Variational Approaches as Fractional Differential Equations Along Theoretical and Numerical Examples

Abstract We study a nonlinear system of Riemann-Liouville fractional differential equations equipped with nonseparated semi-coupled integro-multipoint boundary conditions. We make use of the tools of the fixed-point theory to obtain the desired results, which are well-supported with numerical examples.

Download Full-text

An Integral Operational Matrix of Fractional-Order Chelyshkov Functions and Its Applications

Symmetry ◽

10.3390/sym12111755 ◽

2020 ◽

Vol 12 (11) ◽

pp. 1755

Author(s):

M. S. Al-Sharif ◽

A. I. Ahmed ◽

M. S. Salim

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Fractional Differential Equations ◽

Fractional Integral ◽

Operational Matrix ◽

Operational Matrices ◽

Algebraic Equations ◽

Science And Engineering ◽

Numerical Examples ◽

Fractional Differential

Fractional differential equations have been applied to model physical and engineering processes in many fields of science and engineering. This paper adopts the fractional-order Chelyshkov functions (FCHFs) for solving the fractional differential equations. The operational matrices of fractional integral and product for FCHFs are derived. These matrices, together with the spectral collocation method, are used to reduce the fractional differential equation into a system of algebraic equations. The error estimation of the presented method is also studied. Furthermore, numerical examples and comparison with existing results are given to demonstrate the accuracy and applicability of the presented method.

Download Full-text

A Robust Computational Algorithm of Homotopy Asymptotic Method for Solving Systems of Fractional Differential Equations

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4043617 ◽

2019 ◽

Vol 14 (8) ◽

Cited By ~ 42

Author(s):

Zaid Odibat ◽

Sunil Kumar

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Asymptotic Method ◽

Computational Algorithm ◽

Series Solutions ◽

Numerical Examples ◽

Convergence Of Series ◽

New Ideas ◽

Fractional Differential ◽

Optimal Construction

In this paper, we present new ideas for the implementation of homotopy asymptotic method (HAM) to solve systems of nonlinear fractional differential equations (FDEs). An effective computational algorithm, which is based on Taylor series approximations of the nonlinear equations, is introduced to accelerate the convergence of series solutions. The proposed algorithm suggests a new optimal construction of the homotopy that reduces the computational complexity and improves the performance of the method. Some numerical examples are tested to validate and illustrate the efficiency of the proposed algorithm. The obtained results demonstrate the improvement of the accuracy by the new algorithm.

Download Full-text

A New Scheme for Solving Multiorder Fractional Differential Equations Based on Müntz–Legendre Wavelets

Complexity ◽

10.1155/2021/9915551 ◽

2021 ◽

Vol 2021 ◽

pp. 1-9

Author(s):

Haifa Bin Jebreen ◽

Fairouz Tchier

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Operational Matrix ◽

Pseudospectral Method ◽

Algebraic Equations ◽

Legendre Wavelets ◽

Derivative Operator ◽

Numerical Examples ◽

Linear Algebraic Equations ◽

Fractional Differential

In this study, we apply the pseudospectral method based on Müntz–Legendre wavelets to solve the multiorder fractional differential equations with Caputo fractional derivative. Using the operational matrix for the Caputo derivative operator and applying the Chebyshev and Legendre zeros, the problem is reduced to a system of linear algebraic equations. We illustrate the reliability, efficiency, and accuracy of the method by some numerical examples. We also compare the proposed method with others and show that the proposed method gives better results.

Download Full-text

Oscillatory Behavior of a Type of Generalized Proportional Fractional Differential Equations with Forcing and Damping Terms

Mathematics ◽

10.3390/math8061037 ◽

2020 ◽

Vol 8 (6) ◽

pp. 1037 ◽

Cited By ~ 1

Author(s):

Jehad Alzabut ◽

James Viji ◽

Velu Muthulakshmi ◽

Weerawat Sudsutad

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Oscillatory Behavior ◽

Numerical Examples ◽

Oscillation Criteria ◽

Fractional Differential

In this paper, we study the oscillatory behavior of solutions for a type of generalized proportional fractional differential equations with forcing and damping terms. Several oscillation criteria are established for the proposed equations in terms of Riemann-Liouville and Caputo settings. The results of this paper generalize some existing theorems in the literature. Indeed, it is shown that for particular choices of parameters, the obtained conditions in this paper reduce our theorems to some known results. Numerical examples are constructed to demonstrate the effectiveness of the our main theorems. Furthermore, we present and illustrate an example which does not satisfy the assumptions of our theorem and whose solution demonstrates nonoscillatory behavior.

Download Full-text

General solution of Bernoulli and Riccati fractional differential equations based on conformable fractional derivative

International Journal of Applied Mathematical Research ◽

10.14419/ijamr.v6i2.7014 ◽

2017 ◽

Vol 6 (2) ◽

pp. 49 ◽

Cited By ~ 13

Author(s):

Zainab Ayati ◽

Jafar Biaar ◽

Mousa Ilei

Keyword(s):

General Solution ◽

Differential Equations ◽

Fractional Derivative ◽

Ordinary Differential Equations ◽

Fractional Differential Equations ◽

Riccati Equations ◽

Nonlinear Ordinary Differential Equations ◽

Conformable Fractional Derivative ◽

Numerical Examples ◽

Fractional Differential

This paper is aimed to develop two well-known nonlinear ordinary differential equations, Bernoulli and Riccati equations to fractional form. General solution to fractional differential equations are detected, based on conformable fractional derivative. For each equation, numerical examples are presented to illustrate the proposed approach.

