fa.functional analysis – What is the asymptotics of the Fourier transform of $exp(-x^4)$ for large wave numbers?

The Fourier transform of $exp(-x^4)$ has an analytical expression, it’s the difference of two generalized hypergeometric functions:

$int d x e^{-x^4} e^{ikx} = 2 Gamma(frac{5}{4}) _0F_2(;frac{1}{2},frac{3}{4};frac{k^4}{256}) – frac{k^2}{4} Gamma(frac{3}{4}) _0F_2(;frac{5}{4},frac{3}{2};frac{k^4}{256})$

The asymptotics of the 0F2 is known, but the two series for large k just cancel, so how can one get the asymptotics of the Fourier transform of $exp(-x^4)$ as $k rightarrow infty$?

fourier analysis – I am trying to solve Time evolution of Wave function using Operator Splitting Method. The Do loop in the last line is not working as per plan

`{nmax = 2^7, L = 20.0, dz = L/nmax, dkz = N(2 (Pi)/L), kzmax = N((Pi)/dz)};

zGrid = Table(-L/2 + (n – 1) dz, {n, 1, nmax});

kzGrid = Table(-kzmax + (q – 1) dkz, {q, 1, nmax});

kzpGrid = kzGrid // RotateLeft(#, nmax/2) &;

Vmodel(z_) = VHO(z) E^(-b z^2);

b = 0.1;

VHO(z_) = Vo + z^2/2;

Vo = -2.0;

V = Vmodel(zGrid);

KP = kzpGrid^2/2;

{ntmax = 30, T = 10.0, dt = T/ntmax};

U(dt_)@psi_ := UV(dt) InverseFourier(UK(dt) Fourier(psi))

UV(dt) = E^(-I V dt);

UK(dt) = E^(-I KP dt);

psi(nt_, psi0_) := NestList(U(dt)(#) &, psi0, nt);

psi(0) = E^(I kPo z) E^(-wPo z^2/2) (wPo/(Pi))^(1/4) /. {kPo -> 1.0, wPo -> 2.0} /. z -> zGrid // N;

Do(psi(nt,) = psi(10, psi(nt – 10)), {nt, 10, ntmax, 10});
`

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in the future for the second wave of infections in the USA.

engineering – 1D Wave Equation: Vertical Rod and Displacement vs. Textbook Solution

I am trying to setup Mathematica to analyze a vertical round rod under its own weight, fixed on one end free on the other. I have the 1D wave equation and a distributed load to represent the self weight of the round rod.

Vertical Rod Layout

The problem is when I compare the Mathematica solution to the textbook solution the two do not agree.

Sample problem is given below.

Y = 199*^9; (*Young's modulus in Pa *)
(Rho) = 7860; (* Steel density in kg/m^3*)
dia = 1/39.37; (* 1" dia converted to meters*)
c = Sqrt(Y/(Rho));
len = 1000; (*length in meters*)
tmax = 5; (* Max time for analysis*)
area = (Pi)*dia^2/4; (*Round rod cross sectional area*)
wtfactor = (Rho)*9.81*area/len;

frwt(x_) := (Rho)*
   area*9.81*(1 - 
     x/len); (*Rod Self weight imposed as a distributed load*)
nsol6 = NDSolve({!(
*SubscriptBox(((PartialD)), ({t, 2}))(z(x, t))) == c^2*!(
*SubscriptBox(((PartialD)), ({x, 2}))(z(x, t))) + frwt(x) + 
     NeumannValue(0, x == len),
   z(0, t) == 0}, z(x, t), {x, 0, len}, {t, 0, tmax},
  Method -> {"FiniteElement", 
    "MeshOptions" -> {"MaxCellMeasure" -> 10}}
  )
fnnsol6(x_, t_) = nsol6((1, 1, 2))
Plot3D(fnnsol6(x, t), {x, 0, len}, {t, 0, tmax}, 
 PlotLabels -> Automatic, AxesLabel -> Automatic)

deltaL = (((Rho)*9.81*len^2)/(
 Y*2)) (*Textbook elongation for a vertical rod under self weight*)
calcdeltaL = 
 fnnsol6(len, 
  5) (*Calculated delta Length from PDE solution.  Should match 
textbook*)

deltaLfunc(x_, l_) := (Rho)*9.81*
  x*(2*len - x)/(2*Y) (*Verified Correct*)
xydata = Thread({Range(0, 1000, 100), 
    deltaLfunc(x, 1) /. {x -> Range(0, 1000, 100)}});
xydata2 = 
 Thread({Range(0, 1000, 100), 
   Reverse(a)}); (*Same answer different calc format for debugging*)
Show(Plot(fnnsol6(x, 0), {x, 0, len}, PlotLabels -> {"PDE Val"}, 
  PlotRange -> All
  ),
 ListLinePlot(xydata2, PlotStyle -> Green, PlotLabels -> {"Correct"}))

If you’ve read this far, thank you.

