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Exact separable solutions of delay reaction–diffusion equations and other nonlinear partial functional-differential equations
Accepted Manuscript Exact separable solutions of delay reaction–diffusion equations and other non‐ linear partial functional-differential equations Andrei D. Polyanin, Alexei I. Zhurov PII: DOI: Reference:
S1007-5704(13)00321-3 http://dx.doi.org/10.1016/j.cnsns.2013.07.019 CNSNS 2875
To appear in:
Communications in Nonlinear Science and Numer‐ ical Simulation
Please cite this article as: Polyanin, A.D., Zhurov, A.I., Exact separable solutions of delay reaction–diffusion equations and other nonlinear partial functional-differential equations, Communications in Nonlinear Science and Numerical Simulation (2013), doi: http://dx.doi.org/10.1016/j.cnsns.2013.07.019
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Exact separable solutions of delay reaction–diffusion equations and other nonlinear partial functional-differential equations Andrei D. Polyanina,∗, Alexei I. Zhurovb,a,∗∗ a
Institute for problems in Mechanics, Russian Academy of Sciences, 101 Vernadsky Avenue, bldg 1, 119526 Moscow, Russia b Cardiff University, Heath Park, Cardiff CF14 4XY, UK
Abstract We present a number of exact multiplicative and additive separable solutions to nonlinear delay reaction–diffusion equations of the form ut = kuxx + F (u, w), where u = u(x, t), w = u(x, t − τ ), and τ is the delay time. All of the equations involve one arbitrary function of one argument. We also give a number of exact solutions to more complex nonlinear partial functional-differential equations with a time-varying delay τ = τ (t) as well as to equations with several delay times. We extend the results to nonlinear partial differential-difference equations containing arbitrary linear differential operators of any order in the independent variables x and t; in particular, these equations include the nonlinear delay Klein–Gordon equation. Keywords: exact solutions, separable solutions, reaction-diffusion equations with delay, nonlinear equations, partial differential equations with delay, partial differential-difference equations, partial functional-differential equations, time-varying delay, delay hyperbolic equations, nonlinear Klein–Gordon equation
∗
Principal corresponding author Corresponding author Email addresses: [email protected] (Andrei D. Polyanin), [email protected] (Alexei I. Zhurov) ∗∗
Preprint submitted to Elsevier
July 15, 2013
1. Introduction Delay partial differential equations arise in numerous areas of application, including biology, biophysics, biochemistry, chemistry, medicine, control, climate model theory, and many others (e.g., see the studies [1–6] 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 [7, 8]. The present paper deals with the nonlinear delay reaction–diffusion equation [1, 2, 6, 9] of the form ut = kuxx + F (u, w),
w = u(x, t − τ ).
(1)
A number of exact solutions to the heat equation with a nonlinear source, which is a special case of equation (1) without delay and with F (u, w) = f (u), are listed, for example, in [10–15]. Most comprehensive surveys of exact solutions to this nonlinear equation can be found in [16]. In general, equation (1) admits traveling-wave solutions u = u(αx + βt). Such solutions are analyzed in numerous studies (e.g., see [2–5] and references in them). For a Lie group analysis of the nonlinear differential-difference equation (1), see [6]. The study found four equations of the form (1) that admit invariant solutions; two of these equations have degenerate solutions (linear in x). The present paper lists a number of exact separable solutions to nonlinear delay reaction–diffusion equations and other nonlinear partial functional-differential equations. In what follows, the term ‘exact solution’ with regard to nonlinear partial differential-difference (functional-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 (functional-differential) equations; (iii) the solution is expressible in terms of solutions to linear partial differential equations. Combination of cases (i)–(iii) are also allowed. This definition generalizes the notion of an exact solution used in [16] with regard to nonlinear partial differential equations. 2
Remark 1. For solution methods and various applications of linear and nonlinear ordinary differential-difference and functional-differential equations, which are much simpler than nonlinear partial differential-difference and functional-differential equations, see, for example, [17–22].
The next two sections present multiplicative and additive separable solutions to nonlinear delay reaction–diffusion equations of the form (1) dependent on an arbitrary function f (z) with given z = z(u, w). 2. Equations involving an arbitrary function dependent on w/u Equation 1. Consider equation (1) with the kinetic function F (u, w) = uf (w/u).
(2)
1◦ . Equation (1) with (2) admits the following periodic multiplicative separable solution in the space coordinate x: u = [C1 cos(αx) + C2 sin(αx)]ψ(t),
(3)
where C1 , C2 , and α are arbitrary constants, and ψ(t) is a function satisfying the ordinary differential-difference equation ψt′ (t) = −kα2 ψ(t) + ψ(t)f ψ(t − τ )/ψ(t) . (4) 2◦ . Equation (1) with (2) admits another multiplicative separable solution: u = [C1 exp(−αx) + C2 exp(αx)]ψ(t),
(5)
where C1 , C2 , and α are arbitrary constants and ψ(t) satisfies the ordinary differentialdifference equation ψt′ (t) = kα2 ψ(t) + ψ(t)f ψ(t − τ )/ψ(t) . (6) Remark 2. Equations (4) and (6) admit exponential particular solutions of the form
ψ(t) = Aeλt , where A is an arbitrary constant and λ is a root of the transcendental equations λ = ∓kα2 + f (e−λτ ). Depending on the form of the function f , these equations as well as the other transcendental equations given below can have one, several, infinitely many, or zero solutions.
3
3◦ . Equation (1) with (2) admits the degenerate multiplicative separable solution u = (C1 x + C2 )ψ(t), (7) where C1 and C2 are arbitrary constants and ψ(t) is a function satisfying the ordinary differential-difference equation (4) with α = 0. 4◦ . Equation (1) with (2) admits multiplicative solutions with the factors dependent on different arguments: u = eαx+βt θ(z),
z = λx + γt,
(8)
where α, β, γ, and λ are arbitrary constants, and θ(z) is a function satisfying the ordinary differential-difference equation ′′ kλ2 θzz (z) + (2kαλ − γ)θz′ (z) + (kα2 − β)θ(z) + θ(z)f e−βτ θ(z − σ)/θ(z) = 0, σ = γτ.
Solution (8) can be treated as a nonlinear superposition of two traveling waves. Equation 2. Equation (1) with F (u, w) = au ln u + uf (w/u)
(9)
admits the multiplicative separable solution u = ϕ(x)ψ(t),
(10)
with the functions ϕ(x) and ψ(t) determined by the ordinary differential and differential-difference equations kϕ′′xx = bϕ − aϕ ln ϕ, ψt′ (t) = bψ(t) + ψ(t)f (ψ(t − τ )/ψ(t)) + aψ(t) ln ψ(t),
(11) (12)
and b being an arbitrary constant. Remark 3. Equation (11) is autonomous, and hence its general solution can be written in explicit form. A particular one-parameter solution to this equation is given by a b 1 2 ϕ = exp − (x + C) + + , 4k a 2
where C is an arbitrary constant.
