5 votes

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 ...
Alp Uzman's user avatar
  • 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 $\frac{\partial f}{\partial x}=3y^2+x$ ...
AlgTop1854's user avatar
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 ...
whpowell96's user avatar
  • 3,827
2 votes

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 ...
Lorenzo Pompili's user avatar
1 vote

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 ...
Eli Johnson's user avatar
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^{-\...
Gonçalo's user avatar
  • 4,457
1 vote

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{...
RRL's user avatar
  • 89.2k
1 vote

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 ...
jd27's user avatar
  • 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\...
MaoWao's user avatar
  • 14.7k

Only top scored, non community-wiki answers of a minimum length are eligible