Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation

Abstract

In this paper, the existence and uniqueness of solution for a class of boundary value problems of nonlinear fractional order differential equations involving the Caputo fractional derivative are studied. The estimation of error between the approximate solution and the solution for such equation is presented by employing the quasilinear iterative method, and an example is given to demonstrate the application of our main result.

Semilinear Caputo time-fractional pseudo-parabolic equations

“…Since the fractional dissipation operator (−∆) σ is nonlocal and can be regarded as the infinitesimal generators of Lévy stable diffusion processes [51,38], it describes some physical phenomena more exact than integral differential equations [17,51,46,16,20,32]. Many important physical models and practical problems require us to consider the pseudo-parabolic model with fractional derivative rather than classical one, like the physical model considering memory effects [18,19,13,48] and some corresponding engineering problems [4,5,7,44,46,50,56], especially the power-law memory (non-local effects) in time and space [20,25,51,13,38,39,42,40,53,52,54].…”

Approximation of Iterative Methods for Altering Points Problem With Applications

“…So far, numerous problems in nonlinear analysis involving contraction and nonexpansive mappings have been solved using this fundamental result. In recent past, the fixed point iterative schemes have been examined considerably to study monotone variational inequalities, problems from nonlinear analysis and applied mathematics such as initial and boundary value problems, image recovery problems, image restoration problems, image processing problems, for more details, see [12,13,25,26,32]. Note that, Mann-like iterative algorithms and its variant forms are efficiently applied for solving several nonlinear problems.…”

Controllability of nonlinear fractional evolution systems in Banach spaces: A survey

“…As we all know, control theory is an interdisciplinary branch of economics, engineering and mathematics that investigates and analyses some dynamical behaviors of various systems [78,92,115,130,131]. In addition, control theory of dynamical systems with impulse [7,35,47,74] or time delay [39,48,109,110] have been studied in the last decates.…”