4
3. Equations involving an arbitrary function dependent on u − w Equation 3. Consider equation (1) with the kinetic function F (u, w) = f (u − w).
(13)
1◦ . Equation (1) with (13) admits quadratic additive separable solutions in x: u = ax2 + bx + ψ(t),
(14)
where a and b are arbitrary constants, and ψ(t) is a function satisfying the ordinary differential-difference equation ψt′ (t) = 2ak + f ψ(t) − ψ(t − τ ) . (15) Equation (15) has particular solutions linear in t: ψ(t) = λt + C, where C is an arbitrary constant and λ is a root of the transcendental equation 2ak − λ + f (τ λ) = 0. 2◦ . Equation (1) with (13) has solutions generalizing solution (14): u = ax2 + bx + θ(z),
z = βx + γt,
(16)
where a, b, β, and γ are arbitrary constants, and θ(z) is a function satisfying the ordinary differential-difference equation ′′ γθz′ (z) = kβ 2 θzz (z) + 2ak + f (θ(z) − θ(z − σ)),
σ = γτ.
Equation 4. Now consider equation (1) with the kinetic function F (u, w) = bu + f (u − w).
(17)
1◦ . Equation (1) with (17) and kb > 0 admits periodic additive separable solutions in the space coordinate x: p u = C1 cos(αx) + C2 sin(αx) + θ(t), α = b/k, (18) 5
where C1 and C2 are arbitrary constants and θ(t) is a function satisfying the ordinary differential-difference equation θt′ (t) = bθ(t) + f θ(t) − θ(t − τ ) . (19) 2◦ . Equation (1) with (17) and kb < 0 admits the additive separable solution p u = C1 exp(−αx) + C2 exp(αx) + θ(t), α = −b/k, (20)
where C1 and C2 are arbitrary constants, with the function θ(t) determined by the ordinary differential-difference equation (19). 3◦ . Equation (1) with (17) and kb > 0 has exact solutions generalizing solution (18): p u = C1 cos(αx) + C2 sin(αx) + θ(z), z = βx + γt, α = b/k, (21)
where C1 , C2 , β, and γ are arbitrary constants, with the function θ(z) determined by the ordinary differential-difference equation ′′ γθz′ (z) = kβ 2 θzz (z) + bθ(z) + f (θ(z) − θ(z − σ)),
σ = γτ.
(22)
Unlike (18), solution (21) is not periodic in x; it can be treated as an interaction of a periodic standing wave with a traveling wave. 4◦ . Equation (1) with (17) and kb < 0 admits solutions generalizing solution (20): p u = C1 exp(−αx) + C2 exp(αx) + θ(z), z = βx + γt, α = −b/k, (23) where C1 , C2 , β, and γ are arbitrary constants, and θ(z) is function satisfying the ordinary differential-difference equation (22). 4. General reaction–diffusion equations with variable delay Most of the examples listed above can be generalized by considering more complex nonlinear partial functional-differential equations with time-varying delay of the general form ut = kuxx + F (u, w),
w = u(x, t − τ (t)),
(24)
with the relaxation time τ treated as a given function of t. Equations of the form (24) can be encountered in the literature [7]. 6
Equation 5. Equation (24) with the kinetic function (2) admits a periodic (in the space coordinate x) multiplicative separable solution of the form (3), with the function ψ(t) described by the ordinary differential-difference equation ψt′ (t) = −kα2 ψ(t) + ψ(t)f ψ(t − τ (t))/ψ(t) . (25)
Equation (24) with the kinetic function (2) admits another multiplicative separable solution of the form (5) where the function ψ(t) is determined by the ordinary differential-difference equation ψt′ (t) = kα2 ψ(t) + ψ(t)f ψ(t − τ (t))/ψ(t) . Equation (24) with the kinetic function (2) also admits a multiplicative separable solution of the form (7) with the function ψ(t) described by the ordinary differential-difference equation (25) with α = 0.
Equation 6. Equation (24) with the kinetic function (9) admits a multiplicative separable solution of the form (10) with the functions ϕ(x) and ψ(t) satisfying the ordinary differential and functional-differential equations kϕ′′xx = bϕ − aϕ ln ϕ, ψt′ (t) = bψ(t) + ψ(t)f ψ(t − τ (t))/ψ(t) + aψ(t) ln ψ(t)
and b being an arbitrary constant. The equation for ϕ coincides with (11); for its solutions, see Remark 3. Equation 7. Equation (24) with the kinetic function (13) admits an additive separable solution of the form (14) with the function ψ(t) determined by the ordinary functional-differential equation ψt′ (t) = 2ak + f ψ(t) − ψ(t − τ (t)) . (26)
Equation 8. Equation (24) with the kinetic function (17) admits two additive separable solutions of the form (18) and (20) where the function θ(t) is described by the ordinary functional-differential equation θt′ (t) = bθ(t) + f θ(t) − θ(t − τ (t)) . (27)
7
5. Reaction–diffusion equations with several delay times Consider diffusion–reaction equations of the from ut = kuxx + F (u, w1 , . . . , wn ),
wm = u(x, t − τm ),
(28)
where τm are relaxation times (m = 1, . . . , n), which are assumed to be independent arbitrary constants. Equations of the form (28) can be encountered in the literature [3, 7, 8]. In what follows, f (z1 , . . . , zn ) are arbitrary functions of their arguments. Equation 9. If F (u, w1, . . . , wn ) = uf (w1 /u, . . . , wn /u),
(29)
equation (28) admits a periodic (in the space coordinate x) multiplicative separable solution of the form (3), where the function ψ(t) is described by the ordinary differential-difference equation ψt′ (t) = −kα2 ψ(t) + ψ(t)f ψ(t − τ1 )/ψ(t), . . . , ψ(t − τn )/ψ(t) . (30)
Equation (28) with (29) admits another multiplicative separable solution of the form (5), where the function ψ(t) is determined by the ordinary differentialdifference equation ψt′ (t) = kα2 ψ(t) + ψ(t)f ψ(t − τ1 )/ψ(t), . . . , ψ(t − τn )/ψ(t) . (31) Remark 4. Equations (30) and (31) admit exponential solutions of the form
ψ(t) = Aeλt , where A is an arbitrary constant and λ is a root of the transcendental equations λ = ∓kα2 + f (e−λτ1 , . . . , e−λτn ). Equation (28) with (29) admits a degenerate multiplicative separable solution of the form (7) where the function ψ(t) is determined by the ordinary differential-difference equation (30) with α = 0. Equation (28) with (29) also admits more complex multiplicative solutions of the form (8).
