scholarly journals Residual Power Series Technique for Simulating Fractional Bagley–Torvik Problems Emerging in Applied Physics

2019 ◽  
Vol 9 (23) ◽  
pp. 5029 ◽  
Author(s):  
Alshammari ◽  
Al-Smadi ◽  
Hashim ◽  
Alias
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Nonlinear Modeling ◽  
Test Problems ◽  
Approximate Methods ◽  
Wide Range ◽  
Residual Power Series ◽  
Fractional Differential ◽  
Residual Power

Numerical simulation of physical issues is often performed by nonlinear modeling, which typically involves solving a set of concurrent fractional differential equations through effective approximate methods. In this paper, an analytic-numeric simulation technique, called residual power series (RPS), is proposed in obtaining the numerical solution a class of fractional Bagley–Torvik problems (FBTP) arising in a Newtonian fluid. This approach optimizes the solutions by minimizing the residual error functions that can be directly applied to generate fractional PS with a rapidly convergent rate. The RPS description is presented in detail to approximate the solution of FBTPs by highlighting all the steps necessary to implement the algorithm in addressing some test problems. The results indicate that the RPS algorithm is reliable and suitable in solving a wide range of fractional differential equations applying in physics and engineering.

scholarly journals Computational Optimization of Residual Power Series Algorithm for Certain Classes of Fuzzy Fractional Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Mohammad Alaroud ◽  
Mohammed Al-Smadi ◽  
Rokiah Rozita Ahmad ◽  
Ummul Khair Salma Din
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Optimization Technique ◽  
Fractional Power ◽  
Taylor Formula ◽  
Residual Power Series ◽  
Fractional Differential ◽  
Residual Power ◽  
Fuzzy Fractional Differential Equations

This paper aims to present a novel optimization technique, the residual power series (RPS), for handling certain classes of fuzzy fractional differential equations of order 1<γ≤2 under strongly generalized differentiability. The proposed technique relies on generalized Taylor formula under Caputo sense aiming at extracting a supportive analytical solution in convergent series form. The RPS algorithm is significant and straightforward tool for creating a fractional power series solution without linearization, limitation on the problem’s nature, sort of classification, or perturbation. Some illustrative examples are provided to demonstrate the feasibility of the RPS scheme. The results obtained show that the scheme is simple and reliable and there is good agreement with exact solution.

scholarly journals Toward computational algorithm for time-fractional Fokker–Planck models

2019 ◽  
Vol 11 (10) ◽  
pp. 168781401988103 ◽  
Author(s):  
Asad Freihet ◽  
Shatha Hasan ◽  
Mohammad Alaroud ◽  
Mohammed Al-Smadi ◽  
Rokiah Rozita Ahmad ◽  
...  
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Fractional Power ◽  
Series Method ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Wide Range ◽  
Residual Power Series ◽  
Residual Power ◽  
Fokker Planck

This article describes an efficient algorithm based on residual power series to approximate the solution of a class of partial differential equations of time-fractional Fokker–Planck model. The fractional derivative is assumed in the Caputo sense. The proposed algorithm gives the solution in a form of rapidly convergent fractional power series with easily computable coefficients. It does not require linearization, discretization, or small perturbation. To test simplicity, potentiality, and practical usefulness of the proposed algorithm, illustrative examples are provided. The approximate solutions of time-fractional Fokker–Planck equations are obtained by the residual power series method are compared with those obtained by other existing methods. The present results and graphics reveal the ability of residual power series method to deal with a wide range of partial fractional differential equations emerging in the modeling of physical phenomena of science and engineering.

scholarly journals Construction of Fractional Power Series Solutions to Fractional Boussinesq Equations Using Residual Power Series Method

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Fei Xu ◽  
Yixian Gao ◽  
Xue Yang ◽  
He Zhang
Keyword(s):  
Power Series ◽  
Fractional Power ◽  
Boussinesq Equations ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Time Space ◽  
Residual Power Series ◽  
Fractional Differential ◽  
Fractional Power Series ◽  
Residual Power

This paper is aimed at constructing fractional power series (FPS) solutions of time-space fractional Boussinesq equations using residual power series method (RPSM). Firstly we generalize the idea of RPSM to solve any-order time-space fractional differential equations in high-dimensional space with initial value problems inRn. Using RPSM, we can obtain FPS solutions of fourth-, sixth-, and 2nth-order time-space fractional Boussinesq equations inRand fourth-order time-space fractional Boussinesq equations inR2andRn. Finally, by numerical experiments, it is shown that RPSM is a simple, effective, and powerful method for seeking approximate analytic solutions of fractional differential equations.

scholarly journals Solution of Linear Fuzzy Fractional Differential Equations Using Fuzzy Natural Transform

