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Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
2
votes
1
answer
55
views
Literature on Lateral cauchy problem
Can you please provide me with books that deal with lateral cauchy problem?
Also introductory articles are fine by me.
Is this topic covered in books like Evans', Taylor's, Hilbert-Courant's or othe …
1
vote
1
answer
251
views
A question on theorem 1.1 of Fritz John ultrahyperbolic pde
I have the following paper:
Fritz John, The ultrahyperbolic differential equation with four independent variables, Duke Math. J. 4 (1938), no. 2, pages 300-322
doi:10.1215/S0012-7094-38-00423-5
Now …
2
votes
1
answer
828
views
Lebesgue Riemann Theorem.
Does someone know where may I find the general proof of Riemann Lebesgue theorem which states that
Let $1\leq p \leq \infty $ $M= \prod_{i=1}^{n} (a_i,b_i)$, and $u \in L^p$ , define $u_\nu = u(\nu x …
5
votes
1
answer
641
views
Theorems that tell if an explicit analytical solution is possible for nonlinear PDEs
Are there any theorems that tell if a particular nonlinear PDE can be solved explicitly by analytical methods?
Where analytical methods I refer to methods such as power series or any methods that use …
1
vote
0
answers
101
views
Suggestion for books in Pertubation theory with an emphasis on the theory
As the title suggest I am looking for another good coverage of the theory of Pertubation theory.
Currently I am working through Murodock's book: Pertubations: Theory and Methods.
But I am rest assure …
1
vote
1
answer
2k
views
Lipschitz functions and $W^{1,\infty}$
I am not sure my question is research type, but I am sure I can find here an answer.
So we have the following theorem in the book of Lawrence Evans in PDE, 2nd edition pages 294-295:
Theorem 4 (C …
7
votes
2
answers
562
views
Relativistic Control Theory
I am looking for literature that combines General relativity and control theory.
So far I found a video lecture on "Integrability meets Control Theory: Harmonic maps in GR", other than that not so mu …
-2
votes
1
answer
220
views
Solving a nonlinear PDE numerically
I want to solve numerically the following PDE:
$$ u_x + u_t - (u_{xt})^2 = u(x,t) $$
The boundary conditions are no concern of mine, pick the ones that work.
So which numerical method will work for t …
0
votes
1
answer
247
views
Numerical methods for solving a hyperbolic nonlinear PDE
What type of numercial methods are there to solve PDE of the sorts of:
$$f(x,t,u(x,t))u_{xx} - g(x,t,u(x,t))u_{tt} = F(x,t,u(x,t))$$
$$u(x,0)=G_1(x) , \frac{\partial u(x,0)}{\partial t}=H_1(x) ,u(0,t …
-4
votes
1
answer
571
views
Derivatives of infinite order [closed]
Is there any sense of taking an infinite number of derivatives? Is it discussed in the literature?
For example, can one make sense of
$$\frac{\partial^{\infty}f(x_1,x_2,\cdots)}{\partial x_1 \partia …
7
votes
0
answers
333
views
Methods of variational calculus in analytic number theory
What methods of calculus of variations have been used in analytic number theory?
I mean do Hamilton-Jacobi theory of PDE found usage in analytic number theory, which raises yet another question has P …
1
vote
0
answers
80
views
boundedness of a sequence $ \in L^{\infty}(I,H^1(M))\cap Lip(I,L^2(M))$ implies that its tem... [closed]
Hi I have the next claim which I would like to find a proof of it.
I have a sequence of functions $u_\epsilon(t,x) \in H^1(M)$ where $M$ is a compact manifold, and $u_\epsilon \in L^\infty(I,H^1(M))\ …
6
votes
1
answer
387
views
Can there be an application of discrete mathematics in PDEs, mainly the ones used in hydrody...
Can there be applications of graph theory, combinatorics etc. in PDEs mainly hydrodynamics?
Tried my luck with Google's search engine, didn't show much info.
I guess you can try to use these features …