#### 517.9:532:536 Nonlinear delay reaction-diffusion equations with varying transfer coefficients: generalized and functional separable solutions

##### Polyanin A. D. (Bauman Moscow State Technical University/Ishlinsky Institute for Problems in Mechanics/MEPhI), Zhurov A. I. (Cardiff University/Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences)

###### DELAY REACTION-DIFFUSION EQUATIONS, VARYING TRANSFER COEFFICIENTS, EXACT SOLUTIONS, GENERALIZED SEPARABLE SOLUTIONS, FUNCTIONAL SEPARABLE SOLUTIONS, TIME-VARYING DELAY, NONLINEAR DELAY PARTIAL DIFFERENTIAL EQUATIONS

doi: 10.18698/2309-3684-2015-4-337

We present a number of new simple separable, generalized separable, and functional separable solutions to one-dimensional nonlinear delay reaction-diffusion equations with varying transfer coefficients of the formut = [G(u)ux ]x  F(u,w),where w = u(x,t) and w = u(x,t  ), with  denoting the delay time. All of the equations considered contain one, two, or three arbitrary functions of a single argument. The generalized separable solutions are sought in the form =1 = () () N
n n n u   x  t , withn (x) and n (t) to be determined in the analysis using a new modification of the functional constraints method. Some of the results are extended to nonlinear delay reaction-diffusion equations with time-varying delay  = (t). We also present exact solutions to more complex, three-dimensional delay reactiondiffusion equations of the formut = div[G(u)u] F(u,w).Most of the solutions obtained involve free parameters, so they may be suitable for solving certain problems as well as testing approximate analytical and numerical methods for non-linear delay PDEs.

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Polyanin A., Zhurov A. Nonlinear delay reaction-diffusion equations with varying transfer coefficients: generalized and functional separable solutions. Маthematical Modeling and Coтputational Methods, 2015, №4 (8), pp. 3-37