Tag: ODEs

Classical analysis and Odes – Vasicek model and Sportzins, parameterized by the rate of inversion

By solving an SDE, I want to derive the mean and variance analysis results of the extended Vasicek model process

$ dr = (η-γr) dt + cdX $

whereas $ γ $ is the inversion rate and $ s = η / γ $ the average short rate.

How can I stop $ X (t) = r (t) – s $ and solve by integration on both sides of the SDE using the integration factor $ e ^ {yt} $ and in a second step derive the mean and the variance?

ca.classical analysis and odes – Under what conditions is this family normal?

To let $ mathcal {S} = {s in mathbb {C} , mid , | Im (s) | <1 } $ be a strip of the complex plane. To let $ q (s, z) $ a holomorphic function $ mathcal {S} times mathbb {C} $, Rent $ mathcal {K} $ a CD of $ mathbb {C} $We can accept:

$$
sum_ {j = 1} ^ infty || frac {1} {n} q (s- frac {j} {n}, z) || _ { mathcal {K}} < infty
$$

For all $ n in mathbb {N} $With $ n ge 1 $where the border continues as $ n to infty $ converges evenly into $ s $ on compact sentences of $ mathcal {S} $, Note that $ || h || _ mathcal {K} = sup_ {z in mathcal {K}} | h (z) | $, Also note that the boundary to the integral converges

$$ int _ {- infty} ^ s || q (t, z) || _ { mathcal {K}} , dt $$

Suppose we have a family of holomorphic functions $ F_n (s): mathcal {S} to mathbb {C} $ in which:

$$
F_n (s) to z , , text {as} , , Re (s) to – infty
$$

$$
F_n (s + frac {1} {n}) – F_n (s) = frac {1} {n} q (s, F_n (s))
$$

I want to show that this family approximates the differential equation $ F # (s) = q (s, F (s)) $ with any precision. Or that $ F_n to F $, To get there, I have to show that this family is normal. My guess is that I lack a condition $ q $ that provides for normality, or I miss an obvious fact about this family.

Basically, I wonder if there is a limit anyway $ n to infty $ from $ F_n $ could escape to infinity, or if the fact that its satisfying equation approaches the differential equation $ y # = q (s, y) $ Forces the convergence to uniquely solve the differential equation. Where the unique solution is a function $ ell $ in which $ ell (- infty) = z $ and $ ell & # 39; = q (s, ell) $that can be shown to exist in a neighborhood of and to be unique $ – infty $ by a slight modification of the usual Picard-Lindelöf theorem. Where by the neighborhood of $ – infty $ I mean $ Re (s) <R $. $ | Im (s) | < tau <1 $ to the $ | R | $ big enough.

All in all, the family is $ {F_n } _ {n = 1} ^ infty $ normal? If not, what would be a reasonable condition to ensure this?

Any help would be appreciated.

Thanks, Richard.

Classical Analysis and Odes – On Exponential Polynomials

Suppose we have the following function $ f: mathbb {R} ^ {+} mapsto mathbb {R} $
$$ f (t) = sum_ {i = 1} ^ k P_i (t) exp ( alpha_i t), $$
from where $ alpha_i $s are all algebraic numbers and $ P_i (t) $ are all polynomials with algebraic coefficients and one degree less than $ m $,

There are several questions that interest me.

on. What is the maximum number $ p $ so there is one $ t_0 $ and for everyone $ r leq p $
$$ f ^ {(r)} (t_0) = 0. $$
b. Accept $ t_1, cdots, t_q, cdots $ are the true roots of $ f (t) = 0 $Is it possible to have a convergent sequence of $ t_q $? In other words, it is possible to have a Cauchy sequence $ t_i $ so that $ f (t_i) = 0 $? If not, do we have a lower bound on the distance between different roots?

Is there an algorithm for computing all common real roots of $ f (t) $ and $ f & # 39; (t) $?

