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1
Content available Numerical solution of a fractional coupled system with the Caputo-Fabrizio fractional derivative
Mansouri Ikram, Bekkouche Mohammed Moumen, Ahmed Abdelaziz Azeb
Journal of Applied Mathematics and Computational Mechanics
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2023
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Vol. 22, nr 1
46--56
EN
Within this work, we discuss the existence of solutions for a coupled system of linear fractional differential equations involving Caputo-Fabrizio fractional orders. We prove the existence and uniqueness of the solution by using the Picard-Lindel ̈of method and fixed point theory. Also, to compute an approximate solution of problem, we utilize the Adomian decomposition method (ADM), as this method provides the solution in the form of a series such that the infinite series converge to the exact solution. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method.
2
Content available The implicit numerical method for the one-dimensional anomalous subdiffusion equation with a nonlinear source term
Błasik Marek
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2021
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Vol. 69, nr 6
art. no. e138240
EN
In the paper, the numerical method of solving the one-dimensional subdiffusion equation with the source term is presented. In the approach used, the key role is played by transforming of the partial differential equation into an equivalent integro-differential equation. As a result of the discretization of the integro-differential equation obtained an implicit numerical scheme which is the generalized Crank-Nicolson method. The implicit numerical schemes based on the finite difference method, such as the Carnk-Nicolson method or the Laasonen method, as a rule are unconditionally stable, which is their undoubted advantage. The discretization of the integro-differential equation is performed in two stages. First, the left-sided Riemann-Liouville integrals are approximated in such a way that the integrands are linear functions between successive grid nodes with respect to the time variable. This allows us to find the discrete values of the integral kernel of the left-sided Riemann-Liouville integral and assign them to the appropriate nodes. In the second step, second order derivative with respect to the spatial variable is approximated by the difference quotient. The obtained numerical scheme is verified on three examples for which closed analytical solutions are known.
3
Content available Analytical and numerical study for a fractional boundary value problem with a conformable fractional derivative of Caputo and its fractional integral
Bekkouche M. Moumen, Guebbai H.
Journal of Applied Mathematics and Computational Mechanics
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2020
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Vol. 19, nr 2
31-42
EN
We study the existence and uniqueness of the solution of a fractional boundary value problem with conformable fractional derivation of the Caputo type, which increases the interest of this study. In order to study this problem we have introduced a new definition of fractional integral as an inverse of the conformable fractional derivative of Caputo, therefore, the proofs are based upon the reduction of the problem to a equivalent linear Volterra-Fredholm integral equations of the second kind, and we have built the minimum conditions to obtain the existence and uniqueness of this solution. The analytical study is followed by a complete numerical study.
4
Generalization of different type integral inequalities for generalized (s, m)-preinvex Godunova-Levin functions
Kashuri A., Liko R.
Journal of Applied Analysis
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2018
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Vol. 24, nr 2
211--221
EN
In the present paper, the notion of generalized (s, m)-preinvex Godunova-Levin function of second kind is introduced, and some new integral inequalities involving generalized (s, m)-preinvex Godunova-Levin functions of second kind along with beta function are given. By using a new identity for fractional integrals, some new estimates on generalizations of Hermite-Hadamard, Ostrowski and Simpson type inequalities for generalized (s, m)-preinvex Godunova-Levin functions of second kind via Riemann-Liouville fractional integral are established.
5
Content available remote Hermite-Hadamard type inequalities for MTm-preinvex functions
Kashuri A., Liko R.
Fasciculi Mathematici
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2017
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Nr 58
77--96
EN
In the present paper, the notion of MTm-preinvex function is introduced and some new integral inequalities for the left-hand side of Gauss-Jacobi type quadrature formula involving MTm-preinvex functions along with beta function are given. Moreover, some generalizations of Hermite-Hadamard type inequalities for MTm-preinvex functions via classical integrals and Riemann-Liouville fractional integrals are established. At the end, some applications to special means are given. These results not only extend the results appeared in the literature (see [13]), but also provide new estimates on these types.
