Tag: ODEs

ca.classical analysis and odes – Estimate on an integral involving the Japanese bracket

I’m currently reading the paper Well-posedness for the Zakharov system with the periodic boundary condition by Takaoka. In the proof of Lemma 2.3 about the integral $I_1$ one needs to establish the estimate

$$int_{-infty}^{infty} frac{dtau’}{langle n^2 + |tau’+alpha n^2|rangle^{2a}} leq Cfrac{1}{langle nrangle^{4a-2}},$$

here $langle cdot rangle := (1+|cdot|^2)^{1/2}$ denotes the usual Japanese bracket, $a in (1/2,3/4)$, $alphaneq 0$ is a real parameter and $n$ is an integer.

This estimate is not explicitly mentioned in the proof but an intermediate step in order to arrive at equation (2.1). It is clear to me that the order of the exponent of the RHS is correct, since the Japanese bracket has order one and one expects to gain one order by integration. If $a$ were an integer I’m sure that by substitution one would have to use some $arctan$ properties.

How can I establish this estimate?

differential equations – Solving composite function ODEs but facing syntax difficulties

Today is my first day using Mathematica to solve PDEs and ODEs. I have coded this block:

vAcross(theta_) = vIn Sqrt(Sin(b) Cos(theta) ((Sin(a))^2 Cos(a) Sin(b)
  - 2 Sin(a) Cos(a) Cos(b) + Sin(b) Cos(theta))),

re(vAcross(theta_)) = (rho vAcross(theta) dWire)/mu,

nu(re(vAcross(theta_))) = (0.376 re^(1/2) + 0.057 re^(2/3)) pr^(1/3)
  + 0.92 (Log(7.4055/re) + 4.18 re)^(-1/3)

in order to obtain nu(theta), but I constantly get syntax errors such as

Syntax::tsntxi: “vAcross(theta_)=vIn Sqrt(Sin(b)Costheta),<<1>>,nu(re(vAcross(theta_)))=(0.376re^(1/2)+0.057re^(2/3))pr^(1/3)+0.92(Log(7.4055/re)+4.18re)^(-1/3)” is incomplete; more input is needed.

Please assist me in troubleshooting my coding! Many thanks in advance.

ca.classical analysis and odes – A simple oscillatory integral with a non-smooth phase

Let $phiin C_c^infty(mathbb{R})$ be an even function such that $chi_{(-1/2,1/2)}lephile chi_{(-1,1)}$, where $chi_{(a,b)}$ stands for the indicator function of the interval $(a,b)$. For $lambda>0$ consider the oscillatory integral
$$
I(lambda)=int_mathbb{R} phi(x), exp left(ilambda(x+epsilon|x|^{sqrt{2}})right), dx,
$$
with some fixed (very small) positive constant $epsilon$.

My question is: what is the asymptotic behavior of this integral as $lambdarightarrow infty$? I can show, by essentially doing careful integration by parts, that the upper bound is $lesssim lambda^{-sqrt{2}}$, but I wonder whether $lambda^{-sqrt{2}}$ is also a lower bound?

Note, that if the exponent $sqrt{2}$ is replaced by $2k$ for some positive integer $k$, then the integral decays like $lambda^{-M}$ for any $M>0$ due to the non-stationary phase estimate (the derivative of the function $x+epsilon x^{2k}$ is $gtrsim 1$).

I would appreciate any hints on how to approach this problem.

ca.classical analysis and odes – Increasing concave functions bounded between linear and quadratic

Are there any explicit examples of twice-differentiable functions $g$, defined on the nonnegative real axis $(0,infty)$, such that

  • $g'(x) ge0$
  • $g”(x)le0$,

and such that

  • For $xge K$ for any fixed number $Kge1$ (you have the freedom to choose any $Kge1$ here), we have the property $|g”(x)| > C/x$, preferably $|g”(x)| ge C/x^{1-epsilon}$, for some $epsilon>0$?

Here $C$ is a constant that can be any nonzero number.

ca.classical analysis and odes – Why do the absolute values of functions in Hardy spaces tend to be non-oscillatory?

I’m trying to find a rigorous formulation of an impression/intuitive notion related to Hardy spaces. It seems to me that functions in Hardy spaces tend to have modulus functions which do not oscillate. This apparently generalizes a property of the exponential function. In particular:
$$exp(ikz) = cos(kz) + i*sin(kz)$$
with $sin(kz)$ the harmonic conjugate of $cos(kz)$, and
$$|exp(ikz)|^2 = cos^2(kz) + sin^2(kz) = 1.$$

For many examples of functions $F$ in Hardy spaces, one has
$$F(x) = u(x) + i v(x)$$
with $u(x)^2 + v(x)^2$ very apparently non-oscillatory. For certain specific examples, I am able to prove something. Bessel functions are a good example, I think.

