Boundary Value Problems

Let $p\in C^1\left[a,b\right]$ and $q,w\in C\left[a,b\right]$ with $p>0$ and $w>0$ on $\left[a,b\right],$ and $\lambda\in\mathbb{R}$ be a parameter. A Sturm–Liouville equation is a second-order linear equation of the form

$$-\left(p(x)\,y'\right)'+q(x)\,y=\lambda\,w(x)\,y.$$

The function $w$ is the weight function. Every second-order linear equation $a_2y''+a_1y'+a_0y=\lambda\,y$ with $a_2\neq 0$ can be put in this form by multiplying it by $-\frac{1}{a_2}\exp\int\frac{a_1}{a_2}\,dx.$

Let $\left(\alpha_1,\alpha_2\right)\neq\left(0,0\right)$ and $\left(\beta_1,\beta_2\right)\neq\left(0,0\right).$ Separated boundary conditions on $\left[a,b\right]$ are a pair of conditions, one at each endpoint.

$$\alpha_1y(a)+\alpha_2y'(a)=0,\qquad\beta_1y(b)+\beta_2y'(b)=0$$

Dirichlet conditions are $y(a)=y(b)=0$ and Neumann conditions are $y'(a)=y'(b)=0.$

A regular Sturm–Liouville problem is a Sturm–Liouville equation on a bounded interval $\left[a,b\right]$ together with separated boundary conditions.

A singular Sturm–Liouville problem is a Sturm–Liouville problem on an unbounded interval or with a $p$ or $w$ that vanishes at an endpoint, in which case the condition at that endpoint is replaced by boundedness of $y.$

Let $p(a)=p(b).$ Periodic boundary conditions on $\left[a,b\right]$ are the conditions $y(a)=y(b)$ and $y'(a)=y'(b).$

Let $P$ be a Sturm–Liouville problem. An eigenvalue of $P$ is a value of $\lambda$ for which $P$ has a non-zero solution. An eigenfunction of $P$ for the eigenvalue $\lambda$ is a non-zero solution of $P$ at that value of $\lambda.$ The boundary conditions make the solution set of the equation a subspace, so the eigenfunctions for $\lambda$ together with $0$ form the eigen-space of $\lambda.$

Let $P$ be a Sturm–Liouville problem. The Sturm–Liouville operator of $P$ is the linear transformation $L$ on the $C^2$ functions satisfying the boundary conditions given by

$$L[y]:=\frac{1}{w}\left(-\left(p\,y'\right)'+q\,y\right),$$

so that the equation is $L[y]=\lambda y.$

Let $w\in C\left[a,b\right]$ with $w>0.$ The weighted inner product on $C\left[a,b\right]$ is the inner product

$$\langle f,g\rangle_w:=\int_a^bf(x)\,g(x)\,w(x)\,dx.$$

Let $L$ be a Sturm–Liouville operator and $f,g$ be $C^2$ functions on $\left[a,b\right].$ Lagrange’s identity states that the difference of the inner products in the two orders is a boundary term, which follows from integrating by parts twice.

$$\langle L[f],g\rangle_w-\langle f,L[g]\rangle_w=\Big[p\left(f\,g'-f'\,g\right)\Big]_a^b$$

Let $L$ be a linear transformation on a subspace of $C\left[a,b\right].$ The transformation $L$ is self-adjoint with respect to the weighted inner product if and only if $\langle L[f],g\rangle_w=\langle f,L[g]\rangle_w$ for all $f,g$ in the subspace. The boundary terms in Lagrange’s identity vanish for functions satisfying separated or periodic conditions, so a Sturm–Liouville operator is self-adjoint.

Let $P$ be a regular Sturm–Liouville problem. The problem $P$ has the following properties:

• the eigenvalues are real, countably infinite and simple, each with a one-dimensional eigen-space, and can be ordered as $\lambda_1<\lambda_2<\cdots$ with $\lambda_n\rightarrow\infty;$
• eigenfunctions for distinct eigenvalues are orthogonal with respect to the weighted inner product, $\langle\phi_m,\phi_n\rangle_w=0$ for $m\neq n;$
• the eigenfunction $\phi_n$ has exactly $n-1$ zeros in $\left(a,b\right);$
• if $q\geq 0$ and the boundary conditions are Dirichlet, Neumann or a Robin condition with $\frac{\alpha_1}{\alpha_2}\leq 0\leq\frac{\beta_1}{\beta_2},$ then every eigenvalue is non-negative.