2021 ◽  
pp. 41-65
Author(s):  
Hameeda Oda Al-Humedi ◽  
Shaimaa Abdul-Hussein Kadhim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Wide Range ◽  
Real Functions ◽  
Fractional Models ◽  
Fractional Differential ◽  
Fuzzy Fractional Differential Equations

The purpose of this paper is to apply the fuzzy natural transform (FNT) for solving linear fuzzy fractional ordinary differential equations (FFODEs) involving fuzzy Caputo’s H-difference with Mittag-Leffler laws. It is followed by proposing new results on the property of FNT for fuzzy Caputo’s H-difference. An algorithm was then applied to find the solutions of linear FFODEs as fuzzy real functions. More specifically, we first obtained four forms of solutions when the FFODEs is of order α∈(0,1], then eight systems of solutions when the FFODEs is of order α∈(1,2] and finally, all of these solutions are plotted using MATLAB. In fact, the proposed approach is an effective and practical to solve a wide range of fractional models.

scholarly journals Applications of General Residual Power Series Method to Differential Equations with Variable Coefficients

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Bochao Chen ◽  
Li Qin ◽  
Fei Xu ◽  
Jian Zu
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Variable Coefficients ◽  
Series Solutions ◽  
Series Method ◽  
Power Series Method ◽  
Residual Power Series Method ◽  
Residual Power Series ◽  
Analytical Series ◽  
Residual Power

This paper is devoted to studying the analytical series solutions for the differential equations with variable coefficients. By a general residual power series method, we construct the approximate analytical series solutions for differential equations with variable coefficients, including nonhomogeneous parabolic equations, fractional heat equations in 2D, and fractional wave equations in 3D. These applications show that residual power series method is a simple, effective, and powerful method for seeking analytical series solutions of differential equations (especially for fractional differential equations) with variable coefficients.

scholarly journals Adomian Decomposition and Fractional Power Series Solution of a Class of Nonlinear Fractional Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1070
Author(s):  
Pshtiwan Othman Mohammed ◽  
José António Tenreiro Machado ◽  
Juan L. G. Guirao ◽  
Ravi P. Agarwal
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Power ◽  
Power Series Solution ◽  
True Nature ◽  
Nonlinear Fractional Differential Equations ◽  
Adomian Decomposition ◽  
Fractional Differential ◽  
Fractional Power Series

Nonlinear fractional differential equations reflect the true nature of physical and biological models with non-locality and memory effects. This paper considers nonlinear fractional differential equations with unknown analytical solutions. The Adomian decomposition and the fractional power series methods are adopted to approximate the solutions. The two approaches are illustrated and compared by means of four numerical examples.

scholarly journals Existence, uniqueness, and approximate solutions for the general nonlinear distributed-order fractional differential equations in a Banach space

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Tahereh Eftekhari ◽  
Jalil Rashidinia ◽  
Khosrow Maleknejad
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Collocation Methods ◽  
Systems Of Nonlinear Equations ◽  
Test Problems ◽  
Operational Matrices ◽  
Global Existence Of Solutions ◽  
Fractional Differential ◽  
Distributed Order

AbstractThe purpose of this paper is to provide sufficient conditions for the local and global existence of solutions for the general nonlinear distributed-order fractional differential equations in the time domain. Also, we provide sufficient conditions for the uniqueness of the solutions. Furthermore, we use operational matrices for the fractional integral operator of the second kind Chebyshev wavelets and shifted fractional-order Jacobi polynomials via Gauss–Legendre quadrature formula and collocation methods to reduce the proposed equations into systems of nonlinear equations. Also, error bounds and convergence of the presented methods are investigated. In addition, the presented methods are implemented for two test problems and some famous distributed-order models, such as the model that describes the motion of the oscillator, the distributed-order fractional relaxation equation, and the Bagley–Torvik equation, to demonstrate the desired efficiency and accuracy of the proposed approaches. Comparisons between the methods proposed in this paper and the existing methods are given, which show that our numerical schemes exhibit better performances than the existing ones.

scholarly journals Existence and Uniqueness Results for Two-Term Nonlinear Fractional Differential Equations via a Fixed Point Technique

2021 ◽  
Vol 2021 ◽  
pp. 1-7 ◽  
Author(s):  
H. R. Marasi ◽  
H. Aydi
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Monotone Operators ◽  
Nonlinear Fractional Differential Equations ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Wide Range ◽  
Fractional Differential

The work addressed in this paper is to ensure the existence and uniqueness of positive solutions for initial value problems for nonlinear fractional differential equations with two terms of fractional orders. By virtue of recent fixed point theorems on mixed monotone operators, we get some new straightforward results with a wide range of applications.

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