Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration

Abstract

The paper describes essential reaction–diffusion models with delay arising in population theory, medicine, epidemiology, biology, chemistry, control theory, and the mathematical theory of artificial neural networks. A review of publications on the exact solutions and methods for their construction is carried out. Basic numerical methods for integrating nonlinear reaction–diffusion equations with delay are considered. The focus is on the method of lines. This method is based on the approximation of spatial derivatives by the corresponding finite differences, as a result of which the original delay PDE is replaced by an approximate system of delay ODEs. The resulting system is then solved by the implicit Runge–Kutta and BDF methods, built into Mathematica. Numerical solutions are compared with the exact solutions of the test problems.

Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays

“…Delay PDEs (7) admit traveling-wave solutions, u = u(z), where z = kx + λt (e.g., see [131,[141][142][143]), but do not have self-similar solutions, u = t β ϕ(xt λ ), which non-delay PDEs often have. Reductions and exact solutions with additive, multiplicative, and generalized separation of variables and more complex solutions for delay PDEs are obtained in [144][145][146][147][148][149][150][151][152][153][154][155][156] (see also [157] for a brief review, which, in addition to exact solutions, describes the main numerical methods for solving such equations).…”

Exact Solutions of Reaction–Diffusion PDEs with Anisotropic Time Delay

“…Some specific mathematical models with delay can be found in [14,34,98,99]. Symmetries, linearizations, and some exact solutions of complex first-and second-order ODEs with delay are studied in [100][101][102].…”

“…PDEs with two independent variables, x and t, and constant delay generally admit traveling-wave solutions, u = u(z), where z = kx + λt [55,[106][107][108], and do not have selfsimilar solutions, u = t α U(y), where y = xt β . Additive, multiplicative, and generalized separable solutions and more complex solutions of PDEs with constant or varying delays are obtained in [27,[31][32][33]78,79,[91][92][93][94][95]97,[109][110][111][112][113][114][115][116][117] (for a brief overview of publications on exact solutions, see [97,99]). In contrast, PDEs with one proportional delay do not have traveling-wave solutions but can admit self-similar ones.…”

Separation of Variables and Exact Solutions to Nonlinear PDEs

“…It is easy to check that equation ( 35) also admits an exact solution of the form (33), where the functions ψ 1 = ψ 1 (t) and ψ 2 = ψ 2 (t) satisfy the nonlinear ODE system (34), in which the constant τ must be replaced by an arbitrary function τ (t).…”

“…Exact solutions of nonlinear PDEs are most often constructed using the classical method of symmetry reductions [3][4][5][6], the direct method of symmetry reductions [1,[7][8][9], the nonclassical symmetries methods [10][11][12][13], methods of generalized separation of variables [1,9,[14][15][16], methods of functional separation of variables [1,9,17,18], the method of differential constraints [1,9,19,20], and some other exact analytical methods (see, for example, [21][22][23][24][25][26][27]). On methods for constructing exact solutions of nonlinear delay PDEs and functional PDEs, see, for example, [2,[28][29][30][31][32][33][34]. This article will describe new approaches for constructing exact solutions of nonlinear non-autonomous equations of mathematical physics and other PDEs (including PDEs with delay), based on the use of auxiliary solutions to simpler equations.…”