Contraction analysis for fractional-order nonlinear systems

Numerical schemes for variable exponent fractional‐type integral equations

“…a + be the generalized VO Riemann-Liouville integral operator of order 𝛼(t), where n − 1 < 𝛼(t) ≤ n, n ∈ IN, and g, 𝑓 ∈ C[0, T]. To simulate the VE-FIE (10), over the interval [0, T], we assume that the function g is considered such that a unique solution to the VE-FIE (10) exits on [0, T]. To describe the P-C approach, let us define a uniform grid in the interval [0, T] with N + 1 equispaced nodes t 𝑗 = 𝑗h, 𝑗 = 0, 1, • • • N, where h = T∕N and N ∈ IN.…”

“…This section proposes an improved quadrature-based P-C scheme of the Adams-Bashforth-Moulton method, which in terms of accuracy is superior to the P-C approach presented in Section 3, for the numerical simulation of the VE-FIE (10). Let The suggested improved P-C scheme uses the Simpson's 1/3 rule with respect to the weight function (t k+1 − .)…”

“…Finally, our improved quadrature-based P-C scheme for solving the VE-FIE (10), is completely described by the rules ( 24), ( 26), ( 29), (34), and (37) with the weights defined in ( 21), ( 22), ( 23), ( 28), ( 32), (33), and (36).…”

“…Consequently, studying the dynamics, such as complexity, stability, bifurcation, chaos, and synchronization of fractional-order systems, has become an active area of research in nonlinear sciences. [10][11][12][13][14][15][16][17][18][19][20] Therefore, because of the growing interest in the applications that involve fractional-order differential models, it has become necessary to expand, develop, and improve stable and robust numerical and analytical methods for solving such models.…”

On the formulation of a predictor–corrector method to model IVPs with variable‐order Liouville–Caputo‐type derivatives

“…In some phenomena, the nonlocal feature of fractional operators allows the characterization and description of mathematical models using fractional-order derivatives more sensible and realistic than ordinary derivatives. Recently, the study of dynamics including complexity, bifurcation, stability, chaos, and synchrony of dynamical systems with fractional derivatives has become an important research area [12][13][14][15][16][17][18][19][20]. Moreover, since the study of infectious disease dynamics has become a necessary and encouraging topic for research, there is a growing interest in biological and especially epidemiological models that involve fractional derivatives; see, for example, [21][22][23].…”