scholarly journals Approximate solutions of one-dimensional systems with fractional derivative

2020 ◽  
Vol 31 (07) ◽  
pp. 2050092
Author(s):  
A. Ferrari ◽  
M. Gadella ◽  
L. P. Lara ◽  
E. Santillan Marcus
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Caputo Derivative ◽  
Approximate Solutions ◽  
One Dimensional ◽  
Dissipative Term ◽  
Numerical Resolution ◽  
Fractional Caputo Derivative ◽  
Fractional Differential

The fractional calculus is useful to model nonlocal phenomena. We construct a method to evaluate the fractional Caputo derivative by means of a simple explicit quadratic segmentary interpolation. This method yields to numerical resolution of ordinary fractional differential equations. Due to the nonlocality of the fractional derivative, we may establish an equivalence between fractional oscillators and ordinary oscillators with a dissipative term.

scholarly journals On the Stability of Fractional Differential Equations Involving Generalized Caputo Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Minh Duc Tran ◽  
Vu Ho ◽  
Hoa Ngo Van
Keyword(s):  
Global Existence ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Caputo Derivative ◽  
Fractional Differential ◽  
The Stability ◽  
The Given

This work presents the results of the global existence for fractional differential equations involving generalized Caputo derivative with the case of the fractional order derivative α∈1,2. In addition, the Ulam–Hyers–Mittag-Leffler stability of the given problems is also established.

scholarly journals An Efficient Mechanism to Solve Fractional Differential Equations Using Fractional Decomposition Method

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 984
Author(s):  
Mahmoud S. Alrawashdeh ◽  
Seba A. Migdady ◽  
Ioannis K. Argyros
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Decomposition Method ◽  
Large Scale ◽  
Approximate Solutions ◽  
Science And Engineering ◽  
Large Scale Problems ◽  
Fractional Differential

We present some new results that deal with the fractional decomposition method (FDM). This method is suitable to handle fractional calculus applications. We also explore exact and approximate solutions to fractional differential equations. The Caputo derivative is used because it allows traditional initial and boundary conditions to be included in the formulation of the problem. This is of great significance for large-scale problems. The study outlines the significant features of the FDM. The relation between the natural transform and Laplace transform is a symmetrical one. Our work can be considered as an alternative to existing techniques, and will have wide applications in science and engineering fields.

scholarly journals Ulam stability for delay fractional differential equations with a generalized Caputo derivative

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5265-5274 ◽  
Author(s):  
Raad Ameen ◽  
Fahd Jarad ◽  
Thabet Abdeljawad
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Caputo Derivative ◽  
Ulam Stability ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
Equations With Delay ◽  
Rassias Stability

The objective of this paper is to extend Ulam-Hyers stability and Ulam-Hyers-Rassias stability theory to differential equations with delay and in the frame of a certain class of a generalized Caputo fractional derivative with dependence on a kernel function. We discuss the conditions such delay generalized Caputo fractional differential equations should satisfy to be stable in the sense of Ulam-Hyers and Ulam-Hyers-Rassias. To demonstrate our results two examples are presented.

Analytic approximate solutions for a class of variable order fractional differential equations using the polynomial least squares method

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Constantin Bota ◽  
Bogdan Căruntu
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Fractional Differential Equations ◽  
Least Squares Method ◽  
Approximate Solutions ◽  
Variable Order ◽  
Polynomial Solutions ◽  
Fractional Differential ◽  
Approximate Polynomial

AbstractIn this paper a new way to compute analytic approximate polynomial solutions for a class of nonlinear variable order fractional differential equations is proposed, based on the Polynomial Least Squares Method (PLSM). In order to emphasize the accuracy and the efficiency of the method several examples are included.

scholarly journals New Gronwall-Bellman Type Inequalities and Applications in the Analysis for Solutions to Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Bin Zheng ◽  
Qinghua Feng
Keyword(s):  
Quantitative Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Initial Value ◽  
Liouville Fractional Derivative ◽  
Explicit Bounds ◽  
Fractional Differential ◽  
Differential And Integral Equations