Reality and orthogonality follow from self-adjointness as for symmetric matrices.

Let $P$ be a regular Sturm–Liouville problem and $y$ be a non-zero function satisfying its boundary conditions. The Rayleigh quotient of $y$ is the ratio of the weighted inner product of $L[y]$ with $y$ to the weighted inner product of $y$ with itself.

$$R[y]:=\frac{\langle L[y],y\rangle_w}{\langle y,y\rangle_w}=\frac{\Big[-p\,y\,y'\Big]_a^b+\int_a^b\left(p\,y'^2+q\,y^2\right)dx}{\int_a^by^2w\,dx}$$

The Rayleigh quotient has the following properties:

• the smallest eigenvalue is its minimum, $\lambda_1=\min R[y],$ attained at $\phi_1;$
• $\lambda_n$ is the minimum of $R$ over functions orthogonal to $\phi_1,\ldots,\phi_{n-1}.$

The following are examples of Sturm–Liouville problems:

• $-y''=\lambda y$ on $\left[0,L\right]$ with Dirichlet conditions has $\lambda_n=\left(\frac{n\pi}{L}\right)^2$ and $\phi_n=\sin\frac{n\pi x}{L};$
• $-y''=\lambda y$ on $\left[0,L\right]$ with Neumann conditions has $\lambda_n=\left(\frac{n\pi}{L}\right)^2$ for $n\in\mathbb{N}$ including $\lambda_0=0,$ and $\phi_n=\cos\frac{n\pi x}{L};$
• $-y''=\lambda y$ on $\left[-\pi,\pi\right]$ with periodic conditions has $\lambda_n=n^2$ with the two-dimensional eigen-spaces spanned by $\cos nx$ and $\sin nx,$ so the periodic case is where simplicity of the eigenvalues can fail;
• Legendre’s equation on $\left[-1,1\right]$ with $p=1-x^2,$ $q=0,$ $w=1$ is a singular problem whose eigenfunctions are the Legendre polynomials;
• Bessel’s equation on $\left[0,R\right]$ with $p=x,$ $q=\frac{\nu^2}{x},$ $w=x$ is a singular problem whose eigenfunctions are the Bessel functions.

Let $\phi_1,\phi_2,\ldots$ be the eigenfunctions of a regular Sturm–Liouville problem and $f\in C\left[a,b\right].$ The generalised Fourier coefficients of $f$ are its weighted inner products with the normalised eigenfunctions.

$$c_n:=\frac{\langle f,\phi_n\rangle_w}{\langle\phi_n,\phi_n\rangle_w}$$

Let $\phi_1,\phi_2,\ldots$ be the eigenfunctions of a regular Sturm–Liouville problem and $f\in C\left[a,b\right]$ have generalised Fourier coefficients $c_n.$ The eigenfunction expansion of $f$ is the series of eigenfunctions with these coefficients.

$$f(x)\sim\sum_{n=1}^{\infty}c_n\phi_n(x)$$

Let $\phi_1,\phi_2,\ldots$ be the eigenfunctions of a regular Sturm–Liouville problem. The eigenfunctions form a complete orthogonal system, meaning that the eigenfunction expansion of every function $f$ that is Riemann integrable on $\left[a,b\right]$ converges to $f$ in the weighted mean square.

$$\int_a^b\left(f-\sum_{n=1}^{N}c_n\phi_n\right)^2w\,dx\rightarrow 0$$

Eigenfunction expansions have the following properties:

• Parseval’s identity $\langle f,f\rangle_w=\sum_{n=1}^{\infty}c_n^2\langle\phi_n,\phi_n\rangle_w$ holds;
• if $f$ is piecewise $C^1$ the expansion converges pointwise to the average of the one-sided limits of $f;$
• if $f\in C^2\left[a,b\right]$ satisfies the boundary conditions the expansion converges uniformly and absolutely;
• Fourier, sine and cosine series are the expansions for the periodic, Dirichlet and Neumann problems with $-y''=\lambda y.$

Let $P$ be a regular Sturm–Liouville problem with operator $L$ and eigenfunctions $\phi_n,$ $g\in C\left[a,b\right]$ have generalised Fourier coefficients $g_n$ and $\mu$ not be an eigenvalue. Expanding the unknown as $y=\sum_{n=1}^{\infty}y_n\phi_n$ in the problem $L[y]-\mu y=g$ with the given boundary conditions and comparing coefficients gives

$$y_n=\frac{g_n}{\lambda_n-\mu}.$$

The same step justifies separation of variables for a partial differential equation: expanding the solution in the eigenfunctions of the separated boundary value problem reduces the equation to an uncoupled sequence of ordinary differential equations for the coefficients.

