Dynamic transitions and bifurcations of 1D reaction-diffusion equations: The Self-adjoint case

Second Order Reaction ◽

Kpp Equation ◽

Sivashinsky Equation

This paper deals with the classification of transition phenomena in the most basic dissipative system possible, namely the 1D reaction diffusion equation. The emphasis is on the relation between the linear and nonlinear terms and the effect of the boundaries which influence the first transitions. We consider the cases where the linear part is self-adjoint with 2nd order and 4th order derivatives which is the case which most often arises in applications. We assume that the nonlinear term depends on the function and its first derivative which is basically the semilinear case for the second order reaction-diffusion system. As for the boundary conditions, we consider the typical Dirichlet, Neumann and periodic boundary settings. In all the cases, the equations admit a trivial steady state which loses stability at a critical parameter. We aim to classify all possible transitions and bifurcations that take place. Our analysis shows that these systems display all three types of transitions: continuous, jump and mixed and display transcritical, supercritical bifurcations with bifurcated states such as finite equilibria, circle of equilibria, and slowly rotating limit cycle. Many applications found in the literature are basically corollaries of our main results. We apply our results to classify the first transitions of the Chaffee-Infante equation, the Fisher-KPP equation, the Kuramoto Sivashinsky equation and the Swift-Hohenberg equation.

On Lr-regularity of global attractors generated by strong solutions of reaction-diffusion equations

Applied Mathematics and Nonlinear Sciences ◽

10.21042/amns.2016.2.00033 ◽

2016 ◽

Vol 1 (2) ◽

pp. 375-390 ◽

Cited By ~ 2

Author(s):

José Valero

Keyword(s):

Nonlinear Term ◽

Mild Solutions ◽

Reaction Diffusion ◽

Strong Solutions ◽

Global Attractors ◽

Diffusion Equations ◽

The Cauchy Problem ◽

Weak Attractor

AbstractIn this paper we prove that the global attractor generated by strong solutions of a reaction-diffusion equation without uniqueness of the Cauchy problem is bounded in suitable Lr-spaces. In order to obtain this result we prove first that the concepts of weak and mild solutions are equivalent under an appropriate assumption.Also, when the nonlinear term of the equation satisfies a supercritical growth condition the existence of a weak attractor is established.

Nonlinear solution of the reaction–diffusion equation using a two-step third–fourth-derivative block method

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2019-0309 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Oluwaseun Adeyeye ◽

Ali Aldalbahi ◽

Jawad Raza ◽

Zurni Omar ◽

Mostafizur Rahaman ◽

...

Keyword(s):

Diffusion Equation ◽

Reaction Diffusion ◽

Numerical Approach ◽

Block Method ◽

Wide Range ◽

Higher Derivatives ◽

Fourth Derivative ◽

Improved Accuracy

AbstractThe processes of diffusion and reaction play essential roles in numerous system dynamics. Consequently, the solutions of reaction–diffusion equations have gained much attention because of not only their occurrence in many fields of science but also the existence of important properties and information in the solutions. However, despite the wide range of numerical methods explored for approximating solutions, the adoption of block methods is yet to be investigated. Hence, this article introduces a new two-step third–fourth-derivative block method as a numerical approach to solve the reaction–diffusion equation. In order to ensure improved accuracy, the method introduces the concept of nonlinearity in the solution of the linear model through the presence of higher derivatives. The method obtained accurate solutions for the model at varying values of the dimensionless diffusion parameter and saturation parameter. Furthermore, the solutions are also in good agreement with previous solutions by existing authors.

A Laplace transform finite difference scheme for the Fisher-KPP equation

Journal of Algorithms & Computational Technology ◽

10.1177/1748302621999582 ◽

2021 ◽

Vol 15 ◽

pp. 174830262199958

Author(s):

Colin L Defreitas ◽

Steve J Kane

Keyword(s):

Finite Difference ◽

Laplace Transform ◽

Difference Scheme ◽

Finite Difference Scheme ◽

Finite Difference Methods ◽

Reaction Diffusion ◽

Numerical Approach ◽

Travelling Wave ◽

Kpp Equation

This paper proposes a numerical approach to the solution of the Fisher-KPP reaction-diffusion equation in which the space variable is developed using a purely finite difference scheme and the time development is obtained using a hybrid Laplace Transform Finite Difference Method (LTFDM). The travelling wave solutions usually associated with the Fisher-KPP equation are, in general, not deemed suitable for treatment using Fourier or Laplace transform numerical methods. However, we were able to obtain accurate results when some degree of time discretisation is inbuilt into the process. While this means that the advantage of using the Laplace transform to obtain solutions for any time t is not fully exploited, the method does allow for considerably larger time steps than is otherwise possible for finite-difference methods.

