Nonlinear Reaction-diffusion Systems Research Articles

Homotopy perturbation Laplace transform solution of fractional non-linear reaction diffusion system of Lotka-Volterra type differential equation

This work investigates the fractional non linear reaction diffusion (FNRD) system of Lotka-Volterra type. The system of equations together with the boundary conditions are solved by Homotopy perturbation transform method (HPTM). The series solutions are obtained for the two cases (homogeneous and non-homogeneous) of FNRD system. The effect of fractional parameter on the mass concentration of two species are shown and discussed with the help of 3D graphs.

Numerical methods for a nonlinear reaction–diffusion system modelling a batch culture of biofilm

A biofilm is usually defined as a layer of bacterial cells anchored to a surface. These cells are embedded into a polymer matrix that keeps them attached to each other and to a solid surface. Among a large variety of biofilms, in this paper we consider batch cultures. The mathematical model is formulated in terms of a quasilinear system of diffusion–reaction equations for biomass and nutrients concentrations, which exhibits possible degeneracy and singularities in the nonlinear diffusion coefficient. In the present paper, we propose a set of efficient numerical methods that speeds up the solution of the model. Mainly, Crank–Nicolson finite differences techniques for discretisation are combined with a Newton algorithm for the nonlinearities. Moreover, some numerical examples show the expected behaviour of the biomass and nutrients concentrations and also clearly illustrate some theoretically proved qualitative properties related to exponential decays or convergence to a critical biomass concentration depending on the values of the model parameters.

A Dimensional Splitting of ETD Schemes for Reaction-Diffusion Systems

AbstractNovel dimensional splitting techniques are developed for ETD Schemes which are second-order convergent and highly efficient. By using the ETD-Crank-Nicolson scheme we show that the proposed techniques can reduce the computational time for nonlinear reaction-diffusion systems by up to 70%. Numerical tests are performed to empirically validate the superior performance of the splitting methods.

Analysis of Nonlinear Reaction–Diffusion Systems by the Perturbation Method: Conditions of Application, Construction of Solutions, and Bifurcation Analysis

For nonlinear systems of reaction–diffusion type, we propose a technique for the construction and analysis of solutions based on the method of small parameter. The proposed technique enables us not only to analytically obtain approximate quasiharmonic low-amplitude solutions appearing as a result of bifurcations of spatially homogeneous states of the system but also to determine the type of bifurcation in the system. Examples of application of this approach to the analysis of bifurcations and the construction of solutions of a specific mathematical reaction–diffusion model are given.

Nonlinear delay reaction-diffusion systems with nonlocal initial conditions having affine growth

We consider a class of abstract evolution reaction-diffusion systems with delay and nonlocal initial data of the form $$ \begin{cases} \displaystyle u'(t)\in Au(t)+F(t,u_t,v_t)&\text{for } t\in \mathbb{R}_+, v'(t)\in Bv(t)+G(t,u_t,v_t) & \text{for } t\in \mathbb{R}_+, u(t)=p(u,v)(t)& \text{for } t\in [-\tau_1,0], v(t)=q(u,v)(t)& \text{for } t\in [-\tau_2,0], \end{cases} $$ where $\tau_i\geq 0$, $i=1,2$, $A$ and $B$ are two $m$-dissipative operators acting in two Banach spaces, the perturbations $F$ and $G$ are continuous, while the history functions $p$ and $q$ are nonexpansive functions with affine growth. We prove an existence result of $C^0$-solutions for the above problem and we give an example to illustrate the effectiveness of our abstract theory.

Numerical Investigation of Solutions to a Reaction-diffusion System with Variable Density

In this paper we demonstrate the possibilities of the self-similar and approximately self-similar approaches for studying solutions of a nonlinear mutual reaction-diffusion system. The asymptotic behavior of com- pactly supported solutions and free boundary is studied. Based on established qualitative properties of solutions numerical computation is carried out. The solutions are presented in visualization form, which allows observing evolution of the studied process in time.

Exponential decay towards equilibrium and global classical solutions for nonlinear reaction–diffusion systems

We consider a system of reaction–diffusion equations describing the reversible reaction of two species $${\mathcal{U}}$$ , $${\mathcal{V}}$$ forming a third species $${\mathcal{W}}$$ and vice versa according to mass action law kinetics with arbitrary stoichiometric coefficients (equal or larger than one). Firstly, we prove existence of global classical solutions via improved duality estimates under the assumption that one of the diffusion coefficients of $${\mathcal{U}}$$ or $${\mathcal{V}}$$ is sufficiently close to the diffusion coefficient of $${\mathcal{W}}$$ . Secondly, we derive an entropy entropy-dissipation estimate, that is a functional inequality, which applied to global solutions of these reaction–diffusion systems proves exponential convergence to equilibrium with explicit rates and constants.

