Solvability of Nonlinear Sequential Fractional Dynamical Systems with Damping

Abstract

In this paper, we are concerned with the solvability for a class of nonlinear sequential fractional dynamical systems with damping infinite dimensional spaces, which involves fractional Riemann-Liouville derivatives. The solutions of the dynamical systems are obtained by utilizing the method of Laplace transform technique and are based on the formula of the Laplace transform of the Mittag-Leffler function in two parameters. Next, we present the existence and uniqueness of solutions for nonlinear sequential fractional dynamical systems with damping by using fixed point theorems under some appropriate conditions.

Approximate Analytical Solution of Two-Dimensional Nonlinear Time-Fractional Damped Wave Equation in the Caputo Fractional Derivative Operator

“…Linear and nonlinear fractional diferential equations can successfully simulate fractional derivatives in a range of scientifc and technical domains, including electrical networks, chemical physics, control theory of dynamical systems, reaction-difusion, signal processing, and heat transform [1][2][3][4][5][6][7]. Because fractional diferential equations (FDEs) often exist in several felds of engineering and science, many researchers focus their eforts on obtaining exact/approximate solutions to these dynamic fractional diferential equations utilizing a variety of powerful established approaches, including the fnite diference method [8], Caputo fractional-reduced diferential transform method [9][10][11], Padé-Sumudu-Adomian decomposition method [12], triple Laplace transform method [13][14][15], double Sumudu transform iterative method [16], shifted Chebyshev polynomial-based method [17], Laplace decomposition method [18,19], homotopy analysis method [20], double Laplace transform method [21], homotopy perturbation method [22,23], conformable reduced differential transform method [24], conformable fractionalmodifed homotopy perturbation, Adomian decomposition method [25], diferential transform method [26][27][28], and the new function method based on the Jacobi elliptic functions [29].…”

A modification of the reduced differential transform method for fractional calculus

“…In the last few decades, fractional derivatives have found many applications in various fields of physics and engineering, for example, electrical networks, chemical physics, control theory of dynamical systems, reaction diffusion, signal processing, and heat transform can be successfully modeled in linear and non-linear fractional differential equations [1][2][3][4]. Various definitions of fractional derivatives are available in open literature [5][6][7][8][9].…”