Characterization of Cocycle Attractors for Nonautonomous Reaction–Diffusion Equations

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416501352 ◽

2016 ◽

Vol 26 (08) ◽

pp. 1650135 ◽

Cited By ~ 2

Author(s):

C. A. Cardoso ◽

J. A. Langa ◽

R. Obaya

Keyword(s):

Chaotic Dynamics ◽

Linear Part ◽

Reaction Diffusion ◽

Positive Cone ◽

Diffusion Equations ◽

Minimal Set ◽

Full Characterization ◽

The Stability

In this paper, we describe in detail the global and cocycle attractors related to nonautonomous scalar differential equations with diffusion. In particular, we investigate reaction–diffusion equations with almost-periodic coefficients. The associated semiflows are strongly monotone which allow us to give a full characterization of the cocycle attractor. We prove that, when the upper Lyapunov exponent associated to the linear part of the equations is positive, the flow is persistent in the positive cone, and we study the stability and the set of continuity points of the section of each minimal set in the global attractor for the skew product semiflow. We illustrate our result with some nontrivial examples showing the richness of the dynamics on this attractor, which in some situations shows internal chaotic dynamics in the Li–Yorke sense. We also include the sublinear and concave cases in order to go further in the characterization of the attractors, including, for instance, a nonautonomous version of the Chafee–Infante equation. In this last case we can show exponentially forward attraction to the cocycle (pullback) attractors in the positive cone of solutions.

A Modified Techniques of Fractional-Order Cauchy-Reaction Diffusion Equation via Shehu Transform

Journal of Function Spaces ◽

10.1155/2021/5726822 ◽

2021 ◽

Vol 2021 ◽

pp. 1-15

Author(s):

Mounirah Areshi ◽

A. M. Zidan ◽

Rasool Shah ◽

Kamsing Nonlaopon

Keyword(s):

Fractional Order ◽

Reaction Diffusion ◽

Approximate Solutions ◽

Transformation Method ◽

Closed Form Solutions ◽

Homotopy Perturbation ◽

In Series ◽

The Given

In this article, the iterative transformation method and homotopy perturbation transformation method are applied to calculate the solution of time-fractional Cauchy-reaction diffusion equations. In this technique, Shehu transformation is combined of the iteration and the homotopy perturbation techniques. Four examples are examined to show validation and the efficacy of the present methods. The approximate solutions achieved by the suggested methods indicate that the approach is easy to apply to the given problems. Moreover, the solution in series form has the desire rate of convergence and provides closed-form solutions. It is noted that the procedure can be modified in other directions of fractional order problems. These solutions show that the current technique is very straightforward and helpful to perform in applied sciences.

A PARALLEL SOLVER FOR REACTION–DIFFUSION SYSTEMS IN COMPUTATIONAL ELECTROCARDIOLOGY

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202504003489 ◽

2004 ◽

Vol 14 (06) ◽

pp. 883-911 ◽

Cited By ~ 121

Author(s):

PIERO COLLI FRANZONE ◽

LUCA F. PAVARINO

Keyword(s):

Large Scale ◽

Three Dimensional ◽

Reaction Diffusion ◽

Ionic Currents ◽

Fiber Direction ◽

Axially Symmetric ◽

Reaction Diffusion Systems ◽

Cardiac Model

In this work, a parallel three-dimensional solver for numerical simulations in computational electrocardiology is introduced and studied. The solver is based on the anisotropic Bidomain cardiac model, consisting of a system of two degenerate parabolic reaction–diffusion equations describing the intra and extracellular potentials of the myocardial tissue. This model includes intramural fiber rotation and anisotropic conductivity coefficients that can be fully orthotropic or axially symmetric around the fiber direction. The solver also includes the simpler anisotropic Monodomain model, consisting of only one reaction–diffusion equation. These cardiac models are coupled with a membrane model for the ionic currents, consisting of a system of ordinary differential equations that can vary from the simple FitzHugh–Nagumo (FHN) model to the more complex phase-I Luo–Rudy model (LR1). The solver employs structured isoparametric Q1finite elements in space and a semi-implicit adaptive method in time. Parallelization and portability are based on the PETSc parallel library. Large-scale computations with up to O(107) unknowns have been run on parallel computers, simulating excitation and repolarization phenomena in three-dimensional domains.

