Approximate solutions of Fredholm-type integral equations

in which one is to determine values of X (proper values) such that a continuous solution x(s) ^ 0 exists. Typical problems leading to equations like (1.1) are the Dirichlet and Neumann problems of potential theory; to (1.2), time-dependent problems in elastic vibration and heat flow, by "separating out" the time. I t is now fifty years since Fredholm published his distinguished paper in Acta Mathematica [lO], in which he gave the first detailed account of the existence and multiplicity of solutions of (1.1) and (1.2). Few mathematical publications have stimulated so much further work. Several papers appeared in which physical problems were set up in terms of integral equations (elasticity, gas theory, etc.) ; some were on numerical solutions; most of the many theoretical papers followed the now familiar trend toward greater generality and abstraction. Hubert recognized analogies with Euclidean geometry, except that now space was infinite-dimensional [13] ; through Fréchet and F. Riesz this led to Banach spaces and more rarefied

[1]  George E. Forsythe,et al.  Solving linear algebraic equations can be interesting , 1953 .

[2]  C. Wagner.  On the Solution of Fredholm Integral Equations of Second Kind by Iteration , 1951 .

[3]  A. T. Lonseth.  The propagation of error in linear problems , 1947 .

[4]  F. B. Hildebrand,et al.  A Least Square Procedure for Solving Integral Equations by Polynomial Approximation , 1941 .

[5]  E. N. Oberg.  The approximate solution of integral equations , 1935 .

[6]  E. Nyström.  Über Die Praktische Auflösung von Integralgleichungen mit Anwendungen auf Randwertaufgaben , 1930 .

[7]  Helly.  Integralgleichungen und Gleichungen mit unendlich vielen Unbekannten , 1929 .

[8]  H. Bateman.  On the Numerical Solution of Linear Integral Equations , 1922 .

[9]  E. Schmidt.  Zur Theorie der linearen und nicht linearen Integralgleichungen Zweite Abhandlung , 1907 .

[10]  E. Schmidt,et al.  Zur Theorie der linearen und nicht linearen Integralgleichungen. Zweite Abhandlung: Auflösung der allgemeinen linearen Integralgleichung , 1907 .

[11]  H. Bückner.  Die praktische Behandlung von Integral-Gleichungen , 1952 .

[12]  E. Bodewig.  Konvergenztypen und das Verhalten von Approximationen in der Nähe einer mehrfachen Wurzel einer Gleichung , 1949 .

[13]  A. T. Lonseth.  An Extension of an Algorithm of Hotelling , 1949 .

[14]  S. Banach,et al.  Théorie des opérations linéaires , 1932 .

[15]  D. V. Jonesco.  Sur une classe d'équations fonctionnelles , 1927 .

[16]  N. Kryloff.  Sur différents procédés d'intégration approchée en physique mathématique , 1925 .

[17]  É. Goursat.  Sur un cas élémentaire de l'équation de Fredholm , 1907 .