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Reduction Operators of Linear Second-Order Parabolic Equations
The reduction operators, i.e., the operators of nonclassical (conditional)
symmetry, of (1+1)-dimensional second order linear parabolic partial
differential equations and all the possible reductions of these equations to
ordinary differential ones are exhaustively described. This problem proves to
be equivalent, in some sense, to solving the initial equations. The ``no-go''
result is extended to the investigation of point transformations (admissible
transformations, equivalence transformations, Lie symmetries) and Lie
reductions of the determining equations for the nonclassical symmetries.
Transformations linearizing the determining equations are obtained in the
general case and under different additional constraints. A nontrivial example
illustrating applications of reduction operators to finding exact solutions of
equations from the class under consideration is presented. An observed
connection between reduction operators and Darboux transformations is
discussed.Comment: 31 pages, minor misprints are correcte