Bounds on the critical times for the Fisher-KPP equation

ANZIAM Journal ◽

10.21914/anziamj.v63.16588 ◽

2021 ◽

Vol 63 ◽

pp. 448-468

Author(s):

Marianito Rodrigo

Keyword(s):

Upper And Lower Solutions ◽

Critical Time ◽

Reaction Diffusion ◽

Diffusion Equations ◽

Kpp Equation ◽

Diffusive Process ◽

Reaction Diffusion Systems ◽

Diffusion Systems ◽

Comparison Functions

The Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation is one of the prototypical reaction–diffusion equations and is encountered in many areas, primarily in population dynamics. An important consideration for the phenomena modelled by diffusion equations is the length of the diffusive process. In this paper, three definitions of the critical time are given, and bounds are obtained by a careful construction of the upper and lower solutions. The comparison functions satisfy the nonlinear, but linearizable, partial differential equations of Fisher–KPP type. Results of the numerical simulations are displayed. Extensions to some classes of reaction–diffusion systems and an application to a spatially heterogeneous harvesting model are also presented. doi:10.1017/S1446181121000365

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

Uniqueness of fast travelling fronts in reaction–diffusion equations with delay

10.1098/rspa.2008.0011 ◽

2008 ◽

Vol 464 (2098) ◽

pp. 2591-2608 ◽

Cited By ~ 17

Author(s):

Maitere Aguerrea ◽

Sergei Trofimchuk ◽

Gabriel Valenzuela

Keyword(s):

Reaction Diffusion ◽

Diffusion Equations ◽

Delayed Reaction ◽

Birth Function ◽

Scale Of Banach Spaces ◽

Large Velocity ◽

Travelling Fronts ◽

Equations With Delay

We consider positive travelling fronts, u ( t , x )= ϕ ( ν . x + ct ), ϕ (−∞)=0, ϕ (∞)= κ , of the equation u t ( t , x )=Δ u ( t , x )− u ( t , x )+ g ( u ( t − h , x )), x ∈ m . This equation is assumed to have exactly two non-negative equilibria: u 1 ≡0 and u 2 ≡ κ >0, but the birth function g ∈ C 2 ( , ) may be non-monotone on [0, κ ]. We are therefore interested in the so-called monostable case of the time-delayed reaction–diffusion equation. Our main result shows that for every fixed and sufficiently large velocity c , the positive travelling front ϕ ( ν . x + ct ) is unique (modulo translations). Note that ϕ may be non-monotone. To prove uniqueness, we introduce a small parameter ϵ =1/ c and realize a Lyapunov–Schmidt reduction in a scale of Banach spaces.

Philosophical Transactions of the Royal Society of London Series A Physical and Engineering Sciences ◽

The evolution of travelling waves in reaction-diffusion equations with monotone decreasing diffusivity. II. Abruptly vanishing diffusivity

10.1098/rsta.1995.0020 ◽

1995 ◽

Vol 350 (1694) ◽

pp. 361-378 ◽

Cited By ~ 1

Keyword(s):

Travelling Waves ◽

Reaction Diffusion ◽

Travelling Wave ◽

Wave Form ◽

Initial Development ◽

Finite Concentration ◽

Distinct Process ◽

And Diffusion

In this paper we continue our study of some of the qualitative features of chemical polymerization processes by considering a reaction-diffusion equation for the chemical concentration in which the diffusivity vanishes abruptly at a finite concentration. The effect of this diffusivity cut-off is to create two distinct process zones; in one there is both reaction and diffusion and in the other there is only reaction. These zones are separated by an interface across which there is a jump in concentration gradient. Our analysis is focused on both the initial development of this interface and the large time evolution of the system into a travelling wave form. Some distinct differences from our previous analysis of smoothly vanishing diffusivity are found.

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

An extension of the principle of spatial averaging for inertial manifolds

10.1017/s1446788700036314 ◽

1999 ◽

Vol 66 (1) ◽

pp. 125-142 ◽

Cited By ~ 4

Author(s):

Hyukjin Kwean

Keyword(s):

Diffusion Equation ◽

Reaction Diffusion ◽

Diffusion Equations ◽

Inertial Manifolds ◽

Inertial Manifold ◽

Specific Class ◽

Spatial Averaging ◽

The Laplace Operator

AbstractIn this paper we extend a theorem of Mallet-Paret and Sell for the existence of an inertial manifold for a scalar-valued reaction diffusion equation to new physical domains ωn ⊂ Rn, n = 2,3. For their result the Principle of Spatial Averaging (PSA), which certain domains may possess, plays a key role for the existence of an inertial manifold. Instead of the PSA, we define a weaker PSA and prove that the domains φn with appropriate boundary conditions for the Laplace operator, δ, satisfy a weaker PSA. This weaker PSA is enough to ensure the existence of an inertial manifold for a specific class of scalar-valued reaction diffusion equations on each domain ωn under suitable conditions.