Riccati–Ermakov systems and explicit solutions for variable coefficient reaction–diffusion equations

Abstract

We present several families of nonlinear reaction diffusion equations with variable coefficients including generalizations of Fisher-KPP and Burgers type equations. Special exact solutions such as traveling wave, rational, triangular wave and N-wave type solutions are shown. By means of similarity transformations the variable coefficients are conditioned to satisfy Riccati or Ermakov systems of equations. When the Riccati system is used, conditions are established so that finite-time singularities might occur. We explore solution dynamics across multi-parameters. In the suplementary material, we provide a computer algebra verification of the solutions and exemplify nontrivial dynamics of the solutions.

CMMSE on explicit and numerical solutions for stochastic PDEs: Fisher and Burgers equations

“…By Theorem 1, Equation has a solution I ( t , z )=( I ∗ ) U ( t , X t ), such that U ( t , x ) is the solution of U ( t , x ) is the solution of and X t is the solution of with initial state X 0 = z and for t ∈[0,1]. By the work in Reference 8 , Equation has the solution for . The stochastic differential Equation has a solution given by for t ∈[0,1].…”

Exact and numerical solution of stochastic Burgers equations with variable coefficients

“…with ξ=β(t)x + 2ε(t) and τ (t) = 4γ(t). The functions µ, α, β, γ, δ, ε and κ satisfy a Riccati system, see [17], and they are given explicitly by:…”

Exponential input‐to‐state stability of delay reaction‐diffusion systems

“…RDSs are a kind of spatial-temporal systems, which depend on not only the time but also the spatial position. Many results of RDSs have been published, see [3][4][5][6][7][8][9] and the references therein. Moreover, time delay has been caught in many RDSs, and it can degrade, even destroy, the properties of a considered system.…”