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A note to the Fourier method of solving partial second-order differential equations. (English) Zbl 0703.35039

Consider the linear differential operator \[ Lu(x,y)=a_ 1(x)u_{xx}(x,y)+a_ 2(x)u_ x(x,y)+a_ 3(x)u(x,y)- \]
\[ -b_ 1(y)u_{yy}(x,y)-b_ 2(y)u_ y(x,y)-b_ 3(y)u(x,y) \] for (x,y)\(\in I\times J\) (two bounded or unbounded intervals). The author solves the equation \(Lu(x,y)=0\) by means of the classical Kummer transform.
Reviewer: J.Appell

MSC:

35G05 Linear higher-order PDEs
35A22 Transform methods (e.g., integral transforms) applied to PDEs
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References:

[1] Bochner S., Chandrasekaran K. C.: Fourier Transforms. Princeton, 1949.
[2] Kufner A., Kadlec J.: Fourier series. Praha, Academia, 1969. · Zbl 0215.17901
[3] Laitoch M.: To a problem of ortogonal systems with weight. Acta. Univ. Olom. 3 (1960), 11-28)
[4] Widder D. W.: The Laplace Transform. Princeton, 1946. · Zbl 0060.24801
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