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

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.

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Existence of solutions to initial value problems for nonlinear fractional differential equations on the semi-axis

Fractional Calculus and Applied Analysis ◽

10.2478/s13540-013-0038-3 ◽

2013 ◽

Vol 16 (3) ◽

Cited By ~ 3

Author(s):

Tiberiu Trif

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Initial Value Problems ◽

Existence Of Solutions ◽

Sufficient Conditions ◽

Growth Conditions ◽

Initial Value ◽

Nonlinear Fractional Differential Equations ◽

Global Existence Of Solutions ◽

Fractional Differential

AbstractThe purpose of the paper is to investigate the global existence of solutions to initial value problems for nonlinear fractional differential equations on the semi-axis. More precisely, it deals with the initial value problem (*)$\left\{ \begin{gathered} D_{0 + }^\alpha x(t) = f(t,x(t)),t \in [0,\infty ], \hfill \\ \lim _{t \to 0 + } t^{1 - \alpha } x(t) = x_0 , \hfill \\ \end{gathered} \right. $ where 0 < α < 1, D 0+α denotes the Riemann-Liouville fractional derivative of order α, and f: (0,∞) × ℝ → ℝ is a continuous function. Unlike all the previous papers dealing with the problem of existence of solutions to (*), this problem is solved here by constructing a special locally convex space which is metrizable and complete. Then Schauder’s fixed point theorem enables to provide sufficient conditions on f, ensuring that (*) possesses at least one solution. The growth conditions imposed to f are weaker than other similar conditions already used in the literature.

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Caputo fractional differential equations with non-instantaneous impulses and strict stability by Lyapunov functions

Filomat ◽

10.2298/fil1716217a ◽

2017 ◽

Vol 31 (16) ◽

pp. 5217-5239 ◽

Cited By ~ 5

Author(s):

Ravi Agarwal ◽

Snehana Hristova ◽

Donal O’Regan

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Lyapunov Functions ◽

Sufficient Conditions ◽

Time Intervals ◽

Dini Derivative ◽

Comparison Results ◽

Strict Stability ◽

Caputo Fractional Differential Equations ◽

Fractional Differential

In this paper the statement of initial value problems for fractional differential equations with noninstantaneous impulses is given. These equations are adequate models for phenomena that are characterized by impulsive actions starting at arbitrary fixed points and remaining active on finite time intervals. Strict stability properties of fractional differential equations with non-instantaneous impulses by the Lyapunov approach is studied. An appropriate definition (based on the Caputo fractional Dini derivative of a function) for the derivative of Lyapunov functions among the Caputo fractional differential equations with non-instantaneous impulses is presented. Comparison results using this definition and scalar fractional differential equations with non-instantaneous impulses are presented and sufficient conditions for strict stability and uniform strict stability are given. Examples are given to illustrate the theory.

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Lyapunov Functions and Lipschitz Stability for Riemann–Liouville Non-Instantaneous Impulsive Fractional Differential Equations

Symmetry ◽

10.3390/sym13040730 ◽

2021 ◽

Vol 13 (4) ◽

pp. 730

Author(s):

Ravi Agarwal ◽

Snezhana Hristova ◽

Donal O’Regan

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Lyapunov Functions ◽

Sufficient Conditions ◽

Initial Time ◽

Lipschitz Stability ◽

Comparison Results ◽

Liouville Fractional Derivative ◽

Fractional Differential ◽

Definition Of

In this paper a system of nonlinear Riemann–Liouville fractional differential equations with non-instantaneous impulses is studied. We consider a Riemann–Liouville fractional derivative with a changeable lower limit at each stop point of the action of the impulses. In this case the solution has a singularity at the initial time and any stop time point of the impulses. This leads to an appropriate definition of both the initial condition and the non-instantaneous impulsive conditions. A generalization of the classical Lipschitz stability is defined and studied for the given system. Two types of derivatives of the applied Lyapunov functions among the Riemann–Liouville fractional differential equations with non-instantaneous impulses are applied. Several sufficient conditions for the defined stability are obtained. Some comparison results are obtained. Several examples illustrate the theoretical results.

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Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2019-0030 ◽

2020 ◽

Vol 21 (6) ◽

pp. 571-587 ◽

Cited By ~ 2

Author(s):

Akbar Zada ◽

Sartaj Ali ◽

Tongxing Li

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Fractional Differential Equations ◽

Sufficient Conditions ◽

Uniqueness Of Solutions ◽

Integer Order ◽

New Class ◽

Proposed Model ◽

Fractional Differential ◽

Sequential Fractional Differential Equations

AbstractIn this paper, we study an implicit sequential fractional order differential equation with non-instantaneous impulses and multi-point boundary conditions. The article comprehensively elaborate four different types of Ulam’s stability in the lights of generalized Diaz Margolis’s fixed point theorem. Moreover, some sufficient conditions are constructed to observe the existence and uniqueness of solutions for the proposed model. The proposed model contains both the integer order and fractional order derivatives. Thus, the exponential function appearers in the solution of the proposed model which will lead researchers to study fractional differential equations with well known methods of integer order differential equations. In the last, few examples are provided to show the applicability of our main results.

