Table 1: Laplace and Fourier Transformsorder the Fourier and Laplace transforms are applied to the...
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Transcript of Table 1: Laplace and Fourier Transformsorder the Fourier and Laplace transforms are applied to the...
Table 1: Laplace and Fourier Transforms
u(t) U(s)
δ(t) 1
11
s
e−at1
s+ a
u(t− t0); t0 ≥ 0 e−st0 U(s)
du
dtsU(s)− u(0)∫ t
0
u(t0) dt01
sU(s)∫ t
0
f(t0)h(t− t0) dt0 H(s) · F (s)
u(t) U(jω)
δ(t) 1
11
jω
e−at for t ≥ 0; 0 otherwise1
jω + a
u(t− t0) e−jωt0 U(jω)
du
dtjω U(jω)∫ t
−∞u(t0) dt0
1
jωU(jω)∫ +∞
−∞f(t0)h(t− t0) dt0 H(jω) · F (jω)
Table 2: Green’s Functions for Diffusion in 1-D
B.C. G(x, t;x0, t0)x = 0 x = L t > t0
— — N (x0,√
2D(t− t0)) =1√
4πD(t− t0)exp(− (x− x0)2
4D(t− t0))
u(0, t) = 0 — N (x0,√
2D(t− t0))−N (−x0,√
2D(t− t0))
∂u
∂x(0, t) = 0 — N (x0,
√2D(t− t0)) +N (−x0,
√2D(t− t0))
u(0, t) = 0 u(L, t) = 0∞∑k=1
2
Lsin(
kπ
Lx0) sin(
kπ
Lx) exp(−(
kπ
L)2D(t− t0))
u(0, t) = 0∂u
∂x(L, t) = 0
∞∑k=0
2
Lsin(
(k + 12)π
Lx0) sin(
(k + 12)π
Lx) exp(−(
(k + 12)π
L)2D(t− t0))
∂u
∂x(0, t) = 0 u(L, t) = 0
∞∑k=0
2
Lcos(
(k + 12)π
Lx0) cos(
(k + 12)π
Lx) exp(−(
(k + 12)π
L)2D(t− t0))
∂u
∂x(0, t) = 0
∂u
∂x(L, t) = 0
1
L+∞∑k=1
2
Lcos(
kπ
Lx0) cos(
kπ
Lx) exp(−(
kπ
L)2D(t− t0))
