Nonlinear delay reaction–diffusion equations: Traveling-wave solutions in elementary functions

Polyanin, Andrei D.; Сорокин, В. М.

doi:10.1016/j.aml.2015.01.023

Cited by 21 publications

(11 citation statements)

References 34 publications

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“…Remark 4. Some exact solutions to nonlinear reaction-diffusion equations of Form (6) were obtained, for example, in [3,13,16,22,[35][36][37][38][39][40][41][42][43][44][45][46]; for exact solutions to more complex, nonlinear delay reaction-diffusion equations, see [32,[47][48][49][50][51][52][53][54][55].…”

Section: Using the Methods Of Differential Constraintsmentioning

confidence: 99%

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

Polyanin

2019

Mathematics

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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.

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Section: Using the Methods Of Differential Constraintsmentioning

confidence: 99%

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

Polyanin

2019

Mathematics

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“…where c is an arbitrary parameter. Using Proposition 4, we obtain two more complicated two-parameter families exact solutions of equation (62): It is easy to verify that equation (63) admits translation transformations for both independent variables and has the particular solution…”

Section: Linear Partial Differential Equationsmentioning

confidence: 98%

“…The presence of a delay significantly complicates the analysis of such equations. Although nonlinear PDEs with constant delay allow solutions of the traveling wave type u = u(z), where z = x + λt (see, for example, [59][60][61][62]), they do not allow self-similar solutions of the form u = t β ϕ(xt λ ), which often have simpler PDEs without delay.…”

Section: Partial Differential Equations With Delaymentioning

confidence: 99%

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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“…Eqs (3) can be obtained from the system with finite delay [17,18] as first order approximation with regards to small parameter  t ; the zero order approximation corresponds to equilibrium sorption a(c).The relaxation time is inversely proportional to hydrodynamic dispersion of the components, which in turn is proportional to the velocity [2,4]. The dimensionless group  t is the ratio between the relaxation and flight times.…”

Section: Two-phase Multicomponent Flow With Non-equilibrium Sorptionmentioning

confidence: 99%

Splitting in systems of PDEs for two-phase multicomponent flow in porous media

Borazjani

Roberts

Bedrikovetsky

2016

Applied Mathematics Letters

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The paper investigates the system of PDEs for two-phase n-component flow in porous media consisting of hyperbolic terms for advective transport, parabolic terms of dissipative effects and relaxation nonequilibrium equations. We found that for several dissipative and non-equilibrium systems, using the streamfunction as a free variable instead of time separates the general (n+1)×(n+1) system into an n×n auxiliary system and one scalar lifting equation. In numerous cases, where the auxiliary system allows for exact solution, the general flow problem is reduced to numerical or semi-analytical solution of one lifting equation.

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Nonlinear delay reaction–diffusion equations: Traveling-wave solutions in elementary functions

Cited by 21 publications

References 34 publications

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

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

Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions

Splitting in systems of PDEs for two-phase multicomponent flow in porous media

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