Separation of variables in PDEs using nonlinear transformations: Applications to reaction–diffusion type equations

Polyanin, Andrei D.; Zhurov, Alexei I.

doi:10.1016/j.aml.2019.106055

Cited by 23 publications

(17 citation statements)

References 25 publications

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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%

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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%

“…Remark 1. Exact solutions of nonlinear diffusion and wave type PDEs can be found, for example, in [4,5,9,10,[13][14][15][16][17][18][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44].…”

Section: Concept Of 'Exact Solution' For Nonlinear Pdesmentioning

confidence: 99%

Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions

Aksenov

Polyanin

2021

Mathematics

Self Cite

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“…Here, the upper and lower limits of integral (52) must satisfy the condition (25). Further, the closed solution of horizontal shear stress is also obtained by equations (52) and 11r Obviously, the interfacial shear stress (55) and (56) are symmetric alternating stress of time harmonic, which of the amplitude depends on the amplitude of excitation source. It is worth noting that the upper and lower limits of integral (55) and (56) contain the integrating factors s, which could be calculated by numerical integration methods.…”

Section: Forced Propagation Solutions Of Interfacial Shear Stress Of mentioning

confidence: 99%

“…[31][32][33][34][35][36][37][38][39] Using variational iteration methods, homotopy perturbation methods, integral transformation methods, Green's function methods, separation of variables methods, and so on, wave equations are solved. [40][41][42][43][44][45][46][47][48][49][50][51][52][53] The analytical model of wave motion in laminates is established with the boundary conditions by stiffness-matrix methods, transfer-matrix methods, and global-matrix methods. [54][55][56][57][58][59] By means of classical differential theory, the equilibrium equation is obtained by force analysis of micro elements in solid.…”

Section: Introductionmentioning

confidence: 99%

“…31–39 Using variational iteration methods, homotopy perturbation methods, integral transformation methods, Green’s function methods, separation of variables methods, and so on, wave equations are solved. 40–53 The analytical model of wave motion in laminates is established with the boundary conditions by stiffness-matrix methods, transfer-matrix methods, and global-matrix methods. 54–59…”

Section: Introductionmentioning

confidence: 99%

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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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Derivation of optical solitons of dimensionless Fokas-Lenells equation with perturbation term using Sardar sub-equation method

et al. 2022

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Separation of variables in PDEs using nonlinear transformations: Applications to reaction–diffusion type equations

Cited by 23 publications

References 25 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

Derivation of optical solitons of dimensionless Fokas-Lenells equation with perturbation term using Sardar sub-equation method

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