Local Discontinuous Galerkin Methods Coupled with Implicit Integration Factor Methods for Solving Reaction-Cross-Diffusion Systems

We present a new numerical method for solving nonlinear reaction-diffusion systems with cross-diffusion which are often taken as mathematical models for many applications in the biological, physical, and chemical sciences. The two-dimensional system is discretized by the local discontinuous Galerkin (LDG) method on unstructured triangular meshes associated with the piecewise linear finite element spaces, which can derive not only numerical solutions but also approximations for fluxes at the same time comparing with most of study work up to now which has derived numerical solutions only. Considering the stability requirement for the explicit scheme with strict time step restriction (Δt=O(hmin2)), the implicit integration factor (IIF) method is employed for the temporal discretization so that the time step can be relaxed asΔt=O(hmin). Moreover, the method allows us to compute element by element and avoids solving a global system of nonlinear algebraic equations as the standard implicit schemes do, which can reduce the computational cost greatly. Numerical simulations about the system with exact solution and the Brusselator model, which is a theoretical model for a type of autocatalytic chemical reaction, are conducted to confirm the expected accuracy, efficiency, and advantages of the proposed schemes.

An asymptotic analysis of a self-similar solution for the double nonlinear reaction-diffusion system

An asymptotic analysis of a self-similar solution for the double nonlinear reaction-diffusion system

Q-Conditional Symmetries and Exact Solutions of Nonlinear Reaction–Diffusion Systems

A wide range of reaction–diffusion systems with constant diffusivities that are invariant under Q-conditional operators is found. Using the symmetries obtained, the reductions of the corresponding systems to the systems of ODEs are conducted in order to find exact solutions. In particular, the solutions of some reaction–diffusion systems of the Lotka–Volterra type in an explicit form and satisfying Dirichlet boundary conditions are obtained. An biological interpretation is presented in order to show that two different types of interaction between biological species can be described.

Non-constant positive steady states for a strongly coupled nonlinear reaction-diffusion system arising in population dynamics

We consider a strongly coupled reaction-diffusion system describing three interacting species in a simple food chain structure. Based on the Leray-Schauder degree theory, the existence of non-constant positive steady states is investigated. The results indicate that, when the intrinsic growth rate of the middle species is small, cross-diffusions of the predators versus the preys are helpful to create global coexistence (stationary patterns).

An ADI extrapolated Crank–Nicolson orthogonal spline collocation method for nonlinear reaction–diffusion systems on evolving domains

We consider the approximate solution of nonlinear reaction–diffusion systems on evolving domains that arise in a variety of areas including biology, chemistry, ecology and physics. By mapping a fixed domain onto the evolving domain at each time level, we generalize to evolving domains the ADI extrapolated Crank–Nicolson orthogonal spline collocation technique developed in [8,9] for fixed domains. The new method is tested on the Schnakenberg model and we demonstrate numerically that it preserves the second-order accuracy in time and optimal accuracy in space for piecewise Hermite cubics in various norms. Moreover, the efficacy of the method is demonstrated on several test problems from the literature which involve various types of domain evolution but for which exact solutions are not known.

GPU accelerated solver for nonlinear reaction–diffusion systems. Application to the electrophysiology problem

Solving the electric activity of the heart possess a big challenge, not only because of the structural complexities inherent to the heart tissue, but also because of the complex electric behaviour of the cardiac cells. The multi-scale nature of the electrophysiology problem makes difficult its numerical solution, requiring temporal and spatial resolutions of 0.1 ms and 0.2 mm respectively for accurate simulations, leading to models with millions degrees of freedom that need to be solved for thousand time steps. Solution of this problem requires the use of algorithms with higher level of parallelism in multi-core platforms. In this regard the newer programmable graphic processing units (GPU) has become a valid alternative due to their tremendous computational horsepower. This paper presents results obtained with a novel electrophysiology simulation software entirely developed in Compute Unified Device Architecture (CUDA). The software implements fully explicit and semi-implicit solvers for the monodomain model, using operator splitting. Performance is compared with classical multi-core MPI based solvers operating on dedicated high-performance computer clusters. Results obtained with the GPU based solver show enormous potential for this technology with accelerations over 50× for three-dimensional problems.

Nonlinear reaction–diffusion systems with a non-constant diffusivity: Conditional symmetries in no-go case

Q-conditional symmetries (nonclassical symmetries) for a general class of two-component reaction–diffusion systems with non-constant diffusivities are studied. The work is a natural continuation of our paper “Conditional symmetries and exact solutions of nonlinear reaction–diffusion systems with non-constant diffusivities” (Cherniha and Davydovych, 2012) [1] in order to extend the results on so-called no-go case. Using the notion of Q-conditional symmetries of the first type, an exhaustive list of reaction–diffusion systems admitting such symmetry is derived. The results obtained are compared with those derived earlier. The symmetries for reducing reaction–diffusion systems to two-dimensional dynamical systems (ODE systems) and finding exact solutions are applied. As result, multiparameter families of exact solutions in the explicit form for nonlinear reaction–diffusion systems with an arbitrary power-law diffusivity are constructed and their properties for possible applicability are established.

