The analysis of Homotopy Perturbation Method (HPM) for the solution of fractional partial differential equations (FPDEs) is presented. A unified convergence theorem is given. In order to validate the theory, the solution of fractional-order Burger-Poisson (FBP) equation is obtained. Furthermore, this work presents the method to find the solution of FPDEs, while the same partial differential equation (PDE) with ordinary derivative i.e., for $\alpha =1$ , is not defined in the given domain. Moreover, HPM is applied to a complicated obstacle boundary value problem (BVP) of fractional order.

An innovative concept of water-based Cu–Al₂O₃ hybrid nanofluid has been employed to investigate the behavior of flow and heat transfer inside a rectangular channel whose permeable walls experiences dilation or contraction in height. The transformed set of ordinary differential equations is then solved by a well-known Runge–Kutta–Fehlberg algorithm. The analysis also includes three different shapes of copper nanocomposites, namely, platelet, cylinder and brick- shaped. The impact of various embedded parameters on the flow and heat transfer distributions have been demonstrated through the graphs. All the flow properties, temperature profile and rate of heat transfer at the walls are greatly influenced by the presence of copper nanoparticles. Furthermore, it was observed that the platelet shaped nanocomposites provide a better heat transfer ability as compared to the other shapes of nanoparticles.

This paper is based on finding the exact solutions for Burger’s equation, Zakharov-Kuznetsov (ZK) equation and Kortewegde vries (KdV) equation by utilizing exponential function method that depends on the series of exponential functions. The exponential function method utilizes the homogeneous balancing principle to find the solutions of nonlinear equations. This method is simple, wide-reaching and helpful for finding the exact solution of nonlinear conformable PDEs.