Reaction-Diffusion Equations

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The Experts below are selected from a list of 327 Experts worldwide ranked by ideXlab platform

A. A. M. Arafa - One of the best experts on this subject based on the ideXlab platform.

Saad Zagloul Rida - One of the best experts on this subject based on the ideXlab platform.

Ahmed M. A. El-sayed - One of the best experts on this subject based on the ideXlab platform.

Matthieu Alfaro - One of the best experts on this subject based on the ideXlab platform.

  • Quantitative estimates of the threshold phenomena for propagation in Reaction-Diffusion Equations
    SIAM Journal on Applied Dynamical Systems, 2020
    Co-Authors: Matthieu Alfaro, Arnaud Ducrot, Gregory Faye
    Abstract:

    We focus on the (sharp) threshold phenomena arising in some Reaction-Diffusion Equations supplemented with some compactly supported initial data. In the so-called ignition and bistable cases, we prove the first sharp quantitative estimate on the (sharp) threshold values. Furthermore, numerical explorations allow to conjecture some refined estimates. Last we provide related results in the case of a degenerate monostable nonlinearity "not enjoying the hair trigger effect". AMS Subject Classifications: 35K57 (Reaction-Diffusion Equations), 35K15 (Initial value problems for second-order parabolic Equations), 35B40 (Asymptotic behavior of solutions).

  • Slowing Allee effect versus accelerating heavy tails in monostable reaction diffusion Equations
    Nonlinearity, 2017
    Co-Authors: Matthieu Alfaro
    Abstract:

    We focus on the spreading properties of solutions of monostable Reaction-Diffusion Equations. Initial data are assumed to have heavy tails, which tends to accelerate the invasion phenomenon. On the other hand, the nonlinearity involves a weak Allee effect, which tends to slow down the process. We study the balance between the two effects. For algebraic tails, we prove the exact separation between " no acceleration and acceleration ". This implies in particular that, for tails exponentially unbounded but lighter than algebraic , acceleration never occurs in presence of an Allee effect. This is in sharp contrast with the KPP situation [19]. When algebraic tails lead to acceleration despite the Allee effect, we also give an accurate estimate of the position of the level sets.

A S Vatsala - One of the best experts on this subject based on the ideXlab platform.

  • improved generalized quasilinearization method and rapid convergence for reaction diffusion Equations
    Applied Mathematics and Computation, 2008
    Co-Authors: Tanya G Melton, A S Vatsala
    Abstract:

    Abstract In the method of quasilinearization or the method of generalized quasilinearization we assume that ∂ k f ( t , x , u ) ∂ u k exists, where f ( t , x , u ) is the forcing function and obtain kth order of convergence. In this paper, we assume a weaker condition namely, ∂ k - 1 f ( t , x , u ) ∂ u k - 1 exists and Lipschitzian in u and develop generalized quasilinearization method to reaction diffusion Equations. The iterates will be different depending on ∂ k - 1 f ( t , x , u ) ∂ u k - 1 is nondecreasing or nonincreasing in u and k being even or odd. Finally, we prove that the sequences generated by the generalized quasilinearization method converge to the unique solution of the nonlinear reaction diffusion equation and the convergence is of order k.

  • The generalized quasi-linearization method for reaction diffusion Equations on an unbounded domain
    Journal of Mathematical Analysis and Applications, 1999
    Co-Authors: A S Vatsala, Liwen Wang
    Abstract:

    The method of generalized quasi-linearization has been well developed for ordinary differential Equations. In this paper, we extend the method of generalized quasi-linearization to reaction diffusion Equations on an unbounded domain. The iterates, which are solutions of linear Equations starting from lower and upper solutions, converge uniformly and monotonically to the unique solution of the nonlinear reaction diffusion equation in an unbounded domain. Initially an existence theorem for the linear nonhomogeneous reaction diffusion equation in an unbounded domain has been proved under improved conditions. The quadratic convergence has been proved by using a comparison theorem of reaction diffusion Equations with ordinary differential Equations. This avoids the computational complexity of the quasi-linearization method, since the computation of Green's function at each stage of the iterates is avoided.

  • The quasilinearization method in the system of reaction diffusion Equations
    Applied Mathematics and Computation, 1998
    Co-Authors: Jiahui Jiang, A S Vatsala
    Abstract:

    In this paper the method of generalized quasilinearization has been extended to system of reaction diffusion Equations with initial and boundary conditions. Quadratic convergence of the sequences of lower and upper solutions to the unique solution of the nonlinear system of reaction diffusion Equations has been proved. This has been achieved by using a comparison theorem of the system of reaction diffusion Equations with the system of ordinary differential Equations. Finally, numerical scheme for the generalized quasilinearization method has been developed for reaction diffusion systems.

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