Separating variables in two-way diffusion equations

Nathaniel J. Fisch, Martin D. Kruskal

Research output: Contribution to journalArticlepeer-review

51 Scopus citations


It is shown that solutions to a class of diffusion equations of the two-way type may be found by a method akin to separation of variables. The difficulty with such equations is that the boundary data must be specified partly as initial and partly as final conditions. In contrast to the one-way diffusion equation, where the solution separates only into decaying eigenfunctions, the two-way equations separate into both decaying and growing eigenfunctions. Criteria are set forth for the existence of linear eigenfunctions, which may not be found directly by separating variables. A speculation with interesting ramifications is that the growing and decaying eigenfunctions are separately complete in an appropriate half of the problem domain. This conjecture is not proved, although it does enjoy some numerical support.

Original languageEnglish (US)
Pages (from-to)740-750
Number of pages11
JournalJournal of Mathematical Physics
Issue number4
StatePublished - 1979

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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