A Generalized Henry-Type Integral Inequality and Application to Dependence on Orders and Known Functions for a Fractional Differential Equation

We discuss on integrable solutions for a generalized Henry-type integral inequality in which weak singularity and delays are involved. Not requiring continuity or differentiability for some given functions, we use a modified iteration argument to give an estimate of the unknown function in terms of the multiple Mittag-Leffler function. We apply the result to give continuous dependence of solutions on initial data, derivative orders, and known functions for a fractional differential equation.

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A Coiflets-Based Wavelet Laplace Method for Solving the Riccati Differential Equations

Journal of Applied Mathematics ◽

10.1155/2014/257049 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8

Author(s):

Xiaomin Wang

Keyword(s):

Differential Equation ◽

Iterative Method ◽

Numerical Integration ◽

Fractional Differential Equation ◽

Unknown Function ◽

Nonlinear Fractional Differential Equation ◽

Orthogonal Wavelets ◽

Riccati Differential Equations ◽

Fractional Differential ◽

Approximate Scheme

A wavelet iterative method based on a numerical integration by using the Coiflets orthogonal wavelets for a nonlinear fractional differential equation is proposed. With the help of Laplace transform, the fractional differential equation was converted into equivalent integral equation of convolution type. By using the wavelet approximate scheme of a function, the undesired jump or wiggle phenomenon near the boundary points was avoided and the expansion constants in the approximation of arbitrary nonlinear term of the unknown function can be explicitly expressed in finite terms of the expansion ones of the approximation of the unknown function. Then a numerical integration method for the convolution is presented. As an example, an iterative method which can solve the singular nonlinear fractional Riccati equations is proposed. Numerical results are performed to show the efficiency of the method proposed.

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Continuous dependence of uncertain fractional differential equations with Caputo’s derivative

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-201428 ◽

2020 ◽

pp. 1-10

Author(s):

Ziqiang Lu ◽

Yuanguo Zhu ◽

Jiayu Shen

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Fractional Differential Equation ◽

Fractional Differential Equations ◽

Continuous Dependence ◽

Liu Process ◽

Uncertain Dynamic Systems ◽

Caputo’S Derivative ◽

Fractional Differential ◽

Lipschitz Conditions

Uncertain fractional differential equation driven by Liu process plays an important role in describing uncertain dynamic systems. This paper investigates the continuous dependence of solution on the parameters and initial values, respectively, for uncertain fractional differential equations involving the Caputo fractional derivative in measure sense. Several continuous dependence theorems are obtained based on uncertainty theory by employing the generalized Gronwall inequality, in which the coefficients of uncertain fractional differential equation are required to satisfy the Lipschitz conditions. Several illustrative examples are provided to verify the validity of the obtained results.

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Existence and Continuous Dependence on Initial Data of Solution for Initial Value Problem of Fuzzy Multiterm Fractional Differential Equation

Advances in Fuzzy Systems ◽

10.1155/2019/6457548 ◽

2019 ◽

Vol 2019 ◽

pp. 1-11

Author(s):

Huichol Choi ◽

Kinam Sin ◽

Sunae Pak ◽

Sungryol So

Keyword(s):

Differential Equation ◽

Fixed Point ◽

Fixed Point Theorem ◽

Fractional Differential Equation ◽

Continuous Dependence ◽

Contraction Mapping ◽

Contraction Mapping Principle ◽

Existence Of Solution ◽

Initial Value ◽

Fractional Differential

In this paper, the fuzzy multiterm fractional differential equation involving Caputo-type fuzzy fractional derivative of order 0<α<1 is considered. The uniqueness of solution is established by using the contraction mapping principle and the existence of solution is obtained by Schauder fixed point theorem.

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A New Fractional Integral Inequality with Singularity and Its Application

Abstract and Applied Analysis ◽

10.1155/2012/937908 ◽

2012 ◽

Vol 2012 ◽

pp. 1-12 ◽

Cited By ~ 2

Author(s):

Qiong-Xiang Kong ◽

Xiao-Li Ding

Keyword(s):

Differential Equation ◽

Fractional Differential Equation ◽

Integral Inequality ◽

Fractional Integral ◽

Weighted Space ◽

Initial Condition ◽

Fractional Differential

We prove an integral inequality with singularity, which complements some known results. This inequality enables us to study the dependence of the solution on the initial condition to a fractional differential equation in the weighted space.

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Some New Gronwall-Bellman-Type Inequalities Based on the Modified Riemann-Liouville Fractional Derivative

Journal of Applied Mathematics ◽

10.1155/2013/341706 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 3

Author(s):

Bin Zheng

Keyword(s):

Differential Equation ◽

Fractional Derivative ◽

Fractional Differential Equation ◽

Continuous Dependence ◽

Initial Value ◽

Liouville Fractional Derivative ◽

Explicit Bounds ◽

Fractional Differential

By use of the properties of the modified Riemann-Liouville fractional derivative, some new Gronwall-Bellman-type inequalities are researched. First, we derive some new explicit bounds for the unknown functions lying in these inequalities, which are of different forms from some existing bounds in the literature. Then, we apply the results established to research the boundedness, uniqueness, and continuous dependence on the initial value for the solution to a certain fractional differential equation.

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Existence and Uniqueness of the Mild Solution of an Abstract Semilinear Fractional Differential Equation with State Dependent Nonlocal Condition

Kragujevac Journal of Mathematics ◽

10.46793/kgjmat2106.909h ◽

2021 ◽

Vol 45 (6) ◽

pp. 909-923

Author(s):

MOHAMED A. E. HERZALLAH ◽

◽

ASHRAF H. A. RADWAN

Keyword(s):

Differential Equation ◽

Fractional Differential Equation ◽

Mild Solution ◽

Existence And Uniqueness ◽

Initial Conditions ◽

Nonlocal Condition ◽

Mild Solutions ◽

State Dependent ◽

Fractional Differential ◽

Continuous Dependence Of Solutions

The purpose of this paper is to investigate the existence and uniqueness of mild solutions to a semilinear Cauchy problem for an abstract fractional diﬀerential equation with state dependent nonlocal condition. Continuous dependence of solutions on initial conditions and local ????-approximate mild solution of the considered problem will be discussed.

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A Generalized Nonlinear Gronwall-Bellman Inequality with Maxima in Two Variables

Journal of Applied Mathematics ◽

10.1155/2013/853476 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Yong Yan

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Boundary Value Problem ◽

Integral Inequality ◽

Unknown Function ◽

Boundary Value ◽

Partial Differential ◽

Type Integral ◽

Nonlinear Integrals ◽

Function Of Two Variables

This paper deals with a generalized form of nonlinear retarded Gronwall-Bellman type integral inequality in which the maximum of the unknown function of two variables is involved. This form includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals. Requiring neither monotonicity nor separability of given functions, we apply a technique of monotonization to estimate the unknown function. Our result can be used to weaken conditions for some known results. We apply our result to a boundary value problem of a partial differential equation with maxima for uniqueness.

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