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Results tagged with ap.analysis-of-pdes
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user 134299
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
40
votes
Accepted
Do we lose any solutions when applying separation of variables to partial differential equat...
Consider your purported solution $u(x,t)$ at fixed $t$, i.e., think of it as a function only of $x$. Such a function can be expanded in a complete set of functions $f_n (x)$,
$$
u(x,t)=\sum_{n} u_n f_ …
- 6,696
1
vote
One dimensional heat equation with boundary conditions
In general, there isn't a solution at all, let alone an explicit one. For example, take $f(t)=0$ and $g(x)=(x-L/2)^2 -L^2/4$. Then, at $t=0$, we have $u_t = u_{xx} =2$ at all $x$, including $x=0$ and …
- 6,696
1
vote
What is an "exact solution" to a PDE?
There is no "exact answer" to this question. Answers will contain the words "reasonable" and "appropriate", terms which depend on the context. I'll try to give a reasonable answer.
Given a domain of p …
- 6,696
1
vote
Accepted
Cylindrical coordinates in axis symmetric flow
There's a typo in the last two expressions you write: $u^{\theta } /r$ should be $u^r /r$.
Then, it seems to check out (for any well-behaved $J$): In the first expression, just insert the expressions …
- 6,696
3
votes
Global first integral for certain $3$ dimensional system
One can find a solution of the form $x=y=z$, namely, $x=2$arccot$(\exp (-t-a))$ with the free parameter $a$. Of course, there should be more. Note also the symmetries of the problem: For any solution …
- 6,696
20
votes
Why is resonance such a widespread phenomenon?
Most systems you see around you are subject to a restoring force (otherwise, they'll go find an equilibrium elsewhere). Most restoring forces are linear as long as you're not too violent with the syst …
- 6,696
6
votes
Accepted
Diagonalise self-adjoint operator explicitly?
By noting that $-i\partial_{x_1} $ is diagonalized by $e^{ik_1 x_1} $ and $-i\partial_{x_2} $ by $e^{ik_2 x_2} $, the problem reduces to a $2\times 2$ diagonalization for each $(k_1,k_2)$-block. The r …
- 6,696
6
votes
Accepted
Even and odd solutions for the Schrödinger equation
Because of the uniqueness of the initial value problem, there can be at most two solutions, i.e., if we have one odd $2a$-periodic solution, then there can be at most one more even $2a$-periodic solut …
- 6,696
3
votes
Accepted
On an integral equation
The answer is no. A counterexample is
$$
f(t,x) = x - \frac{1}{2} -\frac{1}{24} t^2
$$
$$
B(t,s,x) = \left( x-\frac{1}{2} \right) (t-s)
$$
(Method: I obtained this by expanding $f$ and $B$ into power …
- 6,696
6
votes
Does there exist an electromagnetic analogue of Einstein's field equations?
There is a profound conceptual shift between Gravitoelectromagnetism and General Relativity that cannot analogously occur between Maxwell's equations and any "???".
In General Relativity, there is not …
- 6,696