Integral equation

An integral equation asks for an unknown function that occurs under an integral. Integral equations with a variable upper limit have a triangular dependence structure that often permits successive approximation.

This square-root-kernel integral equation is inverted by integrating $g(x)/\sqrt{x-z}$ over $z<x<b$ and interchanging integrations. The inner integral over $z<x<y$ equals $\pi$, so ${\int }_z^{b}g(x)/\sqrt{x-z}dx=\pi {\int }_z^{b}f(y)dy$. Differentiating recovers $f(z)=-{\pi }^{-1}d[{\int }_z^{b}g(x)/\sqrt{x-z}dx]/dz$. The changes of variables $x=p^2$, $y={\eta }^2$ put the angular-ray equation into this form.

A linear integral equation depends linearly on its unknown function. For example, $(I+K)f=g$ with $(Kf)(x)=\int K(x,y)f(y)dy$ has a trivial homogeneous null space whenever its inhomogeneous problem has a unique solution: otherwise a nonzero homogeneous solution could be added to that solution. This is the uniqueness step used in dressing constructions of a Lax pair.

A Volterra integral equation has a variable integration limit, so its value at a point depends only on earlier values. Repeated integration produces factorial decay, which often makes its Neumann series converge without a smallness assumption.

The iteration $y_0=x$, $y_n=x-{\int }_0^x(x-s)y_{n-1}(s)/sds$ successively appends terms of the displayed series. Induction verifies the iterates, and factorial denominator bounds give convergence with derivatives on compact sets. Its limit is the power series solution of $x{y}^{\prime \prime }+y=0$, $y(0)=0$, $y^{\prime }(0)=1$.

For time-dependent integral kernels on a Riemannian manifold, composition also integrates over the intermediate point: $(A\ast B)(t,x,y)={\int }_0^t{\int }_{M}A(t-s,x,z)B(s,z,y)d{V}_{z}ds$. It is associative under the usual absolute-integrability hypotheses. The integration simplex gives factorial bounds for iterated convolutions of bounded kernels.

Starting from $\Vert \tau g(t){\Vert }^2\le C^{2}t{\int }_0^t\Vert g(s){\Vert }^{2}ds$, induction integrates $s^{2n-2}$ at the $n$th step. Its denominator accumulates the odd integers $1,3,\dots ,2n-1$. The factorial bound for a Volterra iterate is stronger because $(n!{)}^2\ge (2n-1)!!$.

If $\Vert \tau f(t)\Vert \le C{\int }_0^t\Vert f(s)\Vert ds$, repeated integration produces the time-ordered simplex of volume $t^n/n!$. Thus the Neumann series ${\sum }_{n\ge 0}{\tau }^n$ converges on every finite interval, without requiring the single-step operator norm to be less than one.