Browsing by Author "O'Regan, Donal"

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A Kamenev-type oscillation result for a linear (1+alpha)-order fractional differential equation
(Elsevier Science Inc., 2015) Baleanu, Dumitru; Mustafa, Octavian G.; O'Regan, Donal
We investigate the eventual sign changing for the solutions of the linear equation (x((alpha)))' + q(t)x = t >= 0, when the functional coefficient q satisfies the Kamenev-type restriction lim sup 1/t epsilon integral(t)(to) (t - s)epsilon q(s)ds = +infinity for some epsilon > 2; t(0) > 0. The operator x((alpha)) is the Caputo differential operator and alpha is an element of (0, 1)
Citation - WoS: 4
Citation - Scopus: 4
Analysis of Positivity Results for Discrete Fractional Operators by Means of Exponential Kernels
(Amer inst Mathematical Sciences-aims, 2022) O'Regan, Donal; Brzo, Aram Bahroz; Abualnaja, Khadijah M.; Baleanu, Dumitru; Mohammed, Pshtiwan Othman; O’regan, Donal
In this study, we consider positivity and other related concepts such as alpha-convexity and alpha-monotonicity for discrete fractional operators with exponential kernel. Namely, we consider discrete Delta fractional operators in the Caputo sense and we apply efficient initial conditions to obtain our conclusions. Note positivity results are an important factor for obtaining the composite of double discrete fractional operators having different orders.
Citation - WoS: 1
Citation - Scopus: 1
Analytical Results for Positivity of Discrete Fractional Operators With Approximation of the Domain of Solutions
(Amer inst Mathematical Sciences-aims, 2022) O'Regan, Donal; Baleanu, Dumitru; Hamed, Y. S.; Elattar, Ehab E.; Mohammed, Pshtiwan Othman
We study the monotonicity method to analyse nabla positivity for discrete fractional operators of Riemann-Liouville type based on exponential kernels, where ((CFR)(c0)del F-theta)(t) > -epsilon Lambda(theta - 1) (del F)(c(0) + 1) such that (del F)(c(0) + 1) >= 0 and epsilon > 0. Next, the positivity of the fully discrete fractional operator is analyzed, and the region of the solution is presented. Further, we consider numerical simulations to validate our theory. Finally, the region of the solution and the cardinality of the region are discussed via standard plots and heat map plots. The figures confirm the region of solutions for specific values of epsilon and theta.
Citation - WoS: 14
Citation - Scopus: 17
Approximate Solution for a 2-D Fractional Differential Equation With Discrete Random Noise
(Pergamon-elsevier Science Ltd, 2020) Baleanu, Dumitru; Tran Ngoc Thach; O'Regan, Donal; Nguyen Huu Can; Nguyen Huy Tuan; Thach, Tran Ngoc; Tuan, Nguyen Huy; Can, Nguyen Huu
We study a boundary value problem for a 2-D fractional differential equation (FDE) with random noise. This problem is not well-posed. Hence, we use truncated regularization method to establish regularized solutions for the such problem. Finally, the convergence rate of this approximate solution and a numerical example are investigated. (C) 2020 Elsevier Ltd. All rights reserved.
Citation - WoS: 5
Citation - Scopus: 5
A Filter Method for Inverse Nonlinear Sideways Heat Equation
(Springer, 2020) O'Regan, Donal; Baleanu, Dumitru; Nguyen Hoang Luc; Nguyen Can; Nguyen Anh Triet; Anh Triet, Nguyen; O’Regan, Donal; Hoang Luc, Nguyen; Luc, Nguyen Hoang; Can, Nguyen; Triet, Nguyen Anh
In this paper, we study a sideways heat equation with a nonlinear source in a bounded domain, in which the Cauchy data at x=X are given and the solution in 0 <= x < X is sought. The problem is severely ill-posed in the sense of Hadamard. Based on the fundamental solution to the sideways heat equation, we propose to solve this problem by the filter method of degree alpha, which generates a well-posed integral equation. Moreover, we show that its solution converges to the exact solution uniformly and strongly in L-p(omega,X; L-2 (R)); omega is an element of[0,X) under a priori assumptions on the exact solution. The proposed regularized method is illustrated by numerical results in the final section.
Citation - WoS: 47
Citation - Scopus: 53
Final Value Problem for Nonlinear Time Fractional Reaction-Diffusion Equation With Discrete Data
(Elsevier, 2020) Baleanu, Dumitru; Tran Ngoc Thach; O'Regan, Donal; Nguyen Huu Can; Nguyen Huy Tuan; Thach, Tran Ngoc; Tuan, Nguyen Huy; Can, Nguyen Huu
In this paper, we study the problem of finding the solution of a multi-dimensional time fractional reaction-diffusion equation with nonlinear source from the final value data. We prove that the present problem is not well-posed. Then regularized problems are constructed using the truncated expansion method (in the case of two-dimensional) and the quasi-boundary value method (in the case of multi-dimensional). Finally, convergence rates of the regularized solutions are given and investigated numerically. (C) 2020 Elsevier B.V. All rights reserved.
The Hausdorff-Pompeiu Distance in Gn-Menger Fractal Spaces
(Mdpi, 2022) Saadati, Reza; Li, Chenkuan; Jarad, Fahd; O'Regan, Donal; O’Regan, Donal
This paper introduces a complete Gn-Menger space and defines the Hausdorff-Pompeiu distance in the space. Furthermore, we show a novel fixed-point theorem for Gn-Menger-theta-contractions in fractal spaces.
