Reduction Operators of Linear Second-Order Parabolic Equations

Abstract

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.

31 pages, minor misprints are corrected

References in corpus (8)

• Admissible Transformations and Normalized Classes of Nonlinear Schroedinger Equations
• A precise definition of reduction of partial differential equations
• Symmetry reduction and exact solutions of the Navier-Stokes equations
• Conservation Laws and Potential Symmetries of Linear Parabolic Equations
• Hierarchy of Conservation Laws of Diffusion--Convection Equations
• Potential Nonclassical Symmetries and Solutions of Fast Diffusion Equation
• The structure of the category of parabolic equations
• Equivalence of Q-Conditional Symmetries under Group of Local Transformation

Cited by in corpus (13)

• Extended group analysis of variable coefficient reaction-diffusion equations with exponential nonlinearities
• Group classification of linear evolution equations
• Singular reduction operators in two dimensions
• Extended symmetry analysis of generalized Burgers equations
• Point and generalized symmetries of the heat equation revisited
• Is a nonclassical symmetry a symmetry?
• Singular reduction modules of differential equations
• Reduction operators and exact solutions of generalized Burgers equations
• Conservation laws and hierarchies of potential symmetries for certain diffusion equations
• Classification of reduction operators and exact solutions of variable coefficient Newell-Whitehead-Segel equations
• Reduction operators of Burgers equation
• Extended symmetry analysis of a "nonconservative Fokker-Plank equation"
• Surprising symmetry properties and exact solutions of Kolmogorov backward equations with power diffusivity