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using NDEigensystem to solve the Mathieu equation
Announcing the arrival of Valued Associate #679: Cesar Manara
Unicorn Meta Zoo #1: Why another podcast?How to correctly use DSolve when the force is an impulse (dirac delta) and initial conditions are not zeroIndexing of Large Autonomous System of Equations for Use in NDSolveFEM Solution desired for “Plate with orifice” deflection: Application of Boundary Conditions and use of RegionsSolving an ODE using shooting methodHow to rescale the independent variable?Trouble with shooting method for a 4th-order stiff ODEFinding eigenvalues for Laplacian operator for 3D shape with Neumann boundary conditionsHow do you find the eigenvalues of a PDE (Dynamic Euler-Bernoulli beam)?How to evaluate the PDE solution dependent on the `RegionMarkers"?Using NDEigensystem to solve coupled eigenvalue problem
$begingroup$
To be able to apply the differentialequation capabilities of Mathematica to my graduate thesis, I am trying to apply NDEigensystem to an eigenproblem whose solution I know, but I am having some trouble doing so.
As a test problem, I am using an algebraic version of the Mathieu equation,
$$(1-zeta^2)w^primeprime-zeta w^prime+left(a+2q-4qzeta^2right)w=0$$
For this example I set $q=4/3$ and take only the first three eigenpairs:
m = 3; q = 4/3;
op = -(1 - ζ^2) u''[ζ] + ζ u'[ζ] + 2 q (2 ζ^2 - 1) u[ζ];
bc = DirichletCondition[u[ζ] == 0, True];
λ, fl = NDEigensystem[op, bc, u, ζ, 0, 1, m];
I chose the Mathieu equation as a nontrivial example as Mathematica already has a function for it's evaluation:
λt = Table[MathieuCharacteristicB[2 k, q], k, m];
flt = Table[With[j = j,
MathieuS[MathieuCharacteristicB[2 k, q], q, ArcCos[#]] &], k, m];
The problem is, I do not get the expected eigenvalues!
λ
(* 4.0708, 17.3259, 39.1877 *)
N[λt]
(* 3.85298, 16.0581, 36.0254 *)
And of course, plotting shows that the eigenequation is not satisfied at all:
With[u = fl[[1]], b = λ[[1]],
Plot[(1 - ζ^2) u''[ζ] - ζ u'[ζ] + (b + 2 q - 4 q ζ^2) u[ζ], ζ, 0, 1]]
With[u = flt[[1]], b = λt[[1]],
Plot[(1 - ζ^2) u''[ζ] - ζ u'[ζ] + (b + 2 q - 4 q ζ^2) u[ζ], ζ, 0, 1]]
What was wrong with my attempt? If I can get this example to work, I should be able to apply it to my actual, more complicated problem, so any Good Ideas would be welcome.
differential-equations finite-element-method
New contributor
$endgroup$
add a comment |
$begingroup$
To be able to apply the differentialequation capabilities of Mathematica to my graduate thesis, I am trying to apply NDEigensystem to an eigenproblem whose solution I know, but I am having some trouble doing so.
As a test problem, I am using an algebraic version of the Mathieu equation,
$$(1-zeta^2)w^primeprime-zeta w^prime+left(a+2q-4qzeta^2right)w=0$$
For this example I set $q=4/3$ and take only the first three eigenpairs:
m = 3; q = 4/3;
op = -(1 - ζ^2) u''[ζ] + ζ u'[ζ] + 2 q (2 ζ^2 - 1) u[ζ];
bc = DirichletCondition[u[ζ] == 0, True];
λ, fl = NDEigensystem[op, bc, u, ζ, 0, 1, m];
I chose the Mathieu equation as a nontrivial example as Mathematica already has a function for it's evaluation:
λt = Table[MathieuCharacteristicB[2 k, q], k, m];
flt = Table[With[j = j,
MathieuS[MathieuCharacteristicB[2 k, q], q, ArcCos[#]] &], k, m];
The problem is, I do not get the expected eigenvalues!
