}}{Delta t} = frac{1}{Delta x^2}(U_{j+1}^{n} - 2U_{j}^{n} + U_{j-1}^{n})$
give
$hat{U}^{n+1}(k) - hat{U}^{n}(k) = frac{Delta t}{Delta x^2}(-4sin^2(frac{kDelta X}{2}))hat{U}^n(k)$
I know that applying the fourier transform we get
$frac{1}{2pi} int_{-frac{pi}{Delta x}}^{frac{pi}{Delta x}} frac{hat{U}^{n+1}(k) - hat{U}^{n}(k)}{Delta t} e^{ikj Delta x} dk =frac{1}{2pi} int_{-frac{pi}{Delta x}}^{frac{pi}{Delta x}} hat{U}^n(k)e^{ikj Delta x} frac{e^{ikDelta x} -2 + e^{-ikDelta x}}{Delta x^2} dk $
but im stuck after here
differential-equations fourier-transform
asked Dec 26 '18 at 18:41
pablo_mathscobar
836
if $U_{j}^{n},V_{j}^{n}$ are approximations for $u_{j}^{n} = u(x_j,t_n)$ and $v_{j}^{n} = v(x_j,t_n)$
How does applying the fourier transform defined by:
$U_{j}^{n} = frac{1}{2pi}int_{-frac{pi}{Delta x}}^{frac{pi}{Delta x}} hat{U}^{n}(k)e^{ikjDelta x}dk$
to
$frac{U_{j}^{n+1} - U_{j}^{n}}{Delta t} = frac{1}{Delta x^2}(U_{j+1}^{n} - 2U_{j}^{n} + U_{j-1}^{n})$
give
$hat{U}^{n+1}(k) - hat{U}^{n}(k) = frac{Delta t}{Delta x^2}(-4sin^2(frac{kDelta X}{2}))hat{U}^n(k)$
I know that applying the fourier transform we get
$frac{1}{2pi} int_{-frac{pi}{Delta x}}^{frac{pi}{Delta x}} frac{hat{U}^{n+1}(k) - hat{U}^{n}(k)}{Delta t} e^{ikj Delta x} dk =frac{1}{2pi} int_{-frac{pi}{Delta x}}^{frac{pi}{Delta x}} hat{U}^n(k)e^{ikj Delta x} frac{e^{ikDelta x} -2 + e^{-ikDelta x}}{Delta x^2} dk $
but im stuck after here
differential-equations fourier-transform
differential-equations fourier-transform
asked Dec 26 '18 at 18:41
pablo_mathscobar
836
asked Dec 26 '18 at 18:41
pablo_mathscobar
836
edited Dec 26 '18 at 19:02
asked Dec 26 '18 at 18:41
pablo_mathscobar
836
asked Dec 26 '18 at 18:41
pablo_mathscobar
836
asked Dec 26 '18 at 18:41
pablo_mathscobar
836
836
add a comment |
add a comment |
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