Tagged Questions
3
votes
1answer
57 views
How to solve partial integro-differential equation?
Suppose the following partial integro-differential equation for a function $u(x,t)$ with $t\geq0$, $x \in [0,L]$:
$\partial_t u = \partial_{xx} u + f(u,\lambda)$
$\lambda = B\left(u_0 - \int_{x=0}^L ...
1
vote
1answer
68 views
Laplace transform of convolution with modified limits
I have an expression such as
$\int_0^{x+l}y(z)g(x-z) dz$
and I want to evaluate its Laplace transform w.r.t $x$ in terms of the Laplace transform of $y(x)$. I know that I can substitute $t=x+l$, and ...
2
votes
1answer
110 views
Laplace transform having this unusual property in convolution?
Here is the problem
Solve $y'(t) = 1 - \int_{0}^{t} y(t - v)e^{-2v}dv$
The solution sets $\mathcal{L}(y) = Y(s)$ and does the following
Notice that in step 1, they have $$Y(s)\dfrac{1}{s+2}$$
...
1
vote
1answer
108 views
Proving $\int_0^\infty \frac{w}{1+w^2} \sin wx dw=\frac{\pi}{2}e^{-x}$ with the Laplace transformation (and/or Fourier transformation)
Can anyone help me prove this with the help of the Laplace transformation?
$$\int_0^\infty \frac{w}{1+w^2} \sin wx dw=\frac{\pi}{2}e^{-x}$$
where $x>0$
EDIT: So I was wondering if you could ...
4
votes
1answer
233 views
How to solve $t-2f(t) = \int_0^t(e^\tau- e^{-\tau})f(t-\tau)d\tau$
I want to solve this equation. It reminds me something about Laplace transform.
I am sure that I must use it order to solve it.
$$t-2f(t) = \int_0^t(e^\tau- e^{-\tau})f(t-\tau)d\tau$$
How to do it?
...