λ
(* 4.0708, 17.3259, 39.1877 *)
N[λt]
(* 3.85298, 16.0581, 36.0254 *)
And of course, plotting shows that the eigenequation is not satisfied at all:
With[u = fl[[1]], b = λ[[1]],
Plot[(1 - ζ^2) u''[ζ] - ζ u'[ζ] + (b + 2 q - 4 q ζ^2) u[ζ], ζ, 0, 1]]
With[u = flt[[1]], b = λt[[1]],
Plot[(1 - ζ^2) u''[ζ] - ζ u'[ζ] + (b + 2 q - 4 q ζ^2) u[ζ], ζ, 0, 1]]
What was wrong with my attempt? If I can get this example to work, I should be able to apply it to my actual, more complicated problem, so any Good Ideas would be welcome.
differential-equations finite-element-method
New contributor
$endgroup$
add a comment |
$begingroup$
To be able to apply the differentialequation capabilities of Mathematica to my graduate thesis, I am trying to apply NDEigensystem to an eigenproblem whose solution I know, but I am having some trouble doing so.
As a test problem, I am using an algebraic version of the Mathieu equation,
$$(1-zeta^2)w^primeprime-zeta w^prime+left(a+2q-4qzeta^2right)w=0$$
For this example I set $q=4/3$ and take only the first three eigenpairs:
m = 3; q = 4/3;
op = -(1 - ζ^2) u''[ζ] + ζ u'[ζ] + 2 q (2 ζ^2 - 1) u[ζ];
bc = DirichletCondition[u[ζ] == 0, True];
λ, fl = NDEigensystem[op, bc, u, ζ, 0, 1, m];
I chose the Mathieu equation as a nontrivial example as Mathematica already has a function for it's evaluation:
λt = Table[MathieuCharacteristicB[2 k, q], k, m];
flt = Table[With[j = j,
MathieuS[MathieuCharacteristicB[2 k, q], q, ArcCos[#]] &], k, m];
The problem is, I do not get the expected eigenvalues!
λ
(* 4.0708, 17.3259, 39.1877 *)
N[λt]
(* 3.85298, 16.0581, 36.0254 *)
And of course, plotting shows that the eigenequation is not satisfied at all:
With[u = fl[[1]], b = λ[[1]],
Plot[(1 - ζ^2) u''[ζ] - ζ u'[ζ] + (b + 2 q - 4 q ζ^2) u[ζ], ζ, 0, 1]]
With[u = flt[[1]], b = λt[[1]],
Plot[(1 - ζ^2) u''[ζ] - ζ u'[ζ] + (b + 2 q - 4 q ζ^2) u[ζ], ζ, 0, 1]]
What was wrong with my attempt? If I can get this example to work, I should be able to apply it to my actual, more complicated problem, so any Good Ideas would be welcome.
differential-equations finite-element-method
New contributor
$endgroup$
To be able to apply the differentialequation capabilities of Mathematica to my graduate thesis, I am trying to apply NDEigensystem to an eigenproblem whose solution I know, but I am having some trouble doing so.
As a test problem, I am using an algebraic version of the Mathieu equation,
$$(1-zeta^2)w^primeprime-zeta w^prime+left(a+2q-4qzeta^2right)w=0$$
For this example I set $q=4/3$ and take only the first three eigenpairs:
m = 3; q = 4/3;
op = -(1 - ζ^2) u''[ζ] + ζ u'[ζ] + 2 q (2 ζ^2 - 1) u[ζ];
bc = DirichletCondition[u[ζ] == 0, True];
λ, fl = NDEigensystem[op, bc, u, ζ, 0, 1, m];
I chose the Mathieu equation as a nontrivial example as Mathematica already has a function for it's evaluation:
λt = Table[MathieuCharacteristicB[2 k, q], k, m];
flt = Table[With[j = j,
MathieuS[MathieuCharacteristicB[2 k, q], q, ArcCos[#]] &], k, m];
The problem is, I do not get the expected eigenvalues!
λ
(* 4.0708, 17.3259, 39.1877 *)
N[λt]
(* 3.85298, 16.0581, 36.0254 *)
And of course, plotting shows that the eigenequation is not satisfied at all:
With[u = fl[[1]], b = λ[[1]],
Plot[(1 - ζ^2) u''[ζ] - ζ u'[ζ] + (b + 2 q - 4 q ζ^2) u[ζ], ζ, 0, 1]]
With[u = flt[[1]], b = λt[[1]],
Plot[(1 - ζ^2) u''[ζ] - ζ u'[ζ] + (b + 2 q - 4 q ζ^2) u[ζ], ζ, 0, 1]]
What was wrong with my attempt? If I can get this example to work, I should be able to apply it to my actual, more complicated problem, so any Good Ideas would be welcome.
