An approximate solution of nonlinear fractional reaction–diffusion equation

The article presents a mathematical model of nonlinear reaction diffusion equation with fractional time derivative α (0 < α ? 1) in the form of a rapidly convergent series with easily computable components. Fractional reaction diffusion equation is used for modeling of merging travel solutions in nonlinear system for popular dynamics. The fractional derivatives are described in the Caputo sense. The anomalous behaviors of the nonlinear problems in the form of sub- and super-diffusion due to the presence of reaction term are shown graphically for different particular cases.     

An analytic algorithm for time fractional nonlinear reaction–diffusion equation based on a new iterative method

A new analytic algorithm for highly nonlinear time fractional reaction–diffusion equations is proposed in this paper. The proposed method is an amalgamation of variational iteration method (VIM), Adomian decomposition method (ADM) and further refined by introducing a new correction functional. This new correction functional is obtained from the standard correction functional of VIM by introducing an auxiliary parameter γ and an auxiliary function H(x) in it. Further, a sequence G_n(x, t), with suitably chosen support, is also introduced in the new correction functional. The algorithm is easy to implement and only four to six iterations are sufficient for fairly accurate solutions. The algorithm is tested on Fitzhugh – Nagumo and generalized Fisher equations with nonlinearity ranging from 2 to 5.     

Maximum principles for a time–space fractional diffusion equation

In this paper, we focus on maximum principles of a time–space fractional diffusion equation. Maximum principles for classical solution and weak solution are all obtained by using properties of the time fractional derivative operator and the fractional Laplace operator. We deduce maximum principles for a full fractional diffusion equation, other than time-fractional and spatial-integer order diffusion equations.     

Numerical approximations and solution techniques for the space-time Riesz–Caputo fractional advection-diffusion equation

In this paper, we consider a space-time Riesz–Caputo fractional advection-diffusion equation. The equation is obtained from the standard advection-diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α ∈ (0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of order β ₁ ∈ (0,1) and β ₂ ∈ (1,2], respectively. We present an explicit difference approximation and an implicit difference approximation for the equation with initial and boundary conditions in a finite domain. Using mathematical induction, we prove that the implicit difference approximation is unconditionally stable and convergent, but the explicit difference approximation is conditionally stable and convergent. We also present two solution techniques: a Richardson extrapolation method is used to obtain higher order accuracy and the short-memory principle is used to investigate the effect of the amount of computations. A numerical example is given; the numerical results are in good agreement with theoretical analysis.     

Blow-up and global solutions for a class of time fractional nonlinear reaction–diffusion equation with weakly spatial source

This paper investigates the blow-up of solutions for a time fractional nonlinear reaction–diffusion equation with weakly spatial source. We first derive two sufficient conditions under which the solutions may blow up in finite time. Then, we prove the existence of global solution when the initial data are small enough. Moreover, the long time behavior of bounded solutions will be analyzed.     

Numerical approximation for a variable-order nonlinear reaction–subdiffusion equation

Fractional reaction–subdiffusion equations are widely used in recent years to simulate physical phenomena. In this paper, we consider a variable-order nonlinear reaction–subdiffusion equation. A numerical approximation method is proposed to solve the equation. Its convergence and stability are analyzed by Fourier analysis. By means of the technique for improving temporal accuracy, we also propose an improved numerical approximation. Finally, the effectiveness of the theoretical results is demonstrated by numerical examples.     

An $$H^1$$-Galerkin mixed finite element method for time fractional reaction–diffusion equation

In this article, an \(H^1\)-Galerkin mixed finite element (MFE) method for solving time fractional reaction–diffusion equation is presented. The optimal time convergence order \(O(\varDelta t^{2-\alpha })\) and the optimal spatial rate of convergence in \(H^1\) and \(L^2\)-norms for variable \(u\) and its gradient \(\sigma \) are derived. Moreover, some numerical results are shown to support our theoretical analysis.     

Finite difference approximations for the fractional Fokker–Planck equation

The fractional Fokker–Planck equation has been used in many physical transport problems which take place under the influence of an external force field. In this paper we examine some practical numerical methods to solve a class of initial-boundary value problems for the fractional Fokker–Planck equation on a finite domain. The solvability, stability, consistency, and convergence of these methods are discussed. Their stability is proved by the energy method. Two numerical examples are also presented to evaluate these finite difference methods against the exact analytical solutions.     

Conservation laws for a variable coefficient nonlinear diffusion–convection–reaction equation

We find the Lie point symmetries of a class of second-order nonlinear diffusion–convection–reaction equations containing an unspecified coefficient function of the independent variable t and determine the subclasses of these equations which are nonlinearly self-adjoint. By using a general theorem on conservation laws proved recently by N.H. Ibragimov we establish conservation laws corresponding to the aforementioned Lie point symmetries, one by one, for the simultaneous system of the original equation together with its adjoint equation through a formal Lagrangian. Particularly, for the nonlinearly self-adjoint subclasses, we construct conservation laws for the corresponding equations themselves.     

