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Stability of Exact and Discrete Energy for Non-Fickian Reaction-Diffusion Equations with a Variable Delay
This paper is concerned with the stability of non-Fickian reaction-diffusion equations with a variable delay. It is shown that the perturbation of the energy function of the continuous problems decays exponentially, which provides a more accurate and convenient way to express the rate of decay of energy. Then, we prove that the proposed numerical methods are sufficient to preserve energy stability of the continuous problems. We end the paper with some numerical experiments on a biological model to confirm the theoretical results.
Reaction-diffusion equations with delay are widely proposed as models for biological, physical, and control systems [1–5]. Over the past several years, such equations have been extensively investigated and several important properties such as existence and stability of the travelling wavefront are well understood (e.g., [6–9]). It is known that traveling waves describe transition processes from the physical point of view. If the reaction is very fast, the speed of propagation will become rather large (e.g., [10–12]). This pathologic behavior is not presented in the physical phenomena but introduced by some mathematical models (cf. ). In order to avoid the unphysical behavior, certain memory effects are taken into account in the mathematical models. In [13–15], the time memory flux is proposed by assuming that a flux observed at some time should have something to do with the gradient of the solution at some past time; that is, The first order approximation of the above formula gives Its solution satisfies In fact, the memory term is presented to avoid the infinite propagation speed in flux definition. These ideas yield the study on the following non-Fickian reaction-diffusion equation with a constant delay: where and .
Recently, many researchers have investigated these kinds of equations. The existence and uniqueness of the solutions were derived by Chang . The well posedness of the integrodifferential models was studied by Araújo et al. in . Later, they further studied the effect of the integrodifferential term in the qualitative behavior of the solution of the diffusion equations in . Zhang and Vandewalle  studied the general linear methods for solving the nonlinear integrodifferential equations with memory. Wang et al. investigated the asymptotic stability of exact and discrete solutions of nonlinear integrodifferential equation in [20–22]. Wen et al. considered dissipativity of the functional differential equations . For a more detailed description of this subject as well as its open problems, we refer the readers to the books [1–3, 24, 25], the papers [26–32], and the references therein.
In particular, some continuous and discrete energy estimates, which are determined by the -norm of the solution and the estimates for the -norm of the past in time of its gradient, were derived by Branco et al. in . However, the derived energy estimates rely on time. When time goes to infinity, the energy estimates seem to become boundless. Later, inspired by their instructive work, Li et al. further derived some asymptotic stability of problem in . The stability result implies that the perturbation of the energy estimate tends to zero as approaches infinity. However, it does not imply anything about how long it takes to decay. Moreover, when referring to the impact of various factors (e.g., environment, temperature, and other potential effects) in some real-world problems, the model should be modified to an equation with a variable delay, which makes the numerical simulation and its analysis more complicated. Hence, some further stability results for the non-Fickian equations with a variable delay are much more significant and challenging from both the physical and mathematical points of view.
One aim of the paper is to study the exponential stability of some more general models. Consider the non-Fickian reaction-diffusion equation with a variable delay where , , the boundary conditions and initial condition We find that, the perturbation of the energy estimate decays exponentially. This exponential stability result provides a more accurate and convenient way to express the rate of decay. Meanwhile, the stability result can also be applied to the equations investigated in [13–15]. The given results imply that the perturbations of the energy of the equations are controlled by the initial perturbations from the systems.
The other aim is to investigate the discrete energy stability of (5). Since finite element method can be easily designed for high order of accuracy in space (see, e.g., [34, 35]), we introduce the method to solve the equation. Exponential stability of the semidiscrete system is derived. Then, the implicit Euler method is further applied to discretize the equation. And the integral is approximated by the right rectangular rule and the delay term is approximated by a linear interpolation. It is proved that the proposed numerical methods are sufficient to preserve stability of the underlying systems. Besides, if we discretize the diffusion term by using the centered difference operator, the energy stability of full discrete non-Fickian equation with a variable delay can also be derived. The given result indicates that the perturbations of the numerical solutions vanish eventually. Finally, a numerical simulation on a biological model is presented to confirm the theoretical and numerical results.
The rest of the paper is organized as follows. In Section 2, we discuss energy stability of non-Fickian delay reaction-diffusion equations. Section 3 describes in detail the stability of the numerical methods for the equation. Section 4 shows experimental studies for verifying the proposed results. Finally, conclusions and discussions for this paper are summarized in Section 5.
2. Exponential Stability
The following lemma can be derived from Lemma 2.1 in , where it was used to investigate the dissipativity of delay differential equations. Here it will play a key role in discussing the energy stability of the problem (5).
