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  1. 8 Opuscula Mathematica
  2. 3 Fundamenta Informaticae
  3. 3 Journal of Applied Mathematics and Computational Mechanics
  4. 2 Journal of Applied Analysis
  5. 1 Autobusy : technika, eksploatacja, systemy transportowe
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  1. 3 Wang J.
  2. 2 Ntouyas S. K.
  3. 2 Wei W.
  4. 1 Ababneh M.
  5. 1 Abdo Mohammed S
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  1. 1 2024
  2. 1 2023
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1
Content available Exploring the dynamics of monkeypox: a fractional order epidemic model approach
Baba Isa Abdullahi, Hincal Evren, Rihane Fathalla A.
Journal of Applied Mathematics and Computational Mechanics
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2024
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Vol. 23, nr 1
32-44
EN
We present a mathematical model employing nonlinear fractional differential equations to investigate the transmission of disease from rodents to humans. The existence and uniqueness of the model’s solutions are established through Banach contraction maps, and the local asymptotic stability of equilibrium solutions is confirmed. We calculate a critical parameter, the basic reproduction number, which reflects secondary infection rates. Numerical simulations illustrate dynamic changes over time, showcasing that our model provides a more comprehensive representation of the biological system compared to classical models.
2
Existence of solutions for a three-point Hadamard fractional resonant boundary value problem
Gholami Yousef
Journal of Applied Analysis
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2023
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Vol. 29, nr 1
31--47
EN
This article focuses on the creation of an existence theorem for a fully nonlinear Hadamard fractional boundary value problem subject to special three-point boundary conditions. By making use of the coincidence degree theory, it is proved that our governing problem makes resonance, that is, the linear part of the differential operator is non-invertible (equally, the corresponding linear problem has at least one nontrivial solution). Constructing some hypotheses on the linear part of the differential operator, nonlinearities and boundary conditions, we give an existence criterion for at least one solution of the fractional-order resonant boundary value problem under study. At the end, a numerical example is presented to illustrate the obtained theoretical results.
3
Integral solutions of nondense impulsive conformable-fractional differential equations with nonlocal condition
Bouaouid Mohamed, Hilal Khalid, Hannabou Mohamed
Journal of Applied Analysis
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2021
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Vol. 27, nr 2
187--197
EN
In this paper, a class of nondense impulsive differential equations with nonlocal condition in the frame of the conformable fractional derivative is studied. The abstract results concerning the existence, uniqueness and stability of the integral solution are obtained by using the extrapolation semigroup approach combined with some fixed point theorems.
4
Content available Existence and Ulam-Hyers stability of the implicit fractional boundary value problem with ψ-Caputo fractional derivative
Wahash Hanan A., Abdo Mohammed S, Panchal Satish K.
Journal of Applied Mathematics and Computational Mechanics
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2020
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Vol. 19, nr 1
89--101
EN
In this paper, we investigate the existence, uniqueness and Ulam-Hyers stability of solutions for nonlinear implicit fractional differential equations with boundary conditions involving a ψ-Caputo fractional derivative. The obtained results for the proposed problem are proved under a new approach and minimal assumptions on the function ƒ. The analysis is based upon the reduction of the problem considered to the equivalent integral equation, while some fixed point theorems of Banach and Schauder and generalized Gronwall inequality are employed to obtain our results for the problem at hand. Finally, the investigation is illustrated by providing a suitable example.
5
Content available The Generalized Differential Transform Method for solution of a free vibration linear differential equation with fractional derivative damping
Das Deepanjan
Journal of Applied Mathematics and Computational Mechanics
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2019
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Vol. 18, nr 2
19--29
EN
In the present paper, the Generalized Differential Transform Method (GDTM) is used for obtaining the approximate analytic solutions of a free vibration linear differential equation of a single-degree-of-freedom (SDOF) system with fractional derivative damping. The fractional derivatives are described in the Caputo sense.
6
Content available Niejednoznaczność zapisu dodatniej pochodno-całki Grünwalda-Letnikova
Cioć Radosław
Autobusy : technika, eksploatacja, systemy transportowe
|
2019
|
R. 20, nr 1-2
176--179
PL
W artykule omówiony został problem wynikający z zapisu pochodno-całki niecałkowitych rzędów Grünwalda-Letnikova, w którym to zapis ten może być niejednoznacznie intepretowany jako pochodna wyższych lub niższych rzędów. Biorąc to pod uwagę autor proponuje nowy zapis uwzględniający ten problem.
EN
The paper discussed the problem of Grünwald-Letnikov differintegral notation in which non-integer order can be incorrectly interpreted as a higher or lower order derivative. Taking the problem into consideration the author’s proposal is new notation of differintegrals.
