Nonlinear Reaction-diffusion Equations Research Articles

A Comparative Study of the Numerical Approximations of the Quenching Time for a Nonlinear Reaction-Diffusion Equation

In this paper, we study the numerical methods for solving a nonlinear reaction-diffusion model for the polarization phenomena in ionic conductors. In particular, we propose three types of numerical methods, including the finite difference, cubic B-spline collocation, and local discontinuous Galerkin method, to approximate the quenching time of the model. We prove the conservation properties for all three numerical methods and compare their numerical performance.

A kinetic model for amperometric immobilized enzymes at planar, cylindrical and spherical electrodes: The Akbari-Ganji method

Diffusion and kinetics of amperometric immobilized enzymes on the planar, cylindrical and spherical electrodes are discussed. This model is described as a set of non-linear reaction-diffusion equations consisting of a non-linear term related to Michaelis-Menten kinetics and enzyme-catalyzed reactions. In this paper, we obtain approximately the general analytical solutions for the non-linear equations that describe diffusion-limited reaction within the film by employing the Akbari-Ganji Method (AGM). The obtained analytical results for the concentration of substrate, mediator and current are compared with the available limiting case results and numerical results and found to be in satisfactory agreement. The dependence of the amperometric response on substrate and mediator concentration, enzyme concentration, electrode potential, film thickness and geometry of the electrode is analyzed.

Construction of exact solutions to nonlinear PDEs with delay using solutions of simpler PDEs without delay

We describe new indirect methods for constructing exact solutions of nonlinear PDEs with delay. Features of the proposed methods are illustrated by nonlinear delay reaction-diffusion and wave-type equations with variable coefficients. All the presented equations contain from three to seven arbitrary functions, which depend on the spatial variable and/or unknown quantity. We obtain new generalized traveling-wave solutions and functional separable solutions in explicit or implicit form for equations with constant delay as well as exact solutions with time-varying delay (a special case of which are pantograph-type PDEs with proportional delay). We show that the proposed methods can also be applied to construct exact solutions for nonlinear systems of coupled delay PDEs and higher-order delay PDEs. The considered equations and their exact solutions can be used to formulate test problems to check the adequacy and estimate the accuracy of numerical and approximate analytical methods of solving nonlinear initial-boundary value problems for delay PDEs.

Spontaneous activation under atrial fibrosis: A model using complex order derivatives

The computational modeling of the cardiac electrophysiology allows assertive and quantitative study of the atrial fibrosis under fibrillation conditions. The cardiac electrical propagation is described by the so-called monodomain model, that consists of a nonlinear parabolic reaction-diffusion equation. Fibroblast proliferation, which is an essential component of the fibrotic process, can be modeled by considering the membrane ionic kinetics as a reactive component. However, such a mathematical description does not account the structural feature of fibroblasts. In this work, the electrophysiological properties of fibroblast proliferation and coupling with cardiomyocytes are investigated, using mathematical and computational modelling. The study is focused on the conditions under which spontaneous activations occur in a fibrotic domain. The proposed fibrosis model takes account the electrical and structural interactions of fibroblasts within the myocardium. The electrical component is described through an ionic kinetics formalism, while the structural component is obtained by means of a triplet of complex order derivatives that constructs the diffusion operator. A theoretical analysis determines the model parameters that generate unstable solutions, and numerical simulations illustrate and validate the analytical outcomes. The results evince a strong modulation of the stability conditions of the fibrotic model by the real and imaginary part of the fractional derivative order. The fibrosis structural complexity, controlled by the fractional order, determines the extent of the parameter space that generates spontaneous activation. Moreover, not all the unstable parameter configurations generate electrical propagation. In the cases of electrical conduction after spontaneous activation, the conduction velocity in the fibrotic domain is significantly slower than the one observed in healthy atrial tissue. The results give a new perspective for the development of atrial fibrosis models including the ectopic activity as an initiation factor for fibrillation activity. Indeed, the proposed design exploits the complex order fractional derivatives, to generate a wide set of electrophysiological scenarios.

A boundedness and monotonicity preserving method for a generalized population model

In this work, a nonstandard finite difference (NSFD) method is proposed to approximate the solutions of a nonlinear reaction–diffusion equation which appears in population dynamics. It is well known that the model under study has some travelling-wave solutions, which are positive, bounded and monotone in both space and time. First, a robust NSFD method is presented for the diffusion-free case of original equation. Then, combined with the NSFD method for the diffusion-free equation, an NSFD method is constructed for the full reaction–diffusion equation. It is shown that, under certain conditions on the denominator function of the time-step size, the proposed method is capable of preserving the positivity, boundedness and the spatial and temporal monotonicity of these travelling-wave solutions. Moreover, the nonlinear stability and convergence of this method are also analysed. Finally, some numerical simulations are provided to verify the validity of our analytical results.

