5
votes
Accepted
What difference does it make if many differential equations can't be solved analytically?
When I was in school I learnt that most differential equations cannot be solved analytically and are thus solved using numerical methods.
Perhaps it would be more accurate to say that numerical ...
- 10k
3
votes
Why do we add a function $g(y)$ at the end of a partial differential equation?
It is no different than adding the usual constant $C$ in taking antiderivatives of functions of a single variable. In taking the antiderivative of the expression
$\frac{\partial f}{\partial x}=3y^2+x$
...
- 304
2
votes
linearized operator for ODE system
When you linearize about a point $U_0$, you typically also make a change of variable to something like $\widetilde{U} = U - U_0$ so that the new variables $\widetilde{u}$ and $\widetilde{v}$ have an ...
- 3,827
2
votes
Accepted
Using density to prove an inequality for functions $f \in H^{1}(\mathbb{R}^{d})$
First: Fatou’s lemma is the key. Whenever you have convergence in $L^2$, you have convergence a.e. (up to subsequences). And whenever you have a sequence of functions $\varphi_n$ that converge a.e. to ...
- 3,102
1
vote
Accepted
Laplace equation in an annulus
Yes.
The assumption that your solution is radial implies the boundary values on the inner ring are constant, likewise on the outer ring. So boundary values are determined up to two constants, and can ...
- 425
1
vote
Radial Solution To The Heat Equation $u_t=u_{xx}+u_{yy}+u_{zz}$
Your solution of part 1 has a few typos:
You should do an odd, instead of even, extension for $v(t,r)$, since $v$ must vanish at the origin:
$$
v(t,r)=\frac{1} {\sqrt{4 \pi t}} \int_0^\infty e^{-\...
- 4,457
1
vote
Accepted
Why can we pass limit under integral sign in proof of solving Poisson's equation? (Evans PDE)
We have
$$\left|\frac{u(x+he_i)-u(x)}{h}- \int_{\mathbb{R}^n}\Phi(y)\frac{\partial f}{\partial x_i}(x-y)\, dy\right|\\= \left|\int_{\mathbb{R}^n}\Phi(y)\frac{f(x+he_i-y)-f(x-y)}{h}\, dy- \int_{\mathbb{...
- 89.2k
1
vote
Accepted
How is it justified to take the tensor product of nonlinear functions?
I am going to consider the case where both functions have values in a field instead of a vector space, since i am not sure how the multiplication on the right hand side of the first equation in the ...
- 1,152
1
vote
Semigroup properties of spectral fractional Laplacian
Yes on all counts. The first question should be whether the spectral fractional Laplacian generates a semigroup at all. This is indeed the case. An explicit formula is
$$
P_t f=\sum_{k=1}^\infty e^{-t\...
- 14.7k
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