The generating equations method: Constructing exact solutions to delay reaction–diffusion systems and other non-linear coupled delay PDEs (original) (raw)

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Non-linear delay reaction–diffusion equations

Non-linear delay reaction–diffusion 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] and references in them). It is noteworthy that similar equations occur in the mathematical theory of artificial neural networks and the results are used for signal and image processing as well as in image

Simple modification of the generating equations method

1°. Let Eq. (10) admit a generalized separable solution of the form (5) that satisfies a functional constraint of the second kind (12) and let the functions ψ1n(t) be described by some delay ODEs. Then the more complex non-linear reaction–diffusion system of coupled delay equationsut=kuxx+bu+(au+c)f(z1,z2),wt=kwxx+bw+(aw+c)g(z1,z2),z1=z(u,u¯),z2=z(w,w¯),where f(z1,z2) and g(z1,z2) are arbitrary functions of two arguments, admits an exact solution of the form (5), (6) withφ1n(x)=φ2n(x)=φn(x).

Remark 4

Let

Generating equation contains an arbitrary function f(z) with z=u¯/u

Let us take the delay reaction–diffusion equationut=kuxx+uf(u¯/u)with an arbitrary function f(z) of a single argument, z=u¯/u, to be the generating equation.

1°. Eq. (22) admits, for example, multiplicative separable solutions of the form [30]:u=cos(αx)ψ(t),where α is an arbitrary constant and ψ(t) is a function described by the delay ordinary differential equationψ′(t)=−kα2ψ(t)+ψ(t)f(ψ(t−τ)/ψ(t)).Solution (23) satisfies the functional constraint of the second kind (12) with z=u¯/u=ψ(t−τ)/ψ(t).

Delay reaction–diffusion systems containing two arbitrary functions of three arguments

In this section, we consider three delay reaction–diffusion systems containing two arbitrary functions of three arguments. For the exact solutions obtained below, all of the three arguments satisfy either functional constraints of the second kind (12) or functional constraints of the first kind (11) simultaneously.

Generating equation contains one arbitrary function

Consider the generating equationut=a(eλuux)x+b+e−λuf(eλu−eλu¯),which contains an arbitrary function f(z) of a single argument, z=eλu−eλu¯, and is a special case of Eq. (18).

Eq. (87) admits the functional separable solutionu=1λln[φ(x)+ψ(t)],φ(x)=−bλ2ax2+C1x+C2,where _C_1 and _C_2 are arbitrary constants and ψ(t) is a function satisfying the delay ordinary differential equationψ′(t)=λf(ψ(t)−ψ(t−τ)).Solution (88) satisfies the functional constraint of the second kind (12) with z=eλu−eλu¯=ψ(t)−ψ(t−τ).

Construction of exact solutions to reaction–diffusion systems with time-varying delays

Below we outline a simple procedure for constructing reaction–diffusion systems with time-varying delays that admit generalized separable solutions of the form (5), (6).

Suppose there are generating equations that have exact solutions satisfying functional constraints of the second kind (12). Then the single constant delay time τ in the coupled delay PDEs can be replaced with two different time-varying delays as follows:u¯=u(x,t−τ)⟹u˜=u(x,t−τ1),w¯=w(x,t−τ)⟹w˜=w(x,t−τ2),where τ1=τ1(t) and τ2=τ2(t)

Utilization of modified functional constraints of the second kind

Sometimes, instead of the functional constraints of the second kind (12), one can use modified functional constraints of the second kind of the formz(u,u¯)=q(ζ),u¯=u(x,t−τ),ζ=t+θ(x),where θ(x) is some function. Rather than verbally describing the procedure for utilizing constraints of the form (108), we will give a few illustrative examples.

Example 1

Let us look once again at the delay reaction–diffusion system (34), (35). We will seek its exact solutions in the formu=exp(α1x)U(ζ),w=exp(α2x)W(ζ),ζ=t+βx,

Multicomponent systems of non-linear delay reaction–diffusion equations

The generating equations method is easy to extend to multicomponent systems of delay reaction–diffusion equations. Below we give a few illustrative examples.

Example 1

Consider the system containing three coupled PDEs:(u1)t=k1(u1)xx+u1f1(u¯1/u1,u¯2/u2,u¯3/u3),(u2)t=k2(u2)xx+u2f2(u¯1/u1,u¯2/u2,u¯3/u3),(u3)t=k3(u3)xx+u3f3(u¯1/u1,u¯2/u2,u¯3/u3),where fn=fn(z1,z2,z3) are arbitrary functions of three arguments. System (116) is obtained using the generating equation (22) three times by replacing u with u n, k

Non-linear systems of delay higher-order PDEs

The generating equations method can be generalized to non-linear systems of higher-order delay PDEs. The generating equations can include various-order derivatives with respect to x and t as well as mixed derivatives. In particular, the first derivatives u t and w t in Eqs. (18), (19) and system (20), (21) can be replaced with the second derivatives u tt and w tt.

Example 1

Instead of system (34), (35), one can consider the more complex system containing second-order derivatives in t and fourth-order

Brief conclusions

To summarize, we have presented a new method, called the generating equations method, that utilizes exact solutions of single non-linear delay reaction–diffusion equations to construct exact solutions of more complex systems of coupled delay equations. To illustrate the potential of the method, we have obtained generalized and functional separable solutions to a number of non-linear delay reaction–diffusion systems of the form (3), (4). Importantly, all of the systems considered are of fairly