## machine learning – How to build an ML model from one group of PDE to another PDE?

I have one set of variables A which are the numerical solution of one group of PDEs, say the atmospheric model, and another set of variables B coming from the other PDE, for example, X, X’, X” w.r.t a parameter theta.
So how can I use some fancy techniques to build an ML from A to B and study its mechanisms? Or since the input A comes from PDEs, can I do some feature engineering using these PDEs?

I knew that from Stanford’s paper (https://advances.sciencemag.org/content/5/12/eaay6946) that the computational fluid dynamics equations follow a similar form with RCN. But how can I apply it to our field?

Thanks!

## soft question – Motivations for the term “jet” in the context of vicosity solutions for fully nonlinear PDE

My question is very direct:

What are the motivations for the name “jet”(subjet, superjet) in the context of viscosity solutions for second order fully nonlinear elliptic PDE?

(the definition of which can be seen in Crandall, Ishii, Lions: https://arxiv.org/pdf/math/9207212.pdf).

## Problem

The problem is a heat conduction problem:

$$frac{partial U}{partial t}= frac{partial^2u}{partial x^2}, 00 , ,, (1)$$
$$u(0,t) = 0, , , frac{partial u}{partial x}(1,t) = 0, , , t>0$$
$$u(x,0) = x^2 – 2x, , , 0 le x le 1$$

My task is to derive an explicit numerical method for this heat conduction problem

## My attempt

I used the Finite Difference Method. By using Forward Euler Approximation I arrived at:

$$k = 1/N, h = 1/J$$
$$frac{partial u}{partial t}(x_j, t_n) approx frac{U_{j, n+1}- U_{j, n+1}}{k}$$
$$frac{partial ^2 u}{partial x^2}(x_j, t_n) approx frac{U_{j, n} – 2U_{j,n}+ U_{j-1, n}}{h^2}$$

By using $$(1)$$ I got:
$$U_{j, n+1} = (1-2r)U_{j,n} + r(U_{j+1, n} + U_{j-1, n}), , , r=frac{k}{h^2}$$

The condition become:

$$U_{j, 0} = x_j^2 – 2x_j$$
$$U_{0, n} = 0$$

However how can I use this? $$frac{partial u}{partial x}(1,t) = 0$$

My attempt at using this by centered difference approximation :
$$frac{partial u}{partial x}(1,t) = frac{U_{2,n}-U_{0,n}}{2h} = 0$$

However I do not know how to proceed from here. My goal is to just derive the equations for the difference scheme. I feel like I am only missing the use of the Neumann Condition. All advice will be much appreciated.

## differential equations – Solve first order PDE in mathematica. (Routine economics problem, HJB)

This problem has a known closed form solution, but Mathematica is having trouble solving it.

(*write pde & boundary condition*)
eq2 = rho*V(t, a) == ( D(V(t, a), a) )^((gg - 1)/gg)/(1 - gg) +
D(V(t, a), a)*(y + r*a - ( D(V(t, a), a) )^(-1/gg)) +
D(V(t, a), t);
bc2 = V(T, a) == 0;

(*Try analytical*)
sola = DSolve({eq2, bc2}, V(t, a), {t, a})
(*unable to solve*)

(*plug in param*)
rho = 0.01; gg = 2.0; y = 10.0; r = 0.03; T = 20.0;
sola = DSolve({eq2, bc2}, V(t, a), {t, a})
(*unable to solve*)

(*Try to solve numerically*)
soln = NDSolve({eq2, bc2}, V(t, a), {t, 0, T}, {a, 0, 75})

Power::infy: Infinite expression 1/0.^0.5 encountered.
Power::infy: Infinite expression 1/0.^0.5 encountered.
General::stop: Further output of Power::infy will be suppressed during this calculation.
Infinity::indet: Indeterminate expression 0. ComplexInfinity encountered.
Infinity::indet: Indeterminate expression 0. ComplexInfinity encountered.
Infinity::indet: Indeterminate expression 0. ComplexInfinity encountered.
General::stop: Further output of Infinity::indet will be suppressed during this calculation.
NDSolve::ndnum: Encountered non-numerical value for a derivative at t == 20..
NDSolve::ndnum: Encountered non-numerical value for a derivative at t == 20..


Does anyone know how to solve this?

