Functional Separation of Variables in Nonlinear PDEs: General Approach, New Solutions of Diffusion-Type Equations

Polyanin, Andrei D.

doi:10.3390/math8010090

Cited by 13 publications

(12 citation statements)

References 69 publications

(172 reference statements)

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“…1 • . Solution (18) and Equation (17) retain their form under the scaling transformation x = cx, u = c −2ū . Therefore, by virtue of Proposition 1, Equation 17has a more complex exact solution of the form…”

Section: Example 2 Consider the Boussinesq Equationmentioning

confidence: 99%

“…Let us now return to the Goderley Equation (17). This equation admits the simple exact solution (18), which we write in the form…”

Section: Example 14mentioning

confidence: 99%

“…Substitute it in Equation (17). After combining the functional factors for power-functions y 3n/2 (n = 0, 1, 2), we get…”

Section: Example 16mentioning

confidence: 99%

“…As a result, we obtain a binomial equation, which can be satisfied by setting ϕ xx = 0, ψ xx = 9 8 aϕ 2 . These equations are easily integrated and lead to a four-parameter exact solution of Equation(17):…”

mentioning

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: Example 2 Consider the Boussinesq Equationmentioning

confidence: 99%

“…Let us now return to the Goderley Equation (17). This equation admits the simple exact solution (18), which we write in the form…”

Section: Example 14mentioning

confidence: 99%

“…Substitute it in Equation (17). After combining the functional factors for power-functions y 3n/2 (n = 0, 1, 2), we get…”

Section: Example 16mentioning

confidence: 99%

mentioning

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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“…[66][67][68][69][70][71][72] The free propagation solution of SH waves in plates with unknown coefficients was obtained via separation of variable methods. 73,74 Using variational iteration methods, the solution of wave equation first is approximated with the possible unknowns, and then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally through the variational theory. 75,76 Variational iteration methods can be used to solve homogenous and inhomogeneous partial differential equations in bounded and unbounded domains flexibly.…”

Section: Introductionmentioning

confidence: 99%

Coupling resonance mechanism of interfacial stratification of sandwich plate structures excited by SH waves

Guo

2020

Journal of Low Frequency Noise, Vibration and Active Control

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For interfacial stratification mechanism of sandwich plate structures, the forced propagation solution of interface shear stress excited by SH waves is derived by global matrix methods and integral transformation methods. The necessary condition of interfacial shear delamination excited by alternating stress is analyzed with the interface fatigue failure theory. The exact value of forced propagation solution is calculated by adaptive Gauss–Kronrod quadrature numerical integration methods, which is verified via finite element methods. The coupling resonance mechanism of interface shear stratification is revealed by the forced vibration solution and the mass-spring model. The effects of excitation frequency, structural parameters, accretion, and matrix materials on the interfacial shear delamination are analyzed and discussed by practical cases of vibration de-accretion. For interface shear stratification of sandwich structures, the optimal excitation frequency as well as substrate thickness and accretion thickness is the value at coupling resonance, around which the interfacial shear stratification interval is formed by the interface fatigue failure criteria. In the stratification or/and antistratification design excited by vibrations, the excitation source and structure could be optimized by the method. Therefore, the results have important theoretical value for the extension and application of vibration stratification or/and antistratification technology.

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Symmetries and Separation of Variables

Rosenhaus,

Shankar,

Squellati

2024

J Nonlinear Math Phys

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In this paper, we look at the method of separation of variables of a PDE from its symmetry transformation point of view. Specifically, we discuss the relation between the existence of additively and multiplicatively separated variables of a PDE, and the form of its symmetry operators. We show that solutions in the form of separated variables are in fact, invariant solutions, i.e. solutions invariant under some subalgebra of the symmetry operators of the equation. For the case of two independent variables, we obtain the form of Lie point symmetry operators corresponding to additively and multiplicatively separated solutions, and generalize our results for the case when separated variables are any functions of independent variables. We also discuss the role of contact symmetry transformations and differential invariants for the existence of separated solutions, and outline the role of variational symmetries, as well as conditional (non-classical) symmetry operators. We demonstrate that the symmetry approach is a valuable tool for obtaining information regarding existence of solutions with separated variables.

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Functional Separation of Variables in Nonlinear PDEs: General Approach, New Solutions of Diffusion-Type Equations

Cited by 13 publications

References 69 publications

Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions

Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions

Coupling resonance mechanism of interfacial stratification of sandwich plate structures excited by SH waves

Symmetries and Separation of Variables

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