The Magnus series is an infinite series which arises in the study of linear ordinary differential equations. If the series converges, then the matrix exponential of the sum equals the fundamental solution of the differential equation. The question considered in this paper is: When does the series converge? The main result establishes a sufficient condition for convergence, which improves on several earlier results.
A new technique for deriving the determining equations of nonclassical symmetries associated with a partial differential equation system is introduced. The problem is reduced to computing the determining equations of the classical symmetries associated with a related equation with coefficients which depend on the nonclassical symmetry operator. As a consequence, all the symbolic manipulation programs designed for the latter task can also be used to find the determining equations of the nonclassical symmetries, without any adaptation of the program. The algorithm was implemented as the MAPLE routine GENDEFNC and uses the MAPLE package DESOLV (authors Carminati and Vu). As an example, we consider the Huxley partial differential equation.
We develop an algorithm for computing the solution of a large system of linear ordinary differential equations (ODEs) with polynomial inhomogeneity. This is equivalent to computing the action of a certain matrix function on the vector representing the initial condition. The matrix function is a linear combination of the matrix exponential and other functions related to the exponential (the so-called ϕ-functions). Such computations are the major computational burden in the implementation of exponential integrators, which can solve general ODEs. Our approach is to compute the action of the matrix function by constructing a Krylov subspace using Arnoldi or Lanczos iteration and projecting the function on this subspace. This is combined with time-stepping to prevent the Krylov subspace from growing too large. The algorithm is fully adaptive: it varies both the size of the time steps and the dimension of the Krylov subspace to reach the required accuracy. We implement this algorithm in the MATLAB function phipm and we give instructions on how to obtain and use this function. Various numerical experiments show that the phipm function is often significantly more efficient than the state-of-the-art.
Spatially localized, time-periodic structures are common in pattern-forming systems, appearing in fluid mechanics, chemical reactions, and granular media. We examine the existence of oscillatory localized states in a PDE model with single frequency time dependent forcing, introduced in [A. M. Rucklidge and M. Silber, SIAM J. Appl. Math., 8 (2009), pp. 298-347] as a phenomenological model of the Faraday wave experiment. In this study, we reduce the PDE model to the forced complex Ginzburg-Landau equation in the limit of weak forcing and weak damping. This allows us to use the known localized solutions found in [J. Burke, A. Yochelis, and E. Knobloch, SIAM J. Appl. Dyn. Syst., 7 (2008), pp. 651-711]. We reduce the forced complex Ginzburg-Landau equation to the Allen-Cahn equation near onset, obtaining an asymptotically exact expression for localized solutions. We also extend this analysis to the strong forcing case, recovering the Allen-Cahn equation directly without the intermediate step. We find excellent agreement between numerical localized solutions of the PDE, localized solutions of the forced complex Ginzburg-Landau equation, and analytic (hyperbolic) solutions of the Allen-Cahn equation. This is the first time that a PDE with time dependent forcing has been reduced to the Allen-Cahn equation and its localized oscillatory solutions quantitatively studied. Introduction.Localized patterns arise in a wide range of interesting pattern-forming problems. Much progress has been made on steady problems, where bistability between a steady pattern and the zero state leads to localized patterns bounded by stationary fronts between these two states [6,12]. In contrast, oscillons, which are oscillating localized structures in a stationary background, are relatively less well understood [18,25]. Fluid [3], chemical reaction [21], and granular media [25] problems have been studied experimentally. When the surface of the excited system becomes unstable (the Faraday instability), standing waves are found on the surface of the medium. Oscillons have been found where this primary bifurcation is subcritical [11], and these take the form of alternating conical peaks and craters against a stationary background.Previous studies have averaged over the fast timescale of the oscillation and have focused on PDE models where the localized solution is effectively steady [2,7,11]. Here we will seek localized oscillatory states in a PDE with time dependent parametric forcing. We find
In this paper, we reformulate the Vlasov-Maxwell equations based on the Morrison-Marsden-Weinstein Poisson bracket. In order to get the numerical solutions preserving the Poisson bracket, we split the Hamiltonian of the Vlasov-Maxwell equations into five parts. We construct the numerical methods for the time direction via composing the exact solutions of subsystems. By combining an appropriate spatial discretization, we can prove that the resulting numerical discretization preserves the discrete Poisson bracket. We present numerical simulations for the problems of Landau damping and two-stream stability.
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