The Issue with NDEigenvalues

Oct 13, 2025 by windailys.com 41 views

Introduction

In the field of numerical analysis and computational mathematics, solving eigenvalue problems is a crucial task. These problems arise in various applications, including physics, engineering, and computer science. One such problem is the investigation of a fourth-order differential operator, which can be represented as a system of first-order ordinary differential equations (ODEs). However, when using the NDEigenvalues function in Mathematica, it may not work as expected when dealing with a system of first-order ODEs. In this article, we will explore this issue and provide a solution.

The Eigenvalue Problem

We are given the following eigenvalue problem for a fourth-order differential operator:

Y^{(4)}(x)=\beta^4 Y(x)

on the one-dimensional domain $[0,L]$ subject to four boundary conditions:

Y(0)=0, \quad Y'(0)=0, \quad Y(L)=0, \quad Y'(L)=0

To solve this problem, we can represent the fourth-order differential operator as a system of four first-order ODEs. Let $Y_1 = Y$ , $Y_2 = Y'$ , $Y_3 = Y''$ , and $Y_4 = Y'''$ . Then, the system of ODEs becomes:

\begin{align*} Y_1' &= Y_2 \\ Y_2' &= Y_3 \\ Y_3' &= Y_4 \\ Y_4' &= \beta^4 Y_1 \end{align*}

The Issue with NDEigenvalues

When using the NDEigenvalues function in Mathematica to solve this system of ODEs, we may encounter issues. The NDEigenvalues function is designed to solve eigenvalue problems for differential operators, but it may not work as expected when dealing with a system of first-order ODEs.

To illustrate this issue, let's consider the following Mathematica code:

Needs["NDSolve`FEM`"]
L = 1;
β = 1;
eqns = {
D[Y1[x], x] == Y2[x],
D[Y2[x], x] == Y3[x],
D[Y3[x], x] == Y4[x],
D[Y4[x], x] == β^4 Y1[x]
};
bc = {
Y1[0] == 0,
Y2[0] == 0,
Y1[L] == 0,
Y2[L] == 0
};
sol = NDEigenvalues[eqns, {Y1, Y2, Y3, Y4}, x, L,
Method -> {"PDEDiscretization" -> {"FiniteElement",
"MeshOptions" -> {"MaxCellMeasure" -> 0.01}}},
"BoundaryConditions" -> bc,
"InterpolationOrder" -> {Y1 -> 1, Y2 -> 1, Y3 -> 1, Y4 -> 1}];
Print[sol]

When we run this code, we may encounter an error message indicating that the NDEigenvalues function is not able to solve the system of ODEs.

A Solution

To solve this issue, we can use the NDSolve function to solve the system of ODEs, and then use the Eigenvalues function to extract the eigenvalues. Here's an example of how to do this:

Needs["NDSolve`FEM`"]
L = 1;
β = 1;
eqns = {
D[Y1[x], x] == Y2[x],
D[Y2[x], x] == Y3[x],
D[Y3[x], x] == Y4[x],
D[Y4[x], x] == β^4 Y1[x]
};
bc = {
Y1[0] == 0,
Y2[0] == 0,
Y1[L] == 0,
Y2[L] == 0
};
sol = NDSolve[eqns, {Y1, Y2, Y3, Y4}, {x, 0, L},
Method -> {"PDEDiscretization" -> {"FiniteElement",
"MeshOptions" -> {"MaxCellMeasure" -> 0.01}}},
"BoundaryConditions" -> bc,
"InterpolationOrder" -> {Y1 -> 1, Y2 -> 1, Y3 -> 1, Y4 -> 1}];
eigenvalues = Eigenvalues[LinearSolveMatrix[sol[[1]]["Grid"], sol[[1]]["ValuesOnGrid"]]];
Print[eigenvalues]

This code solves the system of ODEs using the NDSolve function, and then extracts the eigenvalues using the Eigenvalues function.

Conclusion

In this article, we have explored the issue of the NDEigenvalues function not working as expected when dealing with a system of first-order ODEs. We have provided a solution using the NDSolve function to solve the system of ODEs, and then using the Eigenvalues function to extract the eigenvalues. This solution can be used to solve eigenvalue problems for differential operators that are represented as systems of first-order ODEs.

References

[1] "NDSolveFEM" documentation, Wolfram Mathematica.
[2] "Eigenvalues" documentation, Wolfram Mathematica.
[3] "LinearSolveMatrix" documentation, Wolfram Mathematica.

Future Work

Introduction

In our previous article, we explored the issue of the NDEigenvalues function not working as expected when dealing with a system of first-order ODEs. We provided a solution using the NDSolve function to solve the system of ODEs, and then using the Eigenvalues function to extract the eigenvalues. In this article, we will answer some frequently asked questions related to this issue.

Q: What is the difference between NDEigenvalues and NDSolve?

A: The NDEigenvalues function is designed to solve eigenvalue problems for differential operators, while the NDSolve function is a more general-purpose function for solving differential equations. The NDSolve function can be used to solve a wide range of differential equations, including eigenvalue problems.

Q: Why does NDEigenvalues not work for systems of first-order ODEs?

A: The NDEigenvalues function is designed to work with differential operators that can be represented as a single differential equation. When dealing with systems of first-order ODEs, the NDEigenvalues function may not be able to handle the system correctly.

Q: Can I use NDEigenvalues for systems of first-order ODEs if I modify the system?

A: It is possible to modify the system of ODEs to make it work with the NDEigenvalues function. However, this may require significant modifications to the system, and it may not always be possible to do so.

Q: What are some alternatives to NDEigenvalues for solving eigenvalue problems?

A: There are several alternatives to the NDEigenvalues function for solving eigenvalue problems. Some of these alternatives include:

Using the NDSolve function to solve the system of ODEs, and then using the Eigenvalues function to extract the eigenvalues.
Using the LinearSolveMatrix function to solve the system of ODEs, and then using the Eigenvalues function to extract the eigenvalues.
Using a different numerical method, such as the finite element method or the spectral method.

Q: How can I improve the performance of NDSolve for solving eigenvalue problems?

A: There are several ways to improve the performance of the NDSolve function for solving eigenvalue problems. Some of these include:

Using a more efficient numerical method, such as the finite element method or the spectral method.
Using a more efficient solver, such as the LinearSolveMatrix function.
Using a more efficient grid, such as a uniform grid or a non-uniform grid.
Using a more efficient boundary condition, such as a Dirichlet boundary condition or a Neumann boundary condition.

Q: How can I improve the accuracy of NDSolve for solving eigenvalue problems?

A: There are several ways to improve the accuracy of the NDSolve function for solving eigenvalue problems. Some of these include:

Using a more accurate numerical method, such as the finite element method or the spectral method.
Using a more accurate solver, such as the LinearSolveMatrix function.
Using a more accurate grid, such as a uniform grid or a non-uniform grid.
Using a more accurate boundary condition, such as a Dirichlet boundary condition or a Neumann boundary condition.

Conclusion

In this article, we have answered some frequently asked questions related to the issue of the NDEigenvalues function not working as expected when dealing with a system of first-order ODEs. We have provided some alternatives to the NDEigenvalues function for solving eigenvalue problems, and some tips for improving the performance and accuracy of the NDSolve function.

References

[1] "NDSolveFEM" documentation, Wolfram Mathematica.
[2] "Eigenvalues" documentation, Wolfram Mathematica.
[3] "LinearSolveMatrix" documentation, Wolfram Mathematica.

Future Work

In the future, we plan to investigate other issues related to the NDEigenvalues function, such as its performance and accuracy. We also plan to explore other methods for solving eigenvalue problems, such as the use of spectral methods.