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The Solution of One-Dimensional Boundary Value Problems by the Laplace Transformation

Published online by Cambridge University Press:  03 November 2016

J C. Jaeger
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Extract

1. It is well known that the solution of the ordinary linear differential equation with constant coefficients

(Dn + a1Dn−1 + + an−1D+an)y = F(x), .....(1)

where y, Dy, ..., Dn−1y take arbitrary values y0, y1, .., yn−1

when x = 0, ..... (2)

can be obtained by multiplying (1) by e−px, (p > 0) and integrating with regard to x from 0 to ∞

Type
Research Article
Information
The Mathematical Gazette , Volume 23 , Issue 253 , February 1939 , pp. 62 - 67
Copyright
Copyright © Mathematical Association 1939 

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References

page no 62 note * Cf.Doetsch, , Theorie und Anwendung der Laplace-Transformation (1937). Kap. 18. Van der Pol, Phil. Mag. (7), 7, (1929), 1153 Google Scholar.

The approach used here is that of Carslaw, Math. Gazette, 22 (1938). For the verification that solutions obtained in this way satisfy the differential equation, and conditions at x = 0, see Doetsch, loc. cit.

page no 62 note † But see § 7.

page no 62 note ‡ Cf. Doetsch, loc. cit., page 161.

page no 63 note * It is known that the equation can have only on continuous solution F(x). Cf. Doetsch, loc. cit., page 35.

page no 65 note * Concentrated loads are most easily discussed by the use of the Dirac δ function, δ(x − x′), defined as zero if x ǂ x′ and infinite x = x′ in such a way that

This has the property

Then for a concentrated load W at x′ we have to solve D 4 y = Wδ(x − x′)/(EI), which has the subsidiary equation

and introducing the boundary conditions of this section and proceeding as before we easily obtain (9).

page no 66 note * That (if f(p) satisfies certain conditions) when

The line integral is evaluated in this case by completing the contour by a semicircle in the left-hand half plane not passing through any pole of the integrand; it is easy to show that the integral over the semi-circle vanishes as its radius → ∞, and evaluating the residues at the poles of the integrand

(λ = 0 and ±(2n + 1)πi/l, n = 0, 1, ...)

gives the result stated.

See Carslaw, loc. cit., for a detailed discussion of the method.

page no 67 note * Timoshenko and Lascelles, Applied Elasticity, p. 230.