Tag: equations

Equations with Tensor product and Ket in Mathematica:

I tried to express this equation in Mathematica:

enter image description here

I defined necessary things:

P = {{Ket[0], Ket[2], Ket[1], Ket[3], Ket[5], Ket[4], Ket[6], Ket[8], 
   Ket[7]}, {Ket[2], Ket[1], Ket[0], Ket[5], Ket[4], Ket[3], Ket[8], 
   Ket[7], Ket[6]}, {Ket[1], Ket[0], Ket[2], Ket[4], Ket[3], Ket[5], 
   Ket[7], Ket[6], Ket[8]}, {Ket[6], Ket[8], Ket[7], Ket[0], Ket[2], 
   Ket[1], Ket[3], Ket[5], Ket[4]}, {Ket[8], Ket[7], Ket[6], Ket[2], 
   Ket[1], Ket[0], Ket[5], Ket[4], Ket[3]}, {Ket[7], Ket[6], Ket[8], 
   Ket[1], Ket[0], Ket[2], Ket[4], Ket[3], Ket[5]}, {Ket[a], Ket[c], 
   Ket[b], Ket[6], Ket[8], Ket[7], Ket[[Alpha]], Ket[[Gamma]], 
   Ket[[Beta]]}, {Ket[c], Ket[b], Ket[a], Ket[8], Ket[7], Ket[6], 
   Ket[[Gamma]], Ket[[Beta]], Ket[[Alpha]]}, {Ket[b], Ket[a], 
   Ket[c], Ket[7], Ket[6], Ket[8], Ket[[Beta]], Ket[[Alpha]], 
   Ket[[Gamma]]}}

hadamand[[Omega]_, [Omega]2_] := {
  {1, 1, 1, 1, 1, 1, 1, 1, 1},
  {1, [Omega], [Omega]2, 1, [Omega], [Omega]2, 
   1, [Omega], [Omega]2},
  {1, [Omega]2, [Omega], 1, [Omega]2, [Omega], 
   1, [Omega]2, [Omega]},
  {1, 1, 1, [Omega], [Omega], [Omega], [Omega]2, [Omega]2, 
[Omega]2},
  {1, 1, 1, [Omega], [Omega], [Omega], [Omega]2, [Omega]2, 
[Omega]2},
  {1, [Omega]2, [Omega], [Omega], 
   1, [Omega]2, [Omega]2, [Omega], 1},
  {1, 1, 1, [Omega]2, [Omega]2, [Omega]2, [Omega], [Omega], 
[Omega]},
  {1, [Omega], [Omega]2, [Omega]2, 
   1, [Omega], [Omega], [Omega]2, 1},
  {1, [Omega]2, [Omega], [Omega]2, [Omega], 1, [Omega], 
   1, [Omega]2}
  }
H = hadamand[[Omega], [Omega]^2]

My attempt is:

A[u_, j_][n_, P_, Had_] := 
 1/Sqrt[n] Sum[
   TensorProduct[Ket[k], Part[P, k, j]]* Bra[k].Had. Ket[u], {k, 0, 
    n - 1}]

B[1, 2][9, P, H]

I obtain:
enter image description here

Where is issues?

differential equations – Error on DSolve

I’ve been trying to solve this initial value problem using ‘DSolve()’:

$$
frac{dy}{dt}=1+tspace sin(tspace y),quad y(0)=0, quad t=(0,2)
$$

ClearAll(y, t)
eq1 := {y'(t) == 1 + t *Sin(t y(t)), y(0) == 0};
DSolve(eq1, y(t), {t, 0, 2})

All I get is the Inverse function error.


    Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >>

The documentation suggests it has to do with the sine function but I’m not sure how to by-pass it.

Any help would be appreciated.

linear algebra – Estimating Zeta from Hv and Hu , system of equations, estimation problem

Does anyone here know how $beta$ could be estimated in terms of $H_u, H_v $ in the below equations,

$$ H_u = left | zeta; left(frac{ e^{-jcdot 2cdotpi}-e^{-jcdot 2cdotpi(1/T)(T-Delta t)}}{j2pi}right)-zeta-beta right|^2 $$

$$ H_v = left | zeta; left(frac{ e^{jcdot 2cdotpi}-e^{jcdot 2cdotpi(1/T)(Delta t)}}{j2pi}right)+zetabeta right|^2 $$

This is absolute sqauare of $H_v$ and $H_u$

I am working on an estimation problem where from $H_u , H_v $ they estimated $beta$ that is inside $H_u$ and $H_v$.

Anyone who could tell me about any mathematical procedure that how I can we do so?