8
Equation 10. If F (u, w1 , . . . , wn ) = au ln u + uf (w1/u, . . . , wn /u),
(32)
equation (28) admits a multiplicative separable solution of the form (10), with the functions ϕ(x) and ψ(t) determined by the ordinary differential and differentialdifference equations kϕ′′xx = bϕ − aϕ ln ϕ, ψt′ (t) = bψ(t) + ψ(t)f ψ(t − τ1 )/ψ(t), . . . , ψ(t − τn )/ψ(t) + aψ(t) ln ψ(t) (33) and b being an arbitrary constant. The equation for ϕ coincides with (11); for its solutions, see Remark 3. Equation 11. Equation (28) with F (u, w1, . . . , wn ) = f (u − w1 , . . . , u − wn ),
(34)
admits an additive separable solution of the form (14) where the function ψ(t) is described by the ordinary differential-difference equation ψt′ (t) = 2ak + f ψ(t) − ψ(t − τ1 ), . . . , ψ(t) − ψ(t − τn ) . (35) Equation (35) has particular solutions linear in t: ψ(t) = λt + C, where C is an arbitrary constant and λ is a root of the transcendental equation 2ak − λ + f (λτ1 , . . . , λτn ) = 0. Equation (28) with (34) also admits solutions of the form (16). Equation 12. Equation (28) with F (u, w1, . . . , wn ) = bu + f (u − w1 , . . . , u − wn )
(36)
and kb > 0 admits an additive separable solution of the form (18), where the function θ(t) is described by the ordinary differential-difference equation θt′ (t) = bθ(t) + f θ(t) − θ(t − τ1 ), . . . , θ(t) − θ(t − τn ) . (37)
Equation (28) with (36) and kb < 0 admits an additive separable solution of the form (20) where the function θ(t) is described by the ordinary differentialdifference equation (37). Equation (28) with (36) also admits two more complex equations of the form (21) and (23). 9
Remark 5. The constant time delays τm that appear in equation (28) with kinetic functions (29), (32), (34), and (36), which reduce to the ordinary functional-differential equations (30), (31), (33), (35), and (37), as well as in solutions of these equations, can be replaced with time-varying delays τm = τm (t), where τm (t) are arbitrary functions satisfying the conditions 0 ≤ τm (t) ≤ τ0 = const, m = 1, . . . , n.
6. Even more general nonlinear partial differential equations with delay Now consider nonlinear partial differential-difference equations of the more general form L[u] = M[u] + F (t, u, w), (38) w = u(x, t − τ ), x = (x1 , . . . , xm ), where L is an arbitrary linear differential operator with respect to t, whose coefficients can be dependent on t, L[u] =
n X
bi (t)
i=1
∂iu , ∂ti
(39)
and M is a linear differential operator of any (second or higher) order with respect to the space coordinates x1 , . . . , xm , whose coefficients can be dependent on x1 , . . . , xm . Note that in this section, the kinetic function F depends on three arguments. In particular, M can be an elliptic operator of the form M[u] =
m m X X ∂ ∂u ∂u kij (x) + ci (x) ; ∂x ∂x ∂x i j i m=1 i,j=1
(40)
such operators with ci = 0 are used in the studies [7, 8]. Also, M can be a biharmonic operator of the form M[u] = k∆∆u,
∆u ≡
m X ∂2u . 2 ∂x i i=1
Setting L[u] = utt , M[u] = a2 uxx , Ft = 0, and m = 1 in (38) yields the nonlinear delay Klein–Gordon equation utt = a2 uxx + F (u, w).
10
(41)
Such hyperbolic delay partial differential equations are encountered in the literature, for example, [23–25]. Below we list multiplicative and additive separable solutions of nonlinear delay partial differential equations of the form (38) that involve an arbitrary function of two arguments, f (t, z), with given z = z(u, w). The determining equations were derived using following simple properties of the operators L and M: L[ϕ(x)ψ(t)] = ϕ(x)L[ψ(t)], L[ϕ(x) + ψ(t)] = L[ψ(t)], M[ϕ(x)ψ(t)] = ψ(t)M[ϕ(x)], M[ϕ(x) + ψ(t)] = M[ϕ(x)]. Equation 13. The equation L[u] = M[u] + uf (t, w/u),
(42)
admits a multiplicative separable solution u = ϕ(x)ψ(t)
(43)
with the functions ϕ(x) and ψ(t) satisfying the linear partial differential equation and ordinary differential-difference equation M[ϕ] = aϕ,
(44)
L[ψ(t)] = aψ(t) + ψ(t)f t, ψ(t − τ )/ψ(t)
(45)
and a being an arbitrary constant. We specify below three cases where the form of particular solutions of equations (44) and (45) can be found: (i) If a = 0 and M is a linear constant-coefficient operator, equation (44) admits polynomial particular solutions. (ii) If a is any number and M is a linear constant-coefficientP operator, equation m (44) admits exponential solutions of the form ϕ(x) = B exp i=1 βi xi , where B is an arbitrary constant and β1 , . . . , βm are arbitrary constants connected by a single polynomial dispersion relation. (iii) If L is a linear operator of the form (39) with constant coefficients (bi = const) and the kinetic function is explicitly independent of t, f = f (w/u), equation (45) admits exponential solutions, ψ(t) = Aeλt , where A is an arbitrary constant and λ is a root of the transcendental equation n X
bi λi = a + f (e−τ λ ).
i=1
11
Remark 6. If L and M are both linear constant-coefficient operators and the function f is explicitly independent of t, equation (42) admits exact solutions of the form
m X βi xi θ(z), u = exp αt +
z = γt +
i=1
m X
λi x i ,
(46)
i=1
where α, βi , γ, and λi are arbitrary constants, and the function θ(z) is described by an ordinary differential-difference equation. Remark 7. Suppose the conditions listed in Remark 6 hold. Then, solutions of the form (46) will also be solutions of equation (42) with an additional term K[u], where K is an arbitrary constant-coefficient linear differential operator with mixed derivatives.
