Applications to the Study of Differential Equations
Canonical transforms, besides generalizing the Fourier and Bargmann integral transforms, provide a fine tool for the analysis of a class of differential equations. The class consists of up-to-second-order differential operators of parabolic type. These include the diffusion, the Schrödinger free-particle, the linear potential (free-fall), and the attractive and repulsive oscillator equations. It also includes a few others such as the Fokker-Planck equation. Although this class is far from universal, the ease with which solutions and properties are found makes canonical transforms an attractive tool for problems such as these. In Section 10.1 we start with the introduction of inhomogeneous linear canonical transformations and apply the machinery to a deeper study of the diffusion equation: how to find families of solutions out of a known solution (the action of the similarity group of the equation) and the question of separating coordinates, which brings us to generalized normal modes. In Section 10.2 the analysis is applied to a general member of the differential equation class. We show that All computations reduce to, essentially, 2 × 2 matrix algebra. This is in the true spirit of group theory.
KeywordsDiffusion Equation Heat Equation Similarity Group Multiplier Function Projective Transformation
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