The functional constraints method: Application to non-linear delay reaction–diffusion equations with varying transfer coefficients (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], [11], [12] and references in them). 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 recognition

Equations involve arbitrary functions. The functional constraints method

For non-linear delay partial differential equations that involve arbitrary functions, the direct application of the method of generalized separation of variables turns out to be ineffective. It is more reasonable to treat such equations using a modification of the functional constraints method [32], which is briefly described below.

To be specific, we will be considering non-linear delay reaction–diffusion equations of the fairly general formut=[G(u)ux]x+H1(u)+H2(u)f(z),z=z(u,w),where f(z) is an

One-dimensional reaction–diffusion equations involving one arbitrary function of a single argument

The functions G(u), H1(u), and H2(u) in Equations 1–4 below, which have the form (6), are sought in power-law form and those in Equations 5–7 are sought as exponentials or constants.

Equation 1: Consider Eq. (6) of the formut=a(ukux)x+uf(w/u).

Eq. (10) involves an arbitrary function f(z) with _z_=w/u. In this case, 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 multiplicative separable solutionu=φ(x)ψ(

One-dimensional reaction–diffusion equations involving two arbitrary functions of a single argument

Equation 15: Consider the following equation:ut=a(ukux)x+uf(w/u)+uk+1g(w/u),where f(z) and g(z) are arbitrary functions.

Eq. (72) admits multiplicative separable solutions of the formu=eλtφ(x),where λ is a root of the algebraic (transcendental) equation λ=f(e−λτ)and φ(x) is a function described by the ODE a(φkφx′)x′+g(e−λτ)φk+1=0.If k≠−1, the substitution θ=φk+1 leads to a second-order constant-coefficient linear ODE. If k=−1, one should use the substitution θ=lnφ.

Equation 16: The equationut=a(u−

One-dimensional reaction–diffusion equations involving three arbitrary functions of a single argument

Equation 23: Consider the equationut=a[f′(u)ux]x+g(f(u)−f(w))+1f′(u)h(f(u)−f(w)),where f(u), g(z), and h(z) are arbitrary functions and the prime denotes a derivative.

Eq. (89) admits the functional separable solution in implicit formf(u)=At−g(Aτ)2ax2+C1x+C2,where _C_1 and _C_2 are arbitrary constants; the constant A is a root of the algebraic (transcendental) equation A−h(Aτ)=0.

Equation 24: Consider the equationut=a[f′(u)ux]x+f(u)g(f(w)/f(u))+f(u)f′(u)h(f(w)/f(u)),where f(u), g(z), and h(z) are

Three-dimensional reaction–diffusion equations involving one arbitrary function of a single argument

This section describes multidimensional generalizations of some of the one-dimensional equations and their solutions discussed above. We use the notation u=u(x,t)≡u(x1,…,xn,t) and w=u(x,t−τ). Two- and three-dimensional equations correspond to _n_=2 and _n_=3.

Equation 26: Consider the equationut=adiv(uk∇u)+uf(w/u)+buk+1.

1°: If k≠−1, Eq. (94) admits multiplicative separable solutions of the formu=ψ(t)φ1/(k+1)(x)with the function ψ(t) described by the delay ordinary differential equation (17) and φ(x)=

Three-dimensional reaction–diffusion equations involving two arbitrary functions of a single argument

Equation 34: The equationut=adiv(u−1/2∇u)+f(u1/2−w1/2)+u1/2g(u1/2−w1/2)admits generalized separable solutions of the formu=[φ(x)t+ψ(x)]2,where φ=φ(x) and ψ=ψ(x) are functions described by the partial differential equations2aΔφ+φg(τφ)−2φ2=0,2aΔψ+ψg(τφ)−2φψ+f(τφ)=0.Eq. (126) has a particular solution φ=φ0=const, where _φ_0 is a root of the algebraic (transcendental) equation g(τφ0)−2φ0=0. In this case, Eq. (127) becomes the Poisson equation aΔψ+12f(τφ0)=0.

Equation 35: The equationut=adiv(uk∇u)+f(uk+

Three-dimensional reaction–diffusion equations involving three arbitrary functions of a single argument

Equation 40: Consider the equationut=adiv[f′(u)∇u]+g(f(u)−f(w))+1f′(u)h(f(u)−f(w)),where f(u), g(z), and h(z) are arbitrary functions.

Eq. (142) admits the functional separable solution in implicit formf(u)=At+φ(x),where A is a root of the algebraic (transcendental) equation A−h(Aτ)=0 and φ(x) is a function described by the Poisson equationaΔφ+g(Aτ)=0.

Equation 41: Consider the equationut=adiv[f′(u)∇u]+f(u)g(f(w)/f(u))+f(u)f′(u)h(f(w)/f(u)),where f(u), g(z), and h(z) are arbitrary functions.

Eq.

Reaction–diffusion equations with time-varying delay

Most of the results presented above also apply to non-linear reaction–diffusion equations with time-varying delay τ=τ(t). Table 1 lists some of such equations, involving arbitrary functions, along with their solutions. In the determining delay ordinary differential equations, one should assume that τ=τ(t) in the unknown functions dependent on t and t−τ.

Example

Let us look at the first equation in Table 1. In the determining equation (14) for ψ(t), we assume that τ=τ(t) to obtain the delay ODE ψ′(t)=bψk+

Exact solutions as test problems for numerical methods

The exact solutions presented in this paper may be used as practical test problems to check the validity and evaluate the accuracy of numerical and approximate analytical methods for non-linear delay PDEs. Such problems are of particular importance for two reasons: (i) possible instability of solutions to non-linear delay equations and Hadamard ill-posedness of related problems, due to the delay (e.g., see [34], [43], [57]), and (ii) absence of rigorous mathematical validation and precise error

Brief conclusions

To sum up, we have presented exact solutions to one-dimensional non-linear delay reaction–diffusion equations of the form ut=[G(u)ux]x+F(u,w),where u=u(x,t) and w=u(x,t−τ), with τ denoting the delay time. All of the equations considered involve one, two, or three arbitrary functions of a single argument. We have described a number of generalized separable solutions of the form u=∑n=1Nφn(x)ψn(t) as well as functional separable solutions of the form u=U(z) with z=∑n=1Nφn(x)ψn(t). We have also