Differential equations – eigenvalue solution of coupled ODEs

I want an eigenvalue solution from the following paired ODEs: But the code shows errors: BCs are incomplete.
Enter the image description here

                Gamma = 1.67; ep = 0.01; d = 0.05; H = 1.1; Gpara = 0.38;
eqn = {Rho[x]/ Exp[-x/H]* gpara -
1 / gamma / Exp[-x/H]* {D[Rho[Rho[Rho[Rho[x], x]+ D[T[T[T[T[x], x]} +
1,3 * ep * D[V[V[V[V[x], {x, 2}]+ I * lambda * V[x] == 0,
D[Exp[Exp[Exp[Exp[-x/H]* V[x], x]- I * Lambda * Rho[x] == 0,
D[Exp[Exp[Exp[Exp[-x/H]* V[x], x]- V[x]* Rho[x]* gpara -
Gamma * Exp[-x/H]* D[V[V[V[V[x], x]+ gamma * d * D[T[T[T[T[x], {x, 2}]+
I * Lambda * T[x] == 0};
bc = V[0] == 0, V[1] == 0, V & # 39;[0] == 0, V & # 39;[1] == 0, T[0] == 0,
T[1] == 0, T & # 39;[0] == 0, T & # 39;[1] == 0, Rho[0] == 0, Rho[1] == 0;
{vals, funs} =
DEigensystem[{eqn, bc}, {V, Rho, T}, {x, 0, 1}, {lambda}];
vals
plot[Evaluate[funs[V]have fun[Rho]have fun[T]], {x, 0, 1}]

I have to find numerical values ​​of lambda (eigenvalues) and
In the equations above, gpara is considered constant, but it is a variable with 50 rows and 1 column in the gpara.txt file, which is in a different directory. I import gpara by import[“E\ ……\ gpara.txt”] to use in the above equations, but it does not work either. Please help to get the solution.

Differential equations – Solution system of ODEs with additional parameters

I would like to solve one $ 2 times $ 2 System of the form
$$ frac {d} {d theta} T = TA, quad T (0) = Id $$
from where $ theta $ is real and $ A $ is of the form
$$ A = begin {pmatrix} 0 & frac {e ^ {- i theta}} { lambda} \ frac {1} {36} e ^ {- i theta} left (9 Lambda + 2 ( lambda-1) ^ 2 (6 cos { theta} + cos {2 theta} + 6) right) & 0 end {pmatrix}, $$
With $ lambda $ a free parameter in the unit circle,

I am particularly interested in numerical solutions $ theta = 2 pi $ depending on the additional parameters $ lambda $, I'm pretty new with Mathematicaand I've tried it so far:

T[θ_] = {{T11[θ]T12[θ]}, {T21[θ]T22[θ]}};
ON[θ_] = {{0, E ^ (- Iθ) / λ}, {1/36 E ^ (- Iθ) (9λ + 2 (-1 + λ) ^ 2 (6 + 6 Cos[θ] + Cos[2 θ])), 0}};
sys = {T & # 39;[θ] == T[θ].ON[θ]};

The previous code sets the system I want to solve, and now I'm trying to solve numerically. I tried it first

NSol = NDSolve[{sys, T11[0] == 1, T12[0] == 0, T21[0] == 0, T22[0] == 1}, {T11[θ]T12[θ]T21[θ]T22[θ]}, {θ}, {θ, 0, 2 Pi}];

that gives me the output

NDSolve :: dupv: "Double variable θ found in NDSolve[<[<[<[<<1>>], "

I tried too

Nsol2 = ParametricNDSolve[{sys, T11[0] == 1, T12[0] == 0, T21[0] == 0, T22[0] == 1}, {T11, T12, T21, T22}, {θ, 0, 2 Pi}, {λ}];

what gives me as an issue $ T_ {11}, dots, T_ {22} $ as Parametric Functions dependent on each other and on $ lambda $,

I do not know if this is the right approach and if so, how to extract a numeric expression depending on it $ lambda $ from the last issue – all I've seen in the documentation are examples that are recorded for specific parameter values. Any help is greatly appreciated.

Differential Equations – Problem solving the first ODEs with DSolve

The problem I'm facing is the following. I am trying to solve a system of two coupled first order ODEs with DSolve:

eqns = {A * f & # 39;[x] == -x * f[x]/ 2 + B * g[x], A * g & # 39;[x] == x * g[x]/ 2 + B * f[x]};
DSolve[EQNS{f[EQNS{f[eqns{f[eqns{f[x]G[x]}, x]

As output this only gives the same command that means from my collection that DSolve can not solve it.

