Symmetry reductions and new functional separable solutions of nonlinear Klein–Gordon and telegraph type equations

The paper is concerned with different classes of nonlinear Klein–Gordon and telegraph type equations with variable coefficient

Multi-parameter reaction–diffusion systems with quadratic nonlinearity and delays: new exact solutions in elementary functions

The study considers a nonlinear multi-parameter reaction–diffusion system of two Lotka–Volterra-type equations with several delays. It treats both cases of different diffusion coefficients and identical diffusion coefficients. The study describes a few different techniques to solve the system of interest, including (i) reduction to a single second-order linear ODE without delay, (ii) reduction to a system of three second-order ODEs without delay, (iii) reduction to a system of three first-order ODEs with delay, (iv) reduction to a system of two second-order ODEs without delay and a linear Schrödinger-type PDE, and (v) reduction to a system of two first-order ODEs with delay and a linear heat-type PDE. The study presents many new exact solutions to a Lotka–Volterra-type reaction–diffusion system with several arbitrary delay times, including over 50 solutions in terms of elementary functions. All of these are generalized or incomplete separable solutions that involve several free parameters (constants of integration). A special case is studied where a solution contains infinitely many free parameters. Along with that, some new exact solutions are obtained for a simpler nonlinear reaction–diffusion system of PDEs without delays that represents a special case of the original multi-parameter delay system. Several generalizations to systems with variable coefficients, systems with more complex nonlinearities, and hyperbolic type systems with delay are discussed. The solutions obtained can be used to model delay processes in biology, ecology, biochemistry and medicine and test approximate analytical and numerical methods for reaction–diffusion and other nonlinear PDEs with delays

The heat and mass transfer modeling with time delay

Nonlinear hyperbolic reaction-diffusion equations with a delay in time are investigated. All equations considered here contain one arbitrary function. Exact solutions are also presented for more complex nonlinear equations in which delay arbitrarily depends on time. Exact solutions with a generalized separation of variables are found. For special cases, new exact solutions in the form of a traveling waves are obtained, some of which can be represented in terms of elementary functions. All of these solutions contain free (arbitrary) parameters, so that one can use them to solve modeling problems of heat and mass transfer with relaxation phenomena

Comparison of the Effectiveness of Different Methods for Constructing Exact Solutions to Nonlinear PDEs. Generalizations and New Solutions

The paper shows that, in looking for exact solutions to nonlinear PDEs, the direct method of functional separation of variables can, in certain cases, be more effective than the method of differential constraints based on the compatibility analysis of PDEs with a single constraint (or the nonclassical method of symmetry reductions based on an invariant surface condition). This fact is illustrated by examples of nonlinear reaction-diffusion and convection-diffusion equations with variable coefficients, and nonlinear Klein−Gordon-type equations. Hydrodynamic boundary layer equations, nonlinear Schrödinger type equations, and a few third-order PDEs are also investigated. Several new exact functional separable solutions are given. A possibility of increasing the efficiency of the Clarkson−Kruskal direct method is discussed. A generalization of the direct method of the functional separation of variables is also described. Note that all nonlinear PDEs considered in the paper include one or several arbitrary functions