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In this note, we investigate the existence of exact solutions of a nonlinear partial differential equation with time-dependent coefficients that generalizes the well-known nonlinear wave model with damping. The model under consideration generalizes other classical models from physics, like the nonlinear Klein–Gordon equation, the ( 1 + 1 ) -dimensional ϕ 4 -theory, the Fisher–Kolmogorov equation from population dynamics and the Hodgkin–Huxley model used in the description of the propagation of electric signals through the nervous system. An extension of the trial equation method (also known as the direct integral method) for partial differential equations with non-constant coefficients is used in this work in order to derive traveling-wave solutions in exact form.

Departing from a two-dimensional parabolic system that describes the spatial dynamics in a predator–prey system with Michaelis–Menten-type functional response, we investigate a general form of that model using a finite-difference approach. The model under investigation is a hyperbolic nonlinear system consisting of two coupled partial differential equations with generalized reaction terms. We impose initial conditions on a closed and bounded rectangle, and a fully discrete finite-difference methodology is proposed. Among the most important results of this work, we establish analytically the existence and uniqueness of discrete solutions, along with the second-order consistency of our scheme. Moreover, a discrete form of the energy method is employed to prove the stability and the quadratic convergence of the technique. Some numerical simulations obtained through our method show the appearance of Turing patterns in the parabolic case, in agreement with some reports found in the literature. Moreover, our simulations also show that Turing patterns are present in the hyperbolic scenario.

In this work, we investigate numerically the classical Burgers-Fisher equation using a modified Bhattacharya method. The partial differential equation under investigation possesses nonnegative and bounded solutions and, under some suitable parameter conditions, these solutions are traveling waves. It is well known that the use of the Bhattacharya approach leads to the design of numerical techniques that are sensitive to zero solutions. However, in this manuscript, we provide a correction of that technique in order to approximate solutions of the Burgers-Fisher equation that are bounded in 0 , 1 . The proposed methodology is explicit, and we establish thoroughly the capability of the technique to preserve the non-negativity, the boundedness and the monotonicity of the numerical approximations, as well as the constant solutions of the continuous model. The new class of methods introduced in this work considers the presence of a free parameter, and we show that this family tends to an explicit and standard discretization of the Burgers-Fisher equation when the free parameter tends to infinity. Some simulations illustrate the main features of the method.