Equation 14. The equation L[u] = M[u] + bu ln u + uf (t, w/u),
(47)
admits a multiplicative separable solution of the form (43) where the functions ϕ(x) and ψ(t) are determined by the nonlinear partial differential equation and ordinary differential-difference equation M[ϕ] = aϕ − bϕ ln ϕ, L[ψ(t)] = aψ(t) + bψ(t) ln ψ(t) + ψ(t)f (t, ψ(t − τ )/ψ(t)), with a being an arbitrary constant. Equation 15. The equation L[u] = M[u] + bu + f (t, u − w)
(48)
admits an additive separable solution of the form u = ϕ(x) + ψ(t),
(49)
with the functions ϕ(x) and ψ(t) described by the linear partial differential equation and ordinary differential-difference equation M[ϕ] = a − bϕ,
(50)
L[ψ(t)] = a + bψ(t) + f t, ψ(t) − ψ(t − τ )
(51)
and a being an arbitrary constant. Below are two cases where particular solutions of equation (50) can be found. 12
(i) If b = 0 and M is a linear constant-coefficient operator, equation (50) admits polynomial particular solutions. (ii) If a = 0 and M is a linear constant-coefficient P operator, equation (50) m admits exponential solutions of the form ϕ(x) = B exp i=1 βi xi , where B is an arbitrary constant and β1 , . . . , βm are arbitrary constants connected by a single polynomial dispersion relation. Remark 8. A term K[u], where K is an arbitrary linear differential operator containing only mixed derivatives, can be added to equation (48). In this case, solution (49) as well as equations (50) and (51) remain valid. Remark 9. If L and M are linear constant-coefficient operators and the function f is
explicitly independent of t, equation (48) admits exact solutions of the form u = ϕ(x) + θ(z),
z = αt +
m X
βi xi ,
i=1
where α and βi are arbitrary constants, the function ϕ = ϕ(x) satisfies equation (50), and θ(z) is described by an ordinary differential-difference equation. Remark 10. The results presented in this section are easy to extend to more complex nonlinear partial functional-differential equations with a delay arbitrarily dependent on time as well as to equations involving several relaxation times (very similarly to the reaction–diffusion equations in the previous sections).
7. Brief conclusions To summarize, we have presented a number of exact multiplicative and additive separable solutions to nonlinear delay reaction–diffusion equations of the form ut = kuxx + F (u, w), where u = u(x, t), w = u(x, t − τ ), and τ is the relaxation (delay) time. All of the equations involve one arbitrary function of one argument. We have also listed several exact solutions to more complex nonlinear partial functional-differential equations with a time-varying delay arbitrarily dependent on time as well as to nonlinear equations containing several delay times. The results are easy to extend to nonlinear delay partial differential-difference equations of the form L[u] = M[u] + F (t, u, w), w = u(x, t − τ ), x = (x1 , . . . , xm ), 13
where L is an arbitrary linear differential operator of any order with respect to t and M is an arbitrary linear differential operator of any order with respect to the space coordinates x1 , . . . , xm ; this class of equations is quite wide, including, in particular, the nonlinear delay Klein–Gordon equation, which is a hyperbolic-type delay equation. The exact solutions involve several free parameters and can be used, for example, to test approximate analytical and numerical methods for nonlinear partial differential-difference equations. References [1] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. [2] J. Wu, X. Zou. Traveling wave fronts of reaction-diffusion systems with delay. J. Dynamics and Differential Equations, 2001, Vol. 13, No. 3, pp. 651– 687. [3] J. Huang, X. Zou. Traveling wavefronts in diffusive and cooperative Lotka– Volterra system with delays. J. Math. Anal. Appl., 2002, Vol. 271, pp. 455– 466. [4] T. Faria, S. Trofimchuk. Nonmonotone travelling waves in a single species reaction–diffusion equation with delay. J. Differential Equations, 2006, Vol. 228, 357–376. [5] E. Trofimchuk, V. Tkachenko, S. Trofimchuk. Slowly oscillating wave solutions of a single species reaction–diffusion equation with delay. J. Differential Equations, 2008, Vol. 245, pp. 2307–2332. [6] S. V. Meleshko, S. Moyo. On the complete group classification of the reaction–diffusion equation with a delay. J. Math. Anal. Appl., 2008, Vol. 338, pp. 448–466. [7] L. Wang, Y. Gao. Global exponential robust stability of reaction–diffusion interval neural networks with time-varying delays. Physics Letters A, 2006, Vol. 350, pp. 342–348. [8] J. G. Lu. Global exponential stability and periodicity of reaction–diffusion delayed recurrent neural networks with Dirichlet boundary conditions. Chaos, Solitons and Fractals, 2008, Vol. 35, pp. 116–125. 14
[9] V. A. Dorodnitsyn. Applications of Lie Groups to Difference Equations, Chapman & Hall/CRC Press, Boca Raton, 2011. [10] V. A. Dorodnitsyn. On invariant solutions of the nonlinear heat equation with a source [in Russian], Zhurn. vychisl. matem. i matem. fiziki, 1982, Vol. 22, No. 6, pp. 1393–1400. [11] M. C. Nucci, P. A. Clarkson. The nonclassical method is more general than the direct method for symmetry reductions. An example of the Fitzhugh– Nagumo equation. Phys. Lett. A, 1992, Vol. 164, pp. 49–56. [12] N. A. Kudryashov. On exact solutions of families of Fisher equations. Theor. & Math. Phys., 1993, Vol. 94, No. 2, pp. 211–218. [13] N. H. Ibragimov (Editor). CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1, Symmetries, Exact Solutions and Conservation Laws. CRC Press, Boca Raton, 1994. [14] A. D. Polyanin, V. F. Zaitsev, A. I. Zhurov. Solution Methods for Nonlinear Equations of Mathematical Physics and Mechanics [in Russian], Fizmatlit, Moscow, 2005. [15] V. A. Galaktionov, S. R. Svirshchevskii. Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Chapman & Hall/CRC Press, Boca Raton, 2006. [16] A. D. Polyanin, V. F. Zaitsev. Handbook of Nonlinear Partial Differential Equations, Second Edition. Chapman & Hall/CRC Press, Boca Raton, 2012. [17] R. Bellman, K. L. Cooke. Differential-Difference Equations. Academic Press, New York, 1963. [18] J. Hale. Functional Differential Equations. Springer-Verlag, New York, 1977. [19] R. D. Driver. Ordinary and Delay Differential Equations. Springer-Verlag, New York, 1977. [20] V. Kolmanovskii, A. Myshkis. Applied Theory of Functional Differential Equations. Kluwer, Dordrecht, 1992.
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[21] Y. Kuang. Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. [22] H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, 2010. [23] B. S. Lalli, Y. H. Yu, B. T. Cui. Oscillations of hyperbolic equations with functional arguments. Appl. Math. Comput., 1993, Vol. 53, pp. 97–110. [24] B. T. Cui, Y. H. Yu, S. Z. Lin. Oscillations of solutions of delay hyperbolic differential equations. Acta Math. Appl. Sinica, 1996, Vol. 19, pp. 80–88. [25] W. Peiguang. Forced oscillations of a class of nonlinear delay hyperbolic equations. Math. Slovaca, 1999, Vol. 49, No. 4, pp. 495–501.
