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.

Suppose $h$ is continuously differentiable on $[0,L]$. For $0<x\le L$, the improper integral $g(x)={\int }_0^{x}h(t)(x-t{)}^{-1/2}dt$ exists, but substituting directly into the ordinary Leibniz integral rule produces an undefined endpoint term. First use integration by parts to obtainThe new endpoint term at $t=x$ vanishes, and the differentiated kernel is locally integrable. Differentiating gives the displayed formula. One rigorous justification rewrites the second term as ${\int }_0^x[{\int }_0^s{h}^{\prime }(t)(s-t{)}^{-1/2}dt]ds$ by Fubini's theorem; the inner integral is continuous for $s>0$, so the fundamental theorem of calculus applies. Moreover $g(x)=2h(0)\sqrt{x}+O(x^{3/2})$: if $h(0)\ne 0$, the right difference quotient at zero diverges rather than defining a finite derivative.

For an absolutely continuous $f$ with $f(0)=0$, this causal Abel integral equation has inversionUnder Laplace-transform growth hypotheses, the convolution theorem for Laplace transforms converts the original kernel into $\Gamma (1-\alpha )s^{\alpha -1}$. Since $\mathcal{L}[f^{\prime }]=s\mathcal{L}[f]$, the transformed inverse is $s^{-\alpha }\mathcal{L}[f^{\prime }]/\Gamma (1-\alpha )$, giving the formula. It also holds locally in the corresponding integrable convolution class: convolving the two power kernels produces the constant $B(\alpha ,1-\alpha )=\Gamma (\alpha )\Gamma (1-\alpha )$ by the beta-gamma identity, so substitution recovers ${\int }_0^t{f}^{\prime }=f(t)$. The same convolution identity proves uniqueness in that class. For example $f(t)=t$ gives $\Phi =t^{\alpha }/[\Gamma (\alpha +1)\Gamma (1-\alpha )]$.

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 derivative and convolution theorem for Laplace transforms give $[p+1/(p+2)]\bar{x}=1/p$. Partial fractions yield $\bar{x}=2/p-2/(p+1)-1/(p+1{)}^2$, so $x(t)=2-(2+t)e^{-t}$ tends to two. A direct check gives $x^{\prime }=(1+t)e^{-t}$ and $(e^{-2t}\ast x)(t)=1-(1+t)e^{-t}$. Introducing the memory variable $w=e^{-2t}\ast x$ gives the equivalent local system $x^{\prime }+w=1$, $w^{\prime }=x-2w$, $w(0)=0$, explaining why the transform denominator has a repeated pole at $p=-1$.

For causal convolution on $t\ge 0$, the Laplace transform convolution theorem converts $x+\sin \ast x=F$ into $\bar{x}=[(p^2+1)/(p^2+2)]\bar{F}$. Since this multiplier is $1-1/(p^2+2)$, inversion gives the displayed resolvent kernel. The formula remains valid for locally integrable forcing on finite intervals: substitution and the Fubini theorem reduce verification to an identity between the two continuous sine kernels. Thus a global Laplace transform is a convenient derivation, rather than an essential growth restriction on every local forcing.

A function weighting the influence of past values in a history-dependent equation. In a Volterra integral equation its convolution with the history is causal: only earlier times appear. For an integrable kernel, $\tilde{\kappa }(0)={\int }_0^{\infty }\kappa (u)du$ measures its response to a constant input. In a generalized Langevin equation, the friction memory kernel records delayed resistance to the velocity.

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.