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  • 1
    Electronic Resource
    Electronic Resource
    Long-time behavior of Navier-Stokes flow on a two-dimensional torus excited by an external sinusoidal force (1997)
    Springer
    Journal of statistical physics 86 (1997), S. 301-335 
    Details
    ISSN: 1572-9613
    Keywords: Navier-Stokes equations ; bifurcations ; dynamical systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract In this paper we study the Navier-Stokes flow on the two-dimensional torusS 1 ×S 1 excited by the external force (k 2 sinky, 0) and find the long-time behavior for the flow starting from some states, whereS 1=[0,2π](mod 2π). Especially for the casek=2, it follows from an analysis and computation that the Navier-Stokes flow with the initial state cos(mx+ny) or sin(mx+ny) will likely evolve through at most one step bifurcation to either a steady-state solution or a time-dependent periodic solution for any Reynolds number and integersm andn.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02180208
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Finite-difference methods for solving the reaction-diffusion equations of a simple isothermal chemical system (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 10 (1994), S. 435-454 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Numerical methods are proposed for the numerical solution of a system of reaction-diffusion equations, which model chemical wave propagation. The reaction terms in this system of partial differential equations contain nonlinear expressions. Nevertheless, it is seen that the numerical solution is obtained by solving a linear algebraic system at each time step, as opposed to solving a nonlinear algebraic system, which is often required when integrating nonlinear partial differential equations. The development of each numerical method is made in the light of experience gained in solving the system of ordinary differential equations, which model the well-stirred analogue of the chemical system. The first-order numerical methods proposed for the solution of this initialvalue problem are characterized to be implicit. However, in each case it is seen that the numerical solution is obtained explicitly. In a series of numerical experiments, in which the ordinary differential equations are solved first of all, it is seen that the proposed methods have superior stability properties to those of the well-known, first-order, Euler method to which they are compared. Incorporating the proposed methods into the numerical solution of the partial differential equations is seen to lead to two economical and reliable methods, one sequential and one parallel, for solving the travelling-wave problem. © 1994 John Wiley & Sons, Inc.
    Additional Material: 4 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690100404
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    A second-order, chaos-free, explicit method for the numerical solution of a cubic reaction problem in neurophysiology (1993)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 9 (1993), S. 213-224 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A second order explicit method is developed for the numerical solution of the initialvalue problem w′(t) ≡ dw(t)/dt = φ(w), t 〉 0, w(0) = W0, in which the function φ(w) = αw(1 - w) (w - a), with α and a real parameters, is the reaction term in a mathematical model of the conduction of electrical impulses along a nerve axon. The method is based on four first-order methods that appeared in an earlier paper by Twizell, Wang, and Price [Proc. R. Soc. (London) A 430, 541-576 (1990)]. In addition to being chaos free and of higher order, the method is seen to converge to one of the correct steady-state solutions at w = 0 or w = 1 for any positive value of α. Convergence is monotonic or oscillatory depending on W0, α, a, and l, the parameter in the discretization of the independent variable t. The approach adopted is extended to obtain a numerical method that is second order in both space and time for solving the initial-value boundary-value problem ∂u/∂t = κ∂2u/∂x2 + αu(1 - u)(u - a) in which u = u(x,t). The numerical method so developed obtained the solution by solving a single linear algebraic system at each time step. © 1993 John Wiley & Sons, Inc.
    Additional Material: 4 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690090302
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    Paper (German National Licenses)
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