Dimensional Reaction-diffusion Equation Research Articles

Solitons for a generalized reaction–diffusion equation with the higher‐order power‐law nonlinearity in (1+1)‐ and (2+1)‐dimensional systems

For the systems modeled by a generalized reaction–diffusion equation with the higher‐order quintic nonlinearity, we explore the soliton dynamics via the F‐expansion method. By the novel F‐base function ansatz, we first derived the bright soliton and kink soliton solutions for the one‐dimensional case of the reaction–diffusion equation with quintic nonlinearity. Furthermore, we employed self‐similar techniques to analyze the higher‐dimensional dynamics of bright soliton and kink soliton solutions supported by the (2+1)‐dimensional reaction–diffusion equation system with quintic nonlinearity. Additionally, we conducted stability analysis of derived soliton solutions. Our theoretical results demonstrate that under certain parametric setting, the reaction–diffusion equation model with higher‐order nonlinearity supports bright soliton and kink soliton in higher‐dimensional as well as lower dimensional setting, which provides guidance for observing and investigating soliton behavior in systems modeled by the reaction–diffusion equation with higher‐order quintic nonlinearity.

Upper and lower bounds for the speed of fronts of the reaction diffusion equation with Stefan boundary conditions

We establish two integral variational principles for the spreading speed of the one dimensional reaction diffusion equation with Stefan boundary conditions. The first principle is valid for monostable reaction terms and the second principle is valid for arbitrary reaction terms. These principles allow to obtain several upper and lower bounds for the speed. In particular, we construct a generalized Zeldovich–Frank–Kamenetskii type lower bound for the speed and upper bounds in terms of the speed of the standard reaction diffusion problem. We construct asymptotically exact lower bounds previously obtained by perturbation theory.

Soliton solutions of (2+1)-dimensional non-linear reaction-diffusion model via Riccati-Bernoulli approach

In this study, soliton solutions of the (2+1)-dimensional reaction-diffusion equation are investigated by the extended Kudryashov method based on Riccati-Bernoulli approach. Firstly, we obtained the non-linear ordinary differential form of the (2+1)-dimensional non-linear reaction-diffusion equation by implementing the wave transformation. Then, the extended Kudryashov method has been presented and applied to the non-linear ordinary differential form. By applying the extended Kudryashov method the polynomial form has been gained, solution sets have been obtained and soliton solutions have been formed by taking the appropriate sets. Finally, some graphical representations of the gained results for instance bright, dark, kink and singular solutions are presented and commented. Within the scope of the article, the study on investigating the soliton solutions of the (2+1)-dimensional non-linear reaction-diffusion equation via the extended Kudryashov approach has not been studied and the obtained results have not been reported.

Control law in analytic expression of a system coupled by reaction–diffusion equation

Control design of a coupled ODE–PDE boundary control system is considered in this paper. The PDE is one dimensional reaction–diffusion equation, with constant parameters, actuated at one of the boundaries and coupled to a linear and time-invariant ODE at the opposite boundary. By introducing a two-step backstepping design procedure, analytic solutions of the kernel and vector-valued functions in the backstepping transformations are obtained. Therefore, a boundary control law in analytic expression is constructed for this coupled ODE–PDE control system.

Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations

Computational costs of numerically solving multidimensional partial differential equations (PDEs) increase significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points. For multidimensional reaction-diffusion equations, stiffness of the system provides additional challenges for achieving efficient numerical simulations. In this paper, we propose a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions. The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions. Stability and error analysis for the semi-discrete scheme are performed. Numerical examples of both scalar equations and systems in two and three spatial dimensions are provided to demonstrate the accuracy and efficiency of the methods. The stiffness of the reaction-diffusion equations is resolved well and large time step size computations are obtained.

New exact solution for (2+1) and (3+1) dimensional nonlinear partial differential equations

In this paper, dark soliton solutions have been obtained for the (2+1)-dimensional reaction-diffusion equation, the (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili (gBKP) equation and the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation using the solitary wave ansatz. Ansatz approach is utilized to carry out this integration. The constraint relations for each of the equations are given for the existence of dark soliton solutions.

