In this study we propose a general mathematical algorithm for the selection of aux- iliary linear operator () and initial guess 0(), which are the principal parts of Homotopy methods: Homotopy Perturbation Method (HPM) and Homotopy Anal- ysis Method (HAM). We assume the coefficients of derivatives involved in () as a functions of auxiliary roots of () = 0. Based on the residual error minimization we compute unknown roots and thereby obtain the best fitted optimal linear operator. Additionally, from the efficiency standpoint, we suggest discretize the exact square residual using the Simson’s 31 algorithm. We applied our algorithms to six nonlin- ear problems: (i) two nonlinear initial value problem (IVP) (ii) two highly nonlinear BVPs with quadratic and cubic nonlinearity, (iii) Bessel equation of zero-order and (iv) A singular and highly nonlinear BVP (for fluid electrohydrodynamics). We then compare our technique’s accuracy and efficiency to other existing analytical and nu- merical methods. It demonstrates that our best fitted optimal linear operator is much more efficient, important (than the artificial controlling parameters or functions of optimal HAM) and self-sufficient for the convergence of series solutions over the whole domain, specially for IVP. Also, an effort is made to search the best () for different choices of real roots and by means of fastest converges of the solution. Our approach is more effective, straightforward and easy to use when applied to many nonlinear problems arises in science and engineering, and using our propose ap- proach homotopy methods (HAM and HPM) will be more powerful.

This study explores the impact of fear of predators among prey populations in an eco-epidemiological model where an infectious disease infects prey. An incidence delay is introduced for the transition of the susceptible population into the infected population. Further, the dynamical behavior of the non-delay system is studied by modifying Holling type II functional response incorporating the fading memory. This is based on the concept that the predator’s growth rate not only depends on a single moment from the past but also over the whole past (chiefly, on recent history). The conditions for the existence of all the biologically feasible equilibrium points are established. The criterion for the local stability about equilibrium points of both (non-delay and delay) systems as well as for the global stability around the coexistence equilibria of the non-delay system are established here. Sufficient conditions for the existence of Hopf-bifurcation by taking the force of infection and delay as bifurcation parameters are derived. Numerical simulations are performed to verify the analytical results and illuminate the system’s dynamicity. The system’s complex dynamical behaviors are demonstrated using the bifurcation diagram, phase diagram, and spectrum. It is observed that fears reduce predator density and also convert an unstable (periodic or chaotic) system into a stable one. It is recommended that past influence over a short time interval and a higher value of the cost of fears are necessary to persist a sustainable and stable ecosystem. An effort is made to search for the correlation between the cost of fears and other biologically related parameters (viz., the growth rate of prey, the carrying capacity for prey, the force of infection, fading memory, and incidence delay) to understand the dynamicity of the system. The semi-relative as well as logarithmic sensitivities of the system are applied to the proposed model to observe how much change of a state variable occur due to perturbation of the parameters, like, the force of infection and cost of fears.