The closed-form solution by the exponential rational function method for the nonlinear variable-order fractional differential equations

Abstract

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The symmetry features of fractional differential equations allow effective explanation of physical and biological phenomena in nature. The generalized form of the fractional differential equations is the variable-order fractional differential equations that describe the physical and biological applications. This paper discusses the closed-form traveling wave solutions for the nonlinear space–time variable-order fractional modified Kawahara and (2 + 1)-dimensional Burger hierarchy equations. The variable-order fractional differential equation has a derivative operator in the Caputo sense that is converted into the integer-order ordinary differential equation (ODE) by fractional transformation. The obtained ODE is solved by the exponential rational function method, and as a result, new exact solutions are constructed. Two problems are proposed to confirm the solutions of the space-time variable-order fractional differential equations.

Keywords

• variable-order Caputo derivative
• variable-order fractional modified Kawahara equation
• variable-order fractional (2+1)-dimensional Burger hierarchy equation
• exponential rational function method
• closed-form traveling wave solution