The Homotopy Perturbation Method for Solving Nonlocal Initial-Boundary Value Problems for Parabolic and Hyperbolic Partial Differential Equations

Abstract

To obtain approximate-exact solutions to nonlocal initial-boundary value problems (IBVPs) of linear and nonlinear parabolic and hyperbolic partial differential equations (PDEs) subject to initial and nonlocal boundary conditions of integral type, the homotopy perturbation method (HPM) is utilized in this study. The HPM is used to solve the specified nonlocal IBVPs, which are then transformed into local Dirichlet IBVPs. Some examples demonstrate how accurate and efficient the HPM.

Application of homotopy perturbation method to solve a nonlinear mathematical model of depletion of forest resources

“…In 2008, He employed the HPM to solve boundary value problems [20]. In 2007, Javidi and Golbabai used a revised version of the HPM to solve non-linear Fredholm integral equations [21].Recently, HPM with small variations has been applied to study fractal duffing oscillator problems under arbitrary conditions [22], modified HPM for nonlinear oscillators Anjum and He [23], attachment oscillator arising in nanotechnology [24], conservative nonlinear oscillators [25], non-linear oscillator problems in a fractal space [26] and HPM including Aboodh transformation to solve fractional calculus Tao et al [27], vibrating magnetic inverted pendulum Moatimid et al [28], Symmetry-breaking and pull-down motion for the helmholtz-duffing oscillator Niu et al [29], nonlinear fractional Drinfeld-Sokolov-Wilson Equation Nadeem and Alsayaad [30], trajectory analysis of a zero-pitch-angle e-Sail Niccolai et al [31], natural convection between two concentric horizontal circular cylinders Abdulameer and Ali Al-Saif [32], nonlocal initialboundary value problems for parabolic and hyperbolic Al-Hayani and Younis [33], multi-step iterative methods for solving nonlinear equations Saeed et al [34], telegraph equation Moazzzam et al [35], triangular linear diophantine fuzzy system of equations Shams et al [36], condensing coagulation model and Lifshitz-Slyzov equation Arora et al [37], singular nonlinear system of boundary value problems Pathak et al [38], rikitake-yype system Ene and Pop [39], heat and mass transfer with 2D unsteady squeezing viscous flow problem Abdul-Ameer and Ali Al-Saif [40], variable Speed Wind Turbine Control Shalbafian and Ganjefar [41], radial thrust problem Niccolai et al [42], special third grade fluid flow with viscous dissipation effect over a stretching sheet Swain et al [43], and the frequency-amplitude relationship of a nonlinear oscillator with cubic and quintic nonlinearities He et al [44]. The HPM has become a widely-used technique to solve a large variety of problems in different fields and many research papers have been published each year using this method as evidenced by a simple search on Google Scholar.…”