Computational Complexity Study on Krylov Integration Factor WENO Method for High Spatial Dimension Convection–Diffusion Problems

Integration factor (IF) methods are a class of efficient time discretization methods for solving stiff problems via evaluation of an exponential function of the corresponding matrix for the stiff operator. The computational challenge in applying the methods for partial differential equations (PDEs) on high spatial dimensions (multidimensional PDEs) is how to deal with the matrix exponential for very large matrices. Compact integration factor methods developed in Nie et al. (J Comput Phys 227:5238–5255, 2008) provide an approach to reduce the cost prohibitive large matrix exponentials for linear diffusion operators with constant diffusion coefficients in high spatial dimensions to a series of much smaller one dimensional computations. This approach is further developed in Wang et al. (J Comput Phys 258:585–600, 2014) to deal with more complicated high dimensional reaction–diffusion equations with cross-derivatives in diffusion operators. Another approach is to use Krylov subspace approximations to efficiently calculate large matrix exponentials. In Chen and Zhang (J Comput Phys 230:4336–4352, 2011), Krylov subspace approximation is directly applied to the implicit integration factor (IIF) methods for solving high dimensional reaction–diffusion problems. Recently the method is combined with weighted essentially non-oscillatory (WENO) schemes in Jiang and Zhang (J Comput Phys 253:368–388, 2013) to efficiently solve semilinear and fully nonlinear convection–reaction–diffusion equations. A natural question that arises is how these two approaches may perform differently for various types of problems. In this paper, we study the computational power of Krylov IF-WENO methods for solving high spatial dimension convection–diffusion PDE problems (up to four spatial dimensions). Systematical numerical comparison and complexity analysis are carried out for the computational efficiency of the two different approaches. We show that although the Krylov IF-WENO methods have linear computational complexity, both the compact IF method and the Krylov IF method have their own advantages for different type of problems. This study provides certain guidance for using IF-WENO methods to solve general high spatial dimension convection–diffusion problems.

Numerical Study of Fisher’s Equation by Finite Difference Schemes

This research paper represents a numerical approximation to three interesting equations of Fisher, which are linear, non-linear and coupled linear one dimensional reaction diffusion equations from population genetics. We studied accuracy in term of L∞ error norm by random selected grids along time levels for comparison with exact results. The test example demonstrates the accuracy, efficiency and versatility of the proposed schemes. It is shown that the numerical schemes give better solutions. Moreover, the schemes can be easily applied to a wide class of higher dimension non-linear reaction diffusion equations.

Approximate and exact controllability of a reaction–diffusion equation governed by bilinear control

Our goal in this paper is to investigate the global controllability for a reaction–diffusion equation governed in a bounded domain Ω⊂Rn by a multiplicative control in the reaction term. We will show that, under some sufficient conditions on a given couple of initial and desirable states, the considered system can be steered in L2(Ω) from its initial state into any neighborhood of the given desirable target state by means of a static control. This result is further applied to discuss the exact controllability properties for the one dimensional reaction–diffusion equation. Our approaches are based on linear semigroup theory and the null controllability of linear systems with small initial states.

Limiting classification on linearized eigenvalue problems for 1-dimensional Allen–Cahn equation I—asymptotic formulas of eigenvalues

We are interested in classical linearized eigenvalue problems for one dimensional reaction–diffusion equations with small diffusion coefficient. Previously, authors have obtained the precise asymptotic formulas of all eigenvalues and eigenfunctions for f(u)=sin⁡u, which suggest interesting structure of eigenpairs: existence of two special eigenpairs and the limiting classification of the others by special eigenpairs. In this paper we study the Allen–Cahn case f(u)=u−u3 and obtain the asymptotic formulas of eigenvalues, which leads us to the limiting classification of eigenvalues.

Exact Solutions of Nonlinear Evolution Equations in Mathematical Physics Using the Modified Simple Equation Method

The modified simple equation method is employed to construct the exact solutions involving parameters of nonlinear evolution equations via the (1+1)-dimensional modified KdV equation, and the (1+1)-dimensional reaction-diffusion equation. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact solutions. It is shown that the proposed method provides a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

Stochastic Order Methods Applied to Stochastic Travelling Waves

This paper considers some one dimensional reaction diffusion equations driven by a one dimensional multiplicative white noise. The existence of a stochastic travelling wave solution is established, as well as a sufficient condition to be in its domain of attraction. The arguments use stochastic ordering techniques.

Multiplicative controllability for reaction-diffusion equations with target states admitting finitely many changes of sign

We study the global approximate controllability properties of a one dimensional reaction-diffusion equation governed via the coefficient of the reaction term. The traditional (linear operator) controllability methods based on the duality pairing do not apply to such a problem. Instead, we focus on the qualitative study of the diffusion and reaction parts of the evolution process at hand. We consider the case when both the initial and target states admit no more than finitely many changes of sign.

