Generalized separation of variable methods with their comparison: exact solutions of time-fractional nonlinear PDEs in higher dimensions

A systematic investigation of the significance and applicability of two different approaches of generalized separation of variable (GSV) methods for time-fractional nonlinear PDEs in \((2+1)\) and \((m+1)\)-dimensions is presented. Also, this work explicitly shows that while constructing the exact solutions of time-fractional nonlinear PDEs in \((2+1)\) and \((m+1)\)-dimensions without and with delay terms, how to overcome unusual (non-standard) properties of singular kernel fractional derivatives such as chain rule, semigroup property, and the Leibniz rule. Moreover, the importance and effectiveness of the two GSV methods have been discussed through the initial and boundary value problems of the time-fractional nonlinear generalized convection-diffusion equation in \((2+1)\)-dimensions. Additionally, the discussed methods extended to find the exact solutions of time-fractional nonlinear PDEs in \((2+1)\) and \((m+1)\)-dimensions involving multiple linear time-delay terms along with appropriate examples. Also, this work investigates the comparative study of the obtained results and solutions of the underlying equations using the two GSV methods, along with the 2D and 3D graphical representations.