Investigation of the invasion front in heterogeneous environments to study the geometry of tumor border

We develop a model to study how invasion front depends on the relevant properties of a cellular environment. To do so, we use a nonlinear reaction-diffusion equation, the Fisher equation, to model the population dynamics. Our study is intended to understand how heterogeneity in the cellular environment's stiffness, as well as spatial correlations in its morphology, given that the existence of both has been demonstrated by experiments, affects the properties of the invasion front. It is demonstrated that three important factors affect the properties of the front; these are the spatial distribution of the local diffusion coefficients, the correlations between them, and R/D, the ratio of the cells' duplication rate R to the cells' average diffusion coefficient D. Analyzing the scaling properties of the Fisher equation invasion front, we show that , contrary to several previous claims, invasion fronts, including those of tumors and cancerous cells colonies, cannot be described by the well-known model of kinetic growth, such as the Kardar-Parisi-Zhang equation.

Semi-implicit Hermite–Galerkin Spectral Method for Distributed-Order Fractional-in-Space Nonlinear Reaction–Diffusion Equations in Multidimensional Unbounded Domains

In this paper, we construct an efficient Hermite–Galerkin spectral method for the nonlinear reaction–diffusion equations with distributed-order fractional Laplacian in multidimensional unbounded domains. By applying Gauss–Legendre quadrature rule for the distributed integral term, we first approximate the original distributed-order fractional problem by the multi-term fractional-in-space differential equation. Applying Hermite–Galerkin spectral method in space and backward difference method in time, we establish semi-implicit fully discrete scheme. For two- and three-dimensional cases of the original fractional problem, the linear systems are solved by the preconditioned conjugate gradients method. The main advantage of our method is that the original fractional problem is directly solved in the unbounded domains, thus avoiding the errors introduced by the domain truncations. The stability analysis is rigourously established, which shows that our scheme is unconditionally stable under suitable assumption on the nonlinear term. Several numerical examples are presented to validate both stability and accuracy of the numerical method. The numerical results of the fractional Allen–Cahn, Gray–Scott, and Belousov–Zhabotinskii models show that our semi-implicit methods produce good numerical solutions.

Iterative analytic approximation to one‐dimensional nonlinear reaction–diffusion equations

The paper is concerned with a class of nonlinear reaction–diffusion equations with a dissipating parameter. The problem is singularly perturbed from a mathematical perspective. Solutions of these problems are known to exhibit multiscale character. There are narrow regions in which the solution has a steep gradient. To approximate the multiscale solution, we present and analyze an iterative analytic method based on a Lagrange multiplier technique. We obtain closed‐form analytic approximation to nonlinear boundary value problems through iteration. The Lagrange multiplier is obtained optimally, in a general setting, using variational theory and Liouville–Green transforms. The idea of the paper is to overcome the well‐known difficulties associated with numerical methods. Two test examples are taken into account, and rigorous comparative analysis is presented. Moreover, we compare the proposed method with others found in the literature.

DOUBLE-QUASI-WAVELET NUMERICAL METHOD FOR THE VARIABLE-ORDER TIME FRACTIONAL AND RIESZ SPACE FRACTIONAL REACTION–DIFFUSION EQUATION INVOLVING DERIVATIVES IN CAPUTO–FABRIZIO SENSE

Our motive in this scientific contribution is to deal with nonlinear reaction–diffusion equation having both space and time variable order. The fractional derivatives which are used are non-singular having exponential kernel. These derivatives are also known as Caputo–Fabrizio derivatives. In our model, time fractional derivative is Caputo type while spatial derivative is variable-order Riesz fractional type. To approximate the variable-order time fractional derivative, we used a difference scheme based upon the Taylor series formula. While approximating the variable order spatial derivatives, we apply the quasi-wavelet-based numerical method. Here, double-quasi-wavelet numerical method is used to investigate this type of model. The discretization of boundary conditions with the help of quasi-wavelet is discussed. We have depicted the efficiency and accuracy of this method by solving the some particular cases of our model. The error tables and graphs clearly show that our method has desired accuracy.

A novel radial basis function method for 3D linear and nonlinear advection diffusion reaction equations with variable coefficients

The radial basis function method for 3D advection diffusion reaction equations with variable coefficients is presented. The proposed method implements the linear combination of radial basis functions which impose boundary conditions in advance, and thus such a combination with weighted parameters can be used to construct the final approximation. Furthermore, the weighted parameters are solved by substituting the approximation into governing equations. This method leads to crucial improvements in the feasibility and accuracy which can now be easily applied to general 3D nonlinear problems through linearized techniques. Finally, accuracy and efficiency of the proposed method are verified by several examples.