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> Nonlinear delay reaction–diffusion equations are considered > All equations involve one arbitrary function of one argument > Many exact solutions to the delay equations are obtained > Exact solutions to equations with time-varying delay are given > Results are extended to equations with arbitrary linear operators
S1007-5704(13)00321-3 http://dx.doi.org/10.1016/j.cnsns.2013.07.019 CNSNS 2875
To appear in:
Communications in Nonlinear Science and Numer‐ ical Simulation
Please cite this article as: Polyanin, A.D., Zhurov, A.I., Exact separable solutions of delay reaction–diffusion equations and other nonlinear partial functional-differential equations, Communications in Nonlinear Science and Numerical Simulation (2013), doi: http://dx.doi.org/10.1016/j.cnsns.2013.07.019
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Exact separable solutions of delay reaction–diffusion equations and other nonlinear partial functional-differential equations Andrei D. Polyanina,∗, Alexei I. Zhurovb,a,∗∗ a
Institute for problems in Mechanics, Russian Academy of Sciences, 101 Vernadsky Avenue, bldg 1, 119526 Moscow, Russia b Cardiff University, Heath Park, Cardiff CF14 4XY, UK
Abstract We present a number of exact multiplicative and additive separable solutions to nonlinear delay reaction–diffusion equations of the form ut = kuxx + F (u, w), where u = u(x, t), w = u(x, t − τ ), and τ is the delay time. All of the equations involve one arbitrary function of one argument. We also give a number of exact solutions to more complex nonlinear partial functional-differential equations with a time-varying delay τ = τ (t) as well as to equations with several delay times. We extend the results to nonlinear partial differential-difference equations containing arbitrary linear differential operators of any order in the independent variables x and t; in particular, these equations include the nonlinear delay Klein–Gordon equation. Keywords: exact solutions, separable solutions, reaction-diffusion equations with delay, nonlinear equations, partial differential equations with delay, partial differential-difference equations, partial functional-differential equations, time-varying delay, delay hyperbolic equations, nonlinear Klein–Gordon equation
∗
Principal corresponding author Corresponding author Email addresses: [email protected] (Andrei D. Polyanin), [email protected] (Alexei I. Zhurov) ∗∗
Preprint submitted to Elsevier
July 15, 2013
1. Introduction Delay partial differential equations arise in numerous areas of application, including biology, biophysics, biochemistry, chemistry, medicine, control, climate model theory, and many others (e.g., see the studies [1–6] 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 [7, 8]. The present paper deals with the nonlinear delay reaction–diffusion equation [1, 2, 6, 9] of the form ut = kuxx + F (u, w),
w = u(x, t − τ ).
(1)
A number of exact solutions to the heat equation with a nonlinear source, which is a special case of equation (1) without delay and with F (u, w) = f (u), are listed, for example, in [10–15]. Most comprehensive surveys of exact solutions to this nonlinear equation can be found in [16]. In general, equation (1) admits traveling-wave solutions u = u(αx + βt). Such solutions are analyzed in numerous studies (e.g., see [2–5] and references in them). For a Lie group analysis of the nonlinear differential-difference equation (1), see [6]. The study found four equations of the form (1) that admit invariant solutions; two of these equations have degenerate solutions (linear in x). The present paper lists a number of exact separable solutions to nonlinear delay reaction–diffusion equations and other nonlinear partial functional-differential equations. In what follows, the term ‘exact solution’ with regard to nonlinear partial differential-difference (functional-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 (functional-differential) equations; (iii) the solution is expressible in terms of solutions to linear partial differential equations. Combination of cases (i)–(iii) are also allowed. This definition generalizes the notion of an exact solution used in [16] with regard to nonlinear partial differential equations. 2
Remark 1. For solution methods and various applications of linear and nonlinear ordinary differential-difference and functional-differential equations, which are much simpler than nonlinear partial differential-difference and functional-differential equations, see, for example, [17–22].
The next two sections present multiplicative and additive separable solutions to nonlinear delay reaction–diffusion equations of the form (1) dependent on an arbitrary function f (z) with given z = z(u, w). 2. Equations involving an arbitrary function dependent on w/u Equation 1. Consider equation (1) with the kinetic function F (u, w) = uf (w/u).
(2)
1◦ . Equation (1) with (2) admits the following periodic multiplicative separable solution in the space coordinate x: u = [C1 cos(αx) + C2 sin(αx)]ψ(t),
(3)
where C1 , C2 , and α are arbitrary constants, and ψ(t) is a function satisfying the ordinary differential-difference equation ψt′ (t) = −kα2 ψ(t) + ψ(t)f ψ(t − τ )/ψ(t) . (4) 2◦ . Equation (1) with (2) admits another multiplicative separable solution: u = [C1 exp(−αx) + C2 exp(αx)]ψ(t),
(5)
where C1 , C2 , and α are arbitrary constants and ψ(t) satisfies the ordinary differentialdifference equation ψt′ (t) = kα2 ψ(t) + ψ(t)f ψ(t − τ )/ψ(t) . (6) Remark 2. Equations (4) and (6) admit exponential particular solutions of the form
ψ(t) = Aeλt , where A is an arbitrary constant and λ is a root of the transcendental equations λ = ∓kα2 + f (e−λτ ). Depending on the form of the function f , these equations as well as the other transcendental equations given below can have one, several, infinitely many, or zero solutions.
3
3◦ . Equation (1) with (2) admits the degenerate multiplicative separable solution u = (C1 x + C2 )ψ(t), (7) where C1 and C2 are arbitrary constants and ψ(t) is a function satisfying the ordinary differential-difference equation (4) with α = 0. 4◦ . Equation (1) with (2) admits multiplicative solutions with the factors dependent on different arguments: u = eαx+βt θ(z),
z = λx + γt,
(8)
where α, β, γ, and λ are arbitrary constants, and θ(z) is a function satisfying the ordinary differential-difference equation ′′ kλ2 θzz (z) + (2kαλ − γ)θz′ (z) + (kα2 − β)θ(z) + θ(z)f e−βτ θ(z − σ)/θ(z) = 0, σ = γτ.
Solution (8) can be treated as a nonlinear superposition of two traveling waves. Equation 2. Equation (1) with F (u, w) = au ln u + uf (w/u)
(9)
admits the multiplicative separable solution u = ϕ(x)ψ(t),
(10)
with the functions ϕ(x) and ψ(t) determined by the ordinary differential and differential-difference equations kϕ′′xx = bϕ − aϕ ln ϕ, ψt′ (t) = bψ(t) + ψ(t)f (ψ(t − τ )/ψ(t)) + aψ(t) ln ψ(t),
(11) (12)
and b being an arbitrary constant. Remark 3. Equation (11) is autonomous, and hence its general solution can be written in explicit form. A particular one-parameter solution to this equation is given by a b 1 2 ϕ = exp − (x + C) + + , 4k a 2
where C is an arbitrary constant.
