Exact Solutions with Variable Coefficient Function Forms for Conformable Fractional Partial Differential Equations by an Auxiliary Equation Method

Abstract

In this paper, an auxiliary equation method is introduced for seeking exact solutions expressed in variable coefficient function forms for fractional partial differential equations, where the concerned fractional derivative is defined by the conformable fractional derivative. By the use of certain fractional transformation, the fractional derivative in the equations can be converted into integer order case with respect to a new variable. As for applications, we apply this method to the time fractional two-dimensional Boussinesq equation and the space-time fractional (2+1)-dimensional breaking soliton equation. As a result, some exact solutions including variable coefficient function solutions as well as solitary wave solutions for the two equations are found.

Exact Solutions of Three Types of Conformable Fractional-Order Partial Differential Equations

“…The mathematical models of many problems in life can eventually be transformed into solving integer differential equations. The development of integer integral is relatively complete from both theoretical analysis and numerical solution [ 1 ]. Fractional calculus is widely used in biology, control system, and engineering, so it has been highly valued by scientists [ 2 ].…”

Painlevé Analysis and Abundant Meromorphic Solutions of a Class of Nonlinear Algebraic Differential Equations

“…To solve various differential equations, some analytical tools as well as symbolic calculation techniques were established, for instance, fixed point theorems [14][15][16][17], variational methods [18][19][20][21], topological degree method [22][23][24][25], iterative techniques [26][27][28][29], modified Kudryashov method [30,31], exp function method [32,33], sine-Gordon expansion method [34][35][36][37], and complex method [38][39][40][41]. More works about the differential equations can be read in [42][43][44][45][46][47][48][49][50][51][52].…”

Pure Traveling Wave Solutions for Three Nonlinear Fractional Models

“…Meng and Feng [32] used an auxiliary equation method to the space-time fractional ð2 + 1Þ-dimensional breaking soliton equation. Authors of [27,33,34] discussed the spacetime fractional SRLW equation (STFSRLWE) in the following case:…”