Download Full-text

A Fuzzy Method for Solving Fuzzy Fractional Differential Equations Based on the Generalized Fuzzy Taylor Expansion

Mathematics ◽

10.3390/math8122166 ◽

2020 ◽

Vol 8 (12) ◽

pp. 2166

Author(s):

Tofigh Allahviranloo ◽

Zahra Noeiaghdam ◽

Samad Noeiaghdam ◽

Juan J. Nieto

Keyword(s):

Differential Equations ◽

Fractional Derivative ◽

Fractional Differential Equations ◽

Taylor Expansion ◽

Direct Method ◽

Numerical Examples ◽

Euler’S Method ◽

Truncation Errors ◽

Fractional Differential ◽

Fuzzy Fractional Differential Equations

In this field of research, in order to solve fuzzy fractional differential equations, they are normally transformed to their corresponding crisp problems. This transformation is called the embedding method. The aim of this paper is to present a new direct method to solve the fuzzy fractional differential equations using fuzzy calculations and without this transformation. In this work, the fuzzy generalized Taylor expansion by using the sense of fuzzy Caputo fractional derivative for fuzzy-valued functions is presented. For solving fuzzy fractional differential equations, the fuzzy generalized Euler’s method is introduced and applied. In order to show the accuracy and efficiency of the presented method, the local and global truncation errors are determined. Moreover, the consistency, convergence, and stability of the generalized Euler’s method are proved in detail. Eventually, the numerical examples, especially in the switching point case, show the flexibility and the capability of the presented method.

Download Full-text

Bessel Collocation Method for Solving Fredholm–Volterra Integro-Fractional Differential Equations of Multi-High Order in the Caputo Sense

Symmetry ◽

10.3390/sym13122354 ◽

2021 ◽

Vol 13 (12) ◽

pp. 2354

Author(s):

Shazad Shawki Ahmed ◽

Shabaz Jalil MohammedFaeq

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Linear Algebraic Equation ◽

Approximate Solutions ◽

New Approach ◽

Numerical Examples ◽

Time Symmetry ◽

Collocation Points ◽

The Matrix ◽

Fractional Differential

The approximate solutions of Fredholm–Volterra integro-differential equations of multi-fractional order within the Caputo sense (F-VIFDEs) under mixed conditions are presented in this article apply a collocation points technique based completely on Bessel polynomials of the first kind. This new approach depends particularly on transforming the linear equation and conditions into the matrix relations (some time symmetry matrix), which results in resolving a linear algebraic equation with unknown generalized Bessel coefficients. Numerical examples are given to show the technique’s validity and application, and comparisons are made with existing results by applying this process in order to express these solutions, most general programs are written in Python V.3.8.8 (2021).

Download Full-text

A New Attractive Method in Solving Families of Fractional Differential Equations by a New Transform

Mathematics ◽

10.3390/math9233039 ◽

2021 ◽

Vol 9 (23) ◽

pp. 3039

Author(s):

Ahmad Qazza ◽

Aliaa Burqan ◽

Rania Saadeh

Keyword(s):

Differential Equations ◽

Fractional Derivative ◽

Fractional Differential Equations ◽

Caputo Fractional Derivative ◽

Fractional Integral ◽

Integral Transform ◽

Series Solutions ◽

Numerical Examples ◽

Fractional Differential ◽

Expansion Coefficients

In this paper, we use the ARA transform to solve families of fractional differential equations. New formulas about the ARA transform are presented and implemented in solving some applications. New results related to the ARA integral transform of the Riemann-Liouville fractional integral and the Caputo fractional derivative are obtained and the last one is implemented to create series solutions for the target equations. The procedure proposed in this article is mainly based on some theorems of particular solutions and the expansion coefficients of binomial series. In order to achieve the accuracy and simplicity of the new method, some numerical examples are considered and solved. We obtain the solutions of some families of fractional differential equations in a series form and we show how these solutions lead to some important results that include generalizations of some classical methods.

Download Full-text

Efficient Galerkin solution of stochastic fractional differential equations using second kind Chebyshev wavelets

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.v35i1.28262 ◽

2017 ◽

Vol 35 (1) ◽

pp. 195 ◽

Cited By ~ 5

Author(s):

Fakhrodin Mohammadi

Keyword(s):

Approximate Solution ◽

Differential Equations ◽

Fractional Differential Equations ◽

Algebraic Equations ◽

Chebyshev Wavelets ◽

Numerical Examples ◽

Physical Problems ◽

Galerkin Solution ◽

Fractional Differential ◽

Stochastic Fractional Differential Equations

‎Stochastic fractional differential equations (SFDEs) have been used for modeling many physical problems in the fields of turbulance‎, ‎heterogeneous‎, ‎flows and matrials‎, ‎viscoelasticity and electromagnetic theory‎. ‎In this paper‎, ‎an‎ efficient wavelet Galerkin method based on the second kind Chebyshev wavelets are proposed for approximate solution of SFDEs‎. ‎In ‎this ‎app‎roach‎‎, ‎o‎perational matrices of the second kind Chebyshev wavelets ‎are used ‎for reducing SFDEs to a linear system of algebraic equations that can be solved easily‎. ‎C‎onvergence and error analysis of the proposed method is ‎considered‎.‎ ‎Some numerical examples are performed to confirm the applicability and efficiency of the proposed method‎.

Download Full-text