Let $P$ be a regular Sturm–Liouville problem with operator $L$ and $\mu=\lambda_m$ be an eigenvalue with eigenfunction $\phi_m.$ The Fredholm alternative states that the problem $L[y]-\mu y=g$ has a solution if and only if $g$ is orthogonal to $\phi_m,$ and that the solution is then unique up to multiples of $\phi_m.$

Let $L[y]=\frac{1}{w}\left(-\left(p\,y'\right)'+q\,y\right)$ with separated boundary conditions for which $0$ is not an eigenvalue, and $g\in C\left[a,b\right].$ The inhomogeneous problem is the equation with forcing function $g$ and the given boundary conditions.

$$-\left(p(x)\,y'\right)'+q(x)\,y=g(x),\qquad\alpha_1y(a)+\alpha_2y'(a)=0,\qquad\beta_1y(b)+\beta_2y'(b)=0$$

Let the inhomogeneous problem be as above, $y_1$ be a non-zero solution of the homogeneous equation satisfying the condition at $a$ and $y_2$ be a non-zero solution of the homogeneous equation satisfying the condition at $b.$ Since $0$ is not an eigenvalue, $y_1$ and $y_2$ are linearly independent, and Abel’s identity for this equation gives that $p\,W\left(y_1,y_2\right)$ is a non-zero constant $C.$ The Green’s function of the problem is the function

$$G\left(x,\xi\right):=-\frac{1}{C}\begin{cases}y_1(x)\,y_2(\xi) & a\leq x\leq\xi \\ y_1(\xi)\,y_2(x) & \xi\leq x\leq b.\end{cases}$$

The unique solution of the inhomogeneous problem is the integral of $g$ against the Green’s function, which is the inverse of the differential operator.

$$y(x)=\int_a^bG\left(x,\xi\right)g(\xi)\,d\xi$$

The Green’s function has the following properties:

• it is continuous and symmetric, $G\left(x,\xi\right)=G\left(\xi,x\right);$
• as a function of $x$ it satisfies the homogeneous equation for $x\neq\xi$ and both boundary conditions;
• its derivative jumps by $-\frac{1}{p(\xi)}$ at $x=\xi,$ which is the statement that $G\left(\cdot,\xi\right)$ solves the equation with the Dirac delta $\delta\left(x-\xi\right)$ as forcing, so it is the response at $x$ to a unit impulse at $\xi;$
• it has the eigenfunction expansion $G\left(x,\xi\right)=\sum_{n=1}^{\infty}\frac{\phi_n(x)\,\phi_n(\xi)}{\lambda_n\langle\phi_n,\phi_n\rangle_w},$ which is the series solution $y_n=\frac{g_n}{\lambda_n}$ written as an integral;
• for the problem $L[y]-\mu y=\frac{g}{w}$ the same expansion holds with $\lambda_n-\mu$ in place of $\lambda_n;$
• if $0$ is an eigenvalue the Green’s function does not exist and the Fredholm alternative applies instead.

For example, $-y''=g$ on $\left[0,1\right]$ with $y(0)=y(1)=0$ has $y_1=x,$ $y_2=1-x,$ $C=-1$ and

$$G\left(x,\xi\right)=\begin{cases}x\left(1-\xi\right) & x\leq\xi \\ \xi\left(1-x\right) & \xi\leq x,\end{cases}$$

the displacement at $x$ of a string fixed at both ends under a unit point load at $\xi.$ The fundamental solution of Laplace’s equation and the heat kernel are the Green’s functions of Poisson’s equation and the heat equation on $\mathbb{R}^n.$ On a bounded region the Green’s function of the Dirichlet problem is the fundamental solution corrected by a harmonic function so that it vanishes on the boundary.