6
Content available remote Modeling and identification of systems with fractional order integral an differential
Busher V., Yarmolovich V.
Zeszyty Naukowe Politechniki Rzeszowskiej. Elektrotechnika
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2015
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z. 34 [292], nr 3
43--57
EN
A universal model of fractional-order differential equation is proposed. It is derived in form hyper neuron, based on a representation of the solution of the equation by finite increments and a modified form of the Riemann-Liouville. Implemented method for identifying parameters of objects by fractional differential equations is described on the base hyper neuron and modified genetic algorithms. Accuracy of calculations is incased due to excluding of circular references and dynamic correction of the fractional integration error. This allows to use hyper neuron as inlined model of such objects in the digital control systems and in conjunction with genetic algorithm it is used for the identification of their parameters with high accuracy.
PL
Zaproponowano uniwersalny model różniczkowego równania ułamkowego rzędu w postaci hyper neuronów, podstawową którego jest stosowanie metody przyrostów skończonych i modyfikowanej formy Riemann- Liouville do przedstawienia rozwiązania równania. Korzystając z hyper neuronów i modyfikowanych algorytmów genetycznych zrealizowana została metoda identyfikacji parametrów obiektu opisywanego ułamkowymi całkowo-różniczkowymi równaniami. Zaproponowana metoda dynamicznej korekcji obliczenia stanów nieustalonych, dla systemów ze zmiennym rzędem ułamkowego całkowania, zapewnia wyższą wiarygodność wyników. Opracowana metoda pozwala na modelowanie procesów w elektrochemicznych kondensatorach dużej pojemności z wysoką dokładnością.
7
Content available remote Fractional representation formulae under initial conditions and fractional Ostrowski type inequalities
Anastassiou G. A.
Demonstratio Mathematica
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2015
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Vol. 48, nr 3
357--378
EN
Here we present very general fractional representation formulae for a function in terms of the fractional Riemann–Liouville integrals of different orders of the function and its ordinary derivatives under initial conditions. Based on these, we derive general fractional Ostrowski type inequalities with respect to all basic norms.
8
Content available Existence results for random fractional differential equations
Lupulescu V., O’Regan D., Rahman ur G.
Opuscula Mathematica
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2014
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Vol. 34, no. 4
813--825
EN
In this paper, an existence result for a random fractional differential equation is established under a Carathéodory condition. Existence results for extremal random solutions are also proved. Finally, an existence and uniqueness result is given.
9
Content available remote Boundary Value Problems for Differential Equations with Fractional Order and Nonlinear Integral Conditions
Benchohra M., Hamani S.
Commentationes Mathematicae
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2009
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Vol. 49, [Z] 2
147-159
EN
In this paper, we shall establish sufficient conditions for the existence of solutions for a class of boundary value problem for fractional differential equations involving the Caputo fractional derivative and nonlinear integral conditions.
10
Content available remote On the fractional Pettis and Aumann-Pettis integral for multifunctions
Ibrahim A-G., Soliman A.M.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
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2006
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Vol. 46, [Z] 2
181-200
EN
Let L be a positive real number. In the present paper we present the definition of the Aumann Pettis integral and the Pettis integral of order for multifunctions. The properties of these integrals and the relations between them are studied extensively. In particular, a Strassen type theorem in this case and continuation property are proved. Also, we give a version for Fatou’s lemma and dominated convergence theorem for the Aumann-Pettis integral of order and for multifunctions.
11
Content available remote Two-weighted norm inequalities for the fractional integral operator between L^p and Lipschitz spaces
Pradolini G.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
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2001
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[Z] 41
147-169
EN
We characterize the pairs of weights for which the fractional integral of order y, I_y, is bounded from weighted Lebesgue spaces L^ into suitable weighted BMO and Lipschitz integral spaces with a weight w. We also study the properties of the classes of weights that arise in connection with this boundedness of I-y.
12
Content available remote Existence theorems for nonlinear functional integral equations of fractional orders
Cichoń M., El-Sayed A.M., Hussien A.H.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
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2001
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[Z] 41
59-67
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