| Popis: |
In honor of the great Russian mathematician A. N. Kolmogorov, we would like to draw attention in the present paper to a curious mathematical observation concerning fractional differential equations describing physical systems, whose time evolution for integer derivatives has a time-honored conservative form. This observation, although known to the general mathematical community, (Achar et al., 2001; Stanislavsky, 2004; Diethelm and Ford, 2010; Chung and Jung, 2014; Olivar-Romero and Rosas-Ortiz, 2017; Baleanu et al., 2020) has not, in our view, been satisfactorily addressed. More specifically, we follow the recent exploration of Caputo–Riesz time–space-fractional nonlinear wave equation of Macias Diaz (2022), in which two of the present authors introduced an energy-type functional and proposed a finite-difference scheme to approximate the solutions of the continuous model. The relevant Klein–Gordon equation considered here has the form: (0.1) ∂βϕ(x,t)∂tβ−Δαϕ(x,t)+F′(ϕ(x,t))=0,∀(x,t)∈(−∞,∞)where we explore the sine-Gordon nonlinearity F(ϕ)=1−cos(ϕ) with smooth initial data. For α=β=2, we naturally retrieve the exact, analytical form of breather waves expected from the literature. Focusing on the Caputo temporal derivative variation within 1 |