A previous example where we did so… Almost same problem

https://math.stackexchange.com/questions/3984685/estimating-p-from-a1-and-a1-system-of-equations-an-estimation-problem/3984708#3984708

$$ A_1 = left | alpha; left(frac{ 1- e^{-jcdot 2cdotpi rho}}{j2pi rho}right) right|^2 $$

$$ A_2 = left |alpha; left(frac{ 1- e^{-jcdot 2cdotpi rho}}{j2pi+j2pi rho}right) right|^2 $$
$$ rho=frac{A_2+sqrt(A_1A_2)}{A_1-A_2} $$

divided $A_1$ by $A_2$ equations

A1A2=(ρ+1)2ρ2

which is a quadratic in ρ.

I Solved it and select the root you I needed. I found $rho $ in terms of $A_1, A_2 $

open source – Very old doc-file with equations

Please help me if it is possible, in the following problem.

There are certain doc files, made in 1999, possibly on already old software.
These doc files have had equations or other mathematical expressions in them.

If I open these files with notepad, I can see “Equation” on certain places, which leads me to believe that the information is still there, but when I open it with OpenOffice, in these same places, I only see: “µ §”.

The docs in question can be downloaded from here: https://ortvay.elte.hu/regi/regi.html

Please help me with reconstructing the equations as they were. I tried to convert the doc into odt, but it could not do it without error message. I would be very grateful, if anyone can help me.

differential equations – How to add a velocity boundary condition with specific time period

I have a wave equation for displacement and velocity, I want to add this boundary condition $v(x=0,t>0)=1$
My mathematica code is

sol = NDSolve({1/1000 D(u(x, t), {t, 2}) == D(u(x, t), {x, 2}), 
   D(u(x, t), t) == v(x, t), u(x, 0) == 0, v(x, 0) == 0, 
   v(0, t > 0) == 1, v(L, t) == 0}, {u(x, t), v(x, t)}, {x, 0, L}, {t,
    0, 1})

L is a constant.

differential equations – Infinity: Indeterminate expression 0.0389874+Complexinfinity+Complexinfinity encountered

I’d like to solve 2nd order differential equation:
enter image description here

Which I’d like to draw a graph of x_M and h. m, g, and M is a real constant. Cd is well defined function that has a parameter as x_M’
theta is piecewise defined function,
that appears as below:

enter image description here

for this equation, I used code :

(Theta)0 = 0.5236; 
(Omega)1 = 5.2359877; 
(Omega)2 = 1.396;
t1 = (Theta)0 / (Omega)1;
t2 = (Theta)0 / (Omega)2;
T=2*(t1+t2);
f(t_)=Piecewise({{ -Mod(t,T)*(Omega)1 ,0<=Mod(t,T)<t1},{ -(Theta)0+((Mod(t,T)-t1)*(Omega)2), t1<=Mod(t,T)<(t1+2t2)},{(Theta)0-((Mod(t,T)-t1-2t2)*(Omega)1),(t1+2t2)<=Mod(t,T)<T}})

enter image description here

It is clear that derivative of the function is

thetaprime(t_):=Piecewise({{-5.2359877 ,0<=Mod(t,T)<t1},{ 1.396, t1<=Mod(t,T)<(t1+2t2)},{-5.2359877,(t1+2t2)<=Mod(t,T)<=T}});

enter image description here

for differential equation, I used a code

initconds = {x(0) == 0, x'(0) == 0.00001, h(0) == (M + m)/((Rho)*A), h'(0) == 0}
eqns = {m Cos((Theta)(t)) Sin((Theta)(t)) h''(t) + (M + m (Sin((Theta)(t))^2)) x''(t) == - Cd(Derivative(1)(x)(t)) *h(t)* Derivative(1)(x)(t) + m Sin((Theta)(t)) (g Cos((Theta)(t)) + r thetaprime(t)^2),( -M-(m*Cos((Theta)(t))^2))*h''(t) - m Cos((Theta)(t)) Sin((Theta)(t)) x''(t) == -g M + A g (Rho) h(t) - m Cos((Theta)(t)) (g Cos((Theta)(t)) + r thetaprime(t)^2)}

enter image description here

But when I try to solve the Equation,

sol = NDSolve(Append(eqns, initconds), {x, h}, {t, 0, tf}, Method -> {"DiscontinuityProcessing" -> False})

enter image description here

It gives following three errors. I can’t understand why these errors occur.

Infinity::indet: Indeterminate expression 0.0389874 +ComplexInfinity+ComplexInfinity encountered.

Infinity::indet: Indeterminate expression -0.160636+ComplexInfinity+ComplexInfinity+ComplexInfinity+ComplexInfinity+ComplexInfinity encountered.

