Linear Partial Differential Equations
Various physical processes and phenomena are described by partial differential equations (PDEs).
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Linear Partial Differential Equations
Various physical processes and phenomena are described by partial differential equations (PDEs). Their principle part is often linear, which governs the behavior of small amplitude perturbations. The theory of linear PDEs is rather well developed, both in formulation and computation. Its classification falls into three groups: hyperbolic, parabolic and elliptic equations. Illustrative is dynamics of accretion flows. Internal viscosity, possibly augmented by outflows, mediates angular momentum transport outwards, allowing mass flow inwards. The former is diffusive described by a parabolic equation. Instabilities in the disk tend to produce wave motion, described by a hyperbolic equation. Wave motion is of great interest to quasi-periodic oscillations and their propagation may transport angular momentum outwards in accord with the Rayleigh criterion of rotating fluids. The steady-state of parabolic and wave equations satisfies an elliptic equation, that may evolve on a secular time scale of evolution of accretion rate or the central mass. Below, we review the mathematical structure of these three types of equations and some of the solution methods.
4.1 Hyperbolic Equations Hyperbolic equations describe wave motion expressed in terms of an amplitude u = u(t, x, y, z) as a function time and space, e.g., the three space dimensions in Cartesian coordinates (x, y, z). Wave motion may derive as a macroscopic feature of systems with a large number of degrees of freedom,1 like sound waves in air or surface waves in water, or it may be fundamental, as in the theory of electromagnetic waves.
1 Commonly
referred to as emergent.
© Springer Nature Singapore Pte Ltd. 2017 M.H.P.M. van Putten, Introduction to Methods of Approximation in Physics and Astronomy, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-981-10-2932-5_4
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4 Linear Partial Differential Equations
Fig. 4.1 (Left) Initial data on amplitude propagate along the characteristics d x/dt = ±c. The complete past domain of dependence D − of (x, t) is the interval [x − t, x + t], that includes initial data on velocities in (4.7). (Right) Domain of integration in deriving u(x, t) from w(ξ, η) in the Duhamel integral (4.27)
Small amplitude variations propagate along a Monge cone2 at each point (t, x, y, z) in time and space. For small times, the Monge cone [1] is locally generated by straight lines along the directions of wave motion, generalizing the notion of a light cone in the theory of electromagnetic waves. By definition, the principle part—ignoring terms lower order in differentiation— of a hyperbolic PDE is of the form u tt = c2 u,
(4.1)
where c denotes the velocity of wave propagation and denotes the Laplace operator. In Cartesian coordinates for flat space, we have u = u x x + u yy + yzz .
(4.2)
Here, and in what follows, we use a common notation u i = ∂u/∂x i for partial derivatives and x i = (x, y, z). In one dimension, (4.1) reduces to u tt = c2 u x x
(4.3)
for u = u(t, x). The one-dimensional
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