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2003 Analytical solution of a new class of integral equations
A. G. Ramm
Differential Integral Equations 16(2): 231-240 (2003).

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.

Citation

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A. G. Ramm. "Analytical solution of a new class of integral equations." Differential Integral Equations 16 (2) 231 - 240, 2003.

Information

Published: 2003
First available in Project Euclid: 21 December 2012

zbMATH: 1034.45001
MathSciNet: MR1947094

Subjects:
Primary: 45A05
Secondary: 45H05 , 60G60 , 93E10

Rights: Copyright © 2003 Khayyam Publishing, Inc.

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