On the numerical integration of ordinary differential equations of the first order
Author:
Per-Olov Löwdin
Journal:
Quart. Appl. Math. 10 (1952), 97-111
MSC:
Primary 65.0X
DOI:
https://doi.org/10.1090/qam/50996
MathSciNet review:
50996
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Abstract: The difference methods for the numerical integration of ordinary differential equations of the first order are discussed by using operator calculus and symbolic expansions. A new straightforward central difference method is developed, which is based on a formula closely associated with Simpson’s rule. The main features of the method are that, for each step of integration, the largest unknown term is determined by an algebraic equation and that the remaining difference correction is extremely small. The method can directly be applied even to systems of the first order with one-point boundary conditions. A numerical example is given.
Aitken, A. C., 1925, Proc. Roy. Soc. Edinburgh 46, 289; 1936, ibid. 57, 269.
Adams, J. C. and Bashforth, F., 1883, Theories of capillary action, Cambridge.
Bennet, A. A., Milne, W. E., and Bateman, H., 1933, Numerical integration, Nat. Res. Council, Washington.
Bickley, W. G., 1932, Phil. Mag. 13, 1006. 1948, J. Math. Phys. 27, 183.
- W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Philos. Mag. (7) 33 (1942), 1–14. MR 6239
- A. E. Carter and D. H. Sadler, The application of the National accounting machine to the solution of first-order differential equations, Quart. J. Mech. Appl. Math. 1 (1948), 433–441. MR 28687, DOI https://doi.org/10.1093/qjmam/1.1.433
- F. G. Chadaja, On the problem of numerical integration of ordinary differential equations, Bull. Acad. Sci. Georgian SSR [Soobščenia Akad. Nauk Gruzinskoi SSR] 2 (1941), 601–608 (Russian, with Georgian summary). MR 0010069
Charlier, C. L., 1907, Die Mechanik des Himmels, Bd. II, Leipzig.
Collatz, L. and Zurmühl, R., 1942a, Zs. ang. Math. Mech. 22, 42. 1942b, Ing. Arch. 13, 34.
- L. Collatz, Natürliche Schrittweite bei numerischer Integration von Differentialgleichungssystemen, Z. Angew. Math. Mech. 22 (1942), 216–225 (German). MR 8723, DOI https://doi.org/10.1002/zamm.19420220405
- W. J. Duncan, Technique of the step-by-step integration of ordinary differential equations, Philos. Mag. (7) 39 (1948), 493–509. MR 25285
Encke, 1837, Berliner Astr. Jahrbuch; 1867, ibid.
Falkner, V. M., 1936, Phil. Mag. 21, 624.
- Lillian Feinstein and Martin Schwarzschild, Automatic integration of linear second-order differential equations by means of punched card machines, Rev. Sci. Instruments 12 (1941), 405–408. MR 5458
- L. Fox and E. T. Goodwin, Some new methods for the numerical integration of ordinary differential equations, Proc. Cambridge Philos. Soc. 45 (1949), 373–388. MR 30315, DOI https://doi.org/10.1017/s0305004100025007
Gauss, C. F., 1834, Brief an Encke vom 13/10 1834, Gesammelte Werke 7, 433.
- D. R. Hartree, Notes on iterative processes, Proc. Cambridge Philos. Soc. 45 (1949), 230–236. MR 29268, DOI https://doi.org/10.1017/s0305004100024762
- L. F. Hausman and M. Schwarzschild, Automatic integration of linear sixth-order differential equations by means of punched-card machines, Rev. Sci. Instruments 18 (1947), 877–883. MR 23621
- Mark Kormes, A note on the integration of linear second-order differential equations by means of punched cards, Rev. Sci. Instruments 14 (1943), 118. MR 8352
Kryloff, A. N., 1925, Proc. I. Intern. Congr. Appl. Math., Delft, 212.
Levy, H. and Baggot, E. A., 1934, Numerical studies in differential equations, London.