Lemma 1. Suppose that Here and , are continuous functions such that , and for all with constants and . Then where and is defined as
For the stability analysis, we need to consider the perturbed problem, that is, non-Fickian delay reaction-diffusion equation (5) with a different initial condition . Its solution is denoted by and satisfies the following equation:
Besides, let denote the inner product in and by we denote the corresponding induced norm. To simplify the notation, we represent by if is defined in . The energy function is defined by Then, we have the following result.
Theorem 2. Assume that for each If where and are continuous and there exists such that then where and is defined as
Proof. Let ; we can obtain the following equation:
Multiplying each member of (18) by with respect to and integrating by part, we find
As in , we have where we have used the Poincaré inequality.
Now, using assumption (14), we have Therefore,
Then, we derive the following inequality:
Now, applying Lemma 1, we finally have the stability result.
It is remarkable that formula (16) implies that Hence, the perturbation of the solution of the problem also decays exponentially.
When the right-hand side function of the problem (5) does not possess the delay term, the problem degenerates into a non-Fickian reaction-diffusion equation: which is also discussed in . We have the following conclusion.
Corollary 3. Assume that for each If where are continuous and , for . Then there exist constants and such that
Corollary 4. Assume that for each If where and are continuous and there exists such that then there exist constants and such that
If we assume that where and and are continuous functions, there may exist a bounded set and a time , such that for any given initial function , the corresponding energy of the problem is contained in for all . The analogous results and their conclusions for non-Fickian equations with a variable delay can be derived without any difficulty. We do not list them here for brevity.
3. Stability of the Numerical Approximation
In this section, we will concentrate on the stability of two kinds of numerical approximation. Here and below, for the discretization of system (5), we divide the interval with a mesh: with the space stepsize , where is a positive integer. And is a time stepsize, where is a positive integer. We will numerically solve the problem at time , .
3.1. Stability of the Finite Element Approximation
Firstly, the finite element method is introduced to discrete the problems. Let be a finite dimensional subspace of ; then the semidiscrete finite element method is to find , such that, for all test functions, and , satisfying
The exponential stability of semidiscrete approximation is shown as follows.
Theorem 5. Assume that conditions in Theorem 2 hold; then the difference satisfies the following formulae: where the parameters and are defined in Theorem 2 and with satisfying the semidiscrete finite element approximation to the perturbed problem.
Proof. The proof is similar to that of Theorem 2.
If we further apply the right rectangular rule to approximate the integral and implicit Euler method to discrete (36), then the full discrete numerical approximation is to find , such that, for all test functions, and , satisfying for Here denotes an approximation to , which is obtained by the following linear interpolation procedure at the point : where with integer and .
Now, we can establish the discrete version of the energy:
Then, we have the following result.
Proof. The difference satisfies the following equation:
Taking in the above formula, we find that
where we have used condition (14).
As in , we have
Substituting (44) into (43) yields
Applying the following discrete Poincaré inequality and to (45), we arrive at where we have used the inequality .
It follows from (47) that where .
In fact, in view of condition (15), there exists a constant , such that Hence, Together with (49) and (52), we have the final conclusion.
Theorem 5 shows that, for any given initial perturbation, the perturbation of the corresponding discrete energy of the problem decreases in time. Meantime, noting that in the energy function, the difference tends to zero very quickly.
3.2. Stability of the Finite Difference Approximation
In this subsection, we will apply the central finite difference method to discrete the diffusion term and discuss the stability of the full discrete problems.
Let be the time stepsize and denote the numerical approximation to with ; then the full discrete finite difference approximation for (5) can be written as where and the argument denotes an approximation to , which is obtained by the linear interpolation procedure at the point where with integer and .
The discrete version of the energy of the finite difference approximation is defined by where is the backward finite difference operator:
Proof. The proof is similar to the one done in Theorem 6 with the following relation:
In this section, we will provide a numerical experiment to illustrate the given results.
Consider the following Non-Fickian Mackey-Glass equations : where , , , , and are given parameters, is an even positive integer, and is a nonnegative function.
There is no difficulty in verifying that condition (14) holds (). If there exists such that , the perturbation of the energy estimate decays exponentially. For the numerical simulation of the model, we set the parameter , , , the space stepsize , and time stepsize and apply -degree finite element method to solve the problem on with the following different initial conditions: Some statistics of the numerical results with different parameters and are shown in Tables 1 and 2, where the discrete energy and of the problems decreases in time very quickly. Moreover, The numerical differences with the parameters and are shown in Figure 1, where with . The numerical differences (the blue lines) are all bounded by the exponential function (the red line). Clearly, they all confirm the results in this paper.