## physics – Calculation with a distorted heat equation?

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## Is there an R package that can take postal codes and create a heat map of the United States by frequency of occurrence?

I want a heat card that takes the postal codes as input. From a quick search, it looks like the zip code package is out of date. Are there any other solutions?

## finite element method – Solving the nonlinear heat equation with initial conditions

I have the following configuration for the linear heat equation on an infinite rod (see below). This is a fairly standard configuration and the initial condition is $$e ^ {- x ^ 2}$$. This initial condition is not special, it just provides a way to make sure the code works.

Clear(x, t);
With({u = u(t, x)}, eq = D(u, t) == k D(D(u, x), x);
ic = u == Exp(-x^2) /. t -> 0;
bc = D(u, x) == 0 /. x -> 0;)
asol = DSolveValue({eq, ic, bc}, u, {t, x}, Assumptions -> {k > 0});
asol(t, x)


What I want to do is test different ideas so I want to change this linear heat equation to the nonlinear heat equation or another PDE which is similar but different from a linear PDE .

The nonlinear heat equation is defined as follows

$$start {equation} frac { partial u} { partial t} = frac { partial} { partial x} left (g (u) frac { partial u} { partial x} right) end {equation}$$

and more literature on the nonlinear heat equation can be found here.

If we assume that $$g (u) = u$$ then i would expect to be able to change the configuration like:

Clear(x, t);
With({u = u(t, x)}, eq = D(u, t) == k D( u D(u, x), x);
ic = u == Exp(-x^2) /. t -> 0;
bc = D(u, x) == 0 /. x -> 0;)
asol = DSolveValue({eq, ic, bc}, u, {t, x}, Assumptions -> {k > 0});
asol(t, x)


However, Mathematica simply regurgitates the code and does not produce any output.

I also wish to have a different type of installation such as:

$$start {equation} frac { partial u} { partial t} = frac { partial} { partial x} left (h (x) frac { partial u} { partial x} right) end {equation}$$

and also modify the first block of code.

If anyone can help me understand why my last configuration is not working and can show me how to configure it correctly, I would be very grateful.

Nor is it for a class. It is for my own personal understanding.

## dnd 5th – What counts as a sufficient source of heat to survive at the temple of Amber?

in the Curse of Strahd adventure, the box on the temperature of the temple of Amber (p. 181) says:

Extreme cold

The Amber Temple is a cold, dark place carved out of the snowy slope of Mount Ghakis. The temperature throughout the complex is -10 degrees Fahrenheit (-23 degrees Celsius). Characters without heat source, cold weather clothing or magic to protect them are subjected to the effects of extreme cold, as described in the "Weather" section of Chapter 5, "Adventure Environments", of the Dungeon Master Guide. If Kasimir Velikov (see chapter 5, zone N9a) is with the party, his heat ring protects him from the effects of extreme cold.

He specifically mentions "sources of heat", and I wondered what counts as a source of heat so as not to succumb to the cold of the Temple of Amber.

Is a torch good enough? Or are we talking more like a campfire (say, the kind that someone could do during a long rest)? j & # 39; imagine Curse of Strahd itself does not develop much more on this, but there could be generic rules in the DMG or similar that I have not managed to find with regard to temperature?

I ask because there will be at least one member of the group who will not have any (relevant) magic items, cold resistance or cold weather gear by the time they get there. will enter the temple of Amber, and I would like to know what matters as a heat source so I (as MD) know when to ask or give up the need to save the throws from exhaustion .

For reference, the "Extreme Cold" (Dungeon Master Guide, p. 110) section:

Whenever the temperature is 0 degrees Fahrenheit or less, a creature exposed to the cold must make a DC 10 Constitution saving throw at the end of each hour or gain a level of exhaustion. Creatures with
resistance or immunity to cold damage automatically succeeds on the saving throw, as are creatures wearing cold weather clothing (thick coats, gloves, etc.) and creatures naturally adapted to cold climates.

## dnd 5e – Would a sun blade be considered to be made of metal, for the purposes of the Heat Metal spell?

The description of the weapon is silent on the materials of the handle with the exception of the weight statistic: 3 pounds.

This figure does not really help to clarify the composition (pun intended), although it does not exclude the possibility of certain metallic parts. I think the DM should make a call to find out if the metal heat spell has an effect.