Correctors and error estimates for reaction–diffusion processes through thin heterogeneous layers in case of homogenized equations with interface diffusion

In this paper, we construct approximations of the microscopic solution of a nonlinear reaction–diffusion equation in a domain consisting of two bulk-domains, which are separated by a thin layer with a periodic heterogeneous structure. The size of the heterogeneities and thickness of the layer are of order ϵ, where the parameter ϵ is small compared to the length scale of the whole domain. In the limit ϵ→0, when the thin layer reduces to an interface Σ separating two bulk domains, a macroscopic model with effective interface conditions across Σ is obtained. Our approximations are obtained by adding corrector terms to the macroscopic solution, which take into account the oscillations in the thin layer and the coupling conditions between the layer and the bulk domains. To validate these approximations, we prove error estimates with respect to ϵ. Our approximations are constructed in two steps leading to error estimates of order ϵ12 and ϵ in the H1-norm.

Analysis of backward Euler/Spectral discretization for an evolutionary mass and heat transfer in porous medium

This paper presents the unsteady Darcy's equations coupled with two nonlinear reaction-diffusion equations, namely this system describes the mass concentration and heat transfer in porous media. The existence and uniqueness of the solution are established for both the variational formulation problem and for its discrete one obtained using spectral discretization. Optimal a priori estimates are given using the Brezzi-Rappaz-Raviart theorem. We conclude by some numerical tests which are in agreement with our theoretical results.

A variational approach for novel solitary solutions of FitzHugh–Nagumo equation arising in the nonlinear reaction–diffusion equation

PurposeIn the nonlinear model of reaction–diffusion, the Fitzhugh–Nagumo equation plays a very significant role. This paper aims to generate innovative solitary solutions of the Fitzhugh–Nagumo equation through the use of variational formulation.Design/methodology/approachThe partial differential equation of Fitzhugh–Nagumo is modified by the appropriate wave transforms into a dimensionless nonlinear ordinary differential equation, which is solved by a semi-inverse variational method.FindingsThis paper uses a variational approach to the Fitzhugh–Nagumo equation developing new solitary solutions. The condition for the continuation of new solitary solutions has been met. In addition, this paper sets out the Fitzhugh–Nagumo equation fractal model and its variational principle. The findings of the solitary solutions have shown that the suggested method is very reliable and efficient. The suggested algorithm is very effective and is almost ideal for use in such problems.Originality/valueThe Fitzhugh–Nagumo equation is an important nonlinear equation for reaction–diffusion and is typically used for modeling nerve impulses transmission. The Fitzhugh–Nagumo equation is reduced to the real Newell–Whitehead equation if β = −1. This study provides researchers with an extremely useful source of information in this area.

PPINN: Parareal physics-informed neural network for time-dependent PDEs

Physics-informed neural networks (PINNs) encode physical conservation laws and prior physical knowledge into the neural networks, ensuring the correct physics is represented accurately while alleviating the need for supervised learning to a great degree (Raissi et al., 2019). While effective for relatively short-term time integration, when long time integration of the time-dependent PDEs is sought, the time–space domain may become arbitrarily large and hence training of the neural network may become prohibitively expensive. To this end, we develop a parareal physics-informed neural network (PPINN), hence decomposing a long-time problem into many independent short-time problems supervised by an inexpensive/fast coarse-grained (CG) solver. In particular, the serial CG solver is designed to provide approximate predictions of the solution at discrete times, while initiate many fine PINNs simultaneously to correct the solution iteratively. There is a two-fold benefit from training PINNs with small-data sets rather than working on a large-data set directly, i.e., training of individual PINNs with small-data is much faster, while training the fine PINNs can be readily parallelized. Consequently, compared to the original PINN approach, the proposed PPINN approach may achieve a significant speed-up for long-time integration of PDEs, assuming that the CG solver is fast and can provide reasonable predictions of the solution, hence aiding the PPINN solution to converge in just a few iterations. To investigate the PPINN performance on solving time-dependent PDEs, we first apply the PPINN to solve the Burgers equation, and subsequently we apply the PPINN to solve a two-dimensional nonlinear diffusion–reaction equation. Our results demonstrate that PPINNs converge in a few iterations with significant speed-ups proportional to the number of time-subdomains employed.

