scholarly journals Nonexistence results of Caputo-type fractional problem

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammed D. Kassim ◽  
Saeed M. Ali ◽  
Mohammed S. Abdo ◽  
Fahd Jarad
Keyword(s):  
Fractional Derivative ◽  
Differential Inequality ◽  
Global Solutions ◽  
Fractional Integrals ◽  
Function Technique ◽  
Test Function ◽  
Suitable Space ◽  
Fractional Differential ◽  
Low Order

AbstractIn this paper, we deal with Caputo-type fractional differential inequality where there is a low-order fractional derivative with the term polynomial source. We investigate the nonexistence of nontrivial global solutions in a suitable space via the test function technique and some properties of fractional integrals. Finally, we demonstrate three examples to illustrate our results. The presented results are more general than those in the literature, which can be obtained as particular cases.

scholarly journals Nonexistence Results for Higher Order Fractional Differential Inequalities with Nonlinearities Involving Caputo Fractional Derivative

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1866
Author(s):  
Mohamed Jleli ◽  
Bessem Samet ◽  
Calogero Vetro
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
A Priori Estimates ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Higher Order ◽  
Differential Inequalities ◽  
Structure Of Solutions ◽  
Nonlinear Capacity ◽  
Fractional Differential

Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function.

scholarly journals On a Differential Equation Involving Hilfer-Hadamard Fractional Derivative

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
M. D. Qassim ◽  
K. M. Furati ◽  
N.-E. Tatar
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Global Solution ◽  
Differential Inequality ◽  
Source Term ◽  
Existence Of Solutions ◽  
Hadamard Fractional Derivative ◽  
Fractional Differential

This paper studies a fractional differential inequality involving a new fractional derivative (Hilfer-Hadamard type) with a polynomial source term. We obtain an exponent for which there does not exist any global solution for the problem. We also provide an example to show the existence of solutions in a wider space for some exponents.

scholarly journals Nonexistence Results for Some Classes of Nonlinear Fractional Differential Inequalities

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Mohamed Jleli ◽  
Bessem Samet
Keyword(s):  
Function Method ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Differential Inequalities ◽  
Test Function ◽  
Fractional Differential ◽  
Test Function Method

We study the nonexistence of global solutions for new classes of nonlinear fractional differential inequalities. Namely, sufficient conditions are provided so that the considered problems admit no global solutions. The proofs of our results are based on the test function method and some integral estimates.

scholarly journals Blow up results for fractional differential equations and systems

2013 ◽  
Vol 93 (107) ◽  
pp. 173-186 ◽  
Author(s):  
Ali Hakem ◽  
Mohamed Berbiche
Keyword(s):  
Differential Equation ◽  
Blow Up ◽  
Function Theory ◽  
Time Derivative ◽  
Sufficient Conditions ◽  
Fractional Power ◽  
Global Solutions ◽  
Nonlinear Fractional Differential Equation ◽  
Test Function ◽  
Fractional Differential

The aim of this research paper is to establish sufficient conditions for the nonexistence of global solutions for the following nonlinear fractional differential equation D?0|tu + (??)?/2|u|m?1u + a(x)??|u|q?1u = h(x, t)|u|p, (t,x) ? Q, u(0, x) = u0(x), x ? RN where (??)?/2, 0 < ? < 2 is the fractional power of ??, and D?0|t, (0 < ? < 1) denotes the time-derivative of arbitrary ? ? (0; 1) in the sense of Caputo. The results are shown by the use of test function theory and extended to systems of the same type.

scholarly journals On weighted Atangana–Baleanu fractional operators

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mohammed Al-Refai
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Closed Form ◽  
Fractional Differential Equation ◽  
Fractional Integral ◽  
Convergent Series ◽  
Fractional Integrals ◽  
Fractional Operators ◽  
Fractional Differential

AbstractIn this paper, we define the weighted Atangana–Baleanu fractional operators of Caputo sense. We obtain the solution of a related linear fractional differential equation in a closed form, and use the result to define the weighted Atangana–Baleanu fractional integral. We then express the weighted Atangana–Baleanu fractional derivative in a convergent series of Riemann–Liouville fractional integrals, and establish commutative results of the weighted Atangana–Baleanu fractional operators.

scholarly journals On the absence of global solutions to two-times-fractional differential inequalities involving Hadamard-Caputo and Caputo fractional derivatives

2022 ◽  
Vol 7 (4) ◽  
pp. 5830-5843
Author(s):  
Ibtehal Alazman ◽  
◽  
Mohamed Jleli ◽  
Bessem Samet ◽  
Keyword(s):  
Global Solution ◽  
Differential Inequality ◽  
Fractional Derivatives ◽  
Singular Potential ◽  
Global Solutions ◽  
Differential Inequalities ◽  
Caputo Fractional Derivatives ◽  
Potential Term ◽  
Fractional Differential ◽  
Derivatives Of

<abstract><p>In this paper, we consider a two-times nonlinear fractional differential inequality involving both Hadamard-Caputo and Caputo fractional derivatives of different orders, with a singular potential term. We obtain sufficient criteria depending on the parameters of the problem, for which a global solution does not exist. Some examples are provided to support our main results.</p></abstract>

scholarly journals General Fractional Dynamics

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1464
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Fractional Calculus ◽  
Exact Solutions ◽  
Arbitrary Order ◽  
Nonlinear Dynamical Systems ◽  
Fractional Integrals ◽  
Fractional Dynamics ◽  
Fractional Systems ◽  
Nonlinear Dynamical ◽  
Fractional Differential ◽  
Solutions Of Equations

General fractional dynamics (GFDynamics) can be viewed as an interdisciplinary science, in which the nonlocal properties of linear and nonlinear dynamical systems are studied by using general fractional calculus, equations with general fractional integrals (GFI) and derivatives (GFD), or general nonlocal mappings with discrete time. GFDynamics implies research and obtaining results concerning the general form of nonlocality, which can be described by general-form operator kernels and not by its particular implementations and representations. In this paper, the concept of “general nonlocal mappings” is proposed; these are the exact solutions of equations with GFI and GFD at discrete points. In these mappings, the nonlocality is determined by the operator kernels that belong to the Sonin and Luchko sets of kernel pairs. These types of kernels are used in general fractional integrals and derivatives for the initial equations. Using general fractional calculus, we considered fractional systems with general nonlocality in time, which are described by equations with general fractional operators and periodic kicks. Equations with GFI and GFD of arbitrary order were also used to derive general nonlocal mappings. The exact solutions for these general fractional differential and integral equations with kicks were obtained. These exact solutions with discrete timepoints were used to derive general nonlocal mappings without approximations. Some examples of nonlocality in time are described.

scholarly journals On Caputo modification of Hadamard-type fractional derivative and fractional Taylor series

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Rashida Zafar ◽  
Mujeeb ur Rehman ◽  
Moniba Shams
Keyword(s):  
Differential Operator ◽  
Fractional Derivative ◽  
Taylor Series ◽  
General Framework ◽  
Taylor Expansion ◽  
Fractional Integral ◽  
Haar Wavelet ◽  
Wavelet Method ◽  
Fractional Differential Operator ◽  
Fractional Differential

Abstract In this paper a general framework is presented on some operational properties of Caputo modification of Hadamard-type fractional differential operator along with a new algorithm proposed for approximation of Hadamard-type fractional integral using Haar wavelet method. Moreover, a generalized Taylor expansion based on Caputo–Hadamard-type fractional differential operator is also established, and an example is presented for illustration.

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.

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