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 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.

Existence of positive solutions for systems of nonlinear fractional differential equation with p-Laplacian

2019 ◽  
Vol 13 (05) ◽  
pp. 2050089 ◽  
Author(s):  
S. Nageswara Rao ◽  
Meshari Alesemi
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problems ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Nonlinear Fractional Differential Equation ◽  
Existence Of Positive Solutions ◽  
Fractional Differential

In this paper, we establish sufficient conditions for the existence of positive solutions for a system of nonlinear fractional [Formula: see text]-Laplacian boundary value problems under different combinations of superlinearity and sublinearity of the nonlinearities via the Guo–Krasnosel’skii fixed point theorem. Moreover, an example is given to illustrate our results.

scholarly journals Stability of Generalized Proportional Caputo Fractional Differential Equations by Lyapunov Functions

2022 ◽  
Vol 6 (1) ◽  
pp. 34
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Fractional Derivative ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Nonlinear Fractional Differential Equation ◽  
Comparison Results ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
Nonautonomous Equations

In this paper, nonlinear nonautonomous equations with the generalized proportional Caputo fractional derivative (GPFD) are considered. Some stability properties are studied by the help of the Lyapunov functions and their GPFDs. A scalar nonlinear fractional differential equation with the GPFD is considered as a comparison equation, and some comparison results are proven. Sufficient conditions for stability and asymptotic stability were obtained. Examples illustrating the results and ideas in this paper are also provided.

scholarly journals Nonexistence of Global Solutions of Systems of Time Fractional Differential equations posed on the Heisenberg group

2021 ◽  
Author(s):  
Mokhtar Kirane ◽  
alrazi abdeljabbar
Keyword(s):  
Heisenberg Group ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Positive Real ◽  
Nonlinear Capacity ◽  
Linear Damping ◽  
Fractional Differential ◽  
Time Fractional Differential Equations ◽  
Capacity Method

We first consider the nonlinear time fractional diffusion equation D^{1+α}u+D^β u−∆_{H} u=|u|^p posed on the Heisenberg group H, where 1 < p is a positive real nimber to be specified later; D^δ_{0|t} is the Liouville-Caputo derivative of order δ. For 0 < α < 1,0 < β ≤ 1. This equation interpolates the heat equation and the wave equation with the linear damping D^β_{0|t}u. We present the Fujita exponent for blow-up. Then establish sufficient conditions ensuring non-existence of local solutions. We extend the analysis to the case of the system D^{1+α}u+D^β u−∆_{H} u=|v|^q D^{1+δ}v+D^γ v−∆_{H} v=|u|^p. Our method of proof is based on the nonlinear capacity method.

scholarly journals SOLVABILITY FOR A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

2009 ◽  
Vol 80 (1) ◽  
pp. 125-138 ◽  
Author(s):  
YINGXIN GUO
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nontrivial Solutions ◽  
Nonlinear Fractional Differential Equation ◽  
Nonlinear Alternative ◽  
Fractional Differential ◽  
Image Position ◽  
Almost All

AbstractIn this paper, we consider the existence of nontrivial solutions for the nonlinear fractional differential equation boundary-value problem (BVP)where 1<α≤2,η∈(0,1),β∈ℝ=(−∞,+∞),βηα−1≠1,Dαis the Riemann–Liouville differential operator of orderα, andf:[0,1]×ℝ→ℝ is continuous,q(t):[0,1]→[0,+∞) is Lebesgue integrable. We give some sufficient conditions for the existence of nontrivial solutions to the above boundary-value problems. Our approach is based on the Leray–Schauder nonlinear alternative. Particularly, we do not use the nonnegative assumption and monotonicity onfwhich was essential for the technique used in almost all existed literature.

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.

Time Fractional Differential Equation Model With Random Derivative Order

2009 ◽  
Author(s):  
Hongguang Sun ◽  
Yangquan Chen ◽  
Wen Chen
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Diffusion Processes ◽  
Time Derivative ◽  
Equation Model ◽  
Differential Equation Model ◽  
New Model ◽  
Potential Applications ◽  
Fractional Differential ◽  
Derivative Order

This paper proposes a new type of fractional differential equation model, named time fractional differential equation model, in which noise term is included in the time derivative order. The new model is applied to anomalous relaxation and diffusion processes suffering noisy field. The analysis and numerical simulation results show that our model can well describes the feature of these processes. We also find that the scale parameter and the frequency of the noise play a crucial role in the behaviors of these systems. At the end, we recognize some potential applications of this new model.

Sign in / Sign up

Export Citation Format

Share Document