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ee - Chapter 9 Linear and Nonlinear Evolution Equations In...

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Chapter 9 Linearand Nonlinear Evolution Equations In this chapter, we analyze several of the most important evolution equations, both linear and nonlinear, involving a single spatial variable. Our first stop is to revisit the heat equation. We introduce the fundamental solution, which, for dynamical partial dif- ferential equations, assumes the role of the Green’s function, in that its initial condition is a concentrated delta impulse. The fundamental solution leads to an integral superposi- tion formula for the solutions produced by more general initial conditions or by external forcing. For the heat equation on the entire real line, the Fourier transform enables us to construct an explicit formula that identifies its fundamental solution as a Gaussian filter. We next present the Maximum Principle that rigorously justifies the entropic decay of temperature in a heated body, and underlies much of the advanced mathematical analysis of parabolic problems. Finally, we discuss the Black–Scholes equation, the paradigmatic model for investment portfolios, which was first proposed in the early 1970’s and now lies at the heart of the modern financial industry. We learn that the Black–Scholes equation can be transformed into the linear heat equation, whose fundamental solution is applied to establish the celebrated Black–Scholes formula for option pricing. The following section provides a brief introduction to symmetry-based solution tech- niques for linear and nonlinear partial differential equations. Knowing a symmetry of a partial differential equation allows one to readily construct additional solutions from any known solution. Solutions that remain invariant under a one-parameter family of symme- tries can be found by solving a reduced ordinary differential equation. The most important classes are the traveling wave solutions, which are invariant under translation symmetries, and similarity solutions, which are invariant under scaling symmetries. The next evolution equation to be analyzed is a paradigmatic model of nonlinear diffusion known as Burgers’ equation. It can be regarded as a very simplified model of fluid dynamics, combining both nonlinear and viscous effects. We discover a remarkable nonlinear change of variables that maps Burgers’ equation to the linear heat equation, and thereby facilitates its analysis, allowing us to construct explicit solutions, and analyze how they converge to the Rankine–Hugoniot shock wave solutions of the nonlinear transport equation in the inviscid limit. Next, we turn our attention to the simplest third order linear evolution equation, which arises as a model for wave mechanics. Unlike first and second order wave equations, its solutions are not simple traveling waves, but exhibit dispersion, in which oscillatory waves of different frequencies move at different speeds. As a result, initially localized disturbances will spread out or disperse, even while they conserve the underlying energy.
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