Citation - WoS: 12
Citation - Scopus: 12
A Kamenev-Type Oscillation Result for a Linear (1+α)-Order Fractional Differential Equation
(Elsevier Science inc, 2015) Mustafa, Octavian G.; O'Regan, Donal; Baleanu, Dumitru; Bəleanu, Dumitru
We investigate the eventual sign changing for the solutions of the linear equation (x((alpha)))' + q(t)x = t >= 0, when the functional coefficient q satisfies the Kamenev-type restriction lim sup 1/t epsilon integral(t)(to) (t - s)epsilon q(s)ds = +infinity for some epsilon > 2; t(0) > 0. The operator x((alpha)) is the Caputo differential operator and alpha is an element of (0, 1). (C) 2015 Elsevier Inc. All rights reserved.
Citation - WoS: 18
Citation - Scopus: 22
A Nagumo-Like Uniqueness Theorem for Fractional Differential Equations
(Iop Publishing Ltd, 2011) Mustafa, Octavian G.; O'Regan, Donal; Baleanu, Dumitru; ORegan, Donal
We extend to fractional differential equations a recent generalization of the Nagumo uniqueness theorem for ordinary differential equations of first order.
Citation - WoS: 5
On a Fractional Differential Equation With Infinitely Many Solutions
(Springer international Publishing Ag, 2012) Mustafa, Octavian G.; O'Regan, Donal; Baleanu, Dumitru
We present a set of restrictions on the fractional differential equation , , where and , that leads to the existence of an infinity of solutions (a continuum of solutions) starting from . The operator is the Caputo differential operator.
Citation - WoS: 3
Citation - Scopus: 3
On an Inverse Scattering Algorithm for the Camassa-Holm Equation
(Taylor & Francis Ltd, 2008) Mustafa, Octavian G.; O'Regan, Donal
We present a clarification of a recent inverse scattering algorithm developed for the Camassa-Holm equation.
Citation - WoS: 17
Citation - Scopus: 18
On Time Fractional Pseudo-Parabolic Equations With Nonlocal Integral Conditions
(Amer inst Mathematical Sciences-aims, 2022) O'Regan, Donal; Baleanu, Dumitru; Tuan, Nguyen H.; Nguyen Anh Tuan; O’regan, Donal
In this paper, we study the nonlocal problem for pseudo-parabolic equation with time and space fractional derivatives. The time derivative is of Caputo type and of order sigma, 0 < sigma < 1 and the space fractional derivative is of order alpha, beta > 0. In the first part, we obtain some results of the existence and uniqueness of our problem with suitably chosen alpha, beta. The technique uses a Sobolev embedding and is based on constructing a Mittag-Leffler operator. In the second part, we give the ill-posedness of our problem and give a regularized solution. An error estimate in L-p between the regularized solution and the sought solution is obtained.
Citation - WoS: 34
Citation - Scopus: 35
On Well-Posedness of the Sub-Diffusion Equation With Conformable Derivative Model
(Elsevier, 2020) Tran Bao Ngoc; Baleanu, Dumitru; O'Regan, Donal; Nguyen Huy Tuan; Ngoc, Tran Bao; Tuan, Nguyen Huy
In this paper, we study an initial value problem for the time diffusion equation (C)partial derivative(beta)/partial derivative t(beta) u + Au = F, 0 < beta <= 1, on Omega x (0, T), where the time derivative is the conformable derivative. We study the existence and regularity of mild solutions in the following three cases with source term F: F = F (x, t), i.e., linear source term; F = F (u) is nonlinear, globally Lipchitz and uniformly bounded. The results in this case play important roles in numerical analysis. F = F (u) is nonlinear, locally Lipchitz and uniformly bounded. The analysis in this case can be widely applied to many problems such as - Time Ginzburg-Landau equations C partial derivative(beta)u/partial derivative t(beta)+ (-Delta)u = vertical bar u vertical bar(mu-1) u; - Time Burgers equations C partial derivative(beta)u/partial derivative t(beta)-( u center dot del) u + (- Delta)u = 0; etc. (C) 2020 Elsevier B.V. All rights reserved.
Citation - Scopus: 1
Positive Solutions of Some Elliptic Differential Equations With Oscillating Nonlinearity
(Taylor & Francis Ltd, 2012) Mustafa, Octavian G.; O'Regan, Donal; Jarad, Fahd
We discuss the occurrence of positive solutions which decay to 0 as vertical bar xj vertical bar ->+infinity the differential equation Delta u+f(x, u)+g(vertical bar x vertical bar)x . del u=0, vertical bar xj vertical bar>R>0, x is an element of R-n, where n >= 3, g is nonnegative valued and f has alternating sign, by means of the comparison method. Our results complement several recent contributions from Ehrnstrom and Mustafa [ M. Ehrnstrom, O.G. Mustafa, On positive solutions of a class of nonlinear elliptic equations, Nonlinear Anal. TMA 67 (2007), pp. 1147-1154].
Citation - WoS: 9
Citation - Scopus: 8
A Uniqueness Criterion for Fractional Differential Equations With Caputo Derivative
(Springer, 2013) Mustafa, Octavian G.; O'Regan, Donal; Baleanu, Dumitru
We investigate the uniqueness of solutions to an initial value problem associated with a nonlinear fractional differential equation of order alpha a(0,1). The differential operator is of Caputo type whereas the nonlinearity cannot be expressed as a Lipschitz function. Instead, the Riemann-Liouville derivative of this nonlinearity verifies a special inequality.