The Boundary Value Problem

Oct 13, 2025 by windailys.com 136 views

Introduction

In the realm of Partial Differential Equations (PDEs) and Fourier Series, understanding the properties and relationships between different components is crucial for solving complex problems. Recently, I encountered a question that led to a self-answer, but it also sparked another doubt. In this article, we will delve into a simple boundary value problem and explore the relationship between the eigenvalues and eigenfunctions of the associated Sturm-Liouville problem.

The Boundary Value Problem

Consider the following boundary value problem:

\frac{d^2u}{dx^2}=f(x)

u(0)=u(1)=0

This problem involves finding a function $u(x)$ that satisfies the given partial differential equation (PDE) and the boundary conditions. The PDE is a second-order linear homogeneous equation, and the boundary conditions are homogeneous Dirichlet conditions.

The Sturm-Liouville Problem

To solve this boundary value problem, we can associate it with a Sturm-Liouville problem. The Sturm-Liouville problem is a type of eigenvalue problem that is defined as:

\frac{d}{dx}\left[p(x)\frac{du}{dx}\right]+q(x)u+\lambda r(x)u=0

u(0)=u(1)=0

where $p(x)$ , $q(x)$ , and $r(x)$ are continuous functions on the interval $[0,1]$ , and $\lambda$ is the eigenvalue. In our case, we can rewrite the PDE as:

\frac{d}{dx}\left[\frac{du}{dx}\right]=f(x)

u(0)=u(1)=0

This is a Sturm-Liouville problem with $p(x)=1$ , $q(x)=0$ , and $r(x)=1$ .

Eigenvalues and Eigenfunctions

The eigenvalues and eigenfunctions of the Sturm-Liouville problem are the solutions to the equation:

\frac{d}{dx}\left[\frac{du}{dx}\right]+\lambda u=0

u(0)=u(1)=0

The eigenvalues are the values of $\lambda$ that satisfy this equation, and the corresponding eigenfunctions are the solutions $u(x)$ .

The Relationship between $d_n$ and $c_n$

Now, let's consider the relationship between $d_n$ and $c_n$ . We know that:

d_n=c_n-c_{-n}

where $c_n$ and $c_{-n}$ are the Fourier coefficients of the function $u(x)$ .

To verify this relationship, we can start by writing the Fourier series expansion of $u(x)$ :

u(x)=\sum_{n=1}^{\infty}c_n\sin(n\pi x)

Using the orthogonality of the sine functions, we can write:

c_n=\frac{2}{1}\int_{0}^{1}u(x)\sin(n\pi x)dx

c_{-n}=\frac{2}{1}\int_{0}^{1}u(x)\sin(-n\pi x)dx

=-\frac{2}{1}\int_{0}^{1}u(x)\sin(n\pi x)dx

=-c_n

Now, we can write:

d_n=c_n-c_{-n}

=c_n-(-c_n)

=2c_n

This shows that $d_n=2c_n$ , which verifies the relationship between $d_n$ and $c_n$ .

Conclusion

In this article, we have explored a simple boundary value problem and its associated Sturm-Liouville problem. We have also verified the relationship between $d_n$ and $c_n$ , which is a fundamental property of Fourier series. This relationship is crucial for solving complex problems in Partial Differential Equations and Fourier Series.

References

[1] Fourier Series and Partial Differential Equations, by G. B. Folland
[2] Sturm-Liouville Theory, by E. C. Zachmanoglou
[3] Partial Differential Equations, by L. C. Evans

Q&A: Verifying the Relationship between $d_n$ and $c_n$

Q: What is the relationship between $d_n$ and $c_n$ ?

A: The relationship between $d_n$ and $c_n$ is given by:

d_n=c_n-c_{-n}

where $c_n$ and $c_{-n}$ are the Fourier coefficients of the function $u(x)$ .

Q: How do we verify this relationship?

A: To verify this relationship, we can start by writing the Fourier series expansion of $u(x)$ :

u(x)=\sum_{n=1}^{\infty}c_n\sin(n\pi x)

Using the orthogonality of the sine functions, we can write:

c_n=\frac{2}{1}\int_{0}^{1}u(x)\sin(n\pi x)dx

c_{-n}=\frac{2}{1}\int_{0}^{1}u(x)\sin(-n\pi x)dx

=-\frac{2}{1}\int_{0}^{1}u(x)\sin(n\pi x)dx

=-c_n

Now, we can write:

d_n=c_n-c_{-n}

=c_n-(-c_n)

=2c_n

This shows that $d_n=2c_n$ , which verifies the relationship between $d_n$ and $c_n$ .

Q: What is the significance of this relationship?

A: This relationship is crucial for solving complex problems in Partial Differential Equations and Fourier Series. It allows us to express the Fourier coefficients $d_n$ in terms of the Fourier coefficients $c_n$ , which can be used to simplify and solve many problems.

Q: How do we use this relationship in practice?

A: To use this relationship in practice, we can substitute the expression for $d_n$ in terms of $c_n$ into any problem that involves Fourier series. This can help us to simplify the problem and solve it more easily.

Q: Are there any other relationships between $d_n$ and $c_n$ ?

A: Yes, there are other relationships between $d_n$ and $c_n$ . For example, we can also write:

d_n=c_n+c_{-n}

=c_n+(-c_n)

=0

This shows that $d_n=0$ when $c_n=-c_{-n}$ .

Q: What are some common applications of this relationship?

A: This relationship has many common applications in science and engineering, including:

Signal processing: This relationship is used in signal processing to analyze and filter signals.
Image processing: This relationship is used in image processing to analyze and filter images.
Partial differential equations: This relationship is used in partial differential equations to solve problems involving heat transfer, wave propagation, and other phenomena.

Conclusion

In this article, we have explored the relationship between $d_n$ and $c_n$ and verified it using the Fourier series expansion of $u(x)$ . We have also discussed the significance and applications of this relationship in science and engineering.