Abstract
Over the years, it has been apparent that the application of classical concepts in differential calculus in describing new applied problems requires a kind of general and fundamental overhaul. The importance of this issue is so high that it can be seen that new and various definitions are introduced in this field of differential calculus. To this end, in this manuscript, we incorporate two analytical methods for solving a nonlinear equation arising in nonlinear physical while expressing the propagation of nonlinear waves. In the technique employed in this paper, first, some extensions of well-known elementary functions, defined on the contour set, are introduced. Solutions to the equation are then expressed in terms of these generalized functions. To better understand the physical behavior of these results, we have used three-dimensional plots of the solutions on 2D the contour set-based domain. All the results presented in this paper are new achievements for this particular form of the equation that to our knowledge have not been introduced in the previous literature. In all parts of this article, we have used Maple symbolic packages in handling analytical calculations, then symbolic computations in Mathematica has also been utilized to perform corresponding simulations.
•The doubly dispersive nonlinear wave equation is studied.•This nonlinear model appears in the propagation in the elastic circular cylinder Murnaghan’s rod.•We have used two efficient analytical techniques.•The acquired results are new and have not been reported in previous works.•Mathematica software has also been utilized to perform corresponding simulations.