Separation of variables in PDEs using nonlinear transformations: Applications to reaction–diffusion type equations

The paper describes a new approach to constructing exact solutions of nonlinear partial differential equations that employs separation of variables using special (nonlinear integral) transformations and the splitting principle. To illustrate its effectiveness, the method is applied to nonlinear reaction–diffusion type equations that involve variable coefficients and arbitrary functions. New exact functional separable solutions as well as generalized traveling wave solutions are obtained

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

Molecular pathway reconstruction and analysis of disturbed gene expression in depressed individuals who died by suicide

Molecular mechanisms behind the etiology and pathophysiology of major depressive disorder and suicide remain largely unknown. Recent molecular studies of expression of serotonin, GABA and CRH receptors in various brain regions have demonstrated that molecular factors may contribute to the development of depressive disorder and suicide behaviour. Here, we used microarray analysis to examine the expression of genes in brain tissue (frontopolar cortex) of individuals who had been diagnosed with major depressive disorder and died by suicide, and those who had died suddenly without a history of depression. We analyzed the list of differentially expressed genes using pathway analysis, which is an assumption-free approach to analyze microarray data. Our analysis revealed that the differentially expressed genes formed functional networks that were implicated in cell to cell signaling related to synapse maturation, neuronal growth and neuronal complexity. We further validated these data by randomly choosing (100 times) similarly sized gene lists and subjecting these lists to the same analyses. Random gene lists did not provide highly connected gene networks like those generated by the differentially expressed list derived from our samples. We also found through correlational analysis that the gene expression of control participants was more highly coordinated than in the MDD/suicide group. These data suggest that among depressed individuals who died by suicide, wide ranging perturbations of gene expression exist that are critical for normal synaptic connectively, morphology and cell to cell communication