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  • CR: G1.8  (2)
  • diffusion equations with delays  (1)
  • 1
    Electronic Resource
    Electronic Resource
    Numerical solutions for some coupled systems of nonlinear boundary value problems (1987)
    Springer
    Numerische Mathematik 51 (1987), S. 381-394 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N20 ; CR: G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In the well-known Volterra-Lotka model concerning two competing species with diffusion, the densities of the species are governed by a coupled system of reaction diffusion equations. The aim of this paper is to present an iterative scheme for the steady state solutions of a finite difference system which corresponds to the coupled nonlinear boundary value problems. This iterative scheme is based on the method of upper-lower solutions which leads to two monotone sequences from some uncoupled linear systems. It is shown that each of the two sequences converges to a nontrivial solution of the discrete equations. The model under consideration may have one, two or three nonzero solutions and each of these solutions can be computed by a suitable choice of initial iteration. Numerical results are given for these solutions under both the Dirichlet boundary condition and the mixed type boundary condition.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01397542
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Monotone iterative methods for finite difference system of reaction-diffusion equations (1985)
    Springer
    Numerische Mathematik 46 (1985), S. 571-586 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N05 ; CR: G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper presents an existence-comparison theorem and an iterative method for a nonlinear finite difference system which corresponds to a class of semilinear parabolic and elliptic boundary-value problems. The basic idea of the iterative method for the computation of numerical solutions is the monotone approach which involves the notion of upper and lower solutions and the construction of monotone sequences from a suitable linear discrete system. Using upper and lower solutions as two distinct initial iterations, two monotone sequences from a suitable linear system are constructed. It is shown that these two sequences converge monotonically from above and below, respectively, to a unique solution of the nonlinear discrete equations. This formulation leads to a well-posed problem for the nonlinear discrete system. Applications are given to several models arising from physical, chemical and biological systems. Numerical results are given to some of these models including a discussion on the rate of convergence of the monotone sequences.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389659
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Models of genetic control by repression with time delays and spatial effects (1984)
    Springer
    Journal of mathematical biology 20 (1984), S. 39-57 
    Details
    ISSN: 1432-1416
    Keywords: Genetic control models ; reaction ; diffusion equations with delays
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Two models for cellular control by repression are developed in this paper. The models use standard theory from compartmental analysis and biochemical kinetics. The models include time delays to account for the processes of transcription and translation and diffusion to account for spatial effects in the cell. This consideration leads to a coupled system of reactiondiffusion equations with time delays. An analysis of the steady-state problem is given. Some results on the existence and uniqueness of a global solution and stability of the steady-state problem are summarized, and numerical simulations showing stability and periodicity are presented. A Hopf bifurcation result and a theorem on asymptotic stability are given for the limiting case of the models without diffusion.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275860
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