Lindelöf, E., 1938, Acta Soc. Sci. Fennicae, Phys. Mat. (A) 2, no. 13.
- J. G. L. Michel, Central difference formulae obtained by means of operator expansions, J. Inst. Actuar. 72 (1946), 470–480. MR 20341
- Sch. Mikeladze, Verallgemeinerung der Methode der numerischen Integration von Differentialgleichungen mit Hilfe der Formeln der mechanischen Quadratur, Trav. Inst. Math. Tbilissi [Trudy Tbiliss. Mat. Inst.] 7 (1940), 47–63 (Russian, with German summary). MR 0005448
- J. C. P. Miller, The Airy Integral, Giving Tables of Solutions of the Differential Equation $y”=xy$, Cambridge, at the University Press; New York, The Macmillan Company, 1946. MR 0018971
- W. E. Milne, Numerical Integration of Ordinary Differential Equations, Amer. Math. Monthly 33 (1926), no. 9, 455–460. MR 1521023, DOI https://doi.org/10.2307/2299609
v. Mises, R., 1930, Zs. ang. Math. Mech. 10, 81.
Nielsen, H. P., 1908, Arkiv. f. mat. astr. fysik 4, no. 21.
Nystböm, E. J., 1926, Acta Soc. Fennicae 50, no. 13.
Richardson, L. F., 1911, Phil. Trans. 210 A, 307.
Richardson, L. F., and Gaunt, J. A., 1927, Phil. Trans. 226 A, 299.
- Herbert E. Salzer, Coefficients for numerical differentiation with central differences, J. Math. Phys. Mass. Inst. Tech. 22 (1943), 115–135. MR 8720, DOI https://doi.org/10.1002/sapm1943221115
- Paul A. Samuelson, A convergent iterative process, J. Math. Phys. Mass. Inst. Tech. 24 (1945), 131–134. MR 14827, DOI https://doi.org/10.1002/sapm1945241131
Schröder, E., 1870, Math. Ann. 2, 317.
Schulz, G., 1932, Zs. ang. Math. Mech. 12, 44. 1937, Formelsammlung, Samml. Göschen no. 1110.
Sheppard, W. F., 1899, Proc. London Math. Soc. 31, 449.
Sibagaki, W., 1936, Proc. Phys. Math. Soc. Japan 18, 659.
Southwell, R. V., 1940, Relaxation methods in engineering science, Oxford. 1946, Relaxation methods in theoretical physics, Oxford.
Steffensen, J. F., 1922, Skan. Akt. Tidskr. 5, 20. 1924, ibid. 7, 1.
- Karl Stohler, Eine Vereinfachung bei der numerischen Integration gewöhnlicher Differentialgleichungen, Z. Angew. Math. Mech. 23 (1943), 120–122 (German). MR 9516
Theremin, T., 1855, Crelles Journ. 49, 187.
Tollmien, W., 1938, Zs. ang. Math. Mech. 18, 83.
Aitken, A. C., 1925, Proc. Roy. Soc. Edinburgh 46, 289; 1936, ibid. 57, 269.
Adams, J. C. and Bashforth, F., 1883, Theories of capillary action, Cambridge.
Bennet, A. A., Milne, W. E., and Bateman, H., 1933, Numerical integration, Nat. Res. Council, Washington.
Bickley, W. G., 1932, Phil. Mag. 13, 1006. 1948, J. Math. Phys. 27, 183.
Bickley, W. G., and Miller, J. C. P., 1942, Phil. Mag. 33, 1.
Carter, A. E., and Sadler, D. H., 1948, Quart. J. Mech. Appl. Math. 1, 433.
Chadaja, F. G., 1941, Bull. Acad. Sci. Georgian SSR 2, 601. 1942, Trav. Inst. Math. Tbilissi 11, 97.
Charlier, C. L., 1907, Die Mechanik des Himmels, Bd. II, Leipzig.