Approximate solution to the time–space fractional cubic nonlinear Schrodinger equation

By introducing the fractional derivatives in the sense of Caputo, we use the adomian decomposition method to construct the approximate solutions for the cubic nonlinear fractional Schordinger equation with time and space fractional derivatives. The exact solution of the cubic nonlinear Schrodinger equation is given as a special case of our approximate solution. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equation.     

Interface procedures for finite difference approximations of the advection–diffusion equation

We investigate several existing interface procedures for finite difference methods applied to advection–diffusion problems. The accuracy, stiffness and reflecting properties of various interface procedures are investigated.The analysis and numerical experiments show that there are only minor differences between various methods once a proper parameter choice has been made.     

Viability for nonlinear multi-valued reaction–diffusion systems

In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, ${\mathcal{K}}In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, K{\mathcal{K}} be a nonempty and locally closed subset in \mathbbR ×X×Y, A:D(A) í X\rightsquigarrow X, B:D(B) í Y\rightsquigarrow Y{\mathbb{R} \times X\times Y,\, A:D(A)\subseteq X\rightsquigarrow X, B:D(B)\subseteq Y\rightsquigarrow Y} two m-dissipative operators, F:K ? X{F:\mathcal{K} \rightarrow X} a continuous function and G:K \rightsquigarrow Y{G:\mathcal{K} \rightsquigarrow Y} a nonempty, convex and closed valued, strongly-weakly upper semi-continuous (u.s.c.) multi-function. We prove a necessary and a sufficient condition in order that for each (t,x,h) ? K{(\tau,\xi,\eta)\in \mathcal{K}}, the next system

{ lc u￠(t) ? Au(t)+F(t,u(t),v(t))    t 3 tv￠(t) ? Bv(t)+G(t,u(t),v(t))    t 3 tu(t)=x,    v(t)=h, \left\{ \begin{array}{lc} u'(t)\in Au(t)+F(t,u(t),v(t))\quad t\geq\tau \\ v'(t)\in Bv(t)+G(t,u(t),v(t))\quad t\geq\tau \\ u(\tau)=\xi,\quad v(\tau)=\eta, \end{array} \right.      16. Numerical approach for solving fractional relaxation–oscillation equation      Mustafa Gülsu  Yalçın ÖztürkAyşe Anapalı 《Applied Mathematical Modelling》2013 In this study, we will obtain the approximate solutions of relaxation–oscillation equation by developing the Taylor matrix method. A relaxation oscillator is a kind of oscillator based on a behavior of physical system’s return to equilibrium after being disturbed. The relaxation–oscillation equation is the primary equation of relaxation and oscillation processes. The relaxation–oscillation equation is a fractional differential equation with initial conditions. For this propose, generalized Taylor matrix method is introduced. This method is based on first taking the truncated fractional Taylor expansions of the functions in the relaxation–oscillation equation and then substituting their matrix forms into the equation. Hence, the result matrix equation can be solved and the unknown fractional Taylor coefficients can be found approximately. The reliability and efficiency of the proposed approach are demonstrated in the numerical examples with aid of symbolic algebra program, Maple.      17. Numerical study of Fisher's reaction–diffusion equation by the Sinc collocation method      《Journal of Computational and Applied Mathematics》2001,137(2):245-255 Fisher's equation, which describes a balance between linear diffusion and nonlinear reaction or multiplication, is studied numerically by the Sinc collocation method. The derivatives and integrals are replaced by the necessary matrices, and a system of algebraic equations is obtained to approximate solution of the problem. The error in the approximation of the solution is shown to converge at an exponential rate. Numerical examples are given to illustrate the accuracy and the implementation of the method, the results show that any local initial disturbance can propagate with a constant limiting speed when time becomes sufficiently large. Both the limiting wave fronts and the limiting speed are independent of the initial values.      18. On one model problem for the reaction–diffusion–advection equation      M. A. Davydova  S. A. Zakharova  N. T. Levashova 《Computational Mathematics and Mathematical Physics》2017,57(9):1528-1539 The asymptotic behavior of the solution with boundary layers in the time-independent mathematical model of reaction–diffusion–advection arising when describing the distribution of greenhouse gases in the surface atmospheric layer is studied. On the basis of the asymptotic method of differential inequalities, the existence of a boundary-layer solution and its asymptotic Lyapunov stability as a steady-state solution of the corresponding parabolic problem is proven. One of the results of this work is the determination of the local domain of the attraction of a boundary-layer solution.      19. Stability of solutions to extremum problems for the nonlinear convection–diffusion–reaction equation with the Dirichlet condition      R. V. Brizitskii  Zh. Yu. Saritskaya 《Computational Mathematics and Mathematical Physics》2016,56(12):2011-2022 The solvability of the boundary value and extremum problems for the convection–diffusion–reaction equation in which the reaction coefficient depends nonlinearly on the concentration of substances is proven. The role of the control in the extremum problem is played by the boundary function in the Dirichlet condition. For a particular reaction coefficient in the extremum problem, the optimality system and estimates of the local stability of its solution to small perturbations of the quality functional and one of specified functions is established.      20. Lattice Boltzmann model for the bimolecular autocatalytic reaction–diffusion equation      Jianying Zhang  Guangwu Yan 《Applied Mathematical Modelling》2014 A lattice Boltzmann model for the bimolecular autocatalytic reaction–diffusion equation is proposed. By using multi-scale technique and the Chapman–Enskog expansion on complex lattice Boltzmann equation, we obtain a series of complex partial differential equations, complex equilibrium distribution function and its complex moments. Then, the complex reaction–diffusion equation is recovered with higher-order accuracy of the truncation error. This equation can be used to describe the bimolecular autocatalytic reaction–diffusion systems, in which a rich variety of behaviors have been observed. Based on this model, the Fitzhugh–Nagumo model and the Gray–Scott model are simulated. The comparisons between the LBM results and the Alternative Direction Implicit results are given in detail. The numerical examples show that assumptions of source term can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the complex reaction–diffusion equation.

