Abstract
The basis of the finite-difference method is the replacement of all derivatives occuring by the corresponding difference quotients; this is applicable to any problem in differential equations.
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This method of solving systems of linear equations had already been used by GAUSS when it was taken up by PH. L. SEIDEL: Munch. Akad. Abh. 1874, 81–108; its convergence was investigated later by R. V.
Misesand H. Pollaczekgeiringer: Praktische Verfahren der Gleichungsauflösung. Z. Angew. Math. Mech. 9, 62–77 (1929), and in recent years it has been applied to a very wide range of problems and also expounded in two books by R. V.
Southwell: Relaxation methods in engineering science, a treatise on approximate computation. London: Oxford University Press 1943;
Relaxation methods in theoretical physics, a continuation of the treatise. London: Oxford University Press 1946, 248pp.;
D. N. DeG. Allen: Relaxation methods. New York 1954, 257pp. See also our Ch. V, § 1. 6.
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E. Pflanz: Über die Bildung finiter Ausdrücke für die Lösung linearer Differentialgleichungen. Z. Angew. Math. Mech. 17, 296–300 (1937).
Expressions for non-equidistant pivotal points are given by E. Pflanz: Allgemeine Differenzenausdrücke für die Ableitungen einer Funktion y (x) Z. Angew. Math. Mech. 29, 379–381 (1949)•
Householder, A. S.: Principles of numerical analysis, p. 194. New York:
Mcgraw-Hill 1953. We do not imply that the method described here for solving differential equations is directly due to Hermite.
By using the associated Green’s function E. J. NYSTRÖM: Zur numerischen Lösung von Randwertaufgaben bei gewöhnlichen Differentialgleichungen. Acta Math. (Stockh.) 76, 157–184 (1944), has developed a special Hermitian method
A variant of the Hermitian method originally due to Numerovhas been applied to the special equation y“= q (x) y (cf. G. Stracke: Bahnbestimmung der Planeten and Kometen, § 77 Berlin 1929). In this method a system of equations
Following a somewhat differently derived method due to H. Sassenfeld: Ein Summenverfahren für Rand-und Eigenwertaufgaben linearer Differentialgleichungen. Z. Angew. Math. Mech. 31, 240–241 ( 1951
R. ZurmÜhl: Praktische Mathematik für Ingenieure und Physiker, 2nd ed. p. 447 et seq. Berlin-Göttingen-Heidelberg 1957.
These, together with formulae for differential equations of the fourth order, are to be found in R. ZurmÜhl: Praktische Mathematik für Ingenieure und Physiker, 2nd ed. Berlin-Göttingen-Heidelberg 1957.
For many boundary-value problems of monotonic type error estimates can be obtained in a quite different way which does not require knowledge of bounds for the higher derivatives; cf. L. CoLLATZ: Aufgaben monotoner Art. Arch. Math. 3, 375 (1952), in which a numerical example with error limits is also given.
Applications of perturbation methods to eigenvalue problems can be found in the following papers: Meyer Zur Capellen, W.: Methode zur angenäherten Lösung von Eigenwertproblemen mit Anwendungen auf Schwingungsprobleme. Ann. Phys. (5) 8, 297–352 (1931).
Genäherte Berechnung von Eigenwerten. Ing.-Arch. 10, 167–174 (1939).
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Rellich, F.: Störungstheorie der Spektralzerlegung. Math. Ann. 116 555–570 (1939)
Rellich, F.: Störungstheorie der Spektralzerlegung. Math. Ann. 117, 356–382 (1940)
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Nagy, B. v. Sz.: Perturbations des transformations autoadjointes dans l’espace de Hilbert. Comm. Math. Helv. 19, 347–366 (1946).
Perturbations des transformations linéaires fermées. Acta Sci. Math. Szeged 14, 125–137 (1951).
