571 research outputs found

    Group Analysis and New Explicit Solutions of Simplified Modified Kawahara Equation with Variable Coefficients

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    The simplified modified Kawahara equation with variable coefficients is studied by using Lie symmetry method. Then we obtain the corresponding Lie algebra, optimal system, and the similarity reductions. At last, we also give some new explicit solutions for some special forms of the equations

    Studies of Phase Turbulence in the One Dimensional Complex Ginzburg-Landau Equation

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    The phase-turbulent (PT) regime for the one dimensional complex Ginzburg-Landau equation (CGLE) is carefully studied, in the limit of large systems and long integration times, using an efficient new integration scheme. Particular attention is paid to solutions with a non-zero phase gradient. For fixed control parameters, solutions with conserved average phase gradient ν\nu exist only for ∣ν∣|\nu| less than some upper limit. The transition from phase to defect-turbulence happens when this limit becomes zero. A Lyapunov analysis shows that the system becomes less and less chaotic for increasing values of the phase gradient. For high values of the phase gradient a family of non-chaotic solutions of the CGLE is found. These solutions consist of spatially periodic or aperiodic waves travelling with constant velocity. They typically have incommensurate velocities for phase and amplitude propagation, showing thereby a novel type of quasiperiodic behavior. The main features of these travelling wave solutions can be explained through a modified Kuramoto-Sivashinsky equation that rules the phase dynamics of the CGLE in the PT phase. The latter explains also the behavior of the maximal Lyapunov exponents of chaotic solutions.Comment: 16 pages, LaTeX (Version 2.09), 10 Postscript-figures included, submitted to Phys. Rev.

    STARRY: Analytic Occultation Light Curves

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    We derive analytic, closed form, numerically stable solutions for the total flux received from a spherical planet, moon or star during an occultation if the specific intensity map of the body is expressed as a sum of spherical harmonics. Our expressions are valid to arbitrary degree and may be computed recursively for speed. The formalism we develop here applies to the computation of stellar transit light curves, planetary secondary eclipse light curves, and planet-planet/planet-moon occultation light curves, as well as thermal (rotational) phase curves. In this paper we also introduce STARRY, an open-source package written in C++ and wrapped in Python that computes these light curves. The algorithm in STARRY is six orders of magnitude faster than direct numerical integration and several orders of magnitude more precise. STARRY also computes analytic derivatives of the light curves with respect to all input parameters for use in gradient-based optimization and inference, such as Hamiltonian Monte Carlo (HMC), allowing users to quickly and efficiently fit observed light curves to infer properties of a celestial body's surface map.Comment: 55 pages, 20 figures. Accepted to the Astronomical Journal. Check out the code at https://github.com/rodluger/starr

    High-speed shear driven dynamos. Part 2. Numerical analysis

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    This paper aims to numerically verify the large Reynolds number asymptotic theory of magneto-hydrodynamic (MHD) flows proposed in the companion paper Deguchi (2019). To avoid any complexity associated with the chaotic nature of turbulence and flow geometry, nonlinear steady solutions of the viscous-resistive magneto-hydrodynamic equations in plane Couette flow have been utilised. Two classes of nonlinear MHD states, which convert kinematic energy to magnetic energy effectively, have been determined. The first class of nonlinear states can be obtained when a small spanwise uniform magnetic field is applied to the known hydrodynamic solution branch of the plane Couette flow. The nonlinear states are characterised by the hydrodynamic/magnetic roll-streak and the resonant layer at which strong vorticity and current sheets are observed. These flow features, and the induced strong streamwise magnetic field, are fully consistent with the vortex/Alfv\'en wave interaction theory proposed in Deguchi (2019). When the spanwise uniform magnetic field is switched off, the solutions become purely hydrodynamic. However, the second class of `self-sustained shear driven dynamos' at the zero-external magnetic field limit can be found by homotopy via the forced states subject to a spanwise uniform current field. The discovery of the dynamo states has motivated the corresponding large Reynolds number matched asymptotic analysis in Deguchi (2019). Here, the reduced equations derived by the asymptotic theory have been solved numerically. The asymptotic solution provides remarkably good predictions for the finite Reynolds number dynamo solutions

    The finite element analysis of convection heat transfer

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    This thesis reviews the development and current methods of numerical convection heat transfer from available literature, encompassing an analysis of the various finite element formulations available for investigating convection. It further describes the finite element formulation for the primitive variable convection heat transfer equations via a Galerkin weighted residual scheme and using mixed interpolation, and it demonstrates the capability of this method by means of five practical examples, namely natural convection in a thermally driven square cavity, a thermally driven vertical slot, a thermally driven triangular cavity, and a liquid convective diode, and forced convection in a cooling pond. This study also provides the background and framework for the problem of transient convection heat transfer, and for further steady-state studies using parameters outside those considered herein

    Numerical Solution of Kawahara Equation Using Neural Network

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    An artificial neural network technique is proposed in this research to solve the well-known partial differential equations of the types: Kawahara and modified Kawahara equations. The mathematical model of the equation was developed with the help of artificial neural networks. The construction requires imposing certain constrains on the values of the input, bias and output weights, and on the attribution of certain roles of each aforementioned parameters. The results obtained from the proposed technique were very accurate, simple and convenient. Moreover, the comparison between the approximated solutions and the exact one has done. This comparison found them in a good agreement with each other due to of superior properties of the Neural Network
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