Numerical approach for solving fractional relaxation–oscillation equation

In this study, we will obtain the approximate solutions of relaxation–oscillation equation by developing the Taylor matrix method. A relaxation oscillator is a kind of oscillator based on a behavior of physical system’s return to equilibrium after being disturbed. The relaxation–oscillation equation is the primary equation of relaxation and oscillation processes. The relaxation–oscillation equation is a fractional differential equation with initial conditions. For this propose, generalized Taylor matrix method is introduced. This method is based on first taking the truncated fractional Taylor expansions of the functions in the relaxation–oscillation equation and then substituting their matrix forms into the equation. Hence, the result matrix equation can be solved and the unknown fractional Taylor coefficients can be found approximately. The reliability and efficiency of the proposed approach are demonstrated in the numerical examples with aid of symbolic algebra program, Maple.     

Numerical study of Fisher's reaction–diffusion equation by the Sinc collocation method

Fisher's equation, which describes a balance between linear diffusion and nonlinear reaction or multiplication, is studied numerically by the Sinc collocation method. The derivatives and integrals are replaced by the necessary matrices, and a system of algebraic equations is obtained to approximate solution of the problem. The error in the approximation of the solution is shown to converge at an exponential rate. Numerical examples are given to illustrate the accuracy and the implementation of the method, the results show that any local initial disturbance can propagate with a constant limiting speed when time becomes sufficiently large. Both the limiting wave fronts and the limiting speed are independent of the initial values.     

On one model problem for the reaction–diffusion–advection equation

The asymptotic behavior of the solution with boundary layers in the time-independent mathematical model of reaction–diffusion–advection arising when describing the distribution of greenhouse gases in the surface atmospheric layer is studied. On the basis of the asymptotic method of differential inequalities, the existence of a boundary-layer solution and its asymptotic Lyapunov stability as a steady-state solution of the corresponding parabolic problem is proven. One of the results of this work is the determination of the local domain of the attraction of a boundary-layer solution.     

Stability of solutions to extremum problems for the nonlinear convection–diffusion–reaction equation with the Dirichlet condition

The solvability of the boundary value and extremum problems for the convection–diffusion–reaction equation in which the reaction coefficient depends nonlinearly on the concentration of substances is proven. The role of the control in the extremum problem is played by the boundary function in the Dirichlet condition. For a particular reaction coefficient in the extremum problem, the optimality system and estimates of the local stability of its solution to small perturbations of the quality functional and one of specified functions is established.     

Lattice Boltzmann model for the bimolecular autocatalytic reaction–diffusion equation

A lattice Boltzmann model for the bimolecular autocatalytic reaction–diffusion equation is proposed. By using multi-scale technique and the Chapman–Enskog expansion on complex lattice Boltzmann equation, we obtain a series of complex partial differential equations, complex equilibrium distribution function and its complex moments. Then, the complex reaction–diffusion equation is recovered with higher-order accuracy of the truncation error. This equation can be used to describe the bimolecular autocatalytic reaction–diffusion systems, in which a rich variety of behaviors have been observed. Based on this model, the Fitzhugh–Nagumo model and the Gray–Scott model are simulated. The comparisons between the LBM results and the Alternative Direction Implicit results are given in detail. The numerical examples show that assumptions of source term can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the complex reaction–diffusion equation.