Functional constraints method for constructing exact solutions to delay reaction–diffusion equations and more complex nonlinear equations (original) (raw)

Introduction

Nonlinear delay partial differential equations and systems of coupled equations arise in biology, biophysics, biochemistry, chemistry, medicine, control, climate model theory, ecology, economics, and many other areas (e.g. see the studies [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11] and references in them). It is noteworthy that similar equations occur in the mathematical theory of artificial neural networks, whose results are used for signal and image processing as well as in image recognition problems [12], [13], [14], [15], [16], [17], [18], [19], [20], [21].

The present paper deals with nonlinear delay reaction–diffusion equations [1], [3], [11], [22] of the formut=kuxx+F(u,w),w=u(x,t-τ).A number of exact solutions to the heat equation with a nonlinear source, which is a special case of Eq. (1) without delay and with F(u,w)=f(u), are listed, for example, in [23], [24], [25], [26], [27], [28], [29]. A comprehensive survey of exact solutions to this nonlinear equation can be found in the handbook [30]; it also describes a considerable number of generalized and functional separable solutions to nonlinear reaction–diffusion systems of two coupled equations without delay.

The list of known exact solutions to Eq. (1) is quite limited.

In general, Eq. (1) admits traveling-wave solutions, u=u(αx+βt). Such solutions are dealt with in many studies (e.g. see the papers [2], [3], [4], [5], [6], [7] and references in them). Simple separable solutions of Eq. (1) were studied in Ref. [22].

A complete group analysis of the non-linear differential–difference equation (1) was carried out in [11]. Four equations of the form (1) were found to admit invariant solutions; two of these equations are of limited interest, since they have degenerate solutions (linear in x). There was only one equation that involved an arbitrary function and had a non-degenerate solution:ut=kuxx+u-alnu+f(wu-b),b=eaτ,where f(z) is an arbitrary function. The exact solution to this equation found in [11] wasu=exp(Cxe-at)φ(t),where C is an arbitrary constant and φ(t) is a function satisfying the delay ordinary differential equationφ′(t)=φ(t)C2ke-2at-alnφ(t)+f(φ(t-τ)φ-b(t)).The other equation obtained in Ref. [11] that had a non-degenerate solution coincides, up to notation, with a special case of Eq. (4), at f(z)=c1+c2lnz.

Remark 1

It is noteworthy that Eq. (2) is explicitly dependent on the delay time τ, which corresponds to a more general kinetic function F(u,w,τ) than in Eq. (1). It is only at a=0 in (2) that we have a kinetic function explicitly independent of τ:F(u,w)=uf(w/u); in this case, (3) represents a separable solution, u=eCxφ(t).

In what follows, the term ‘exact solution’ with regard to nonlinear partial differential–difference equations, including delay partial differential equations, is used in the following cases:

(i) the solution is expressible in terms of elementary functions or in closed form with definite or indefinite integrals;
(ii) the solution is expressible in terms of solutions to ordinary differential or ordinary differential–difference equations (or systems of such equations);
(iii) the solution is expressible in terms of solutions to linear partial differential equations.

Combinations of cases (i)–(iii) are also allowed.

This definition generalizes the notion of an exact solution used in Ref. [30] with regard to nonlinear partial differential equations.

Remark 2

Solution methods and various applications of linear and nonlinear ordinary differential–difference equations, which are much simpler than nonlinear partial differential–difference equations, can be found, for example, in Refs. [31], [32], [33], [34], [35], [36].

Remark 3

For interesting biomedical and numerical applications of nonlinear delay partial differential equations and systems of such equations, see, for example, the studies [37], [38], [39] and references in them.

Section snippets

General description of the functional constraints method

Consider a wide class of nonlinear delay reaction–diffusion equations:ut=kuxx+uf(z)+wg(z)+h(z),w=u(x,t-τ),z=z(u,w),where f(z),g(z), and h(z) are arbitrary functions

and z=z(u,w) is a given function. In addition, we will sometimes consider more complex equations where f,g, and h can additionally depend on the independent variables x or/and t explicitly.

We look for generalized separable solutions of the formu=∑n=1NΦn(x)Ψn(t),where the functions Φn(x) and Ψn(t) are to be determined in the analysis.

