One approach to the solution of non-homogeneous boundary value problems is by means of the construction of functions known as Green’s functions. Historically, the concept originated with work on potential theory published by Green in 1828. Green’s work has provided the germs of a much wider formulation for solving a variety of eigenvalue, boundary value, and inhomogeneous problems, particularly since the advent of generalized functions. We shall not attempt a systematic treatment in this book; rather we will discuss problems and methods where integral transform techniques are useful.1 In particular, we will discuss in this section problems where the Fourier transform in one variable is applicable.
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- 1.Excellent accounts are given in STAKGOLD (1968) and MORSE, FESHBACH (1953), Ch. 7.Google Scholar
- 2.See GELFAND 6 SHILOV (1964), pp. 39ff.Google Scholar
- 3.The difference between any two solutions of V2g = d(r-r’) satisfies V2¢ = 0; therefore we may write g(r,r’) = e(r-r’) + (f)(r,r’).Google Scholar
- 4.The normal derivative of (47) is a Fourier transform which is given in Problem 7.26.Google Scholar
- 5.This problem is adapted from a paper by W. E. Williams, Q. J. Mech. Appl. Math., (1973), 26, 397, where some more general results may be found.Google Scholar
- 6.Problems 9–11 are based on results given by G. S. Argawal, A. J. Devaney and D. N. Pattenayak, J. Math. Phys. (1973), 14, 906.Google Scholar