Exact solutions and qualitative features of nonlinear hyperbolic reaction—diffusion equations with delay

Polyanin, Andrei D.; Сорокин, В. М.; Vyazmin, A. V.

doi:10.1134/s0040579515050243

Cited by 16 publications

(14 citation statements)

References 31 publications

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“…where A and B are some constants. Substituting (74) in Equation 71, we obtain a solution of the form (74), where A and B are arbitrary constants, and the function ψ = ψ(t) satisfies the nonlinear ODE:…”

Section: Nonlinear Partial Differential Equationsmentioning

confidence: 99%

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Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions

Aksenov

Polyanin

2021

Mathematics

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This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for constructing complex solutions from simple solutions using translation and scaling. We show that in some cases, rather complex solutions can be obtained by adding one or more terms to simpler solutions. There are situations where nonlinear superposition allows us to construct a complex composite solution using similar simple solutions. We also propose a few methods for constructing complex exact solutions to linear and nonlinear PDEs by introducing complex-valued parameters into simpler solutions. The effectiveness of the methods is illustrated by a large number of specific examples (over 30 in total). These include nonlinear heat equations, reaction–diffusion equations, wave type equations, Klein–Gordon type equations, equations of motion through porous media, hydrodynamic boundary layer equations, equations of motion of a liquid film, equations of gas dynamics, Navier–Stokes equations, and some other PDEs. Apart from exact solutions to `ordinary’ partial differential equations, we also describe some exact solutions to more complex nonlinear delay PDEs. Along with the unknown function at the current time, u=u(x,t), these equations contain the same function at a past time, w=u(x,t−τ), where τ>0 is the delay time. Furthermore, we look at nonlinear partial functional-differential equations of the pantograph type, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments, w=u(px,qt), where p and q are scaling parameters. We propose an efficient approach to construct exact solutions to such functional-differential equations. Some new exact solutions of nonlinear pantograph-type PDEs are presented. The methods and examples in this paper are presented according to the principle “from simple to complex”.

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Section: Nonlinear Partial Differential Equationsmentioning

confidence: 99%

“…Substituting (70) into (74) and 75, we arrive at a new solution containing already trigonometric functions in x,…”

Section: Nonlinear Partial Differential Equationsmentioning

confidence: 99%

Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions

Aksenov

Polyanin

2021

Mathematics

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“…A τ -periodic solution of model (30) is a solution w(t, x) such that w(t − τ, x) = w(t, x) for some τ > 0. Similar to [33], the following theorem shows that model (30) admits a τ -periodic solution.…”

mentioning

confidence: 91%

“…Given that the diffusion has a stabilizing effect, the constant equilibria of models (33) and (34) preserve their stability and the model dynamics remain the same with respect to constant equilibria. The new dynamics of models (33) and (34) compared to those of the previous models (28) - (32) are the existence of traveling and stationary wave solutions. The traveling wave solutions of these models have been studied by several researchers (see [52] for a review).…”

mentioning

confidence: 98%

“…The traveling wave solutions of these models have been studied by several researchers (see [52] for a review). Also, the existence of stationary wave solutions of models (33) and (34) has been proven in [3,6] and their stability has been numerically examined. Formation of a traveling wavefront may resemble the spatial dispersal of invasive species, whereas a stationary wavefront or wave pulse relates to space-dependent extinction-survival (see Table 3 of [6]).…”

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confidence: 99%

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Delay-induced instabilities of stationary solutions in a single species nonlocal hyperbolic-parabolic population model

Bani-Yaghoub¹,

Ou²,

Yao³

2020

Discrete &Amp; Continuous Dynamical Systems - S

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The present work investigates the effects of maturation and dispersal delays on dynamics of single species populations. Both delays have been incorporated in a single species nonlocal hyperbolic-parabolic population model, which admits traveling and stationary wave solutions. We reduce the model into various forms and obtain the corresponding analytical solutions. Analysis of the reduced models indicates that the dispersal delay can result in loss of monotonicity, where the solutions oscillate as they converge to a positive equilibrium. The stability analysis of the general model reveals that the maturation time delay admits a Hopf bifurcation threshold, which is expressed as a function of the dispersal delay. The numerical simulations of the general model suggest that the global stability of the stationary wave solutions is lost when the dispersal delay is increased from zero. In conclusion, population models with maturation and dispersal delays can give new insights into the complex dynamics of single species.

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New exact solutions of nonlinear wave type PDEs with delay

Polyanin

Сорокин

2020

Applied Mathematics Letters

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Exact solutions and qualitative features of nonlinear hyperbolic reaction—diffusion equations with delay

Cited by 16 publications

References 31 publications

Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions

Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions

Delay-induced instabilities of stationary solutions in a single species nonlocal hyperbolic-parabolic population model

New exact solutions of nonlinear wave type PDEs with delay

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