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Fehlerabschätzungen zur Störungsrechnung für lineare Eigenwertprobleme bei gewöhnlichen Differentialgleichungen. Z. Angew. Math. Mech. 34, 140–149 (1954) (with a summary of results in a directly applicable form).
Schafke, Fr. W.: Über Eigenwertprobleme mit 2 Parametern. Math. Nachr. 6, 109–124 (1951).
Verbesserte Konvergenz- und Fehlerabschätzungen für die Störungsrechnung. Z. Angew. Math. Mech 33, 255–259 (1953).
Walter Ritz, born 22 February 1878 in Sion (Switzerland) in the Rhone valley, son of the artist Raphael Ritz, studied in Zürich, then in Göttingen, where in 1902 he obtained his doctor’s degree under Voigt; he then worked in. Leyden under H. A. Lorentz, in Paris under A. Cotton, in Tübingen under F. Paschen and in 1908 went back to Göttingen, where he died on the 7 July 1909. After a poorly healed pleurisy in 1900 his zeal for his scientific work was continually in conflict with consideration for his state of health, until eventually his health was sacrificed. (Obituary by Pierre Weiszin W. Ritz: Gesammelte Werke. Paris 1911.)
Habilitationsschrift von Ritz über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik. J. Reine Angew. Math. 135, H. 1 (1908).
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Cf., for example, G. GrÜsz: Variationsrechnung, p. 11. Leipzig 1938
R. Courant: Differential and Integral Calculus Vol. II, p. 495 et seq. London: Blackie 1936.
Error estimates for RITZ’S method can be found in Nicolas Kryloff: Les méthodes de solution approchée des problèmes de la physique mathématique. Mémorial. Sci. Math., Paris 49 (1931).
Of the numerous shorter papers by KRYLOFF on error estimates of this kind we mention without giving their titles just a few published in the C. R. Acad. Sci., Paris 180, 1316–1318 (1925)
Of the numerous shorter papers by KRYLOFF on error estimates of this kind we mention without giving their titles just a few published in the C. R. Acad.Sci., Paris 181, 86–88 (1925)
Of the numerous shorter papers by Kryloffon error estimates of this kind we mention without giving their titles just a few published in the C. R. Acad.Sci., Paris 183, 476–479 (1926)
Of the numerous shorter papers by KRYLOFF on error estimates of this kind we mention without giving their titles just a few published in the C. R. Acad.Sci., Paris 186, 298–300
Of the numerous shorter papers by KRYLOFF on error estimates of this kind we mention without giving their titles just a few published in the C. R. Acad.Sci., Paris 422–425 (1928).
See, for example, T. Pöschl: Über eine mögliche Verbesserung der Ritzschen Methode. Ing.-Arch. 23, 365–372 (1955).
Kamke, E.: Math. Z. 48, 70 (1942).
Weser, C.: Z. Angew. Math. Mech. 21, 310–311 (1941).
An example of the application of this technique to a non-linear problem can be found in U. T. BÖDewadt: Die Drehströmung über festem Grunde. Z. Angew. Math. Mech. 20, 241–253 (1940).
For example, R. Courantand D. Hilbert: Methods of mathematical physics, Vol. I, 1st English ed. New York: Interscience Publishers, Inc. 1953
Collatz, L.: Eigenwertaufgaben mit technischen Anwendungen. Leipzig 1949.
A more general theorem has been established by H. Wlelandt: Das Iterations-verfahren bei nicht selbstadjungierten Eigenwertaufgaben. Math. Z. 50, 93–143 (1944).
Fewer restrictive assumptions are made by E. Stiefeland H. Ziegler: Natürliche Eigenwertprobleme. Z. Angew. Math. Phys. 1, 111–138 (1950).
Cf. the comparison theorem in L. Collatz: Eigenwertaufgaben, § 8. Leipzig 1949; this deals with the case in which the differential expression M is the same in both problems but the extension to the more general case considered here is immediate.