NDSolve::ndnum: Encountered non-numerical value for a derivative at t == 0..

enter image description here

Please help me to solve the problem.

diophantine equations – Does the formulation of a proof involve substitution of numerical solutions into the problem?

Does the formulation of a proof involve substitution of numerical solutions into the problem ?

I asked this because I was trying to find a way I can prove Erdos right in Brocard’s problem using Diophantine equations (something I am not acquainted with except problems involving a system of linear Diophantine equations). My approach is like this : if we assume that the next solution $(m,n)$ is $(71 + x, 7 + y)$ do the following to get a new equation:
$$(7 + y)! + 1 = (71 + x)^2 longrightarrow(1)$$
$$7! + 1 = 71^2 longrightarrow(2)$$
$$(1) – (2) = (7 + y)! – 5040 = x^2 + 142xlongrightarrow(3)$$
And then prove that only a finite number of integer solutions exist for (3) and out of them, only 3 satisfy the constraints in Brocard’s problem. I doubt whether what I am doing is proof-writing or not. I got a few numerical solutions using Wolfram Alpha and those are what I am using.

differential equations – NDSolve evolution in a specific point

m = 0; n = 0; B = 0.7; W = 4;

T1(m_,n_):= NIntegrate((-Log(t))^(m – 1)*LaguerreL(n, m, -Log(t))*LaguerreL(n, m, -Log(t)), {t, 0, 1});

T2(m_,n_):= NIntegrate((-Log(t))^(m – 1)*LaguerreL(n – 1, m, -Log(t))*LaguerreL(n – 1, m, -Log(t)), {t, 0, 1});

T3(m_,n_):= NIntegrate((-Log(t))^(m – 1)*LaguerreL(n, m, -Log(t))*LaguerreL(n – 1, m, -Log(t)), {t, 0, 1});

T4(m_,n_):= NIntegrate((-Log(t))^(m + 1)LaguerreL(n, m, -Log(t))^2Exp(-B/(f0(z))^2*(-Log(t))^mtLaguerreL(n, m, -Log(t))^2)(-2n*(-Log(t))^(m – 1)tLaguerreL(n, m, -Log(t))^2 +2*(m + n)(-Log(t))^(m – 1)tLaguerreL(n, m, -Log(t))LaguerreL(n – 1, m, -Log(t)) – (-Log(t))^mtLaguerreL(n, m, -Log(t))^2 -m*(-Log(t))^(m – 1)tLaguerreL(n, m, -Log(t))^2), {t, 0, 1});

s(m_,n_,B_,W_,z):=NDSolve({f0”(z) + 1/f0(z)(f0′(z))^2 == n!/((n + m)!(2n + m + 1))1/(f0(z))^3(((n + m)!)/n!(-2n – m + 1) + T1(m, n)(2n + m)^2 + 4T2(m, n)(m + n)^2 – 4T3(m, n)(m + n)(m + 2n) + BW^2*T4(m, n, B, z)), f0(0) == 1, f0′(0) == 0},f0(z), {z, 0, 1});

f0(z_)/.s

f0(0.1)

My code is given above. I have some integrations T1,T2,T3,T4 and one differentiations. I want to find the result of NDSolve at a specific value z=0.1, but I am getting some error.
Can you suggest me to how to write the code.

differential equations – Neumann boundary condition ignored

Im trying to solve a system of two PDE (im(t,x) and ia(t,x)) that are dependent on time and distance. I’m using my own anisotropic mesh (thanks to @TimLaska) that is aimed to represent an interface between a membrane and a liquid. The interface is located at L/2 where L = thickness of the system. “im” takes positive values in the interval 0 <= x <= L/2 and is equal to 0 at x > L/2. “ia” behaves similarly.

The problem that I’m encountering is that when I specify a Neumann boundary value at L/2, Mathematica doesn’t find this value in the mesh and the Neumann value is effectively ignored.

These are the functions for creating anisotropic meshes (kindly provided by @TimLaska in one of my previous questions)