differential-equations finite-element-method
differential-equations finite-element-method
New contributor
New contributor
edited 49 mins ago
user21
21k55998
21k55998
New contributor
asked 2 hours ago
宮川園子宮川園子
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2 Answers
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$begingroup$
If you refine the mesh, you will get closer:
m = 3; q = 4/3;
op = -(1 - [Zeta]^2) u''[[Zeta]] + [Zeta] u'[[Zeta]] +
2 q (2 [Zeta]^2 - 1) u[[Zeta]];
bc = DirichletCondition[u[[Zeta]] == 0, True];
[Lambda], fl =
NDEigensystem[op, bc, u, [Zeta], 0, 1, m,
Method -> "PDEDiscretization" -> "FiniteElement", "MeshOptions"
-> "MaxCellMeasure" -> 0.00001];
[Lambda]
3.855, 16.074, 36.064
[Lambda]t = Table[MathieuCharacteristicB[2 k, q], k, m];
flt = Table[
With[j = j,
MathieuS[MathieuCharacteristicB[2 k, q], q, ArcCos[#]] &], k, m];
[Lambda]t // N
3.852, 16.058, 36.025
$endgroup$
add a comment |
$begingroup$
It looks to me like NDEigensystem
is struggling with the singularity at $zeta=1$, as does the method that I'm going to show. But perhaps it'll be useful for you, at least as a cross-check.
I have a package for numerically calculating solutions of eigenvalue problems using the Evans function via the method of compound matrices, which is hosted on github. See my answers to other questions, the example notebook on the github or this introduction for some more details.
First we install the package (only need to do this the first time):
Needs["PacletManager`"]
PacletInstall["CompoundMatrixMethod",
"Site" -> "http://raw.githubusercontent.com/paclets/Repository/master"]
Then we first need to turn the ODEs into a matrix form $mathbfy'=mathbfA cdot mathbfy$, using my function ToMatrixSystem
:
Needs["CompoundMatrixMethod`"]
sys[ζend_] = ToMatrixSystem[op == a u[ζ], u[0] == 0, u[ζend] == 0, u, ζ, 0, ζend, a]
Now the function Evans will calculate the Evans function (also known as the Miss-Distance function) for any given value of $a$ and $zeta_end$; this is an analytic function whose roots coincide with eigenvalues of the original equation.
Plugging in $zeta_end = 1$ fails due to the singularity, but you can try moving the endpoint slightly away:
FindRoot[Evans[a, sys[1 - 10^-3]], a, 3]
(* a -> 4.00335 *)
Moving the endpoint closer approaches the correct value, but I can't get the exact value with this method.
FindRoot[Evans[a, sys[1 - 10^-10], WorkingPrecision -> 30], a, 3,
WorkingPrecision -> 30] // Quiet
(* a -> 3.85301 *)
You can see the same effect for the other roots.
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
If you refine the mesh, you will get closer:
m = 3; q = 4/3;
op = -(1 - [Zeta]^2) u''[[Zeta]] + [Zeta] u'[[Zeta]] +
2 q (2 [Zeta]^2 - 1) u[[Zeta]];
bc = DirichletCondition[u[[Zeta]] == 0, True];
[Lambda], fl =
NDEigensystem[op, bc, u, [Zeta], 0, 1, m,
Method -> "PDEDiscretization" -> "FiniteElement", "MeshOptions"
-> "MaxCellMeasure" -> 0.00001];
[Lambda]
3.855, 16.074, 36.064
[Lambda]t = Table[MathieuCharacteristicB[2 k, q], k, m];
flt = Table[
With[j = j,
MathieuS[MathieuCharacteristicB[2 k, q], q, ArcCos[#]] &], k, m];
[Lambda]t // N
3.852, 16.058, 36.025
$endgroup$
add a comment |
$begingroup$
If you refine the mesh, you will get closer:
m = 3; q = 4/3;
op = -(1 - [Zeta]^2) u''[[Zeta]] + [Zeta] u'[[Zeta]] +
2 q (2 [Zeta]^2 - 1) u[[Zeta]];
bc = DirichletCondition[u[[Zeta]] == 0, True];
[Lambda], fl =
NDEigensystem[op, bc, u, [Zeta], 0, 1, m,
Method -> "PDEDiscretization" -> "FiniteElement", "MeshOptions"
-> "MaxCellMeasure" -> 0.00001];
[Lambda]
3.855, 16.074, 36.064
[Lambda]t = Table[MathieuCharacteristicB[2 k, q], k, m];
flt = Table[
With[j = j,
MathieuS[MathieuCharacteristicB[2 k, q], q, ArcCos[#]] &], k, m];
[Lambda]t // N
3.852, 16.058, 36.025
$endgroup$
add a comment |
$begingroup$
If you refine the mesh, you will get closer:
m = 3; q = 4/3;
op = -(1 - [Zeta]^2) u''[[Zeta]] + [Zeta] u'[[Zeta]] +