One point to remember here: we are dealing with a magic fantasy game. So, while real-world attorneys can help inform the imagination, they shouldn't limit its scope.

## Free mobile app heat map?

I need to analyze my applications in terms of usability. As I know, there are many free solutions for the web, but I haven't found any for mobile apps. Is there a free one?

## macbook pro – 16 "MBP: relative ambient heat of i7 v. i9

So, for me, a very important aspect of the devices is their ambient heat. I have some sort of annoying eczema which causes rashes on my hands / skin if I have been in constant contact with something hot, and this can be triggered by computers etc.

I wonder if anyone has figures on the warmth of the i7 chips compared to the i9 of the new 16 "MacBook Pro.

I bought the upgrade to get the cheaper i9 (2.3 GHz) and I find the device very very hot even when the processors are just sitting at 1-3% (and I'm in a cold winter room, so the internal heat is overcoming the outside air too).

I'm not sure what makes it so hot (iStat menus show batteries, palm rests, and Thunderbolt ports as the hottest parts), but I'm curious how this computer would compare to the 39; i7. I seriously considered buying the i7 just to get a cooler computer, but the upgrade was pretty cheap and I wanted the best GPU, so I went for it.

If it turns out that the i7 IS much cooler than this i9, I could trade it in for the long-term well-being of my skin.

So the question is again: Is there any data on relative ambient temperatures and relative underload of the i7 v versions. MacBook Pro I9. If so, is the i7 significantly cooler under normal circumstances.

## pde – Heat equation that depends on the viscosity constant $epsilon$

I want to solve the following heat equation, for $$epsilon> 0$$:

$$begin {cases} u_t- epsilon u_ {xx} = 0 & x <0, ; t in mathbb {R ^ {+ *}} \ u (0, x) = e ^ {- frac {-x} {2 epsilon}} & x <0 \ u (t, 0) = 1 & t in mathbb {R ^ {+ *}} end {cases}$$
And I want to make sure that the explicit solution is:

$$u (t, x) = frac {1} { sqrt {4t epsilon}} int _ { mathbb {R ^ {-}}} u (0, s) e ^ { frac { xs) ^ 2} {4 epsilon}} ds$$

## pde – Derivative $L ^ p$ – $L ^ q$ Estimates of the linear equation of heat

Let $$f: mathbb {R} ^ {N} to mathbb {R}$$ be the initial data of the following linear heat equation:
begin {align *} tag {1.1} begin {cases} partial_ {t} u (t, x) = Delta u (t, x) quad x in mathbb {R} ^ {N}, t> 0 \ u (0, x) = f (x) quad x in mathbb {R} ^ {N} end {cases} end {align *}
Then we leave $$G_ {t} (x): = frac {1} {(4 pi t) ^ {N / 2}} exp big ( frac {- | x | ^ {2}} {4t} big)$$ and we can write the solution of (1.1) as follows
begin {align *} tag {1.2} u (t, x): = int _ { mathbb {R} ^ {N}} G_ {t} (x-y) f (y) dy end {align *}

I want to prove the following theorem using Young's inequality for convolution.

Theorem ($$L ^ {p} -L ^ {q}$$ estimates)
Let $$u$$ to be the solution of (1.1) with the initial data $$f$$. For $$1 leq q leq p leq infty$$, there is a constant $$C = C (p, q, N)$$ such as
begin {align *} tag {1.3} | partial_ {x_ {j}} u (t) | _ {p} leq Ct ^ {- frac {N} {2} big ( frac {1} {q} – frac {1} {q} big) + frac {1} {2} } || f || _ {q} quadri = 1,2, …, N, , t> 0 end {align *}
begin {align *} tag {1.4} | partial_ {t} u (t) | _ {p} leq Ct ^ {- frac {N} {2} big ( frac {1} {q} – frac {1} {q} big) +1} || f || _ {q} quadri = 1,2, …, N, , t> 0 end {align *}
In addition, for higher derivatives, there is a constant $$C = C (p, q, N, k, alpha)$$ such as
begin {align *} tag {1.5} || partial_ {t} ^ {k} partial_ {x} ^ { alpha} u (t) || _ {p} leq Ct ^ {- frac {N} {2} big ( frac {1} {q} – frac {1} {p} large) + k + frac {| alpha |} {2}} | f | _ {q} end {align *}
or $$k in mathbb {N}$$ and $$alpha$$ is a multi-index $$( alpha_ {1}, alpha_ {2}, …, alpha_ {N})$$ with $$| alpha | = alpha_ {1} + alpha_ {2} + … + alpha_ {N}$$.