4
3. Equations involving an arbitrary function dependent on u − w Equation 3. Consider equation (1) with the kinetic function F (u, w) = f (u − w).
(13)
1◦ . Equation (1) with (13) admits quadratic additive separable solutions in x: u = ax2 + bx + ψ(t),
(14)
where a and b are arbitrary constants, and ψ(t) is a function satisfying the ordinary differential-difference equation ψt′ (t) = 2ak + f ψ(t) − ψ(t − τ ) . (15) Equation (15) has particular solutions linear in t: ψ(t) = λt + C, where C is an arbitrary constant and λ is a root of the transcendental equation 2ak − λ + f (τ λ) = 0. 2◦ . Equation (1) with (13) has solutions generalizing solution (14): u = ax2 + bx + θ(z),
z = βx + γt,
(16)
where a, b, β, and γ are arbitrary constants, and θ(z) is a function satisfying the ordinary differential-difference equation ′′ γθz′ (z) = kβ 2 θzz (z) + 2ak + f (θ(z) − θ(z − σ)),
σ = γτ.
Equation 4. Now consider equation (1) with the kinetic function F (u, w) = bu + f (u − w).
(17)
1◦ . Equation (1) with (17) and kb > 0 admits periodic additive separable solutions in the space coordinate x: p u = C1 cos(αx) + C2 sin(αx) + θ(t), α = b/k, (18) 5
where C1 and C2 are arbitrary constants and θ(t) is a function satisfying the ordinary differential-difference equation θt′ (t) = bθ(t) + f θ(t) − θ(t − τ ) . (19) 2◦ . Equation (1) with (17) and kb < 0 admits the additive separable solution p u = C1 exp(−αx) + C2 exp(αx) + θ(t), α = −b/k, (20)
where C1 and C2 are arbitrary constants, with the function θ(t) determined by the ordinary differential-difference equation (19). 3◦ . Equation (1) with (17) and kb > 0 has exact solutions generalizing solution (18): p u = C1 cos(αx) + C2 sin(αx) + θ(z), z = βx + γt, α = b/k, (21)
where C1 , C2 , β, and γ are arbitrary constants, with the function θ(z) determined by the ordinary differential-difference equation ′′ γθz′ (z) = kβ 2 θzz (z) + bθ(z) + f (θ(z) − θ(z − σ)),
σ = γτ.
(22)
Unlike (18), solution (21) is not periodic in x; it can be treated as an interaction of a periodic standing wave with a traveling wave. 4◦ . Equation (1) with (17) and kb < 0 admits solutions generalizing solution (20): p u = C1 exp(−αx) + C2 exp(αx) + θ(z), z = βx + γt, α = −b/k, (23) where C1 , C2 , β, and γ are arbitrary constants, and θ(z) is function satisfying the ordinary differential-difference equation (22). 4. General reaction–diffusion equations with variable delay Most of the examples listed above can be generalized by considering more complex nonlinear partial functional-differential equations with time-varying delay of the general form ut = kuxx + F (u, w),
w = u(x, t − τ (t)),
(24)
with the relaxation time τ treated as a given function of t. Equations of the form (24) can be encountered in the literature [7]. 6
Equation 5. Equation (24) with the kinetic function (2) admits a periodic (in the space coordinate x) multiplicative separable solution of the form (3), with the function ψ(t) described by the ordinary differential-difference equation ψt′ (t) = −kα2 ψ(t) + ψ(t)f ψ(t − τ (t))/ψ(t) . (25)
Equation (24) with the kinetic function (2) admits another multiplicative separable solution of the form (5) where the function ψ(t) is determined by the ordinary differential-difference equation ψt′ (t) = kα2 ψ(t) + ψ(t)f ψ(t − τ (t))/ψ(t) . Equation (24) with the kinetic function (2) also admits a multiplicative separable solution of the form (7) with the function ψ(t) described by the ordinary differential-difference equation (25) with α = 0.
Equation 6. Equation (24) with the kinetic function (9) admits a multiplicative separable solution of the form (10) with the functions ϕ(x) and ψ(t) satisfying the ordinary differential and functional-differential equations kϕ′′xx = bϕ − aϕ ln ϕ, ψt′ (t) = bψ(t) + ψ(t)f ψ(t − τ (t))/ψ(t) + aψ(t) ln ψ(t)
and b being an arbitrary constant. The equation for ϕ coincides with (11); for its solutions, see Remark 3. Equation 7. Equation (24) with the kinetic function (13) admits an additive separable solution of the form (14) with the function ψ(t) determined by the ordinary functional-differential equation ψt′ (t) = 2ak + f ψ(t) − ψ(t − τ (t)) . (26)
Equation 8. Equation (24) with the kinetic function (17) admits two additive separable solutions of the form (18) and (20) where the function θ(t) is described by the ordinary functional-differential equation θt′ (t) = bθ(t) + f θ(t) − θ(t − τ (t)) . (27)
7
5. Reaction–diffusion equations with several delay times Consider diffusion–reaction equations of the from ut = kuxx + F (u, w1 , . . . , wn ),
wm = u(x, t − τm ),
(28)
where τm are relaxation times (m = 1, . . . , n), which are assumed to be independent arbitrary constants. Equations of the form (28) can be encountered in the literature [3, 7, 8]. In what follows, f (z1 , . . . , zn ) are arbitrary functions of their arguments. Equation 9. If F (u, w1, . . . , wn ) = uf (w1 /u, . . . , wn /u),
(29)
equation (28) admits a periodic (in the space coordinate x) multiplicative separable solution of the form (3), where the function ψ(t) is described by the ordinary differential-difference equation ψt′ (t) = −kα2 ψ(t) + ψ(t)f ψ(t − τ1 )/ψ(t), . . . , ψ(t − τn )/ψ(t) . (30)
Equation (28) with (29) admits another multiplicative separable solution of the form (5), where the function ψ(t) is determined by the ordinary differentialdifference equation ψt′ (t) = kα2 ψ(t) + ψ(t)f ψ(t − τ1 )/ψ(t), . . . , ψ(t − τn )/ψ(t) . (31) Remark 4. Equations (30) and (31) admit exponential solutions of the form
ψ(t) = Aeλt , where A is an arbitrary constant and λ is a root of the transcendental equations λ = ∓kα2 + f (e−λτ1 , . . . , e−λτn ). Equation (28) with (29) admits a degenerate multiplicative separable solution of the form (7) where the function ψ(t) is determined by the ordinary differential-difference equation (30) with α = 0. Equation (28) with (29) also admits more complex multiplicative solutions of the form (8).