Global existence for fully nonlinear reaction–diffusion systems describing multicomponent reactive flows

We consider combustion problems in the presence of complex chemistry and nonlinear diffusion laws leading to fully nonlinear multi-species reaction–diffusion equations. The nonlinear diffusion coefficients are obtained by resolution of the so-called Stefan–Maxwell equations that we discuss thoroughly. The reaction diffusion equations are coupled with the Navier–Stokes equations and the heat equation in the context of a typical combustion problem. We establish results of existence of solutions in space dimensions two and three.

The locally extrapolated exponential time differencing LOD scheme for multidimensional reaction–diffusion systems

This paper introduces the local extrapolation of first order locally one-dimensional exponential time differencing scheme for numerical solution of multidimensional nonlinear reaction–diffusion systems. This novel scheme has the benefit of solving multidimensional problems in locally one dimensional fashion by implementing sequences of tridiagonal matrix solvers instead of solving a banded system. The storage size needed for solving systems in higher dimensions with this scheme is similar to that needed for one spatial dimension systems. The stability, monotonicity, and convergence of the locally extrapolated exponential time differencing scheme have been examined. Stability analysis shows that the scheme is strongly stable (L-stable) and is particularly beneficial to nonlinear partial differential equations with irregular initial data or discontinuity involving initial and boundary conditions due to its ability to damp spurious oscillations caused by high frequency components in the solution. The performance of the novel scheme has been investigated by testing it on a two-dimensional Schnakenberg model, two and three-dimensional Brusselator models, and a three-dimensional enzyme kinetics of Michaelis–Menten type reaction–diffusion model. Numerical experiments demonstrate the efficiency, accuracy, and reliability of the scheme.

The generating equations method: Constructing exact solutions to delay reaction–diffusion systems and other non-linear coupled delay PDEs

We present a new method for constructing exact solutions to non-linear systems of coupled delay reaction–diffusion equations of the formut=k1uxx+F(u,u¯,w,w¯),wt=k2wxx+G(u,u¯,w,w¯),where u=u(x,t), w=w(x,t), u¯=u(x,t−τ), and w¯=w(x,t−τ), and τ is the delay time. The method relies on using generalized and functional separable solutions, obtained for single delay reaction–diffusion equations (called generating equations), to construct exact solutions of more complex systems of coupled delay equations. All of the systems considered contain two or more arbitrary functions of two or three arguments. We show that the generating equations method can also be used for (i) non-linear reaction–diffusion systems with time-varying delay τ=τ(t), (ii) delay reaction–diffusion systems with varying transfer coefficients, (iii) multicomponent delay reaction–diffusion systems, (iv) multidimensional delay reaction–diffusion systems, and (v) non-linear delay systems of higher-order PDEs. The results may be suitable for solving certain model problems and testing approximate analytical and numerical methods for some classes of similar and more complex non-linear coupled delay systems.

A Class of Reaction-Diffusion Systems with Nonlocal Initial Conditions

Abstract We consider an abstract nonlinear multi-valued reaction-diffusion system with delay and, using some compactness arguments coupled with metric fixed point techniques, we prove some sufficient conditions for the existence of at least one C0-solution.

Qualitative Properties of Solutions of a Doubly Nonlinear Reaction-Diffusion System with a Source

In this paper, we study properties of solutions to doubly nonlinear reaction-diffusion systems with variable density and source. We demonstrate the possibilities of the self-similar approach to studying the qualitative properties of solutions of such reaction-diffusion systems. We also study the finite speed of propagation (FSP) properties of solutions, an asymptotic behavior of the compactly supported solutions and free boundary asymptotic solutions in quick diffusive and critical cases.

Construction of Lyapunov Functions for Interconnected Parabolic Systems: An iISS Approach

This paper is devoted to two issues. The first one is to provide Lyapunov-based tools to establish integral input-to-state stability (iISS) and input-to-state stability (ISS) for some classes of nonlinear parabolic equations. The other one is to provide a stability criterion for interconnections of iISS parabolic systems. The results addressing the former problem allow us to overcome obstacles arising in tackling the latter one. The results for the latter problem are a small-gain condition and a formula of Lyapunov functions which can be constructed for interconnections whenever the small-gain condition holds. It is demonstrated that for interconnections of partial differential equations, the choice of a right state and input spaces is crucial, in particular for iISS subsystems which are not ISS. As illustrative examples, stability of two highly nonlinear reaction-diffusion systems is established by the proposed small-gain criterion.