7
Content available remote Differential equations with random Gamma distributed moments of non-instantaneous impulses and p-moment exponential stability
Agarwal R., Hristova S., O’Regan D., Kopanov P.
Demonstratio Mathematica
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2018
|
Vol. 51, nr 1
151--170
EN
Nonlinear differential equations with impulses occurring at random time and acting noninstantaneously on finite intervals are studied. We consider the case when the time where the impulses occur is Gamma distributed. Lyapunov functions are applied to obtain sufficient conditions for the p-moment exponential stability of the trivial solution of the given system.
8
Content available remote Solvability of a Coupled System of Fractional Differential Equations with Nonlocal and Integral Boundary Conditions
Ahmad B., Ntouyas S. K., Alsaedi A., Hobiny A.
Fundamenta Informaticae
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2017
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Vol. 151, nr 1/4
91--108
EN
This paper is concerned with the existence and uniqueness of solutions for a coupled system of fractional differential equations with nonlocal and integral boundary conditions. The existence and uniqueness of solutions is established by Banach’s contraction principle, while the existence of solutions is derived by using Leray-Schauder’s alternative. The results are explained with the aid of examples. The case of nonlocal strip conditions is also discussed.
9
Content available remote On General Solution for Fractional Differential Equations with Not Instantaneous Impulses
Zhang X., Zhang X., Cao H.
Fundamenta Informaticae
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2017
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Vol. 151, nr 1/4
355--369
EN
In this paper we mainly study a kind of fractional differential equations with not instantaneous impulses, and find the equivalent equations of the impulsive system. The obtained result discovers that there exist general solution for the impulsive system. Next, an example is given to illustrate the obtained result.
10
Content available remote A New Optimization Method Based on Generalized Polynomials for Fractional Differential Equations
Hassani H., Dahaghin M. S., Heydari M. H., Eshkaftaki A. B.
Fundamenta Informaticae
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2017
|
Vol. 151, nr 1/4
443--457
EN
In this paper, we present a new optimization method based on a new class of functions, namely generalized polynomials (GPs) for solving linear and nonlinear fractional differential equations (FDEs). In the proposed method, the solution of the problem under study is expanded in terms of the GPs with fixed coefficients, free coefficients and control parameters. The initial conditions are employed to compute the fixed coefficients. The residual function and its ǁ.ǁ2 are employed for converting the problem under consideration to an optimization one and then choosing the unknown free coefficients and control parameters optimally. As a useful result, the necessary conditions of optimality are derived as a system of nonlinear algebraic equations with unknown free coefficients and control parameters. The validity and accuracy of the approach are illustrated by some numerical examples. The obtained results show that the proposed method is very efficient and accurate.
11
Content available On fractional random differential equations with delay
Vu H., Phung N. N., Phuong N.
Opuscula Mathematica
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2016
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Vol. 36, no. 4
541--556
EN
In this paper, we consider the existence and uniqueness of solutions of the fractional random differential equations with delay. Moreover, some kind of boundedness of the solution is proven. Finally, the applicability of the theoretical results is illustrated with some real world examples.
12
Content available Integral and fractional equations, positive solutions, and Schaefer’s fixed point theorem
Becker L. C., Burton T. A., Purnaras I. K.
Opuscula Mathematica
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2016
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Vol. 36, no. 4
431--458
EN
This is the continuation of four earlier studies of a scalar fractional differential equation of Riemann-Liouville type [formula] in which we first invert it as a Volterra integral equation [formula] and then transform it into [formula] where R is completely monotone with [formula] and J is an arbitrary positive constant. Notice that when x is restricted to a bounded set, then by choosing J large enough, we can frequently change the sign of the integrand in going from (b) to (c). Moreover, the same kind of transformation will produce a similar effect in a wide variety of integral equations from applied mathematics. Because of that change in sign, we can obtain an a priori upper bound on solutions of (b) with a parameter λ ∈ (0, 1] and then obtain an a priori lower bound on solutions of (c). Using this property and Schaefer’s fixed point theorem, we obtain positive solutions of an array of fractional differential equations of both Caputo and Riemann-Liouville type as well as problems from turbulence, heat transfer, and equations of logistic growth. Very simple results establishing global existence and uniqueness of solutions are also obtained in the same way.
13
Content available Application of the multi-step differential transform method to solve a fractional human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells
Zurigat M., Ababneh M.