(*Import required FEM package*)
 Needs("NDSolve`FEM`");
(*Define Some Helper Functions For Structured Meshes*)
 pointsToMesh(data_) := MeshRegion(Transpose({data}), Line@Table({i, i + 1}, {i,Length(data)- 1}));
 unitMeshGrowth(n_, r_) :=  Table((r^(j/(-1 + n)) - 1.)/(r - 1.), {j, 0, n - 1})
 meshGrowth(x0_, xf_, n_, r_) := (xf - x0) unitMeshGrowth(n, r) + x0
 firstElmHeight(x0_, xf_, n_, r_) := Abs@First@Differences@meshGrowth(x0, xf, n, r)
 lastElmHeight(x0_, xf_, n_, r_) := Abs@Last@Differences@meshGrowth(x0, xf, n, r)
 findGrowthRate(x0_, xf_, n_, fElm_) :=  Quiet@Abs@ FindRoot(firstElmHeight(x0, xf, n, r) -    fElm, {r, 1.0001, 100000},Method -> "Brent")((1, 2))
 meshGrowthByElm(x0_, xf_, n_, fElm_) := N@Sort@Chop@meshGrowth(x0, xf, n, findGrowthRate(x0, xf, n, fElm))
 meshGrowthByElm0(len_, n_, fElm_) := meshGrowthByElm(0, len, n, fElm)
 flipSegment(l_) := (#1 - #2) & @@ {First(#), #} &@Reverse(l);
 leftSegmentGrowth(len_, n_, fElm_) := meshGrowthByElm0(len, n, fElm)
 rightSegmentGrowth(len_, n_, fElm_) := Module({seg}, seg = leftSegmentGrowth(len, n, fElm);
 flipSegment(seg))
 extendMesh(mesh_, newmesh_) := Union(mesh, Max@mesh + newmesh)

Here I define a couple of constants and create my own mesh

(* Constants *)
L = 0.1; dim = dia = 1.*^-6; kf = kr = 1; kr = 
 1./kf; 
(* Mesh generation *)
seg1 = rightSegmentGrowth(L/2., 100, L/1000);
seg2 = leftSegmentGrowth(L/2., 100, L/1000);
totalseg = extendMesh(seg1, seg2);
rh = pointsToMesh@totalseg ;
crd = MeshCoordinates(rh);
mesh = ToElementMesh(crd);

And here is the simple code I use to find the solution alongside the error I get

(*PDE system*)

eqsim = {D(im(t, x), t) - dim D(im(t, x), x, x) == NeumannValue(kf*ia(t, x), x == L/2.) + NeumannValue(0, x == L/2.),im(0, x) == 0.1};
eqsia = {D(ia(t, x), t) - dia D(ia(t, x), x, x) ==  NeumannValue(kr*im(t, x), x == L/2.) + NeumannValue(0, x ==   L/2.), ia(0, x) == 0.1};
sol = NDSolve({eqsim, eqsia}, {im, ia},x (Element) mesh, {t, 0, 60}, Method ->{"MethodOfLines", "SpatialDiscretization" -> {"FiniteElement"}})
(*NDSolve::bcnop: No places were found on the boundary where x==0.05 was True, so NeumannValue(ia,x==0.05) will effectively be ignored.*)
(*NDSolve::bcnop: No places were found on the boundary where x==0.05 was True, so NeumannValue(1. im,x==0.05) will effectively be ignored.*)
(* NDSolve::bcnop: No places were found on the boundary where x==0.05 was True, so NeumannValue(0,x==0.05) will effectively be ignored.*)
(*General::stop: Further output of NDSolve::bcnop will be suppressed during this calculation.*)

I first thought that the discretisation in the mesh didn’t include the point L/2 but after double checking I found out that it’s there

Position(crd, L/2.)
(*{{100, 1}}*)

Similar problems to this have been reported when using Dirichlet conditions here DiscretizeRegion does not include the boundary specified in ImplicitRegion (10.1)
, here PeriodicBoundaryConditions: missing points (a simpler example)
 and here Solving Laplace’s equation in 2D using region primitives
, but after reading the solutions provided I haven’t been able to find a solution to my issue.

Any help, feedback or advice is more than welcome (I’m using v 12.2.0.0).

differential equations – Solve PDE with contraints and unknown boundary conditions

I have a 2D PDE

mu = -0.1;
lambda = -1;
x10 = 0;
x20 = 0;
eq = {mu*x1 + D(h1(x1, x2), x1)*mu*x1 + 
     D(h1(x1, x2), x2)*lambda*(x2 - x1^2) == mu*(x1 + h1(x1, x2)), 
   lambda*(x2 - x1^2) + D(h2(x1, x2), x1)*mu*x1 + 
     D(h2(x1, x2), x2)*lambda*(x2 - x1^2) == 
    lambda*(x2 + h2(x1, x2))};

The problem is that I don’t know the boundary condition of the PDE, but I do have a constraint that the gradient of h1 and h2 at point (x10, x20) is zero.

D(h1(x1, x2), x1)/.{x1 -> x10, x2 -> x20} == 0
D(h1(x1, x2), x2)/.{x1 -> x10, x2 -> x20} == 0
D(h2(x1, x2), x1)/.{x1 -> x10, x2 -> x20} == 0
D(h2(x1, x2), x2)/.{x1 -> x10, x2 -> x20} == 0

How can I approximate h1(x1, x2) and h2(x1, x2) numerically in Mathematica?