The spherical Hankel function $h_n(z)$ of the first kind admits
the representation
$$h_n(z) = exp(iz) C_n int_0^infty exp(-zt) P_n(1+it) dt$$ with $C_n$ a constant depending on $n$ and $P_n$ the Legendre function of the first kind of order n. This can can be verified for integers n by direct substitution of the (finite) series expansions for each of these functions. This holds for noninteger orders as well, but proving it is a bit harder. It follows from the equivalent formula
$$h_n(z) = C_n int_1^infty exp(izt) P_n(t) dt$$
that $h_n$ is in a Hardy space of functions analytic on the upper half of the complex plane
(not $H^p$ for any $p>0$, though, since $P_n(t)$ belows up at infinity).

It can be shown by direct expansion of the relevant series that
$$|h_n(x)|^2 = 1+ int_0^infty exp(-xt) d/dt P_n(1+t^2/2) dt.$$
The derivative of $P_n(1+t^2/2)$ is positive on $(0,infty)$, so $|h_n(x)|^2$ is completely monotone on $(0,infty)$. An equivalent formula is
$$|h_n(x)|^2 = z int_0^infty exp(-xt) P_n(1+t^2/2) dt.$$

Obviously, it is too much to ask for |F(z)|^2 to be completely monotone for all F in a Hardy space. But is there a general principal here? It is not enough for F to be
in a Hardy space because there are obvious examples where |F| is highly oscillatory.
For example, take
$$F(z) = int_0^infty exp(izt) phi(t) dt$$
with $phi(t)$ a smooth function which is equal to 1 on (1,100000). The function |F(x)|^2
will oscillate rapidly on $(0,infty)$.

I have the strong impression that this must related to a well-known, standard result, but I don’t know where to look for it.

ca.classical analysis and odes – Slick proof of Stirling’s Formula?

In Upper Limit on the Central Binomial Coefficient, Noam Elkies and David Speyer have given a nice proof that the central binomial coefficient $binom{2n}{n} sim frac{4^n}{sqrt{pi n}}$. This can be used to derive Stirling’s formula
$$ a_n = frac{n!e^n}{n^n sqrt{n}} sim sqrt{2pi} $$
by showing that $a_n$ is decreasing, hence convergent to some positive real number $c$, and computing $c$ via
$$ c = lim_{n to infty} frac{a_{2n}}{a_n^2} = sqrt{2pi}. $$
My question is whether we can do without this roundabout tour and prove Stirling’s formula directly along the lines of the proofs quoted above.

differential equations – How to numerically solve ODEs with a complex variable?

I want to solve the following ODEs:
3(a'(z))^2 == 1/2 a(z)^2 (p'(z))^2 – 1 ;
a(z)p”(z) + 3a'(z)p'(z) + z^2 == 0 ;
where z is generally a complex variable and a prime denotes differentiation with respect to z. The initial conditions are
a(0) == 0 ; a'(0) == i/Sqrt(3) ; z(0) == 5 ; z'(0) == 0 .
However, when I use NDSolve for this problem, it returns that the endpoint of the variable z must be a real number. Should I always decompose the variable z into z = x + iy and transfer the problem to a PDE system with two real variable x and y?
Besides, the solution to the ODEs should be a curve on the complex plane. How should I input the endpoint of z in NDSolve, before actually get any information from the numerical solution?

ca.classical analysis and odes – Recursive formula for integral of Chebyshev-type integral

Define
$$
I_{m,n}(x,y,r) = int_a^b T_m(x + r sin(gamma)) T_n(y-r cos(gamma)) dgamma
$$
where $T_m(x)$ are the Chebyshev polynomials of the first kind, and $a$ and $b$ are constants. Assume that the constants are appropriately chosen so that arguments to Chebyshev polynomials are between -1 and 1.

I was hoping to find a recurrence relation for $I_{m,n}$ in terms of $I_{m-1,n},I_{m,n-1},I_{m-1,n-1}$, but so far I cannot. I have tried using the normal recurrence for $T_m$, the derivative formula, and many different integrations by parts. Any simplification would be interesting (perhaps Bessel functions might apply?) If it makes a difference, the case $a=0, b=2 pi$ is of interest.

(Here is a related idea on SE, which did not seem to apply directly:
https://math.stackexchange.com/questions/2324185/integration-of-chebyshev-polynomials-of-first-kind-with-an-exponential-funcion

Background: This is related to calculating spherical mean averages after a Chebyshev interpolation. I am currently doing this numerically and this is quite poor in terms of numerical stability and efficiency.