8
Equation 10. If F (u, w1 , . . . , wn ) = au ln u + uf (w1/u, . . . , wn /u),
(32)
equation (28) admits a multiplicative separable solution of the form (10), with the functions ϕ(x) and ψ(t) determined by the ordinary differential and differentialdifference equations kϕ′′xx = bϕ − aϕ ln ϕ, ψt′ (t) = bψ(t) + ψ(t)f ψ(t − τ1 )/ψ(t), . . . , ψ(t − τn )/ψ(t) + aψ(t) ln ψ(t) (33) and b being an arbitrary constant. The equation for ϕ coincides with (11); for its solutions, see Remark 3. Equation 11. Equation (28) with F (u, w1, . . . , wn ) = f (u − w1 , . . . , u − wn ),
(34)
admits an additive separable solution of the form (14) where the function ψ(t) is described by the ordinary differential-difference equation ψt′ (t) = 2ak + f ψ(t) − ψ(t − τ1 ), . . . , ψ(t) − ψ(t − τn ) . (35) Equation (35) has particular solutions linear in t: ψ(t) = λt + C, where C is an arbitrary constant and λ is a root of the transcendental equation 2ak − λ + f (λτ1 , . . . , λτn ) = 0. Equation (28) with (34) also admits solutions of the form (16). Equation 12. Equation (28) with F (u, w1, . . . , wn ) = bu + f (u − w1 , . . . , u − wn )
(36)
and kb > 0 admits an additive separable solution of the form (18), where the function θ(t) is described by the ordinary differential-difference equation θt′ (t) = bθ(t) + f θ(t) − θ(t − τ1 ), . . . , θ(t) − θ(t − τn ) . (37)
Equation (28) with (36) and kb < 0 admits an additive separable solution of the form (20) where the function θ(t) is described by the ordinary differentialdifference equation (37). Equation (28) with (36) also admits two more complex equations of the form (21) and (23). 9
Remark 5. The constant time delays τm that appear in equation (28) with kinetic functions (29), (32), (34), and (36), which reduce to the ordinary functional-differential equations (30), (31), (33), (35), and (37), as well as in solutions of these equations, can be replaced with time-varying delays τm = τm (t), where τm (t) are arbitrary functions satisfying the conditions 0 ≤ τm (t) ≤ τ0 = const, m = 1, . . . , n.
6. Even more general nonlinear partial differential equations with delay Now consider nonlinear partial differential-difference equations of the more general form L[u] = M[u] + F (t, u, w), (38) w = u(x, t − τ ), x = (x1 , . . . , xm ), where L is an arbitrary linear differential operator with respect to t, whose coefficients can be dependent on t, L[u] =
n X
bi (t)
i=1
∂iu , ∂ti
(39)
and M is a linear differential operator of any (second or higher) order with respect to the space coordinates x1 , . . . , xm , whose coefficients can be dependent on x1 , . . . , xm . Note that in this section, the kinetic function F depends on three arguments. In particular, M can be an elliptic operator of the form M[u] =
m m X X ∂ ∂u ∂u kij (x) + ci (x) ; ∂x ∂x ∂x i j i m=1 i,j=1
(40)
such operators with ci = 0 are used in the studies [7, 8]. Also, M can be a biharmonic operator of the form M[u] = k∆∆u,
∆u ≡
m X ∂2u . 2 ∂x i i=1
Setting L[u] = utt , M[u] = a2 uxx , Ft = 0, and m = 1 in (38) yields the nonlinear delay Klein–Gordon equation utt = a2 uxx + F (u, w).
10
(41)
Such hyperbolic delay partial differential equations are encountered in the literature, for example, [23–25]. Below we list multiplicative and additive separable solutions of nonlinear delay partial differential equations of the form (38) that involve an arbitrary function of two arguments, f (t, z), with given z = z(u, w). The determining equations were derived using following simple properties of the operators L and M: L[ϕ(x)ψ(t)] = ϕ(x)L[ψ(t)], L[ϕ(x) + ψ(t)] = L[ψ(t)], M[ϕ(x)ψ(t)] = ψ(t)M[ϕ(x)], M[ϕ(x) + ψ(t)] = M[ϕ(x)]. Equation 13. The equation L[u] = M[u] + uf (t, w/u),
(42)
admits a multiplicative separable solution u = ϕ(x)ψ(t)
(43)
with the functions ϕ(x) and ψ(t) satisfying the linear partial differential equation and ordinary differential-difference equation M[ϕ] = aϕ,
(44)
L[ψ(t)] = aψ(t) + ψ(t)f t, ψ(t − τ )/ψ(t)
(45)
and a being an arbitrary constant. We specify below three cases where the form of particular solutions of equations (44) and (45) can be found: (i) If a = 0 and M is a linear constant-coefficient operator, equation (44) admits polynomial particular solutions. (ii) If a is any number and M is a linear constant-coefficientP operator, equation m (44) admits exponential solutions of the form ϕ(x) = B exp i=1 βi xi , where B is an arbitrary constant and β1 , . . . , βm are arbitrary constants connected by a single polynomial dispersion relation. (iii) If L is a linear operator of the form (39) with constant coefficients (bi = const) and the kinetic function is explicitly independent of t, f = f (w/u), equation (45) admits exponential solutions, ψ(t) = Aeλt , where A is an arbitrary constant and λ is a root of the transcendental equation n X
bi λi = a + f (e−τ λ ).
i=1
11
Remark 6. If L and M are both linear constant-coefficient operators and the function f is explicitly independent of t, equation (42) admits exact solutions of the form
m X βi xi θ(z), u = exp αt +
z = γt +
i=1
m X
λi x i ,
(46)
i=1
where α, βi , γ, and λi are arbitrary constants, and the function θ(z) is described by an ordinary differential-difference equation. Remark 7. Suppose the conditions listed in Remark 6 hold. Then, solutions of the form (46) will also be solutions of equation (42) with an additional term K[u], where K is an arbitrary constant-coefficient linear differential operator with mixed derivatives.
