Abstract
Abstract The main purpose of this article is to obtain the new solutions of fractional bad and good modified Boussinesq equations with the aid of auxiliary equation method, which can be considered as a model of shallow water waves. By using the conformable wave transform and chain rule, nonlinear fractional partial differential equations are converted into nonlinear ordinary differential equations. This is an important impact because both Caputo definition and Riemann–Liouville definition do not satisfy the chain rule. By using conformable fractional derivatives, reliable solutions can be achieved for conformable fractional partial differential equations.
Highlights
In 1695, since L’ Hospital asked the question, what might be a derivative order 1/2
Phenomena related to nonlinear partial differential equations (NLPDEs) have emerged in many areas such as physics, mechanics and chemistry to investigate the exact solutions for NLPDEs
There are a lot of workings with NLPDEs
Summary
In 1695, since L’ Hospital asked the question, what might be a derivative order 1/2. Many researchers tried to find a definition of fractional derivative after this question. Khalil et al [5] introduced a new definition of the integral and conformable fractional derivatives. Let us give the definition and some properties of conformable fractional derivative and integral. Phenomena related to nonlinear partial differential equations (NLPDEs) have emerged in many areas such as physics, mechanics and chemistry to investigate the exact solutions for NLPDEs. In recent years, there are a lot of workings with NLPDEs. For example, Whitham [9] studied variational methods and applications on water waves. Eslami and Mirzazadeh [4] used the first integral method to obtain the exact solutions of the nonlinear Schrödinger equation The properties of this new definition [5] are given below.
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