An optimal method for approximating the delay differential equations of noninteger order

Abstract

The main purpose of this paper is to use a method with a free parameter, named the optimal asymptotic homotopy method (OHAM), in order to obtain the solution of delay differential equations, delay partial differential equations, and a system of coupled delay equations featuring fractional derivative. This method is preferable to others since it has faster convergence toward homotopy perturbation method, and the convergence rate can be set as a controlled area. Various examples are given to better understand the use of this method. The approximate solutions are compared with exact solutions as well. MSC: 35R11; 65F10; 26A33

Application of fractional differential equations to heat transfer in hybrid nanofluid: modeling and solution via integral transforms

“…In this section, the dimensional system is first transformed to dimensionless form using non-similarity variables to reduce the number of variables and get rid of units. The dimensionless system is then artificially converted to time-fractional form or generalized form using the Caputo-Fabrizio fractional operator (see [2], p. [1][2][3][4][5][6][7][8][9][10][11][12][13]. It is worth to mention here that the fractional models are more general and convenient in the description of flow behavior and memory effect.…”

“…Atangana and Baleanu developed fractional operators in the Caputo and Riemann-Liouville sense using the generalized Mittag-Leffler law E α (-φx α ) as a kernel (see [3], p. 763-769). All these fractional operators have some shortcomings and challenges but at the same time this area is growing fast, and researchers devoted their attention to this field (see [4][5][6][7][8][9][10] and the references therein).…”

Study of time fractional proportional delayed multi‐pantograph system and integro‐differential equations

“…Being transcendental aspect of the delay concept makes these models more complex to reliably determine the behavior of numerical solutions. In the present, the scholars have special attention in the study of delayed differential equations, spectral collocation/waveform relaxation approaches, 14 iterated pseudospectral approach, 15 reproducing kernel Hilbert space method, 16 and optimal asymptotic homotopy method, 17 although the approximate solutions of time fractional Burger‐type PDEs with proportional delay has been evaluated via homotopy perturbation method, 18 homotopy perturbation transform method, 19 variational iteration technique, 20 reduced differential transform method for caputo/conformable fractional derivatives with proportinal delay, 21,22 homotopy analysis transform method, 23 modified finite difference method, 24 mixed Euler method, 25 and homotopy analysis method (HAM) 26 . Delayed volterra integral equations were studied via collocation methods, 27 fractional delay integro‐differential equation by spectral collocation method with Chebyshev polynomials as base functions, 28 and fractional differential transform method (FDTM) 29 .…”

A New Optimal Homotopy Asymptotic Method for Fractional Optimal Control Problems

“…A recent approximate analytical methods are: modified ADM for fractional optimal control problems (FOCPs) [20], optimal control of a constrained fractionally damped elastic beam [21], analyses of an optimal solutions of optimization problems from fractional gradient based system using VIM [22], conformable fractional optimal control problem of an heat conduction equations using Laplace and finite Fourier sine transforms [23], spectral Galerkin approximation [24], and direct transcription methods for FOCPs [25], but the aforementioned methods lack convergence norm that will guarantee the convergence of the series solution. In 1992, Liao proposed the homotopy analysis method (HAM) [26] for solving the nonlinear problem which was later advanced to optimal homotopy asymptotic method for noninteger order [27], new fractional homotopy method for optimal control problems (OCPs) [28], and comparisons of OHAM [29]. But OHAM has never been used to solve FOCPs, which drives this research work.…”