For convenience, I will also include the following theorem.
Theorem (Young Inequality)
Let $$1 leq p, q, r leq infty$$ such as $$frac {1} {p} = frac {1} {r} + frac {1} {q} -1$$. Then for everything $$h in L ^ {r} ( mathbb {R} ^ {N})$$ and $$f in L ^ {q} ( mathbb {R} ^ {N})$$, we have $$h * f in L ^ {p} ( mathbb {R} ^ {N})$$ and
begin {align *} tag {Y-in} | h * f | _ {p} leq | h | _ {r} | f | _ {q} end {align *}

This is my attempt until here:
First, we will prove (1.3) and (1.4). Observe that we have the following calculations
begin {align *} partial_ {x_ {j}} u & = partial_ {x_ {j}} G_ {t} * f \ partial_ {t} u & = partial_ {t} G_ {t} * f end {align *}
It means that
begin {align *} tag {1.6a} | partial_ {x_ {j}} G_ {t} (x) | & = bigg | (4 pi t) ^ {- N / 2} big ( frac {-2x_ {j}} {4t} big) exp big ( frac {- | x | ^ {2}} {4t } big) bigg | \ & leq (4 pi t) ^ {- N / 2} frac {| x |} {2t} exp big ( frac {- | x | ^ {2}} {4t} big) end {align *}
and
begin {align *} tag {1.6b} | partial_ {t} G_ {t} (x) | & = (4 pi t) ^ {- N / 2} bigg | – frac {N} {2} (4 pi t) ^ {- 1} 4 pi + frac {| x | ^ {2}} {4t ^ {2}} bigg | exp big ( frac {- | x | ^ {2}} {4t} big) \ & = 4 pi (4 pi t) ^ {- N / 2-1} bigg | – frac {N} {2} + frac {| x | ^ {2}} {4t} bigg | exp big ( frac {- | x | ^ {2}} {4t} big) \ & leq 4 pi (4 pi t) ^ {- N / 2-1} bigg ( frac {N} {2} + frac {| x | ^ {2}} {4t} bigg exp big ( frac {- | x | ^ {2}} {4t} big) end {align *}
Then we start by examining the case $$p = infty$$ and $$q = 1$$. From (1.6a), (1.6b) and $$z = frac {x} {2 sqrt {t}}$$we see that
begin {align *} | partial_ {x_ {j}} G_ {t} (x) | & leq frac {(4 pi t) ^ {- N / 2}} { sqrt {t}} | z | exp (- | z | ^ {2}) \ | partial_ {x_ {j}} G_ {t} | _ { infty} & leq (4 pi t) ^ {- N / 2- frac {1} {2}} frac {1} { sqrt {2e}} end {align *}
and
begin {align *} | partial_ {t} G_ {t} (x) | & leq 4 pi (4 pi t) ^ {- N / 2 -1} bigg ( frac {N} {2} + | z | ^ {2} bigg) exp (- | z | ^ {2}) \ | partial_ {t} G_ {t} | _ { infty} & leq 4 pi (4 pi t) ^ {- N / 2 -1} ( frac {N} {2} + frac {1} {e}) end {align *}
To choose $$C (p, q, N) = max bigg { frac {(4 pi) ^ {- N / 2- frac {1} {2}} { sqrt {2e}}, 4 pi (4 pi) ^ {- N / 2 -1} ( frac {N} {2} + frac {1} {e}) bigg$$we obtain (1.3) and (1.4) for the case $$p = infty$$ and $$q = 1$$ since (Y-in) implies
begin {align *} | partial_ {x_ {j}} u (t) | _ { infty} & leq | partial_ {x_ {j}} G_ {t} | _ { infty} | f | _ {1} \ | partial_ {t} u (t) | _ { infty} & leq | partial_ {t} G_ {t} | _ { infty} | f | _ {1} end {align *}

