I wonder if it is possible to compute statistics from a stochastic differential equation. I start with a simple question about the stochastic linear differential equation. Leave $ dot {w} = z $Y $ z $ a normal distribution function, how can I calculate the expected value and standard deviation? I suspect that its standard deviation is 0, but I'm not sure about its stddev.

# Tag: differential

## Differential Geometry – Commutation Covariant Derivatives of a Variation

I am trying to understand the proof of the first and second variation of the Arclength formulas for Riemannian manifests. I want a check that the following covariant derivatives switch. I find it intuitive but I also want to have a formal test.

Some notation: Let $ gamma (t, s): overbrace {(a, b)} ^ {t} times overbrace {(- epsilon, epsilon)} ^ {s} rightarrow M $ be a variation of $ gamma_0 (t) $ with $ | dot { gamma_ {0} |} = lambda forall t in (a, b) $ Y $ V = gamma _ {*} left ( frac { partial} { partial s} right) $ The variational vector field.

So I would like to demonstrate that $ frac {D} {ds} ( dot { gamma_t}) = frac {D} {dt} V $ or in other words that $ frac {D} {ds}, frac {D} {dt} $ commute. Note that $ frac {D} {ds} $ is the covariant derivative throughout the map $ gamma $ therefore we will use the property for connections across maps that $ nabla ^ { gamma} _X (Z circ gamma (p)) = nabla _ { gamma _ * (X)} Z | _p $.

Indeed $ frac {D} {ds} left ( dot { gamma_t} right) = frac {D} {ds} ( gamma _ * ( frac {d} {dt}) circ ( gamma (s, t))) = D _ { gamma _ * left ( frac {d} {dt} right)} gamma _ * ( frac {d} {ds}) = D _ { gamma _ * left ( frac {d} {ds} right)} gamma _ * ( frac {d} {dt}) $.

The last equality follows from $ D _ { gamma _ * left ( frac {d} {dt} right)} gamma _ * ( frac {d} {ds}) – D _ { gamma _ * left ( frac {d} {ds} right)} gamma _ * ( frac {d} {dt}) = left ( gamma _ * ( frac {d} {dt}), gamma _ * ( frac {d} {ds}) right) = gamma _ * (( frac {d} {ds}, frac {d} {dt})) = 0 $ as $ frac {d} {ds}, frac {d} {dt} $ they are vector coordinate fields.

## smooth collectors: when does the retracement of differential forms preserve the cohomology classes: $ F ^ *[alpha]=[F^* alpha]$

Leave $ M ^ n $ be a smooth, compact collector without limit. Consider a dipheomorphism $ F: M rightarrow M. $

The retrocess $ F ^ * $ defines a linear isomorphism in the de Rham cohomogy spaces $ F ^ *: H ^ k_ {dR} (M) rightarrow H ^ k_ {dR} (M) $ where $ 0 leq k leq n. $

My question is: under what conditions is this the identity map in $ H ^ k_ {dR} (M) $? Also if $ M $ it is a complex closed complex, under which it conditions a holomorphic dipheomorphism $ F $ would give the identity in $ H ^ r, r} (M) $?

Any help is really appreciated!

## differential equations: how to model diffusion across a membrane?

This is a follow-up to How to handle the discontinuity in the diffusion coefficient?

Consider dissemination of $ u (t, x) $ in the domain $ x in (0.2) $ with some simple boundary conditions like $ u (0) = 2, u (1) = 1 $.

Our domain is divided into two parts: $ (0.1) $ to the left and $ (1,2) $ on the right, with different diffusion coefficients, p. $ D ^ text {left} = 1, D ^ text {right} = 3 $.

The diffusion equation is:

$$

partial_t u = partial_x (D partial_x u)

$$

**So far, this is the summary of the linked question.**

This time we also have a membrane in $ x = 1 $, imposing the following condition on the flows in $ x = 1 $:

$$

D ^ text {left} partial_x u ^ text {left} =

D ^ text {right} partial_x u ^ text {right} =

d ^ text {membrane} (u ^ text {right} – u ^ text {left})

$$

What is the cleanest way to model this with `NDSolve`

? Is there a way to preserve severe conditions in $ x = 1 $? Perhaps an approach that could be used is to consider a membrane of finite thickness, which has a very high diffusion coefficient. However, this is really a trick. Is it possible to solve the equation in the two half domains "separately" and couple the boundary conditions in $ x = 1 $?

## general solution of the first-order differential equation dy / dx = (x + y – 1) ^ 2, where x, y are real?

Could someone help me solve this.

