Exact Solutions And Fundamental Solution Of Several Kinds Of Conformable Fractional Partial Differential Equations

The research object of this paper is Conformable fractional partial differential equations.And this kind of equation is mainly studied from two aspects.On the one hand,solutions of fractional Burgers equation,fractional coupled Boussinesq-Whitham-Broer-Kaup(BWBK)equations,fractional modified equal width(MEW)equation and(2+1)-dimensional generalized Conformable time fractional Burgers equations are established by some specific mathematical methods.Among them,we solve the Conformable time fractional Burgers equation and(2+1)-dimensional generalized Conformable time fractional Burgers equations using the functional separation variable method.The first integral method is utilized to solve the Conformable space-time fractional coupled BWBK equations and Conformable space-time fractional MEW equation.Based on the Conformable Adomian Decomposition Method and Laplace Adomian Decomposition Method,solutions of the above equations in the sense of Conformable and Caputo fractional derivative except(2+1)dimensional generalized Burgers equations are solved.Furthermore,with the help of Maple software,we draw the 3D graphics,2D graphics and density graphics of partial solutions of the studied equations,and carry out a brief analysis according to the graphics.Through above analysis,we find that the Conformable fractional derivative can be appropriately used instead of Caputo fractional derivative for modeling.On the other hand,we also establish the formula of the fundamental solution and Mean-value formula of Conformable time fractional heat equation.Based on the obtained fundamental solution and Mean-value formula,the existence and uniqueness,regularity and the estimates on derivatives of the classical solution are given.