## fa.functional analysis – Reference request: PDE of the form $(Delta – |u|^2)f = F(u)$

I am interested in equations of the form
$$(Delta -|u|^2)f = F(u)$$
where $$F$$ depends on $$u$$ and preferably on its derivative, too. $$u$$ is supposed to be given and $$f$$ the unknown. More precisely I am interested in how to obtain a priori bounds on $$f$$, i.e.
$$|f|_X leq |(Delta-|u|^2)^{-1} F(u)|_Y$$
Of course, if we were to replace $$(Delta-|u|^2)^{-1}$$ by, say, $$Delta^{-1}$$ we could use something like Hardy-Littlewood-Sobolev to obtain bounds which is not possible if you take the term $$|u|^2$$ into account. If you can help me out with references on this kind of equation I would be very grateful.

## partial differential equations – How to solve this PDE by direct integration?

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## Discovery of norm in PDE

We have seen so many norms we need for PDE. For example, for elliptic PDE, we require a continuous version of $$C^k$$, i.e. $$C^{k,alpha}$$. Roughly speaking, under appropriate norm, we could capture the topological information we want. But a question (maybe too vague), how can we know what kind of norm we want in PDE?How can we invent the norm we want? I am just asking for a general idea.

## fa.functional analysis – Solution space of first order PDE

We consider the following first order PDE

$$(partial_x + ipartial_y) u(x,y)+ begin{pmatrix} 0 & cos(2x+y) \ cos(2x-y) & 0 end{pmatrix}u(x,y) =0,$$

where $$u in mathbb C^2$$ is vector-valued and periodic on $$(0,2pi)^2.$$

I ask: What is the dimension of the solution space to that equation?-My conjecture is that it is two-dimensional.

The reason is that if
the equation was

$$(partial_x + ipartial_y) u(x,y)+ begin{pmatrix} 0 & cos(2x+y) \ cos(2x+y) & 0 end{pmatrix}u(x,y) =0,$$

then we could explicitly state the two-dimensional solution space

$$u(x,y)=operatorname{exp}left( frac{i}{2i-1} begin{pmatrix} 0 & sin(2x+y) \ sin(2x+y) & 0 end{pmatrix}right)x_0$$

for any $$x_0 in mathbb{C}^2.$$

## analysis – Solving a specific linear parabolic PDE analytically

I have the following parabolic PDE on the domain $$(0,T)times mathbb{R}$$:
$$partial_t u = F(t,x) + (x+mu(t)) u + (a+bx) partial_x u +cpartial_x^2 u,$$
where $$F$$ is a function, $$mu$$ only depends on $$t$$, $$a,b$$ and $$c>0$$ are real numbers and the final condition is $$u(T,x)=f(x)$$. Assume all regularity on $$F$$, $$mu$$ and $$f$$, if needed.

I have the feeling this equation should be easily solved. I managed to somehow come to an analytic solution applying Fourier transforms which transforms it to a first order PDE and use the method of characteristics. But somehow I believe one should be able to solve it “at once” without Fourier transforms in a much easier way.

I read: http://www1.maths.leeds.ac.uk/~kersale/Teach/M3414/Notes/chap3.pdf but in my case: B^2 – AC=0 with B=0 so I can not carry out the change of variables suggested in that paper.

Does anyone know how to solve it or what is the smartest way? Thanks a lot!

## PDE with boundary condition (differential equation)

I am trying to find a solution to this boundary value problem

h_t= 1/2 sigma^2 x^2 h_xx + rx h_x s.t.
Ah(b,t)+Bh_x(b,t)=g(t),
where A,B,sigma,r,b are constants and g(t) is a given function of time (h_x is the partial derivative w.r.t. the first component of h(x,t)).

I need an explicit solution for it, whatever it is (any solutions satisfying the PDE and the boundary condition will be fine). I tried to use these commands, but it did not give anything:

eqn = 1/2 x^2 (Sigma)^2 D(h(x, t), {x, 2}) + r x D(h(x, t), x) – D(h(x, t), {t}) == 0;
ibc = {A h(b, t) + B Derivative(1, 0)(h)(b, t) == g(t)};
sol = DSolveValue({eqn, ibc}, h(t, x), {t, x}) // FullSimplify

I think I need to use something else, but I am just a beginner in Mathematica.
Thank you in advance!