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- In stability theory, the Briggs–Bers criterion is a criterion for determining whether the trivial solution to a linear partial differential equation with constant coefficients is stable, or . This is often useful in applied mathematics, especially in fluid dynamics, because linear PDEs often govern small perturbations to a system, and we are interested in whether such perturbations grow or decay. The Briggs–Bers criterion is named after and . Suppose that the PDE is of the form , where is a function of space and time( and ). The partial differential operator has constant coefficients, which do not depend on and . Then a suitable ansatz for is the normal mode solution Making this ansatz is equivalent to considering the problem in Fourier space – the solution may be decomposed into its Fourier components in space and time. Making this ansatz, the equation becomes or, more simply, This is a dispersion relation between and , and tells us how each Fourier component evolves in time. In general, the dispersion relation may be very complicated, and there may be multiple which satisfy the relation for a given value of , or vice versa. The solutions to the dispersion relation may be complex-valued. Now, an initial condition can be written as a superposition of Fourier modes of the form . In practice, the initial condition will have components of all frequencies. Each of these components evolves according to the dispersion relation, and therefore the solution at a later time may be obtained by Fourier inversion. In the simple case where is first-order in time, the dispersion relation determines a unique value of for each given value of , and so where is the Fourier transform of the initial condition. In the more general case, the Fourier inversion must be performed by contour integration in the complex and planes. While it may not be possible to evaluate the integrals explicitly, asymptotic properties of as may be obtained from the integral expression, using methods such as the method of stationary phase or the method of steepest descent. In particular, we can determine whether decays or grows exponentially in time, by considering the largest value that may take. If the dispersion relation is such that always, then any solution will decay as , and the trivial solution is stable. If there is some mode with , then that mode grows exponentially in time. By considering modes with zero group velocity and determining whether they grow or decay, we can determine whether an initial condition which is localised around moves away from as it grows, with (convective instability); or whether (absolute instability). (en)
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| rdfs:comment
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- In stability theory, the Briggs–Bers criterion is a criterion for determining whether the trivial solution to a linear partial differential equation with constant coefficients is stable, or . This is often useful in applied mathematics, especially in fluid dynamics, because linear PDEs often govern small perturbations to a system, and we are interested in whether such perturbations grow or decay. The Briggs–Bers criterion is named after and . or, more simply, where (en)
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