In this study, a new modification of the homotopy perturbation method (HPM) is introduced for various order boundary value problems (BVPs). In this modification, HPM is hybrid with least square optimizer and named as the least square homotopy perturbation method (LSHPM). The proposed scheme is tested against various linear and nonlinear BVPs (second to seventh order DEs). Validity of the obtained solutions is confirmed by finding absolute errors. To analyze the efficiency of the proposed scheme, tested problems have also been solved through HPM and results are compared with LSHPM. Furthermore, obtained results are also compared with other numerical schemes available in literature. Analysis reveals that LSHPM is a consistent and effective scheme which can be used for more complex BVPs in science and engineering.

Squeezing flow has many applications in different fields including chemical, mechanical, and electrical engineering as these flows can be observed in many hydrodynamical tools and machines. Due to importance of squeezing flow, in this paper, an unsteady squeezing flow of a viscous magnetohydrodynamic (MHD) fluid which is passing through porous medium has been modeled and analyzed with and without slip effects at the boundaries. The least squares homotopy perturbation method (LSHPM) has been proposed to determine the solutions of nonlinear boundary value problems. To check the validity and convergence of the proposed scheme (LSHPM), the modeled problems are also solved with the Fehlberg–Runge–Kutta method (RKF45) and homotopy perturbation method (HPM) and residual errors are compared with LSHPM. To the best of the authors’ knowledge, the current problems have not been attempted before with LSHPM. Moreover, the impact of different fluid parameters on the velocity profile has been examined graphically in slip and no-slip cases. Analysis shows that the Reynolds number, MHD parameter, and porosity parameter have opposite effects in case of slip and no slip at the boundaries. It is also observed that nonzero slip parameter accelerates the velocity profile near the boundaries. Analysis also reveals that LSHPM provides better results in terms of accuracy as compared to HPM and RKF45 and can be effectively used for the fluid flow problems.

In this manuscript, modification of homotopy perturbation method (HPM) is proposed for integro-differential equations by coupling the least square method (LSM) with HPM. Improved accuracy in a very few iterations is the general advantage of this technique. The proposed method is applied to different higher order integro-differential equations of linear and nonlinear nature, and results are compared with exact as well as available solutions from the literature. Numerical and graphical analysis reveal that the proposed algorithm is reliable for integro-differential equations and hence can be utilized for more complex problems.