It is well-known that discovering and then calculating all branches of solutions of fractional order nonlinear differential equations with boundary conditions can be difficult even by numerical methods. To overcome this difficulty, in this chapter two semianalytic methods are presented to predict and obtain multiple solutions of nonlinear boundary value problems. These methods are based on the homotopy analysis method (HAM) and Picard method namely, predictor HAM and controlled Picard method. The used techniques are capable of predicting and calculating all branches of the solutions simultaneously. Four problems are solved, three of them are practical problems which are generalized in fractional order domain to show the efficiency and importance of these methods. And the solutions are calculated by simple procedures without any need for special transformations or perturbation techniques. © 2018 Elsevier Inc. All rights reserved.
Controlled Picard Method for Solving Nonlinear Fractional Reaction–Diffusion Models in Porous Catalysts
This paper discusses the diffusion and reaction behaviors of catalyst pellets in the fractional-order domain as well as the case of nth-order reactions. Two generic models are studied to calculate the concentration of reactant in a porous catalyst in the case of a spherical geometric pellet and a flat-plate particle with different examples. A controlled Picard analytical method is introduced to obtain an approximated solution for these systems in both linear and nonlinear cases. This method can cover a wider range of problems due to the extra auxiliary parameter, which enhances the convergence and is suitable for higher-order differential equations. Moreover, the exact solution in the linear fractional-order system is obtained using the Mittag–Leffler function where the conventional solution is a special case. For nonlinear models, the proposed method gives matched responses with the homotopy analysis method (HAM) solutions for different fractional orders. The effect of fractional-order parameter on the dimensionless concentration of the reactant in a porous catalyst is analyzed graphically for different cases of order reactions and Thiele moduli. Moreover, the proposed method has been applied numerically for different cases to predict and calculate the dual solutions of a nonlinear fractional model when the reaction order n = ?1. © 2017, Copyright © Taylor & Francis Group, LLC.
Single and dual solutions of fractional order differential equations based on controlled Picard’s method with Simpson rule
This paper presents a semi-analytical method for solving fractional differential equations with strong terms like (exp, sin, cos,Â…). An auxiliary parameter is introduced into the well-known Picard’s method and so called controlled Picard’s method. The proposed approach is based on a combination of controlled Picard’s method with Simpson rule. This approach can cover a wider range of integer and fractional orders differential equations due to the extra auxiliary parameter which enhances the convergence and is suitable for higher order differential equations. The proposed approach can be effectively applied to Bratu’s problem in fractional order domain to predict and calculate all branches of problem solutions simultaneously. Also, it is tested on other fractional differential equations like nonlinear fractional order Sine-Gordon equation. The results demonstrate reliability, simplicity and efficiency of the approach developed. © 2017 University of Bahrain
Modified methods for solving two classes of distributed order linear fractional differential equations
This paper introduces two methods for the numerical solution of distributed order linear fractional differential equations. The first method focuses on initial value problems (IVPs) and based on the ?th Caputo fractional definition with the shifted Chebyshev operational matrix of fractional integration. By applying this method, the IVPs are converted into simple linear differential equations which can be easily handled. The other method focuses on boundary value problems (BVPs) based on Picard’s method frame. This method is based on iterative formula contains an auxiliary parameter which provides a simple way to control the convergence region of solution series. Several numerical examples are used to illustrate the accuracy of the proposed methods compared to the existing methods. Also, the response of mechanical system described by such equations is studied. © 2017 Elsevier Inc.
Realization of fractional-order capacitor based on passive symmetric network
In this paper, a new realization of the fractional capacitor (FC) using passive symmetric networks is proposed
Fundamentals of fractional-order LTI circuits and systems: number of poles, stability, time and frequency responses
This paper investigates some basic concepts of fractional-order linear time invariant systems related to their physical and non-physical transfer functions, poles, stability, time domain, frequency domain, and their relationships for different fractional-order differential equations
An optimal linear system approximation of nonlinear fractional-order memristor-capacitor charging circuit
The analysis of nonlinear fractional-order circuits is a challenging problem
The minimax approach for a class of variable order fractional differential equation
This paper introduces an approximate solution for Liouville-Caputo variable order fractional differential equations with order 0 < ?(t) ? 1