A precise definition of reduction of partial differential equations
1999
We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial differential equations. Using the introduced definition of reduction of differential equations we establish equivalence of the non-classical (conditional symmetry) and direct (Ansatz) approaches to reduction of partial differential equations. As an illustration we give an example of non-classical reduction of the nonlinear wave equation in 1 + 3 dimensions. The conditional symmetry approach when applied to the equation in question yields a number of non-Lie reductions which are far-reaching generalizations of the well-known symmetry reductions of the nonlinear wave equations.
Keywords:
- Method of characteristics
- Stochastic partial differential equation
- First-order partial differential equation
- Mathematical analysis
- Differential equation
- Mathematics
- Hyperbolic partial differential equation
- Numerical partial differential equations
- Separable partial differential equation
- d'Alembert's formula
- Differential algebraic equation
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