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Green–Haar wavelets method for generalized fractional differential equations

Advances in Difference Equations ◽

10.1186/s13662-020-02974-6 ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 1

Author(s):

Mujeeb ur Rehman ◽

Dumitru Baleanu ◽

Jehad Alzabut ◽

Muhammad Ismail ◽

Umer Saeed

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Haar Wavelet ◽

Haar Wavelets ◽

Computational Time ◽

Operational Matrices ◽

Numerical Techniques ◽

Fractional Differential ◽

Fractional Boundary Value Problems ◽

Better Than

Abstract The objective of this paper is to present two numerical techniques for solving generalized fractional differential equations. We develop Haar wavelets operational matrices to approximate the solution of generalized Caputo–Katugampola fractional differential equations. Moreover, we introduce Green–Haar approach for a family of generalized fractional boundary value problems and compare the method with the classical Haar wavelets technique. In the context of error analysis, an upper bound for error is established to show the convergence of the method. Results of numerical experiments have been documented in a tabular and graphical format to elaborate the accuracy and efficiency of addressed methods. Further, we conclude that accuracy-wise Green–Haar approach is better than the conventional Haar wavelets approach as it takes less computational time compared to the Haar wavelet method.

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Stability Analysis of Distributed Order Fractional Differential Equations

Abstract and Applied Analysis ◽

10.1155/2011/175323 ◽

2011 ◽

Vol 2011 ◽

pp. 1-12 ◽

Cited By ~ 12

Author(s):

H. Saberi Najafi ◽

A. Refahi Sheikhani ◽

A. Ansari

Keyword(s):

Stability Analysis ◽

Characteristic Function ◽

Differential Equations ◽

Density Function ◽

Robust Stability ◽

Fractional Differential Equations ◽

Stability Condition ◽

Fractional Differential ◽

The Stability ◽

Distributed Order

We analyze the stability of three classes of distributed order fractional differential equations (DOFDEs) with respect to the nonnegative density function. In this sense, we discover a robust stability condition for these systems based on characteristic function and new inertia concept of a matrix with respect to the density function. Moreover, we check the stability of a distributed order fractional WINDMI system to illustrate the validity of proposed procedure.

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Numerical solution of nonlinear fractional differential equations by spline collocation methods

Journal of Computational and Applied Mathematics ◽

10.1016/j.cam.2013.04.049 ◽

2014 ◽

Vol 255 ◽

pp. 216-230 ◽

Cited By ~ 50

Author(s):

Arvet Pedas ◽

Enn Tamme

Keyword(s):

Differential Equations ◽

Numerical Solution ◽

Fractional Differential Equations ◽

Collocation Methods ◽

Spline Collocation ◽

Nonlinear Fractional Differential Equations ◽

Fractional Differential

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Wavelet Collocation Method for Solving Multiorder Fractional Differential Equations

Journal of Applied Mathematics ◽

10.1155/2012/542401 ◽

2012 ◽

Vol 2012 ◽

pp. 1-19 ◽

Cited By ~ 30

Author(s):

M. H. Heydari ◽

M. R. Hooshmandasl ◽

F. M. Maalek Ghaini ◽

F. Mohammadi

Keyword(s):

Differential Equations ◽

Collocation Method ◽

Fractional Differential Equations ◽

Initial Conditions ◽

General Procedure ◽

Wavelet Expansion ◽

Operational Matrices ◽

Chebyshev Wavelets ◽

Block Pulse Functions ◽

Fractional Differential

The operational matrices of fractional-order integration for the Legendre and Chebyshev wavelets are derived. Block pulse functions and collocation method are employed to derive a general procedure for forming these matrices for both the Legendre and the Chebyshev wavelets. Then numerical methods based on wavelet expansion and these operational matrices are proposed. In this proposed method, by a change of variables, the multiorder fractional differential equations (MOFDEs) with nonhomogeneous initial conditions are transformed to the MOFDEs with homogeneous initial conditions to obtain suitable numerical solution of these problems. Numerical examples are provided to demonstrate the applicability and simplicity of the numerical scheme based on the Legendre and Chebyshev wavelets.

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