(Non-) Convergence of solutions in a family of linear ODE’s

I’m trying to solve an optimal stopping problem which led me to an obstacle problem involving the following family of ODE’s

$$(x^2+d)y'(x)-2xy(x) = 1.$$

For simplicity I first considered the case $d = 0$ before moving to the much more relevant case $d > 0$. This revealed a somewhat unexpected phenomenon to me: The solutions are of the form
$C(x^2+d) + f(x)$, where $C$ is some constant. For $d = 0$ we have $f(x) = -frac{1}{3x}$ which is negative for $x > 0$. On the other hand, for $d > 0$ we have
$$f(x) = frac{x}{2 d}+ frac{(x^2+d)tan^{-1}left(frac x {sqrt d}right)}{2 d^{3/2}},$$
which is positive and $f(x)$ diverges as $dto 0$; instead of converging to, say $xmapsto -frac 1 {3x}$.

Is there some theoretic result that shows that this must be the case? If not is there some result that would show convergence, but we can verify that its assumptions are violated?

ca.classical analysis and odes – Are there “natural” sequences with “exotic” growth rates? What metatheorems are there guaranteeing “elementary” growth rates?

A thing that consistently surprises me is that many “natural” sequences $f(n)$, even apparently very complicated ones, have growth rates which can be described by elementary functions $g(n)$ (say, to be precise, functions expressible using a bounded number of arithmetic operations, $exp$, and $log$), in the sense that $lim_{n to infty} frac{f(n)}{g(n)} = 1$. I’ll write $f sim g$ for this equivalence relation. Here are some examples roughly in increasing order of how surprising I think it is that the growth rate is elementary (very subjective, of course).

  • The Fibonacci sequence $F_n$ satisfies $F_n sim frac{phi^n}{sqrt{5}}$.
  • The factorial $n!$ satisfies $n! sim sqrt{2pi n} left( frac{n}{e} right)^n$.
  • The prime counting function $pi(n)$ satisfies $pi(n) sim frac{n}{log n}$.
  • The partition function $p(n)$ satisfies $p(n) sim frac{1}{4n sqrt{3}} exp left( pi sqrt{ frac{2n}{3} } right)$.
  • The Landau function $log g(n)$ satisfies $log g(n) sim sqrt{n log n}$. I don’t know whether or not it’s expected that $g(n) sim exp(sqrt{n log n})$.
  • For $p$ a prime, the number $G(p^n)$ of groups of order $p^n$ satisfies $log_p G(p^n) sim frac{2}{27} n^3$.

I know some metatheorems guaranteeing elementary asymptotics for some classes of sequences. The simplest one involves sequences with meromorphic generating functions; this gives the Fibonacci example above as well as more complicated examples like the ordered Bell numbers. I have the impression that there are analogous theorems for Dirichlet series involving tauberian theorems that produce the PNT example and other similar number-theoretic examples. There’s a metatheorem involving saddle point bounds which gives the factorial example and at least heuristically gives the partition function example. And I don’t know any metatheorems relevant to the Landau function or $p$-group examples. So, questions:

Q1: What are some “natural” sequences $f(n)$ which (possibly conjecturally) don’t have elementary asymptotics, in the sense that there are no elementary functions $g(n)$ such that $f(n) sim g(n)$?

Right off the bat I want to rule out two classes of counterexamples that don’t get at what I’m interested in: $f(n)$ may oscillate too wildly to have an elementary growth rate (for example $f(n)$ could be the number of abelian groups of order $n$), or it may grow too fast to have an elementary growth rate (for example $f(n)$ could be the busy beaver function). Unfortunately I’m not sure how to rigorously pose a condition that rules these and other similar-flavored counterexamples out. At the very least I want $f(n)$ to be monotonic, and I also want it to be bounded from above by an elementary function.

The kinds of sequences I’m interested in as potential counterexamples are sequences like the Landau function above, as well as combinatorial sequences like the Bell numbers $B_n$. The Bell numbers themselves might be a potential counterexample. Wikipedia gives some elementary bounds but expresses the growth rate in terms of the Lambert W function; it seems that the Lambert W function has elementary growth but I’m not sure if that implies that $B_n$ itself does.

Q2: What are some more metatheorems guaranteeing elementary growth rates? Are there good organizing principles here?

Q3: What are some “natural” sequences known to have elementary growth rates but by specific arguments that don’t fall under cases covered by any metatheorems?

Apologies for the somewhat open-ended questions, I’d ask a tighter question if I knew how to state it.