Partial Differential Equations in Applied Mathematics (Jun 2023)
A reliable analytic technique for the modified prototypical Kelvin–Voigt viscoelastic fluid model by means of the hyperbolic tangent function
Abstract
Inspired by the extensive applicability of solitons in nonlinear optics, advanced telecommunication industry, trans-continental, and trans-oceanic systems, coupled with the copious functionality of viscoelastic models in highway engineering pavement theory, civil engineering, and solid mechanics, unique closed-form solutions have been obtained for a highly nonlinear model built on a highly nonlinear partial differential equation, which describes the dynamics of the incompressible viscoelastic Kelvin–Voigt fluid viz: the modified (1＋1) dimension Oskolkov equation. Abundant solitary and traveling wave solutions to the prototypical Kelvin–Voigt incompressible model have been elucidated and put forward by means of the hyperbolic tangent method using the tangent function as an ansatz on the transformed nonlinear ordinary differential equation, thus, depicted breather, cusp, rouge, compacton, and kink waveforms via graphical illustrations for arbitrary intervals x∈a1,a2, t∈b1,b2, a1,a2,b1,b2∈Z.This present research yields new traveling wave profiles and structures with holomorphic terms, precise with minimized computational work, capturing all solitonic features of the Kelvin–Voigt model, contrary to the existing results with cumbersome and longwinded solution profiles, thus, the present research novelty established. The dynamical analyses, the mathematical, and solution framework presented here will be of substantial benefit to the geosciences, ocean, plasma, and, material sciences, civil engineering, nanotechnology, signaling, and signal processing to effectively tackle and convey theoretical explorations on cogent equations arising in these afore-mentioned fields. Notably, the impact of the diffusive parameter Ω<0on the incompressible model induces a significant translation of the solution profiles whilst the methodology of this study is unprecedented for the model with no existence in the literature.