Exact solutions and generalized conditional symmetries to (1 + 1)-dimensional reaction-diffusion equations

For the (1 + 1)-dimensional nonlinear reaction-diffusion equations, we obtain conditions under which the equations admit a class of second-order generalized conditional symmetries η ( u ) = u xx - H ( u ) u x 2 - A ( u ) u x - B ( u ) and first-order sign-invariants J ( u ) = u t - F ( u ) u x 2 - P ( u ) u x - T ( u ) on the solutions u ( x , t ) . Several different generalized conditional symmetries and first-order sign-invariants for equations are presented. Exact solutions to the resulting equations corresponding to the generalized conditional symmetries and first-order sign-invariants are constructed.

Numerical investigation of the solution of Fisher's equation via the B‐spline Galerkin method

AbstractThe Galerkin method is used with quadratic B‐spline base functions to obtain the numerical solutions of Fisher's equation which is a one dimensional reaction‐diffusion equation. To observe the effects of reaction and diffusion, four test problems related to pulse disturbance, step disturbance, super‐speed wave and strong reaction are studied. A comparison is performed between the obtained numerical results and some earlier studies. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

Numerical approximation of characteristic values of partial retarded functional differential equations

The stability of an equilibrium point of a dynamical system is determined by the position in the complex plane of the so-called characteristic values of the linearization around the equilibrium. This paper presents an approach for the computation of characteristic values of partial differential equations of evolution involving time delay, which is based on a pseudospectral method coupled with a spectral method. The convergence of the computed characteristic values is of infinite order with respect to the pseudospectral discretization and of finite order with respect to the spectral one. However, for one dimensional reaction diffusion equations, the finite order of the spectral discretization is proved to be so high that the convergence turns out to be as fast as one of infinite order.

Local structure of solutions of the Dirichlet problem for $N$-dimensional reaction-diffusion equations in bounded domains

We consider the problem \begin{alignat}{2} u_t-\Delta u^m+bu^\beta & =0, &\quad& (x,t)\ \epsilon\ Q_T =\Omega\times(0,T]\\ u(x,t) & =0, &\quad& (x,t)\ \epsilon\ S_T=\partial\Omega\times(0,T]\\ u(x,0) & =u_0(x), &\quad& x\ \epsilon\ \Omega, \end{alignat} where $m\geq 1,$ $ b \in R^1,$ $ \beta>0, $ $ T>0;$ $ \Omega$ is a bounded, connected domain in $R^N$ with compact boundary, which is assumed to be piecewise smooth and to satisfy the exterior sphere condition. Let $u_0 \in C(\bar\Omega),$ $ u_0>0$ for $x \in \Omega$ and $u_0=0$ for $x\in \partial\Omega$. Assume also that $u_0$ is smooth near $\partial\Omega$. We show that the small time behaviour of the solution near the boundary $\partial\Omega$ depends on the relative strength of the diffusion and reaction terms near the boundary, that is to say on the function $$ \gamma(\bar x)=\lim\limits_{x\to\bar x} \bigl(\Delta u_0^m\big/bu^\beta_0\bigr),\quad\bar x \in \partial\Omega,\quad x\in \Omega. $$ We are essentially interested in the initial development of the interface between the dead core and the positive $u(x,t)$ field. In all possible cases, the small time behaviour of the interface is found, together with the local solution.

Images of Ca 2+ flux in astrocytes: evidence for spatially distinct sites of Ca 2+ release and uptake

In this study, we have developed a mathematical method to derive the Ca 2+ fluxes underlying agonist-evoked Ca 2+ waves in cultured rat cortical astrocytes. Astrocytes were stimulated with norepinephrine (100 nM) to evoke Ca 2+ waves, which were recorded by measuring FIuo-3 fluorescence changes with high spatial and temporal resolution. Normalized fluorescence (ΔF/F) was analyzed in discrete cellular spaces in a series of successive slices along the length of the cell. From these data, Ca 2+ flux was then calculated using a one dimensional reaction-diffusion equation which utilizes the temporal and spatial derivatives of the fluorescence data and the diffusion coefficient of Ca 2+ in the cytosol. This method identified distinct sites of positive flux (Ca 2+ release into the cytosol) and of negative flux (Ca 2+ removal from cytosol) and showed that in astrocytes, sites of Ca 2+ release from stores regularly alternate with sites of Ca 2+ removal from the cytosol. Cross correlation analysis of the two distribution patterns gave positive correlation at 2 μm out of phase and a negative correlation in phase. Thapsigargin-induced Ca 2+ waves were analyzed to determine if the negative flux was due to Ca 2+ uptake via thapsigargin-sensitive Ca 2+ pumps. Negative flux sites were still found under these conditions, suggesting that multiple mechanisms of Ca 2+ removal from the cytosol may contribute to negative flux sites. This method of calculation of flux may serve as a means to describe the distribution of functional ion channels and pumps participating in cellular Ca 2+ signalling.