Dual conformable derivative: Variational approach and nonlinear equations

¹ Universidade Federal Rural do Rio de Janeiro, UFRRJ- IM/DTL - Av. Governador Roberto Silveira s/n, Nova Iguaçú, Rio de Janeiro, 695014, Brazil
² Departamento de Física, Universidade Federal Rural do Rio de Janeiro (UFRRJ) - BR 465-07, 23890-971, Seropédica, RJ, 695014, Brazil
³ Hohai University, Institute of Soft Matter Mechanics, College of Mechanics and Materials Nanjiing, Jiangsu 211100, China

^(a) josewebe@gmail.com
^(b) crgodinho@gmail.com
^(c) liangyj@hhu.edu.cn

Received: 14 October 2019
Accepted: 19 November 2019

Abstract

In this letter, we present a variational approach with a recently proposed form of local deformed derivative, the dual conformable derivative, which leads us to obtain a class of nonlinear equations. The ansatz on the solutions can be mathematically inferred by this dual conformable derivative eigenvalue equation and the q-exponential family functions appear naturally. Also, a clear and natural justification for the appearance of the q → 2 - q-symmetry is given. To show the potential of our variational approach with this type of dual deformed derivative, we obtain the porous medium equation and present insight for the solution in terms of the q-Gaussian. Also, a new dual conformable wave equation, which is nonlinear, is proposed and a solution is built up in terms of q-plane waves. A dual conformable harmonic oscillator equation is also obtained and promptly solved by the natural ansatz. Aspects of the nonlinear Schroedinger equation are also contemplated and one shows that it can be obtained without the need of an additional ϕ-field, from a simple Lagrangian density. The solution to the nonlinear Schroedinger is also expressed in terms of the q-exponential family of functions.