2 q (2 [Zeta]^2 - 1) u[[Zeta]];
bc = DirichletCondition[u[[Zeta]] == 0, True];
[Lambda], fl =
NDEigensystem[op, bc, u, [Zeta], 0, 1, m,
Method -> "PDEDiscretization" -> "FiniteElement", "MeshOptions"
-> "MaxCellMeasure" -> 0.00001];
[Lambda]
3.855, 16.074, 36.064
[Lambda]t = Table[MathieuCharacteristicB[2 k, q], k, m];
flt = Table[
With[j = j,
MathieuS[MathieuCharacteristicB[2 k, q], q, ArcCos[#]] &], k, m];
[Lambda]t // N
3.852, 16.058, 36.025
$endgroup$
If you refine the mesh, you will get closer:
m = 3; q = 4/3;
op = -(1 - [Zeta]^2) u''[[Zeta]] + [Zeta] u'[[Zeta]] +
2 q (2 [Zeta]^2 - 1) u[[Zeta]];
bc = DirichletCondition[u[[Zeta]] == 0, True];
[Lambda], fl =
NDEigensystem[op, bc, u, [Zeta], 0, 1, m,
Method -> "PDEDiscretization" -> "FiniteElement", "MeshOptions"
-> "MaxCellMeasure" -> 0.00001];
[Lambda]
3.855, 16.074, 36.064
[Lambda]t = Table[MathieuCharacteristicB[2 k, q], k, m];
flt = Table[
With[j = j,
MathieuS[MathieuCharacteristicB[2 k, q], q, ArcCos[#]] &], k, m];
[Lambda]t // N
3.852, 16.058, 36.025
answered 45 mins ago
user21user21
21k55998
21k55998
add a comment |
add a comment |
$begingroup$
It looks to me like NDEigensystem
is struggling with the singularity at $zeta=1$, as does the method that I'm going to show. But perhaps it'll be useful for you, at least as a cross-check.
I have a package for numerically calculating solutions of eigenvalue problems using the Evans function via the method of compound matrices, which is hosted on github. See my answers to other questions, the example notebook on the github or this introduction for some more details.
First we install the package (only need to do this the first time):
Needs["PacletManager`"]
PacletInstall["CompoundMatrixMethod",
"Site" -> "http://raw.githubusercontent.com/paclets/Repository/master"]
Then we first need to turn the ODEs into a matrix form $mathbfy'=mathbfA cdot mathbfy$, using my function ToMatrixSystem
:
Needs["CompoundMatrixMethod`"]
sys[ζend_] = ToMatrixSystem[op == a u[ζ], u[0] == 0, u[ζend] == 0, u, ζ, 0, ζend, a]
Now the function Evans will calculate the Evans function (also known as the Miss-Distance function) for any given value of $a$ and $zeta_end$; this is an analytic function whose roots coincide with eigenvalues of the original equation.
Plugging in $zeta_end = 1$ fails due to the singularity, but you can try moving the endpoint slightly away:
FindRoot[Evans[a, sys[1 - 10^-3]], a, 3]
(* a -> 4.00335 *)
Moving the endpoint closer approaches the correct value, but I can't get the exact value with this method.
FindRoot[Evans[a, sys[1 - 10^-10], WorkingPrecision -> 30], a, 3,
WorkingPrecision -> 30] // Quiet
(* a -> 3.85301 *)
You can see the same effect for the other roots.
$endgroup$
add a comment |
$begingroup$
It looks to me like NDEigensystem
is struggling with the singularity at $zeta=1$, as does the method that I'm going to show. But perhaps it'll be useful for you, at least as a cross-check.
I have a package for numerically calculating solutions of eigenvalue problems using the Evans function via the method of compound matrices, which is hosted on github. See my answers to other questions, the example notebook on the github or this introduction for some more details.
First we install the package (only need to do this the first time):
Needs["PacletManager`"]
PacletInstall["CompoundMatrixMethod",
"Site" -> "http://raw.githubusercontent.com/paclets/Repository/master"]
Then we first need to turn the ODEs into a matrix form $mathbfy'=mathbfA cdot mathbfy$, using my function ToMatrixSystem
:
Needs["CompoundMatrixMethod`"]
sys[ζend_] = ToMatrixSystem[op == a u[ζ], u[0] == 0, u[ζend] == 0, u, ζ, 0, ζend, a]
Now the function Evans will calculate the Evans function (also known as the Miss-Distance function) for any given value of $a$ and $zeta_end$; this is an analytic function whose roots coincide with eigenvalues of the original equation.
