On the convergence of a finite-difference discretization à la Mickens of the generalized Burgers–Huxley equation

Interdiffusion in many dimensions: mathematical models, numerical simulations and experiment

“…It follows immediately from the form of equations (25) to (27), (29), (30), (37) and (38) that our difference method is consistent and the truncation errors r, R = O(τ + h).…”

“…Proof. Taking into account the regularity assumptions, the following relations for the terms that generate the difference method (equations (29) and (30)) in the internal points hold:…”

“…Numerical methods that preserve some futures of a continuous model are very important and interesting from a physical and mathematical point of view. For example, boundedness-, positivity-or monotonicity-preserving numerical methods were studied in [25][26][27][28][29][30][31][32][33][34][35] and in the references therein. We prove that the implicit difference schemes given have a unique solution and that the difference methods are consistent.…”

Existence and Uniqueness of Positive and Bounded Solutions of a Discrete Population Model with Fractional Dynamics

“…and we only need to show that +1 ≤ or, equivalently, that V +1 = − +1 ≥ 0, where is the vector of the same Discrete Dynamics in Nature and Society 5 dimension as +1 , all of whose entries are equal to 1. Note that the recursive identity in (15) is equivalent to…”

“…To solve this problem, various computational approaches have been designed in order to approximate the solutions of continuous (fractional or integral) Fisher's equations. Some approximation techniques have been proposed in the literature to that effect, including finite-difference schemes [15], differential and integral quadrature techniques [16,17], Legendre spectral collocation methods [18], discrete local discontinuous Galerkin methods [19], finite volume schemes with preconditioned Lanczos techniques [20], and homotopy perturbation methods [21]. In most of the cases, the methods proposed are capable of approximating the solutions of the fractional Fisher's equation with a high degree of precision, but the preservation of the most important features of the solutions of interest is neglected.…”