Global boundedness, hair trigger effect, and pattern formation driven by the parametrization of a nonlocal Fisher-KPP problem

The global boundedness and the hair trigger effect of solutions for the nonlinear nonlocal reaction-diffusion equation∂u∂t=Δu+μuα(1−κJ⁎uβ),inRN×(0,∞),N≥1 with α≥1, β,μ,κ>0 and u(x,0)=u0(x) are investigated. Under appropriate assumptions on J, it is proved that for any nonnegative and bounded initial condition, if α∈[1,α⁎) with α⁎=1+β for N=1,2 and α⁎=1+2βN for N>2, then the problem has a global bounded classical solution. Under further assumptions on the initial datum, the solutions satisfying 0≤u(x,t)≤κ−1β for any (x,t)∈RN×[0,+∞) are shown to converge to κ−1β uniformly on any compact subset of RN, which is known as the hair trigger effect. 1D numerical simulations of the above nonlocal reaction-diffusion equation are performed and the effect of several combinations of parameters and convolution kernels on the solution behavior is investigated. The results motivate a discussion about some conjectures arising from this model and further issues to be studied in this context. A formal deduction of the model from a mesoscopic formulation is provided as well.

Adaptive ADI Numerical Analysis of 2D Quenching-Type Reaction: Diffusion Equation with Convection Term

An adaptive high-order difference solution about a 2D nonlinear degenerate singular reaction-diffusion equation with a convection term is initially proposed in the paper. After the first and the second central difference operator approximating the first-order and the second-order spatial derivative, respectively, the higher-order spatial derivatives are discretized by applying the Taylor series rule and the temporal derivative is discretized by using the Crank–Nicolson (CN) difference scheme. An alternating direction implicit (ADI) scheme with a nonuniform grid is built in this way. Meanwhile, accuracy analysis declares the second order in time and the fourth order in space under certain conditions. Sequentially, the high-order scheme is performed on an adaptive mesh to demonstrate quenching behaviors of the singular parabolic equation and analyse the influence of combustion chamber size on quenching. The paper displays rationally that the proposed scheme is practicable for solving the 2D quenching-type problem.

Nonnegative controllability for a class of nonlinear degenerate parabolic equations with application to climate science

We consider a nonlinear degenerate reaction-diffusion equation. First we prove that if the initial state is nonnegative, then the solution remains nonnegative for all time. Then we prove the approximate controllability between nonnegative states via multiplicative controls, this is done using the reaction coefficient as control.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/59/abstr.html

Dynamical analysis on traveling wave of a reaction–diffusion model

In this paper, a class of nonlinear reaction–diffusion equations is studied, and Abelian integral method is applied to show the existence of a unique periodic traveling wave solution. Simulation is presented to verify the theoretical prediction.

Numerical solutions for the reaction–diffusion, diffusion‐wave, and Cattaneo equations using a new operational matrix for the Caputo–Fabrizio derivative

In this research work, we study the model of nonlinear reaction–diffusion equation, diffusion‐wave equation, and Cattaneo equation with the help of a numerical method in which time‐fractional derivative is of Caputo–Fabrizio type (C‐F). First, we derive a formula of the fractional‐order C‐F derivative of xk. Using this formula and some properties of shifted Chebyshev polynomials, we find out the operational matrix of the C‐F derivative. We solve a new class of fractional partial differential equations (FPDEs) using the collocation method and C‐F operational matrix. We depict the feasibility and validity of this numerical method by giving the solution of some numerical examples. We take the diffusion equation, nonlinear reaction–diffusion equation, diffusion‐wave equation, and Cattaneo equation as a particular case of our model. By comparing the obtained numerical results with the given exact solution, we draw an error table depicting that our results are so accurate and desirable accuracy. In the authors' knowledge, the C‐F shifted Chebyshev operational matrix is derived for the first time in this work.

Interpolatory super-convergent discontinuous Galerkin methods for nonlinear reaction diffusion equations on three dimensional domains

An error analysis of a super-convergent discontinuous Galerkin method formulated in mixed form and applied to a general class of semi-linear equations is presented. To reduce the computational cost at each time step, the nonlinear term is approximated with a Lagrange interpolatory operator. Optimal convergence of order O(hk+1), for both, the primary and auxiliary variables, is obtained for polynomial approximations of degree k. Using a well known element-by-element post-processing procedure, super-convergence of the primary variable is proven. Computational aspects for an efficient implementation are discussed in detail. To avoid solving a large global nonlinear problem at each time step the symmetric Strang operator splitting method is considered as time marching scheme. To validate our error estimates, a series of numerical experiments are carried out using unstructured meshes of three dimensional domains.