Journal of Mathematics and Applications
|
2015
|
Vol. 38
171--180
EN
Human T-cell Lymphotropic Virus I (HTLV-I) infection of CD4+ T-Cells is one of the causes of health problems and continues to be one of the significant health challenges. In this article, a multi-step differential transform method is implemented to give approximate solutions of fractional modle of HTLV-I infection of CD4+ T-cells. Numerical results are compared to those obtained by the fourth-order Runge-Kutta method in the case of intger-order derivatives. The suggested method is efficient as the Runge-Kutta method. Some plots are presented to show the reliability and simplicity of the method.
14
Content available Remarks for one-dimensional fractional equations
Ferrara M., Bisci G. M.
Opuscula Mathematica
|
2014
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Vol. 34, no. 4
691--698
EN
n this paper we study a class of one-dimensional Dirichlet boundary value problems involving the Caputo fractional derivatives. The existence of infinitely many solutions for this equations is obtained by exploiting a recent abstract result. Concrete examples of applications are presented.
15
Content available Boundary value problems for nonlinear fractional differential equations and inclusions with nonlocal and fractional integral boundary conditions
Ntouyas S. K.
Opuscula Mathematica
|
2013
|
Vol. 33, no. 1
117-138
EN
This paper studies the boundary value problem of nonlinear fractional differential equations and inclusions of order q ∈ (1, 2] with nonlocal and integral boundary conditions. Some new existence and uniqueness results are obtained by using fixed point theorems.
16
Content available On the controllability of fractional dynamical systems
Balachandran K., Kokila J.
International Journal of Applied Mathematics and Computer Science
|
2012
|
Vol. 22, no. 3
523-531
EN
This paper is concerned with the controllability of linear and nonlinear fractional dynamical systems in finite dimensional spaces. Sufficient conditions for controllability are obtained using Schauder's fixed point theorem and the controllability Grammian matrix which is defined by the Mittag-Leffler matrix function. Examples are given to illustrate the effectiveness of the theory.
17
Content available Positive solutions of arbitrary order nonlinear fractional differential equations with advanced arguments
Ntouyas S. K, Wang G., Zhang L.
Opuscula Mathematica
|
2011
|
Vol. 31, no. 3
433-442
EN
In this paper, we investigate the existence and uniqueness of positive solutions to arbitrary order nonlinear fractional differential equations with advanced arguments. By applying some known fixed point theorems, sufficient conditions for the existence and uniqueness of positive solutions are established.
18
Content available On nonlocal problems for fractional differential equations in Banach spaces
Dong X., Wang J., Zhou Y.
Opuscula Mathematica
|
2011
|
Vol. 31, no. 3
341-357
EN
In this paper, we study the existence and uniqueness of solutions to the nonlocal problems for the fractional differential equation in Banach spaces. New sufficient conditions for the existence and uniqueness of solutions are established by means of fractional calculus and fixed point method under some suitable conditions. Two examples are given to illustrate the results.
19
Content available Existence and uniqueness results for fractional differential equations with boundary value conditions
Lv L., Wang J., Wei W.
Opuscula Mathematica
|
2011
|
Vol. 31, no. 4
629-643
EN
In this paper, we study the existence and uniqueness of fractional differential equations with boundary value conditions. A new generalized singular type Gronwall inequality is given to obtain important a priori bounds. Existence and uniqueness results of solutions are established by virtue of fractional calculus and fixed point method under some weak conditions. An example is given to illustrate the results.
20
Content available remote Application of the fractional oscillator equation to simulations of granular flow in a silo
Błaszczyk T., Kotela E., Leszczyński J.
Computer Methods in Materials Science
|
2011
|
Vol. 11, No. 1
64-67
EN
In this paper we consider a problem of continuous operation of silos. We have focused our attention on the two extreme experimental cases. The first one is considered as a mass flow and the second one is known as a blockade of particles in a silo. In our present work we have analysed in more detail the outflow of granular material in the intermediate regimes between the above mentioned extreme cases. Considering the transition between stable and unstable operation we proposed a no vel mathematical model of the silo emptying. This model involves a differential equation with fractional derivatives.
PL
W pracy rozpatruje się problem stabilnej pracy silosu. Eksperymentalnie rozważa się dwa skrajne przypadki. W pierwszym analizuje się wypływ masowy granulatu, natomiast w drugim rozważa się blokadę zbiornika. W celu bardziej szczegółowej analizy wypływu granulatu z silosu rozpatruje się dodatkowo stany pośrednie pomiędzy wypływem masowym a blokadą zbiornika. W oparciu o analizę przejścia pomiędzy stabilną i niestabilną pracą proponuje się nowy matematyczny model opróżniania silosu. Model ten zawiera równanie różniczkowe z pochodnymi niecałkowitego rzędu.
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