But if I explicitly use the equations, it can suddenly be solved:

Fun = f[x] /. To solve[A*g&#39;[A*g'[A*g'[A*g'[x] == x * g[x]/ 2 + B * f[x]f[x]];
dfun = D[fun, x];
DSolve[A*f#39;[A*f'[A*f'[A*f'[x] == -x * f[x]/ 2 + B * g[x] /. {f[x] -> fun, f & # 39;[x] -> dfun},
G[x], x]

Which gives

{{G[x] ->
C[2] ParabolicCylinderD[B^2/A, (I x)/Sqrt[A]]+
C[1] ParabolicCylinderD[(-A - B^2)/A, x/Sqrt[A]]}}

That makes me think that I'm just doing something wrong on the first line, which is probably a stupid notation error. If someone could help me, I would be very grateful.

Differential equations – How do you solve these ODEs with NDSolve?

The calculation changes t = 0 there sin[ψ

eq1 = ω1
eq2 = ω2

and construct the linear combinations,

eq1n = Simplify[eq1Sin[eq1Sin[eq1Sin[eq1Sin[ψ
(* Cos[ψ

(If this were not possible, the equations themselves could not be solved in principle.)

Now replace eq1, eq2 by eq1n, eq2n,

I1 = 2; I2 = 3; I3 = 4;
s = NDSolveValue[{I1*ω1&#39;[{I1*ω1'[{I1*ω1'[{I1*ω1'
I2 * ω2 & # 39;
I3 * ω3 & # 39;
eq1n == 0, eq2n == 0,
ω3
ω1[0] == 2, ω2[0] == 3, ω3[0] == 4, ψ[0] == 0, φ[0] == 0, θ[0] == Pi / 6},
{ω1
plot[Evaluate@s[[1 ;; 3]], {t, 0, 120}, ImageSize -> Large]plot[Evaluate@s[[4 ;; 6]], {t, 0, 120}, ImageSize -> Large]

Enter the image description here

Enter the image description here

Incidentally, the original equations can be solved by slightly changing the initial state ψ[0] == 0 to ψ[0] == 10 ^ -6,

Another approach is to use the option.

Method -> {"EquationSimplification" -> "Residual"}

Everyone gives the same answer.

How do I solve these Odes with NDSolve?

I have six odes and can not use DSolve. So I tried NDSolve. But it does mean that there may be mistakes. The code looks like this:

I1 = 2; I2 = 3; I3 = 4;
NDSolve[{I1*ω1'
I2 * ω2 & # 39;
I3 * ω3 & # 39;
0, ω1[
t] == φ & # 39;
sin[ψ[ψ[ψ[ψ
t] == φ & # 39;
cos[ψ[ψ[ψ[ψ
t] == φ & # 39;
0] == 2, ω2[0] == 3, ω3[0] == 4, ψ[0] ==
0, φ[0] == 0, θ[0] ==
Pi / 6}, {ω1, ω2, ω3, ψ, φ,
θ}, {t, 0, 120}]

I want to know how to avoid this mistake.

Classical analysis and Odes – Algebraic Riccati and WKB

It is a one-liner to show that the algebraic Riccati equation (ARE) and the lowest-order form of WKB are the same for a linear ode. But I've looked everywhere on the Internet and it seems that this connection, despite its triviality is not recognized. It seems that there is a total separation between physicists who make many WKB and control people who do a lot of Riccati.

Another trivial point is that a linear ode in psi has the Lie symmetry psi -> lambda psi, so the Riccati variable is the Lie invariant, and this leads to a reduction in order (or more precisely, a reduced one Order) followed by a quadrature.) Is this so obvious that it can never be mentioned? This finding means that not only linear equations are used, but every equation that is homogeneous in degree 1 and whose order is reduced by the Riccati transformation.

Classical analysis and odes – Failure of a Falconer distance problem in one dimension

I am told that the Falconer distance guess in one dimension is trivial, but I really can not find any clue. I ask exactly the following question:

For a compact set $ E subseteq mathbb R ^ n $we define the set distance $ Delta (E) subseteq[0infty)$[0infty)$[0infty)$[0infty)$ his:
$$
Delta (E) = {| x-y |: x, y in E }.
$$
Then, when $ n = 1 $do we ask the following questions?

  1. Can we find a compact set? $ E subseteq [0,1]$ so that $ mathrm {dim} _H (E)> 1/2 $ but $ mathcal L ^ 1 ( Delta (E)) = 0 $?

  2. Can we find a compact set? $ E subseteq [0,1]$ so that $ mathrm {dim} _H (E) = 1 $ but $ mathcal L ^ 1 ( Delta (E)) = 0 $?