In this article we consider an even entire function of order one as the solution of the classical wave equation in one-dimensional space. We suggest a conjecture that this function has only purely real zeros in the entire complex plane. This is given as a new prospective on a nice connection with number theory and wave equation.

A good initial guess and an appropriate homotopy equation are two main factors in applications of the homotopy perturbation method. For a nonlinear oscillator, a cosine function is used in an initial guess. This article recommends a general approach to construction of the initial guess and the homotopy equation. Duffing oscillator is adopted as an example to elucidate the effectiveness of the method.

We view random walks as the paths of foraging animals, perhaps searching for food or avoiding predators while forming a mental map of their surroundings. The formation of such maps requires them to memorise the locations they have visited. We model memory using a kernel, proportional to the number of locations recalled as a function of the time since they were first observed. We give exact analytic expressions relating the elongation of the memorised walk to the structure of the memory kernel, and confirm these by simulation. We find that more slowly decaying memories lead to less elongated mental maps.

In this paper, we discuss the competition of two species for a single essential growth-limiting nutriment with viral infection that affects only the first species. Although the classical models without viral infection suggest competitive exclusion, this model exhibits the stable coexistence of both species. We reduce the fourth-dimension proposed model to a three-dimension one. Thus, the coexistence of the two competing species is demonstrated using the theory of uniform persistence applied to the three-variable reduced system. We prove that there is no coexistence of both species without the presence of the virus and the satisfaction of some assumptions on the growth rates of species. Finally, we give some numerical simulations to confirm the obtained theoretical findings.

In the paper, we consider some fractional integral formulas in terms of the Riemann–Liouville, Erdélyi–Kober type, and Weyl fractional integral operators and present the general fractional kinetic model involving the hypergeometric superhyperbolic sine function via the Gauss hypergeometric series.