Finite difference reaction-diffusion equations with nonlinear diffusion coefficients

Wang, Jingyuan; Pao, C. V.

doi:10.1007/s002110000140

Cited by 15 publications

(8 citation statements)

References 14 publications

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“…Such a problem also arises in the theories of diffusion, chemical kinetics, etc. (see, e.g., [1,[6][7][8]24,25]). In a recent article [28] the author showed that by a proper transformation, Numerov's method is also valid for the above strongly nonlinear problem.…”

Section: Introductionmentioning

confidence: 99%

The extrapolation of Numerov's scheme for nonlinear two-point boundary value problems

Wang

2007

Applied Numerical Mathematics

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Section: Introductionmentioning

confidence: 99%

The extrapolation of Numerov's scheme for nonlinear two-point boundary value problems

Wang

2007

Applied Numerical Mathematics

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“…The main concerns in the above works are for the existence of a global solution and the blow-up property of the solution. In particular, the works in [20,21] deal with the Cauchy problem in the one-spatial dimensional case with D(u) = u σ and f (u) = u β , for some positive constants σ and β; those in [5,24,40,41,46] treat the Cauchy problem with more general equations; those in [6,13,42,44] consider initial-boundary value problems in a bounded domain with either the Neumann type or the mixed type boundary conditions; and that in [31] deals with the Dirichlet boundary condition. The global existence and blow-up problem has been extended in [11,12,16,17,22,23,28,45] Recently the authors treated a general coupled system of N equations in the form of (1.1) but with non-linear boundary conditions (cf.…”

Section: Introductionmentioning

confidence: 99%

“…[5,13,[19][20][21]24,31,[40][41][42]44,46]). The main concerns in the above works are for the existence of a global solution and the blow-up property of the solution.…”

Section: Introductionmentioning

confidence: 99%

Positive solutions of quasilinear parabolic systems with Dirichlet boundary condition

Pao¹,

Ruan²

2010

Journal of Differential Equations

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MSC:primary 35K50, 35J55 secondary 35K57 Keywords:Quasilinear parabolic and elliptic equations Degenerate reaction-diffusion system Maximal and minimal solutions Asymptotic behavior of solution Method of upper and lower solutions Coupled systems for a class of quasilinear parabolic equations and the corresponding elliptic systems, including systems of parabolic and ordinary differential equations are investigated. The aim of this paper is to show the existence, uniqueness, and asymptotic behavior of time-dependent solutions. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients D i (u i ) may have the property D i (0) = 0 for some or all i = 1, . . . , N, and the boundary condition is u i = 0. Using the method of upper and lower solutions, we show that a unique global classical time-dependent solution exists and converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a scalar polynomial growth problem, a coupled system of polynomial growth problem, and a two component competition model in ecology.

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“…An attractive group of linearization schemes represents the method of upper and lower solutions, e.g., Pao [12, p. 155], Wang and Pao [13], Barrett and Knabner [14]. The linearization of a nonlinear problem relies on the ordering properties of solutions.…”

Section: Introductionmentioning

confidence: 99%

Approximation of a nonlinear degenerate parabolic equation via a linear relaxation scheme

Slodička

2004

Numerical Methods Partial

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We study a nonlinear degenerate parabolic equation of the typeaccompanied by an initial datum and mixed boundary conditions. The symbol [ ⅐ ] ϩ denotes the usual cutoff function. The problem represents a model of a reactive solute transport in porous media. The exponent p fulfills p ʦ (0, 1). This limits the regularity of a solution and leads to inconveniences in the error analysis. We design a new robust linear numerical scheme for the time discretization. This is based on a suitable combination of the backward Euler method and a linear relaxation scheme. We prove the convergence of relaxation iterations on each time point t i . We derive the error estimates in suitable function spaces for all values of p ʦ (0, 1).

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Finite difference reaction-diffusion equations with nonlinear diffusion coefficients

Cited by 15 publications

References 14 publications

The extrapolation of Numerov's scheme for nonlinear two-point boundary value problems

The extrapolation of Numerov's scheme for nonlinear two-point boundary value problems

Positive solutions of quasilinear parabolic systems with Dirichlet boundary condition

Approximation of a nonlinear degenerate parabolic equation via a linear relaxation scheme

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