Differential and Integral Equations

Analytical solution of a new class of integral equations

A. G. Ramm

Abstract

Let $(1) Rh=f$, $0\leq x\leq L$, $Rh=\int^L_0 R(x,y)h(y)\,dy$, where the kernel $R(x,y)$ satisfies the equation $QR=P\delta(x-y)$. Here, $Q$ and $P$ are formal differential operators of order $n$ and $m <n$, respectively, $n$ and $m$ are nonnegative even integers, $n>0$, $m\geq 0$, $Qu:=q_n(x)u^{(n)} + \sum^{n-1}_{j=0} q_j(x) u^{(j)}$, $Ph:=h^{(m)} +\sum^{m-1}_{j=0} p_j(x) h^{(j)}$, $q_n(x)\geq c>0$, the coefficients $q_j(x)$ and $p_j(x)$ are smooth functions defined on $\mathbb R$, $\delta(x)$ is the delta-function, $f\in H^\alpha(0,L)$, $\alpha:=\frac{n-m}{2}$, $H^\alpha$ is the Sobolev space. An algorithm for finding analytically the unique solution $h\in\dot H^{-\alpha} (0,L)$ to (1) of minimal order of singularity is given. Here, $\dot H^{-\alpha}(0,L)$ is the dual space to $H^\alpha(0,L)$ with respect to the inner product of $L^2(0,L)$. Under suitable assumptions it is proved that $R:\dot H^{-\alpha}(0,L) \to H^\alpha(0,L)$ is an isomorphism. Equation (1) is the basic equation of random processes estimation theory. Some of the results are generalized to the case of multidimensional equation (1), in which case this is the basic equation of random fields estimation theory.

Article information

Source
Differential Integral Equations, Volume 16, Number 2 (2003), 231-240.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060686

Mathematical Reviews number (MathSciNet)
MR1947094

Zentralblatt MATH identifier
1034.45001

Citation

Ramm, A. G. Analytical solution of a new class of integral equations. Differential Integral Equations 16 (2003), no. 2, 231--240. https://projecteuclid.org/euclid.die/1356060686