Remark 4

The equation contains one arbitrary function dependent on w/u

Equation 1. Consider the equationut=kuxx+uf(w/u),which is a special case of Eq. (5) with g=h=0 and z=w/u.

1.1. The functional constraint of the second kind (8) becomesw/u=q(t),w=u(x,t-τ).It is clear that the difference equation (11) can be satisfied with a simple separable solutionu=φ(x)ψ(t),which gives q(t)=ψ(t-τ)/ψ(t). Substituting (12) into (10) and separating the variables, we arrive at the following equations for φ(x) and ψ(t):φ″=aφ,ψ′(t)=akψ(t)+ψ(t)f(ψ(t-τ)/ψ(t)),where a is an arbitrary

Equations with one arbitrary function dependent on a linear combination of u and w

Equation 2. Consider the equationut=kuxx+bu+f(u-w),which is a special case of Eq. (5) with f(z)=b,g=0, and z=u-w (the function h has been renamed f).

2.1. In this case, the functional constraint of the second kind (8) has the formu-w=q(t),w=u(x,t-τ).It is clear that the additive separable solutionu=φ(x)+ψ(t)satisfies the difference equation (28). We have q(t)=ψ(t)-ψ(t-τ). Substituting (29) into (27) and separating the variables, one arrives at equations for determining φ(x) and ψ(t):kφxx″+bφ=a,ψt

Equations with two arbitrary functions dependent on a linear combination of u and w

Equation 5. Now consider the more general equationut=kuxx+uf(u-w)+wg(u-w)+h(u-w),where f(z),g(z), and h(z) are arbitrary functions; in this case, either function f or g can be set equal to zero without loss of generality.

5.1. The difference constraint (7) for Eq. (47) has the form (34). The linear difference equation (34) can be satisfied, as previously, with a generalized separable solution of the form (35). As a result, one can obtain equations for determining φ(x) and ψ(x); these equations

Equation with two arbitrary functions dependent on u2+w2

Equation 8. Now consider the equationut=kuxx+ufu2+w2+wgu2+w2,which contains a nonlinear (quadratic) argument, z=u2+w2.

The difference constraint (7) for Eq. (55) has the formu2+w2=p(x),w=u(x,t-τ).The nonlinear difference equation (53) can be satisfied by settingu=φn(x)cos(λnt)+ψn(x)sin(λnt),λn=π(2n+1)2τ,n=0,±1,±2,….It is not difficult to verify thatw=(-1)nφn(x)sin(λnt)+(-1)n+1ψn(x)cos(λnt),andu2+w2=φn2(x)+ψn2(x)=p(x).Substituting (57) into (55) and splitting the resulting expression with respect

More general nonlinear delay partial differential equations

Now consider nonlinear partial differential–difference equations of the more general formL[u]=M[u]+uf(z)+wg(z)+h(z),w=u(x,t-τ),z=z(u,w),where L and M are arbitrary constant–coefficient linear differential operators with respect to t and x:L[u]=∑i=1kbi∂iu∂ti,M[u]=∑i=1mai∂iu∂xi.As before, f(z),g(z), and h(z) are arbitrary functions and z=z(u,w) is a given function.

By setting L[u]=utt and M[u]=a2uxx in (58), we get the nonlinear delay Klein–Gordon equation (a delay hyperbolic PDE)utt=a2uxx+uf(z)+wg

General partial differential equations with time-varying delay

In very much the same manner, one can obtain exact solutions to some nonlinear partial functional differential equations with time-varying delay of general form.

Below we give a few simple examples to illustrate the aforesaid.

Equation 17. Consider nonlinear partial functional differential equations of the formL[u]=M[u]+uf(w/u),w=u(x,ζ(t)),where ζ(t) is a given function. In applications (e.g. see Ref. [20]), it is conventional to write ζ(t)=t-τ(t), where τ(t) is a time-varying delay such that 0⩽τ(

Brief conclusions

To summarize, we have proposed a new method, which we call the functional constraints method, for constructing exact solutions of nonlinear delay reaction–diffusion equations of the formut=kuxx+F(u,w),where u=u(x,t),w=u(x,t-τ), and τ is the relaxation time. The method is based on searching for generalized separable solutions of the formu=∑n=1Nξn(x)ηn(t),with the functions ξn(x) and ηn(t) determined from additional functional constraints (difference or functional equations) and the original