Equation 14. The equation L[u] = M[u] + bu ln u + uf (t, w/u),
(47)
admits a multiplicative separable solution of the form (43) where the functions ϕ(x) and ψ(t) are determined by the nonlinear partial differential equation and ordinary differential-difference equation M[ϕ] = aϕ − bϕ ln ϕ, L[ψ(t)] = aψ(t) + bψ(t) ln ψ(t) + ψ(t)f (t, ψ(t − τ )/ψ(t)), with a being an arbitrary constant. Equation 15. The equation L[u] = M[u] + bu + f (t, u − w)
(48)
admits an additive separable solution of the form u = ϕ(x) + ψ(t),
(49)
with the functions ϕ(x) and ψ(t) described by the linear partial differential equation and ordinary differential-difference equation M[ϕ] = a − bϕ,
(50)
L[ψ(t)] = a + bψ(t) + f t, ψ(t) − ψ(t − τ )
(51)
and a being an arbitrary constant. Below are two cases where particular solutions of equation (50) can be found. 12
(i) If b = 0 and M is a linear constant-coefficient operator, equation (50) admits polynomial particular solutions. (ii) If a = 0 and M is a linear constant-coefficient P operator, equation (50) m admits exponential solutions of the form ϕ(x) = B exp i=1 βi xi , where B is an arbitrary constant and β1 , . . . , βm are arbitrary constants connected by a single polynomial dispersion relation. Remark 8. A term K[u], where K is an arbitrary linear differential operator containing only mixed derivatives, can be added to equation (48). In this case, solution (49) as well as equations (50) and (51) remain valid. Remark 9. If L and M are linear constant-coefficient operators and the function f is
explicitly independent of t, equation (48) admits exact solutions of the form u = ϕ(x) + θ(z),
z = αt +
m X
βi xi ,
i=1
where α and βi are arbitrary constants, the function ϕ = ϕ(x) satisfies equation (50), and θ(z) is described by an ordinary differential-difference equation. Remark 10. The results presented in this section are easy to extend to more complex nonlinear partial functional-differential equations with a delay arbitrarily dependent on time as well as to equations involving several relaxation times (very similarly to the reaction–diffusion equations in the previous sections).
7. Brief conclusions To summarize, we have presented a number of exact multiplicative and additive separable solutions to nonlinear delay reaction–diffusion equations of the form ut = kuxx + F (u, w), where u = u(x, t), w = u(x, t − τ ), and τ is the relaxation (delay) time. All of the equations involve one arbitrary function of one argument. We have also listed several exact solutions to more complex nonlinear partial functional-differential equations with a time-varying delay arbitrarily dependent on time as well as to nonlinear equations containing several delay times. The results are easy to extend to nonlinear delay partial differential-difference equations of the form L[u] = M[u] + F (t, u, w), w = u(x, t − τ ), x = (x1 , . . . , xm ), 13
where L is an arbitrary linear differential operator of any order with respect to t and M is an arbitrary linear differential operator of any order with respect to the space coordinates x1 , . . . , xm ; this class of equations is quite wide, including, in particular, the nonlinear delay Klein–Gordon equation, which is a hyperbolic-type delay equation. The exact solutions involve several free parameters and can be used, for example, to test approximate analytical and numerical methods for nonlinear partial differential-difference equations. References [1] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. [2] J. Wu, X. Zou. Traveling wave fronts of reaction-diffusion systems with delay. J. Dynamics and Differential Equations, 2001, Vol. 13, No. 3, pp. 651– 687. [3] J. Huang, X. Zou. Traveling wavefronts in diffusive and cooperative Lotka– Volterra system with delays. J. Math. Anal. Appl., 2002, Vol. 271, pp. 455– 466. [4] T. Faria, S. Trofimchuk. Nonmonotone travelling waves in a single species reaction–diffusion equation with delay. J. Differential Equations, 2006, Vol. 228, 357–376. [5] E. Trofimchuk, V. Tkachenko, S. Trofimchuk. Slowly oscillating wave solutions of a single species reaction–diffusion equation with delay. J. Differential Equations, 2008, Vol. 245, pp. 2307–2332. [6] S. V. Meleshko, S. Moyo. On the complete group classification of the reaction–diffusion equation with a delay. J. Math. Anal. Appl., 2008, Vol. 338, pp. 448–466. [7] L. Wang, Y. Gao. Global exponential robust stability of reaction–diffusion interval neural networks with time-varying delays. Physics Letters A, 2006, Vol. 350, pp. 342–348. [8] J. G. Lu. Global exponential stability and periodicity of reaction–diffusion delayed recurrent neural networks with Dirichlet boundary conditions. Chaos, Solitons and Fractals, 2008, Vol. 35, pp. 116–125. 14
[9] V. A. Dorodnitsyn. Applications of Lie Groups to Difference Equations, Chapman & Hall/CRC Press, Boca Raton, 2011. [10] V. A. Dorodnitsyn. On invariant solutions of the nonlinear heat equation with a source [in Russian], Zhurn. vychisl. matem. i matem. fiziki, 1982, Vol. 22, No. 6, pp. 1393–1400. [11] M. C. Nucci, P. A. Clarkson. The nonclassical method is more general than the direct method for symmetry reductions. An example of the Fitzhugh– Nagumo equation. Phys. Lett. A, 1992, Vol. 164, pp. 49–56. [12] N. A. Kudryashov. On exact solutions of families of Fisher equations. Theor. & Math. Phys., 1993, Vol. 94, No. 2, pp. 211–218. [13] N. H. Ibragimov (Editor). CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1, Symmetries, Exact Solutions and Conservation Laws. CRC Press, Boca Raton, 1994. [14] A. D. Polyanin, V. F. Zaitsev, A. I. Zhurov. Solution Methods for Nonlinear Equations of Mathematical Physics and Mechanics [in Russian], Fizmatlit, Moscow, 2005. [15] V. A. Galaktionov, S. R. Svirshchevskii. Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Chapman & Hall/CRC Press, Boca Raton, 2006. [16] A. D. Polyanin, V. F. Zaitsev. Handbook of Nonlinear Partial Differential Equations, Second Edition. Chapman & Hall/CRC Press, Boca Raton, 2012. [17] R. Bellman, K. L. Cooke. Differential-Difference Equations. Academic Press, New York, 1963. [18] J. Hale. Functional Differential Equations. Springer-Verlag, New York, 1977. [19] R. D. Driver. Ordinary and Delay Differential Equations. Springer-Verlag, New York, 1977. [20] V. Kolmanovskii, A. Myshkis. Applied Theory of Functional Differential Equations. Kluwer, Dordrecht, 1992.
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[21] Y. Kuang. Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. [22] H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, 2010. [23] B. S. Lalli, Y. H. Yu, B. T. Cui. Oscillations of hyperbolic equations with functional arguments. Appl. Math. Comput., 1993, Vol. 53, pp. 97–110. [24] B. T. Cui, Y. H. Yu, S. Z. Lin. Oscillations of solutions of delay hyperbolic differential equations. Acta Math. Appl. Sinica, 1996, Vol. 19, pp. 80–88. [25] W. Peiguang. Forced oscillations of a class of nonlinear delay hyperbolic equations. Math. Slovaca, 1999, Vol. 49, No. 4, pp. 495–501.
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> Nonlinear delay reaction–diffusion equations are considered > All equations involve one arbitrary function of one argument > Many exact solutions to the delay equations are obtained > Exact solutions to equations with time-varying delay are given > Results are extended to equations with arbitrary linear operators