Differential and Integral Equations

Analytical solution of a new class of integral equations

A. G. Ramm

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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

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

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 45A05: Linear integral equations
Secondary: 45H05: Miscellaneous special kernels [See also 44A15] 60G60: Random fields 93E10: Estimation and detection [See also 60G35]


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

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