Some new Gronwall-Bellman type inequalities are presented in this paper. Based on these inequalities, new explicit bounds for the related unknown functions are derived. The inequalities established can also be used as a handy tool in the research of qualitative as well as quantitative analysis for solutions to some fractional differential equations defined in the sense of the modified Riemann-Liouville fractional derivative. For illustrating the validity of the results established, we present some applications for them, in which the boundedness, uniqueness, and continuous dependence on the initial value for the solutions to some certain fractional differential and integral equations are investigated.

scholarly journals Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Khalid Hattaf
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Direct Method ◽  
Lyapunov Direct Method ◽  
Science And Engineering ◽  
Fractional Differential ◽  
The Stability

This paper aims to study the stability of fractional differential equations involving the new generalized Hattaf fractional derivative which includes the most types of fractional derivatives with nonsingular kernels. The stability analysis is obtained by means of the Lyapunov direct method. First, some fundamental results and lemmas are established in order to achieve the goal of this study. Furthermore, the results related to exponential and Mittag–Leffler stability existing in recent studies are extended and generalized. Finally, illustrative examples are presented to show the applicability of our main results in some areas of science and engineering.

scholarly journals Conformable fractional derivative and its application to fractional Klein-Gordon equation

2019 ◽  
Vol 23 (6 Part B) ◽  
pp. 3745-3749
Author(s):  
Kangle Wang ◽  
Shaowen Yao
Keyword(s):  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Gordon Equation ◽  
Variational Iteration Method ◽  
Conformable Fractional Derivative ◽  
Fractional Complex Transform ◽  
Klein Gordon ◽  
Fractional Differential ◽  
Fractional Equation ◽  
Klein Gordon Equation

This paper adopts conformable fractional derivative to describe the fractional Klein-Gordon equations. The conformable fractional derivative is a new simple well-behaved definition. The fractional complex transform and variational iteration method are used to solve the fractional equation. The result shows that the proposed technology is very powerful and efficient for fractional differential equations.

scholarly journals On the Convergence of LHAM and its Application to Fractional Generalised Boussinesq Equations for Closed Form Solutions

2021 ◽  
pp. 25-47
Author(s):  
S. O. Ajibola ◽  
E. O. Oghre ◽  
A. G. Ariwayo ◽  
P. O. Olatunji
Keyword(s):  
Differential Equations ◽  
Closed Form ◽  
Fractional Differential Equations ◽  
Closed Form Solution ◽  
Boussinesq Equations ◽  
Approximate Solutions ◽  
Form Solution ◽  
Restrictive Assumption ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential

By fractional generalised Boussinesq equations we mean equations of the form \begin{equation} \Delta\equiv D_{t}^{2\alpha}-[\mathcal{N}(u)]_{xx}-u_{xxxx}=0, \: 0<\alpha\le1,\label{main}\nonumber \end{equation} where $\mathcal{N}(u)$ is a differentiable function and $\mathcal{N}_{uu}\ne0$ (to ensure nonlinearity). In this paper we lay emphasis on the cubic Boussinesq and Boussinesq-like equations of fractional order and we apply the Laplace homotopy analysis method (LHAM) for their rational and solitary wave solutions respectively. It is true that nonlinear fractional differential equations are often difficult to solve for their {\em exact} solutions and this single reason has prompted researchers over the years to come up with different methods and approach for their {\em analytic approximate} solutions. Most of these methods require huge computations which are sometimes complicated and a very good knowledge of computer aided softwares (CAS) are usually needed. To bridge this gap, we propose a method that requires no linearization, perturbation or any particularly restrictive assumption that can be easily used to solve strongly nonlinear fractional differential equations by hand and simple computer computations with a very quick run time. For the closed form solution, we set $\alpha =1$ for each of the solutions and our results coincides with those of others in the literature.

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