Which of the following is the general solution of the first-order differential equation dy / dx = (x + y – 1) ^ 2, where x, y are real?

(A) Y = 1 + x + tan ^ -1 (x + c), where c is a constant.

(B) Y = 1 + x + tan (x + c), where c is a constant.

(C) Y = 1-x + tan ^ -1 (x + c), where c is a constant

(D) Y = 1-x + tan (x + c), where c is a constant.

## dg. differential geometry: how sectional curvature is affected by multiplying the Riemann metric by constant

I have the following short question: The section curvature of a plain is defined in terms of the Riemann curvature tensor, which is defined in terms of the Levi-Chivita connection, which is defined in terms of the Riemann matrix.

My question is, how does multiplying the Riemann metric by a constant affect the sectional curvature?

Thank you.

## differential equations: how to define the boundary condition in 1D heat transfer

I am trying to calculate the head transfer between a 1-D rod, with an insulated end while the right end is submerged on a surface of constant temperature T = 0. Suppose the initial temperature of the rod is T = 1. The bar length is 5. I configured the equation like this:

$$

frac { partial ^ 2 u (x, t)} { partial x ^ 2} – frac { partial u (x, t)} { partial t} = 0 \

u (x, 0) = 1 \

frac { partial u (5, t)} { partial x} = 0 \

u (0, t) = 0

$$

```
sol = NDSolve({
eqn = D(u(t, x), t) - D(u(t, x), {x, 2}) == 0,
u(0, x) == 1,
u(t, 0) == 0,
(D(u(t, x), x) /. x -> 5) == 0
}, u, {t, 0, 50}, {x, 0, 5})
Plot3D(Evaluate(u(t, x) /. %), {t, 0, 50}, {x, 0, 5},
PlotRange -> All)
```

Unfortunately, I got something like this:

NDSolve :: ibcinc: Warning: Initial limits and conditions are inconsistent.

Can anyone help me with the limit value problem?

## differential equations: using WhenEvent to bind variables

I would like to use the `WhenEvent`

(link here) to set a limit on a dynamic variable in an EDO set. That is, when the variables `k21(t)`

Y `k12(t)`

reach above the specific value `a`

I would like to put them back at that value. Seems like a simplified version of the problem

```
w1 = 6/24.5;
w2 = 6/23.5;
a = 0.1;
Eqs = {
x1'(t) == w1 + (k21(t)/2)*Sin(x2(t) - x1(t)),
x2'(t) == w2 + (k12(t)/2)*Sin(x1(t) - x2(t)),
k21'(t) == a*(Cos(x2(t) - x1(t) + (Pi)) + 1),
k12'(t) == a*(Cos(x1(t) - x2(t) + (Pi)) + 1)};
ICs = {x1(0) == 3/2, x2(0) == 3/4, k21(0) == 0.0001, k12(0) == 0.0001};
events = {WhenEvent(Abs(k21(t)) > a, k21(t) -> a), WhenEvent(Abs(k12(t)) > a, k12(t) -> a)};
EqsICs = Join(Eqs, ICs, events);
SolutionValue(t_) = NDSolveValue(EqsICs, {x1(t), x2(t), k21(t), k12(t)}, {t, 0, 10^6});
```

And I trace the solutions as:

```
Show(
Plot(SolutionValue(t)((3)), {t, 0, tmax}, PlotRange -> {{0, tmax}, {-0.1, 0.1}}, AxesOrigin -> {0, 0}),
Plot(SolutionValue(t)((4)), {t, 0, tmax}, PlotRange -> {{0, tmax}, {-0.1, 0.1}}, AxesOrigin -> {0, 0})
)
```

However, my `WhenEvent`

does not work The graph shows that the values โโof `kij`

continues to grow beyond `a`

. Is my syntax incorrect? Thank you ๐

## differential equations: analytical resolution of nonlinear ODE

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## Riemannian geometry: when is this differential form harmonic?

Leave $ (M ^ 3, g) $ be a Riemannian variety (closed) and leave $ u: M a S $ be a harmonic function where $ S $ It is a closed steerable surface. Yes $ omega $ is a $ 2 $-form in $ S $What are the sufficient conditions for $ omega $ in order $ u ^ * omega $ be a harmonic $ 2 $-form in $ M $?

The specific case I am analyzing is the following: we have $ u_1, u_2: M to mathbb {S} ^ 1 $ harmonic functions, $ u = (u_1, u_2): M to mathbb {S} ^ 1 times mathbb {S} ^ 1 $ Y $ omega = d theta_1 wedge d theta_2 $, where $ d theta_i $ denotes the shape of the volume in each $ mathbb {S} ^ 1 $.