Study perturbation theory and methods for some differential equations with stability analysis

Partial differential equations can be used to describe a variety of physical and mechanical issues, including heat transfer by conduction in both stable and unstable states. The physical phenomena that are reflected in the magnitude of turbulence can be understood analytically using turbulence techniques. Where the solution of partial differential equations is effective using HPM. The infinite series of functions represent the exact solution produced by HPM, which converges to the solution. Two basic aspects are the selection of an effective prime approximation estimate and a suitable symmetric formula. In the current work, the HPM was developed and a new HPM was obtained, which we called the NHPM. The new method was compared with the old method. The results of the NHPM were compared with the exact solution. The results were compared numerically and by plotting, in order to find out which is more accurate and the current method was more accurate than the HPM method.