Volume 25, Issue 3-4

In this study, we devise a high-order numerical scheme to approximate the Caputo–Prabhakar derivative of order α ∈ (0, 1), using an r th-order time stepping Lagrange interpolation polynomial, where 3 ≤ r ∈ N $3\le r\in \mathbb{N}$ . The devised scheme is a generalization of the existing schemes developed earlier. Further, we adopt the discussed scheme for solving a linear time fractional advection–diffusion equation and a nonlinear time fractional reaction–diffusion equation with Dirichlet type boundary conditions. We show that the discussed method is unconditionally stable, uniquely solvable and convergent with convergence order O ( τ r +1− α , h 2 ), where τ and h are the temporal and spatial step sizes, respectively. Without loss of generality, applicability of the discussed method is established by illustrative examples for r = 4, 5.

This manuscript concentrates on the problem of designing a sampled data controller (SDC) for the consensus of a fractional-order multi-agent system (FOMAS) with Lipschitz non-linearity via an algebraic approach. The solution of the FOMAS is represented by using the Laplace transform approach. An upper bound of the sampling period is determined through various integral inequality techniques. Distinguished from the existing works, the estimate for an upper bound is more accurate which involves the Lipschitz constant of the non-linear function. Finally, numerical examples are given to validate the correctness of results. Furthermore, the comparison results are presented to show the proposed method determines a better upper bound of the sampling period.

Random initial-time placed on the real line enables the evolution of the governing law defined by the Caputo derivative, which represents an equation of motion and enables the motion of the path of trajectories to many different unknown places. In order to track them to reach desired locations with or without technology, it is often necessary to use some sound, reasonable theory to understand the motion of evolution. In particular, a very challenging goal is to find reasonable conditions and Lyapunov theorems that provide potential significance to obtain reasonable conclusions about symbolic V -stability. In this paper, we address some new Lyapunov stability theorems for real-order systems associated with random initial time placed on the real axis. We introduce ideas of V -asymptotic stability and V -stability and formulate Lyapunov stability theorems. A typical inequality that we establish to find Lyapunov functions in such systems has been proposed. We develop a theoretical estimate that gives the Mittag–Leffler bound of such systems under reasonable assumptions. We present a few illustrative examples.

This paper introduces a computational spectral collocation approach utilizing orthonormal Chelyshkov polynomials to estimate solutions for both linear and nonlinear stochastic differential equations containing fractional Brownian motion characterized by the Hurst parameter H . The method employs Newton–Cotes nodes to transform these integral equations into manageable linear and nonlinear algebraic forms. The reliability of the proposed method is established through theorems on error estimation and convergence analysis. Moreover, some examples are given to demonstrate the viability, credibility, coherence, and dependability of the approach. Also, a comparative evaluation is conducted to contrast the results obtained from the suggested approach with those produced by the spectral collocation method utilizing orthonormal Bernoulli polynomials. Furthermore, a 95 % confidence interval is constructed for the error mean, revealing that the approximations maintain high accuracy.