On dynamical behavior for approximate solutions sustained by nonlinear fractional damped Burger and Sharma–Tasso–Olver equation

Abstract

In this paper, we study a nonlinear fractional Damped Burger and Sharma–Tasso–Olver equation using a new novel technique, called homotopy perturbation transform method (FHPTM). There are three examples used to demonstrate and validate the proposed algorithm’s efficiency. This nonlinear model depicts nonlinear wave processes in fluid dynamics, ecology, solid-state physics, shallow-water wave propagation, optical fibers, fluid mechanics, plasma physics, and other applied science, engineering, and mathematical physics disciplines, as well as other phenomena. Numerous algebraic properties of the fractional derivative Caputo–Fabrizio operator are illustrated concerning the Laplace transformation to demonstrate their utility. Different graphs and tables compare the results obtained by R. Nawaz et al. [Alex. Eng. J. 60, 3205 (2021)] and M. S. Rawashdeh [Appl. Math. Inform. Sci. 9, 1239 (2015)]. The proposed scheme accelerates the convergence of the series solution and guarantees the convergence associated with the homotopy parameter. Furthermore, the physical nature of various fractional orders has been captured in plots. The obtained results demonstrate that the employed solution procedure is dependable and methodical in investigating the behaviors of nonlinear models of both integer and fractional orders.

A Fixed Point Theorem for Time-Fractional Fornberg-Whitham Equation Arising in Atmospheric Science

“…In this section, the author provides definitions of important terms and concepts related to the topic being studied [3][4].…”

“…The Homotopy perturbation transform method (HPTM) gives a straightforward explanation for changing and managing the series (63) (64) (65) solution's convergence. It is worth noting that the proposed method might also be utilised to solve the TFFWEs, which incorporates Caputo-Fabrizio fractional derivative (CFFD) of order 0 < α ≤ 1, as in the HPTM [3][4]. The investigation has shown that the offered methodologies and the problem's exact answers are most closely related.…”

“…Around the same time, the German mathematician Felix Klein also made contributions to the development of fractional calculus. In the early 20th century, the Italian mathematician Umberto Grunwald and the Russian mathematician Mikhail Letnikov independently developed the Grunwald-Letnikov definition of fractional derivatives, which paved the way for the study of FDEs [3][4]. The same definition was independently rediscovered and developed further by the German mathematician Gustav Riesz.…”

Data assimilation method for improving the global spatiotemporal predictions of epidemic dynamics yielded by an ensemble Kalman filter and Metropolis–Hastings sampling

“…The first contribution was provided by Daniel Bernoulli [1], and the canonical susceptible-infected-recovered (SIR) model was proposed by Kermack and McKendrick [2]. Subsequent studies used either complex networks [3,4] or fractional differential equations [5][6][7][8] to extend the SIR model by incorporating more heterogeneities and correlations into the transmission processes [9]. However, due to the remarkable spatiotemporal variations exhibited by related factors [10], such as non-pharmaceutical interventions [11], population movements [12], and social contacts [13], as well as the uncertainties caused by virus mutations and vaccinations [14], mathematical models have long faced challenges in capturing epidemic dynamics in terms of multiple factors.…”