Then we can consider another case. From (1.6a) and (1.6b), we can again obtain
begin {align *} | partial_ {x_ {j}} G_ {t} (x) | ^ {r} & leq frac {2 ^ {r}} {(4 pi t) ^ {Nr / 2}} big | frac {| x |} {4t} big | ^ {r} exp big ( frac {- | x | ^ {2} r} {4t} big) \ | partial_ {x_ {j}} G_ {t} | _ {r} ^ {r} & leq frac {2 ^ {r}} {(4 pi t) ^ {Nr / 2}} int _ { mathbb {R} ^ {N}} t ^ {- r / 2} (2 sqrt {t}) ^ {N} | z | ^ {r} exp (- | z | ^ {2} r) dz \ | partial_ {x_ {j}} G_ {t} | _ {r} ^ {r} & leq C_ {1} t ^ {- frac {N} {2} (r-1) – frac {r} {2}} int _ { mathbb {R } ^ {N}} | z | ^ {r} exp (- | z | ^ {2} r) dz \ | partial_ {x_ {j}} G_ {t} | _ {r} & leq C_ {A} t ^ {- frac {N} {2} (1- frac {1} {r}) – frac {1} {2}} \ | partial_ {x_ {j}} G_ {t} | _ {r} & leq C_ {A} t ^ {- frac {N} {2} ( frac {1} {q} – frac {1} {p}) – frac {1} {2 }} end {align *}
and similarly
begin {align *} | G_ {t} (x) | _ {r} & leq (4 pi t) ^ {- N / 2 -1} 4 pi bigg { frac {N} {2} | exp (- frac {- | , cdot , | ^ {2}} {4t}) | _ {r} + | frac {- | , cdot , | ^ {2}} {4t} exp (- frac {- | , cdot , | ^ {2}} {4t}) | _ {r} bigg) \ & leq C_ {B} t ^ {- frac {N} {2} -1+ frac {N} {2r}} \ & = C_ {B} t ^ {- frac {N} {2} ( frac {1} {q} – frac {1} {p}) – 1} end {align *}
Using (Y-in), again, we obtain (1.3) and (1.4) for the other case of $$p$$ and $$q$$.

Now, my question is how to proceed to get the upper derivative case in (1.5)? I've tried using induction and strong induction but it does not work very well with manual calculation. Any help or suggestion allowing me to move to (1.5) is very appreciated!

Thank you!

## Mathematical physics – Transport of heat in an axi-symmetric flow through a thick cylinder [BVP]

A fluid passes through a cylindrical cavity with thick walls. A constant heat flux acts on the outer wall when the fluid is in thermal interaction with the inner wall. $$T$$ represents the temperature of the solid and the fluid counterpart is designated by $$T_f$$.
$$T$$ is coupled with $$T_f$$ in the sense that they influence each other (in other words, the fluid cools the solid along its flow). By modeling this situation, I reduced the phenomenon to the following problem:

In the field, $$r in (a_1, a_2)$$ and $$z in (0, L)$$ (given the axymetry, $$frac { partial T} { partial theta} = 0$$) so $$T = T (r, z)$$

$$a_1, a_2$$ are the inner and outer rays of the cavity, whereas $$L$$ is the length of the cylinder. The limit condition to $$r = a_1$$ takes into account the coupling between the two mediums.
$$frac { partial ^ 2 T} { partial r ^ 2} + frac {1} {r} frac { partial T} { partial r} + frac { partial ^ 2 T} { partial z ^ 2} = 0$$

is subject to

$$frac { T partial} { z partial} green_ {z = 0} = frac { partial T} { z partial} green_ {z = L} = 0$$

$$frac { T partial} { partial r} green_ {r = a_2} = gamma$$

$$frac { partial T} { partial r} green_ {r = a_1} = delta Bigg (T- alpha e ^ {- alpha z} bigg ( int_0 ^ ze ^ { alpha s} T (r, s) mathrm {d} s + frac {A} { alpha} bigg) Bigg)$$

$$A, alpha, gamma, delta$$ are constants.

In Cartesian coordinates, I could have taken some form of solution involving $$cos (f (z))$$ because of homogeneous Neumann boundaries along the $$z$$-direction. Anyone can suggest such an alternative in the cylindrical coordinate system. In addition, how to deal with the integral type boundary conditions $$r = a_1$$ ?