For the special eigenvalue problems the enclosure theorem for the first eigen-value was proved by ‘G. Temple: The computatipn of characteristic numbers and characteristic functions. Proc. Lond. Math. Soc. (2) 29, 257–280 (1929).
This method is due to F. Kieszling: Eine Methode zur approximativen Berechnung einseitig eingespannter Druckstäbe mit veränderlichem Querschnitt. Z. Angew. Math. Mech. 10, 594–599 (1930).
Wielandt, H.: Ein Einschließungssatz für charakteristische Wurzeln normaler Matrizen. Arch. Math. 1, 348-352 (1949)
Fiat-Review, Naturforschung und Medizin in Deutschland 1939–1946, 2, 98 (1948).
An older formulation was given by K. Friedrichsand G. Horvay: The finite Stieltjes momentum problem. Proc. Nat. Acad. Sci., Wash. 25, 528— 534 (1939)
H. BÜckner: Die praktische Behandlung von Integralgleichur}gen (Ergebnisse der Angew. Mathematik, H. 1). Berlin-Göttingen-Heidelberg 1952.
A comprehensive presentation based on the techniques of functional analysis is given by N. Aronszajn: Study of eigenvalue problems. The Rayleigh-Ritz and the Weinstein methods for approximation of eigenvalues. Dept. of Math. Oklahoma Agricultural and Mechanical College. Stillwater 1949. 214 pp.
Error estimates for self-adjoint full-definite eigenvalue problems have been obtained by G. Bertram: Zur Fehlerabschätzung für das Ritzsche Verfahren bei Eigenwertaufgaben. Diss. 56pp. Hannover 1950
Other estimates and investigations of the rate of convergence can be found in L. V. Kantorovichand V. I. Krylov: Näherungsmethoden der Höheren Analysis, pp. 226–329. Berlin 1956.
Kamke, E.: Über die definiten selbstadjungierten Eigenwertaufgaben IV. Math. Z. 48, 67–100 (1942).
Collatz, L.: Z. Angew. Math. Mech. 19, 228 (1939).
Gramme, R.: Ein neues Verfahren zur Lösung technischer Eigenwertprobleme. Ing.-Arch. 10, 35-46 (1939). GRAMMEL derives the equations in a different way.
Practical examples are worked out by E. Maier: Biegeschwingungen von spannungslos verwundenen Stäben, insbesondere von Luftschraubenblättern. Ing.Arch. 11, 73–98 (1940).
See also R. Grammel: Viper die Lösung technischer Eigenwertprobleme. VDI-Forsch.-Heft, Gebiet Stahlbau 6, 36–42 (1943).
See L. Collarz: Eigenwertaufgaben. Leipzig 1949. pp. 144 and 191.
N. J. Lehmann: Beiträge zur numerischen Lösung linearer Eigenwertprobleme. Z. Angew. Math. Mech. 29, 341— 356 (1949)
N. J. Lehmann: Beiträge zur numerischen Lösung linearer Eigenwertprobleme. Z. Angew. Math. Mech. 30, 1–16 (1950).
Another method is given by A. Fraenkle: Ing.-Arch. 1, 499-526 (1930), specifically “Methode II”, p. 510 et seq.
A method of minimized iterations has been devised by C. Lanczos: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Nat. Bur. Stand. 45, 255–282 (1950).
Developed for integral equations by G. Wiarda: Integralgleichungen unter besonderer Berücksichtigung der Anwendungen, p. 126. Leipzig 1930.
Developed for integral equations by H. BÜCkner: Ein unbeschränkt anwendbares Iterationsverfahren für Fredholmsche Integralgleichungen. Math. Nachr. 2, 304–313 (1949).
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Collatz, L. (1960). Boundary-value problems in ordinary differential equations. In: The Numerical Treatment of Differential Equations. Die Grundlehren der Mathematischen Wissenschaften, vol 60. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-88434-4_3
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