The \((3 + 1)\) -dimensional time-fractional KdV–Zakharov–Kuznetsov equation and modified KdV equation with variable coefficients, which have very crucial applications in various areas, such as fluid dynamics, plasma physics and so on, are studied via unified and generalised unified method in this work. Solitary, soliton, elliptic, singular (periodic type) and non-singular (soliton-type) solutions of these two equations are extracted using the unified method. Otherwise, polynomial solutions in double-wave form and rational solutions in double-soliton form of the modified KdV equation with variable coefficients are acquired by exploiting the generalised unified method. The dynamical demeanours of these solutions help to comprehend the physical phenomena reflected by the equations, are depicted and analysed graphically for specific values of randomly undetermined parameters, which are diverse in each solution. The outcomes show that these two methods are quite trustworthy and effective to explore numerous solutions of nonlinear partial differential equations. We recognise that two approaches have never been utilised to study these two equations and work carried out in this paper is fresh and handy. Compared to previous methods, more comprehensive solutions can be obtained using them in this paper.
