Linear Integral Equations

An integral equation asks for a function whose weighted aggregate values match known data. Writing it as an operator equation exposes the decisive difference: first-kind problems try to invert a smoothing operator directly, while second-kind problems add the identity and are often stably invertible except at isolated spectral parameters.

Classification before solution #

A Fredholm equation integrates over fixed limits:

$$\text{first kind:}\quad \int_a^b k(x,t)\varphi(t)dt=f(x),$$

$$\text{second kind:}\quad \varphi(x)-\lambda\int_a^b k(x,t)\varphi(t)dt=f(x).$$

A Volterra equation uses a variable upper limit:

$$\varphi(x)-\lambda\int_a^x k(x,t)\varphi(t)dt=f(x).$$

With $(K\varphi)(x)=\int k(x,t)\varphi(t)dt$, these are $K\varphi=f$ and $(I-\lambda K)\varphi=f$. The fixed triangular support $a\le t\le x$ makes Volterra iteration much better behaved than a general Fredholm kernel.

Neumann iteration and resolvent kernel #

If $K$ is bounded and $|\lambda|\|K\|<1$, then

$$(I-\lambda K)^{-1}=\sum_{n=0}^{\infty}\lambda^nK^n$$

in operator norm. Picard iteration

$$\varphi_{n+1}=f+\lambda K\varphi_n$$

is the same geometric series viewed as an algorithm.

Define iterated Fredholm kernels by

$$k_1(x,t)=k(x,t),\qquad k_{n+1}(x,t)=\int_a^b k(x,s)k_n(s,t)ds.$$

Where the series converges, the resolvent kernel is

$$r_\lambda(x,t)=\sum_{n=1}^{\infty}\lambda^{n-1}k_n(x,t),$$

and the solution is

$$\varphi(x)=f(x)+\lambda\int_a^b r_\lambda(x,t)f(t)dt.$$

The condition $|\lambda|\|K\|<1$ is sufficient, not necessary. Outside it, spectral/Fredholm analysis may still prove invertibility.

Why Volterra series converges for every parameter #

Suppose $|k(x,t)|\le M$ on $a\le t\le x\le b$. Nested triangular integration gives a factorial denominator:

$$|k_n(x,t)|\le M^n\frac{(x-t)^{n-1}}{(n-1)!}.$$

Therefore the resolvent-kernel series converges uniformly for every finite $\lambda$. This is stronger than the generic geometric condition. It also explains why the Volterra operator has spectral radius zero even though it is not nilpotent.

For $k\equiv1$,

$$\varphi(x)=f(x)+\lambda\int_a^x\varphi(t)dt.$$

If $f$ is differentiable, differentiation gives $\varphi'-\lambda\varphi=f'$ with $\varphi(a)=f(a)$. The integral equation already encodes the initial condition.

Degenerate kernels reduce to linear algebra #

If

$$k(x,t)=\sum_{j=1}^m a_j(x)b_j(t),$$

set $c_j=\int_a^b b_j(t)\varphi(t)dt$. Then

$$\varphi(x)=f(x)+\lambda\sum_{j=1}^m a_j(x)c_j,$$

and multiplying by each $b_i$ and integrating produces an $m\times m$ system for $c$. The infinite-dimensional equation has finite-rank coupling.

Fredholm solvability #

For a compact $K$ on a Hilbert space, $(I-\lambda K)\varphi=f$ is uniquely solvable for every $f$ unless $1/\lambda$ is a nonzero eigenvalue of $K$. At a singular parameter, solvability requires

$$\langle f,\psi\rangle=0 \quad\text{for every }\psi\in \ker(I-\overline\lambda K^*).$$

For an integral kernel, the adjoint kernel is $k^*(x,t)=\overline{k(t,x)}$. If compatibility holds, solutions differ by a homogeneous solution.

First-kind equations are inverse problems #

A compact integral operator sends high-frequency or localized differences to small outputs. If $K$ has infinite-dimensional range, its inverse on that range is generally unbounded. Small data noise can therefore create large solution error. A tiny residual $\|K\varphi-f\|$ is not evidence that $\varphi$ is close to the true solution.

Stable recovery needs additional information, commonly Tikhonov regularization:

$$\min_\varphi\ \|K\varphi-f_\delta\|^2 +\alpha\|L\varphi\|^2.$$

$L$ expresses the permitted solution structure, and $\alpha$ trades data fit against noise amplification. Parameter choice is part of the inference, not a cosmetic numerical option.

Discretization without losing the operator problem #

Nyström methods replace the integral by quadrature; Galerkin methods project the equation onto a finite-dimensional trial/test space; degenerate-kernel methods explicitly approximate $K$ by finite rank. In every case verify:

• quadrature/basis convergence in the relevant operator or solution norm;
• conditioning as resolution grows;
• the residual in the continuous equation, not only the discrete system;
• compatibility near a Fredholm singularity;
• regularization and noise sensitivity for first-kind data.

The matrix is an approximation to the operator problem. Its solvability and condition number must be interpreted through the function spaces and norms from which it came.