A detailed study on a new ( 2 + 1 ) $(2 + 1)$ -dimensional mKdV equation involving the Caputo–Fabrizio time-fractional derivative

Abstract

Abstract The present article aims to present a comprehensive study on a nonlinear time-fractional model involving the Caputo–Fabrizio (CF) derivative. More explicitly, a new ( 2 + 1 ) $(2 + 1)$ -dimensional mKdV (2D-mKdV) equation involving the Caputo–Fabrizio time-fractional derivative is considered and an analytic approximation for it is retrieved through a systematic technique, called the homotopy analysis transform (HAT) method. Furthermore, after proving the Lipschitz condition for the kernel ψ ( x , y , t ; u ) $\psi (x,y, t;u)$ , the fixed-point theorem is formally utilized to demonstrate the existence and uniqueness of the solution of the new 2D-mKdV equation involving the CF time-fractional derivative. A detailed study finally is carried out to examine the effect of the Caputo–Fabrizio operator on the dynamics of the obtained analytic approximation.

Keywords

• ( 2 + 1 ) $(2 + 1)$ -dimensional mKdV equation
• Caputo–Fabrizio time-fractional derivative
• Homotopy analysis transform method
• Analytic approximation
• Fixed-point theorem
• Existence and uniqueness of the solution