In this paper, the convective-radiative fins of rectangular profile with temperature-dependent thermal conductivity are considered. By studying the conventional heat transfer equation, its modified fractal form, which can describe the problem in the porous medium, is presented based on He?s fractal derivative for the first time. The fractal two-scale transform method together with the Taylor series are applied to deal with fractal model, and an analytical approximate solution is obtained. The impact of the different fractal orders on the thermal behavior of the fins is also elaborated in detail. In addition, a comparison between our solution and the existing one is given to prove the correctness of the proposed method, which shows that the proposed method is easy but effective, and are expected to shed a bright light on practical applications of fractal calculus.

The current study aims to present detailed qualitative and quantitative research on the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation (BLMPE), which plays a key role in the field of incompressible fluids. First, we constructed the variational principle using the semi-inverse method (SIM) and developed the Hamiltonian based on the variational principle. Then, we derived the planar dynamical system (PDS), depicted the phase portraits, and carried out the bifurcation analysis to expound the existence of wave solutions with the different wave shapes. In addition, chaotic behaviors were elaborated by imposing the perturbed term, and the sensitivity analysis was conducted by taking the different initial conditions. Finally, the invariant algebraic curve approach (based on the PDS), the direct mapping method (from the unified equation), and the Hamiltonian based method were used to investigate the diverse wave solutions, including the anti-kink solitary, kink solitary, periodic, and singular periodic wave solutions. The profiles of these different solutions are graphically illustrated by assigning reasonable parameters. The results of this paper are novel and can provide a better insight into the dynamics of the equation under consideration.