The Approximate Solution of Convolution-Type Integral Equations

• Published in: SIAM journal on mathematical analysis Vol. 4; no. 3; pp. 536 - 555
• Main Author: Stenger, Frank
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.05.1973
• Subjects: Approximation; Fourier transforms; Integral equations; Linear equations
• ISSN: 0036-1410; 1095-7154
• DOI: 10.1137/0504047

A result on the convergence of an approximate factorization of $\exp \phi $, where $\phi \in L^2 ( - \infty ,\infty ) \cap L^\infty ( - \infty ,\infty )$, is obtained and then used to get an explicit approximate solution $f^{(h)} $ of the problem $f(x) = \int _0^\infty K(x - t)f(t)dt + g(x),x > 0$, where $K$ and $g$ are in $L^2 ( - \infty ,\infty )$. The approximation , $f^{(h)} $ depends on a parameter $h$ and satisfies $\| {f - f^{(h)} } \|_2 \to 0$ as $h \to 0$. A computationally more accessible explicit approximation $f_k^{(h)} $ is also obtained, which depends on a parameter $k$, and satisfies $| {f^{(h)} (x) - f_k^{(h)} (x)} | \to 0$ as $k \to 0$ for all $x \geqq 0$. Explicit bounds are obtained, for $| {f(x) - f^{(h)} (x)} |$ and also for $| {f^{(h)} (x) - f_k^{(h)} (x^{(h)} )} |$.