New generalized and functional separable solutions to non-linear delay reaction–diffusion equations (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). 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

Reaction–diffusion equations containing one arbitrary function of one variable

Equation

1: Consider Eq. (1) of the formut=kuxx−culnu+uf(w/ua),a>0,where f(z) is an arbitrary function.

1. Eq. (5) with c=(lna)/τ admits a multiplicative separable solution:u=exp(Ae−ct)φ(x),where A is an arbitrary constant
  and φ(x) is a function described by the non-linear autonomous ordinary differential equation kφxx″−cφlnφ+φf(φ1−a)=0.
1. Eq. (5) with c=(lna)/τ admits a functional separable solution:u=exp(Axe−ct)ψ(t),where A is an arbitrary constant
  and ψ(t) is a function described by the non-linear

Reaction–diffusion equations containing two arbitrary functions of one variable

Now consider non-linear delay reaction–diffusion equations with the kinetic function containing two arbitrary functions of one variable.

Equation

3: Consider Eq. (1) of the formut=kuxx+u[f(u−aw)+g(w/u)],a>0,where f(z1) and g(z2) are arbitrary functions.

Below are three exact solutions, wherec=(lna)/τ,b=f(0)+g(1/a)−c.

1. Eq. (10) with b>0 admits a multiplicative separable solution:u=ect[C1cos(λx)+C2sin(λx)],λ=(b/k)1/2,where _C_1 and _C_2 are arbitrary constants.
1. Eq. (10) with b<0 admits another

Reaction–diffusion equations containing one arbitrary function of two variables

Now consider non-linear delay reaction–diffusion equations containing one arbitrary function of two variables of special form.

Equation

7: The equationut=kuxx+uf(u−w,w/u),where f(z1,z2) is an arbitrary function, admits a _τ_-periodic solution of the formu=V1(x,t;b),b=f(0,1),with the function V1(x,t;b) given by formulas (15), (16).

Equation

8: Consider the equationut=kuxx+uf(u−aw,w/u),a>0.

Below are three exact solutions, in whichc=(lna)/τ,b=f(0,1/a)−c.

1. Eq. (31) with b>0 admits a multiplicative

More complex delay reaction–diffusion equations

Equation

13: Eq. (29) corresponds to a special case of the equationut=kuxx+uf(η,ζ),η=φ(u,w)−φ(w,u),ζ=ψ(u,w)/ψ(w,u),where φ(u,w) and ψ(u,w) are arbitrary functions; it admits the same _τ_-periodic solution u=V1(x,t;b) with b=f(0,1).

Remark 7

The more complex equationut=kuxx+uf(η1,…,ηp,ζ1,…,ζq),ηi=φi(u,w)−φi(w,u),ζj=ajψj(u,w)/ψj(w,u),p,q=1,2,…,i=1,…,p,j=1,…,q,where φi(u,w) and ψj(u,w) are arbitrary functions

and a j are arbitrary constants, also admits a _τ_-periodic solution u=V1(x,t;b) with b=f(0,…,0,a1,…,aq).

Some non-linear higher-order PDEs with delay

Equation

16: Consider the equationL[u]=uf(u−aw,w/u),a>0,where L[u] is an arbitrary linear constant-coefficient differential operator with respect to x and _t_L[u]=∑m+n=1Mbmn∂m+nu∂mx∂nt,bmn=const.

Eqs. (49), (50) admit exact solutionu=ectv(x,t),c=(lna)/τ.The function v(x,t) is _τ_-periodic and has the formv(x,t)=∑n=0Nexp(−λnx)[Ancos(βnt−γnx)+Bnsin(βnt−γnx)],where A n, B n, C n, and D n are arbitrary constants, N is an arbitrary nonnegative integer, βn=(2πn)/τ, and γ n and λ n are constants determined by