Plugging in $zeta_end = 1$ fails due to the singularity, but you can try moving the endpoint slightly away:
FindRoot[Evans[a, sys[1 - 10^-3]], a, 3]
(* a -> 4.00335 *)
Moving the endpoint closer approaches the correct value, but I can't get the exact value with this method.
FindRoot[Evans[a, sys[1 - 10^-10], WorkingPrecision -> 30], a, 3,
WorkingPrecision -> 30] // Quiet
(* a -> 3.85301 *)
You can see the same effect for the other roots.
$endgroup$
add a comment |
$begingroup$
It looks to me like NDEigensystem
is struggling with the singularity at $zeta=1$, as does the method that I'm going to show. But perhaps it'll be useful for you, at least as a cross-check.
I have a package for numerically calculating solutions of eigenvalue problems using the Evans function via the method of compound matrices, which is hosted on github. See my answers to other questions, the example notebook on the github or this introduction for some more details.
First we install the package (only need to do this the first time):
Needs["PacletManager`"]
PacletInstall["CompoundMatrixMethod",
"Site" -> "http://raw.githubusercontent.com/paclets/Repository/master"]
Then we first need to turn the ODEs into a matrix form $mathbfy'=mathbfA cdot mathbfy$, using my function ToMatrixSystem
:
Needs["CompoundMatrixMethod`"]
sys[ζend_] = ToMatrixSystem[op == a u[ζ], u[0] == 0, u[ζend] == 0, u, ζ, 0, ζend, a]
Now the function Evans will calculate the Evans function (also known as the Miss-Distance function) for any given value of $a$ and $zeta_end$; this is an analytic function whose roots coincide with eigenvalues of the original equation.
Plugging in $zeta_end = 1$ fails due to the singularity, but you can try moving the endpoint slightly away:
FindRoot[Evans[a, sys[1 - 10^-3]], a, 3]
(* a -> 4.00335 *)
Moving the endpoint closer approaches the correct value, but I can't get the exact value with this method.
FindRoot[Evans[a, sys[1 - 10^-10], WorkingPrecision -> 30], a, 3,
WorkingPrecision -> 30] // Quiet
(* a -> 3.85301 *)
You can see the same effect for the other roots.
$endgroup$
It looks to me like NDEigensystem
is struggling with the singularity at $zeta=1$, as does the method that I'm going to show. But perhaps it'll be useful for you, at least as a cross-check.
I have a package for numerically calculating solutions of eigenvalue problems using the Evans function via the method of compound matrices, which is hosted on github. See my answers to other questions, the example notebook on the github or this introduction for some more details.
First we install the package (only need to do this the first time):
Needs["PacletManager`"]
PacletInstall["CompoundMatrixMethod",
"Site" -> "http://raw.githubusercontent.com/paclets/Repository/master"]
Then we first need to turn the ODEs into a matrix form $mathbfy'=mathbfA cdot mathbfy$, using my function ToMatrixSystem
:
Needs["CompoundMatrixMethod`"]
sys[ζend_] = ToMatrixSystem[op == a u[ζ], u[0] == 0, u[ζend] == 0, u, ζ, 0, ζend, a]
Now the function Evans will calculate the Evans function (also known as the Miss-Distance function) for any given value of $a$ and $zeta_end$; this is an analytic function whose roots coincide with eigenvalues of the original equation.
Plugging in $zeta_end = 1$ fails due to the singularity, but you can try moving the endpoint slightly away:
FindRoot[Evans[a, sys[1 - 10^-3]], a, 3]
(* a -> 4.00335 *)
Moving the endpoint closer approaches the correct value, but I can't get the exact value with this method.
FindRoot[Evans[a, sys[1 - 10^-10], WorkingPrecision -> 30], a, 3,
WorkingPrecision -> 30] // Quiet
(* a -> 3.85301 *)
You can see the same effect for the other roots.
answered 26 mins ago
KraZugKraZug
3,49821130
3,49821130
add a comment |
add a comment |
宮川園子 is a new contributor. Be nice, and check out our Code of Conduct.
宮川園子 is a new contributor. Be nice, and check out our Code of Conduct.
宮川園子 is a new contributor. Be nice, and check out our Code of Conduct.
宮川園子 is a new contributor. Be nice, and check out our Code of Conduct.
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StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
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Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown