Classification of reduction operators and exact solutions of variable coefficient Newell–Whitehead–Segel equations

Vaneeva, Olena; Boyko, Vyacheslav M.; Zhalij, Alexander; Sophocleous, Christodoulos

doi:10.1016/j.jmaa.2019.01.044

Cited by 8 publications

(4 citation statements)

References 58 publications

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“…For constructing the leading term of the semiclassical asymptotic solution to Equation ( 7), we consider a set of moments of the form (13), including σ u (t, D), x u (t, D), and the secondorder moments α ν u (t, D), |ν| = 2, which can be represented in the form of a n-dimensional symmetric matrix…”

Section: The Einstein-ehrenfest System Of the Second Ordermentioning

confidence: 99%

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Semiclassical Approach to the Nonlocal Kinetic Model of Metal Vapor Active Media

Шаповалов

Kulagin

2021

Mathematics

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A semiclassical approach based on the WKB–Maslov method is developed for the kinetic ionization equation in dense plasma with approximations characteristic of metal vapor active media excited by a contracted discharge. We develop the technique for constructing the leading term of the semiclassical asymptotics of the Cauchy problem solution for the kinetic equation under the supposition of weak diffusion. In terms of the approach developed, the local cubic nonlinear term in the original kinetic equation is considered in a nonlocal form. This allows one to transform the nonlinear nonlocal kinetic equation to an associated linear partial differential equation with a given accuracy of the asymptotic parameter using the dynamical system of moments of the desired solution of the equation. The Cauchy problem solution for the nonlinear nonlocal kinetic equation can be obtained from the solution of the associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation. Within the developed approach, the plasma relaxation in metal vapor active media is studied with asymptotic solutions expressed in terms of higher transcendental functions. The qualitative analysis of such the solutions is given.

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Section: The Einstein-ehrenfest System Of the Second Ordermentioning

confidence: 99%

“…where a( x, t) = q i ( x, t)n neut ( x, t), q tr ( x, t), and D a (t) are given functions. For D a = const , a = const , q tr = const , the Equation ( 2) is termed the Newell-Whitehead equation [12,13].…”

Section: Introductionmentioning

confidence: 99%

Semiclassical Approach to the Nonlocal Kinetic Model of Metal Vapor Active Media

Шаповалов

Kulagin

2021

Mathematics

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“…To approximate the solutions of NWS equation from fluid mechanics, Macías-Díaz and Ruiz-Ramírez [34] proposed a method called the finite-difference method. The class of NWS equations with Lie and "nonclassical" symmetry points of view was studied in [35]. Some papers used the variational iteration method (VIM) or modified VIM to solve the NWS equations [36,37].…”

Section: Introductionmentioning

confidence: 99%

Numerical solutions of nonlinear fractional model arising in the appearance of the strip patterns in two-dimensional systems

Kumar

Kumar²,

Momani

et al. 2019

Adv Differ Equ

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The main aim of this paper is to present a comparative study of modified analytical technique based on auxiliary parameters and residual power series method (RPSM) for Newell–Whitehead–Segel (NWS) equations of arbitrary order. The NWS equation is well defined and a famous nonlinear physical model, which is characterized by the presence of the strip patterns in two-dimensional systems and application in many areas such as mechanics, chemistry, and bioengineering. In this paper, we implement a modified analytical method based on auxiliary parameters and residual power series techniques to obtain quick and accurate solutions of the time-fractional NWS equations. Comparison of the obtained solutions with the present solutions reveal that both powerful analytical techniques are productive, fruitful, and adequate in solving any kind of nonlinear partial differential equations arising in several physical phenomena. We addressed $L_{2}$ L 2 and $L_{\infty }$ L ∞ norms in both cases. Through error analysis and numerical simulation, we have compared approximate solutions obtained by two present aforesaid methods and noted excellent agreement. In this study, we use the fractional operators in Caputo sense.

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“…where a( x, t) = q i ( x, t)n neut ( x, t), q tr ( x, t), and D a (t) are given functions. For D a = const , a = const , q tr = const , the equation (1.2) is termed the Newell-Whitehead equation [12,13].…”

Section: Introductionmentioning

confidence: 99%

Semiclassical approach to the nonlocal kinetic model of metal vapor active media

Shapovalov,

Kulagin

2021

Preprint

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A semiclassical approach based on the WKB-Maslov method is developed for the kinetic ionization equation in dense plasma with approximations characteristic of metal vapor active media excited by a contracted discharge. We develop the technique for constructing the leading term of the semiclassical asymptotics of the Cauchy problem solution for the kinetic equation under the supposition of weak diffusion. In terms of the approach developed, the local cubic nonlinear term in the original kinetic equation is considered in a nonlocal form. This allows one to transform the nonlinear nonlocal kinetic equation to an associated linear partial differential equation with a given accuracy of the asymptotic parameter using the dynamical system of moments of the desired solution of the equation. The Cauchy problem solution for the nonlinear nonlocal kinetic equation can be obtained from the solution of the associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation. Within the developed approach, the plasma relaxation in metal vapor active media is studied with asymptotic solutions expressed in terms of higher transcendental functions. The qualitative analysis of such the solutions is given.

show abstract

Classification of reduction operators and exact solutions of variable coefficient Newell–Whitehead–Segel equations

Cited by 8 publications

References 58 publications

Semiclassical Approach to the Nonlocal Kinetic Model of Metal Vapor Active Media

Semiclassical Approach to the Nonlocal Kinetic Model of Metal Vapor Active Media

Numerical solutions of nonlinear fractional model arising in the appearance of the strip patterns in two-dimensional systems

Semiclassical approach to the nonlocal kinetic model of metal vapor active media

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