In summary my question is: Is this a Mathematica issue or a PDE setup problem? The PDE is right out of a textbook so I don’t think that’s the problem but Mathematica gives no errors and I am out of troubleshooting ideas so looking for some help.

Thank You

dnd 5e – How does the Leviathan’s Tidal Wave interact with objects?

The ability in question is mechanically similar to the Tsunami spell, which only affects creatures.
However, the Leviathan is a siege monster, and its description states:

Siege Monster. The leviathan deals double damage to objects and structures (included in Tidal Wave).

Tidal Wave (Recharge 6). While submerged, the leviathan magically creates a wall of water centered on itself. The wall is up 250 feet long, up to 250 feet high, and up to 50 feet thick.
    When the wall appears, all other creatures within its area must each make a DC 24 Strength saving throw. A creature takes 33 (6d10) bludgeoning damage on failed save, or half as much damage on a successful one.
    At the start of each of the leviathan’s turns after the wall appears, the wall, a long with any other creatures in it, moves 50 feet away from the leviathan. Any Huge or smaller creature inside the wall or whose space the wall enters when it moves must succeed on a DC 24 Strength saving throw or take 27 (5d10) bludgeoning damage. A creature takes this damage no more than once on a turn. At the end of each turn the wall moves, the wall’s height is reduced by 50 feet, and the damage creatures take from the wall on subsequent rounds is reduced by 1d10. When the wall reaches 0 feet in height, the effect ends.
    A creature caught in the wall can move by swimming. Because of the force of the wave, though, the creature must make a successful DC 24 Strength (Athletics) check to swim at all during that turn.

1. Does the wave move through walls? What happens when the tidal wave encounters a wall, or any object whose size is significant compared to the wave? Does it clip through the object, like bad Roblox physics? Possibly killing everyone inside? I have a proposition: the wave damages the object, and if the damage is enough to collapse the wall, then it moves through the object. Perhaps if it is a ship, then you could describe it as the ship being capsized. However, if the wave isn’t enough to destroy the object, then the wave curves around it. Forget conservation of mass, if the object is bigger than the wave, then perhaps it could ghost through it (move through it but without existing inside it) and reappear behind it, if the object is small enough.

2. What happens if a creature, caught by the wave, hits an object ? Let’s say that you hit an object big enough not to be destroyed by the damage, what happens then? Surely you wouldn’t phase through the object to continue following the wave, nor would you curve around it to do so (it might not always even be possible).

3. How much damage does Tidal Wave do to objects? The rules for siege monster would indicate that the damage (6d10, 5d10…) is doubled for an object. So 2×(6d10) …?

This is about a CR20 monster whose description states that it destroys coastal settlements. So surely it can damage buildings, but in-game, how does it work?

Perhaps this would help:

Water Form. The leviathan can enter a hostile creature’s space and stop there. It can move through a space as narrow as 1 inch wide without squeezing.

Surely a wave is more so water than a monster? So perhaps the wave can also enter a closed space, so long as there is at least 1 inch of space? Does this make sense?

2d: How is the use of square and wave function collapse different in an automatic mosaic?

I am discovering procedural generation techniques.
I find it difficult to understand the use of these two techniques:

  • Wave function collapse
  • Squares marching

As I understand them both can be used to populate the neighborhood of a mosaic according to the rules.

In the case of a game map editor where a user can "paint" a dungeon. When would you use one against the other?

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Why is my DSolve solution for a wave equation not plotting and what are some of the best practices for verifying the solution?

1. Request for tips to plot DSolve
It is necessary to discover once and for all how to correctly plot the solution to a second order linear ordinary differential equation with constant coefficients. The harmonic oscillator is a very simple common case. Provide tips for doing so when two or more solutions are returned.

2. General solution
Although the plot seems correct, I am not sure I have used the best practices. Please also check my plot. I got it through trial and error, so I'm not so sure it's correct or uses best practices.

3. Plotting a particular solution
Why is my particular solution not plotting? How would you go about verifying the solution so that it returns True, False, or Not Evaluated?

(* harmonic oscillator *)
diagnosticDSolve(eqn_ : (x''(t) + (Omega)^2 x(t) == 0), 
   ic1_ : (x(0) == -1), ic2_ : (x'(0) == 0)) := 
  Module({eqs, solg, solp, versolg, versolp, plotg, plotp},
   eqs = Flatten( {eqn, ic1, ic2});
   solg = Flatten(FullSimnplify(DSolve(eqn, x(t), t)));
   solp = Flatten(FullSimplify(DSolve({eqn, ic1, ic2}, x(t), t)));(* 
   with ic's *)
   plotg = 
    Plot(solg /. C(1) -> 1 /. C(2) -> 1 /. (Omega) -> 1, {t, 0, 
      3 (Pi)}, ImageSize -> Small);
   plotp = 
    Plot(solp /. (Omega) -> 1, {t, 0, 3 (Pi)} ImageSize -> Small);
   Grid({{Column ({eqn, solg, plotg}), Column({eqns, solp, plotp})}}));
diagnosticDSolve()

tracing – Mathematica code for plane wave transmission

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