Collatz, L. and Zurmühl, R., 1942a, Zs. ang. Math. Mech. 22, 42. 1942b, Ing. Arch. 13, 34.
Collatz, L., 1942, Zs. ang. Math. Mech. 22, 216. 1948, FIAT Rev. German Sci. 1939-46, Appl. Math. I, 1. 1949, Zs. ang. Math. Mech. 29, 199.
Duncan, W. J., 1948, Phil. Mag. 39, 493.
Encke, 1837, Berliner Astr. Jahrbuch; 1867, ibid.
Falkner, V. M., 1936, Phil. Mag. 21, 624.
Feinstein, L. and Schwarzschild, M., 1941, Rev. Sci. Instr. 12, 405.
Fox, L., and Goodwin, E. T., 1949, Proc. Camb. 45, 373.
Gauss, C. F., 1834, Brief an Encke vom 13/10 1834, Gesammelte Werke 7, 433.
Hartree, D. R., 1928, Proc. Camb. 24, 89. 1949, Proc. Camb. 45, 230.
Hausman, L. F., and Schwarzschild, M., 1947, Rev. Sci. Instr. 18, 877.
Kormes, M., 1943, Rev. Sci. Instr. 14, 118.
Kryloff, A. N., 1925, Proc. I. Intern. Congr. Appl. Math., Delft, 212.
Levy, H. and Baggot, E. A., 1934, Numerical studies in differential equations, London.
Lindelöf, E., 1938, Acta Soc. Sci. Fennicae, Phys. Mat. (A) 2, no. 13.
Michel, J. G. L., 1946, J. Inst. Actuar. 72, 470.
Mikeladze, S. E., 1939, Bull. Acad. Sci. URSS Sci. Math., 627. 1940, Trav. Inst. Math. Tbilissi 7, 47. 1942, Bull. Acad. Sci. Georgian SSR 3, 633. 1943, ibid. 4, 215. 1948, Doklady Akad. Nauk SSR 61, 613, 789. 1948, Uspchi Matem. Nauk 3, no. 6 (28), 3. 1949, Doklady Akad. Nauk SSR 65, 125.
Miller, J. C. P., 1946, B. A. Math. Tables, The Airy integral, Cambridge.
Milne, W. E., 1926, Amer. Math. Monthly 33, 455. 1949, Numerical calculus, Princeton.
v. Mises, R., 1930, Zs. ang. Math. Mech. 10, 81.
Nielsen, H. P., 1908, Arkiv. f. mat. astr. fysik 4, no. 21.
Nystböm, E. J., 1926, Acta Soc. Fennicae 50, no. 13.
Richardson, L. F., 1911, Phil. Trans. 210 A, 307.
Richardson, L. F., and Gaunt, J. A., 1927, Phil. Trans. 226 A, 299.
Salzer, H. E., 1943, J. Math. Phys. 22, 115. 1944, Phil. Mag. 35, 262. 1945, Phil. Mag. 36, 216.
Samuelson, P. A., 1945, J. Math. Phys. 24, 131.
Schröder, E., 1870, Math. Ann. 2, 317.
Schulz, G., 1932, Zs. ang. Math. Mech. 12, 44. 1937, Formelsammlung, Samml. Göschen no. 1110.
Sheppard, W. F., 1899, Proc. London Math. Soc. 31, 449.
Sibagaki, W., 1936, Proc. Phys. Math. Soc. Japan 18, 659.
Southwell, R. V., 1940, Relaxation methods in engineering science, Oxford. 1946, Relaxation methods in theoretical physics, Oxford.
Steffensen, J. F., 1922, Skan. Akt. Tidskr. 5, 20. 1924, ibid. 7, 1.
Stohler, K., 1943, Zs. ang. Math. Mech. 23, 120.
Theremin, T., 1855, Crelles Journ. 49, 187.
Tollmien, W., 1938, Zs. ang. Math. Mech. 18, 83.
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© Copyright 1952
American Mathematical Society
