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    Tytuł:
    Solutions to fractional diffusion-wave equation in a circular sector
    Autorzy:
    Povstenko, Y.
    Powiązania:
    https://bibliotekanauki.pl/articles/121674.pdf
    Data publikacji:
    2013
    Wydawca:
    Uniwersytet Humanistyczno-Przyrodniczy im. Jana Długosza w Częstochowie. Wydawnictwo Uczelniane
    Tematy:
    Mittag-Leffler function
    Caputo derivative
    funkcja Mittag-Lefflera
    pochodna Caputo
    Opis:
    The time-fractional diffusion-wave equation with the Caputo derivative of the order 0 < α ≤ 2 is considered in a domain 0 ≤ r < R, 0 < ϕ < ϕ0 under different boundary conditions. The Laplace integral transform with respect to time, the finite Fourier transforms with respect to the angular coordinate, and the finite Hankel transforms with respect to the radial coordinate are used. The fundamental solutions are expressed in terms of the Mittag-Leffler function. The particular cases of the obtained solutions corresponding to the diffusion equation (α = 1) and the wave equation (α = 2) coincide with those known in the literature.
    Źródło:
    Scientific Issues of Jan Długosz University in Częstochowa. Mathematics; 2013, 18; 41-54
    2450-9302
    Pojawia się w:
    Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Time-fractional heat conduction in a finite composite cylinder with heat source
    Autorzy:
    Kukla, Stanisław
    Siedlecka, Urszula
    Powiązania:
    https://bibliotekanauki.pl/articles/1839814.pdf
    Data publikacji:
    2020
    Wydawca:
    Politechnika Częstochowska. Wydawnictwo Politechniki Częstochowskiej
    Tematy:
    ractional heat conduction
    Caputo derivative
    composite cylinder
    pochodna Caputo
    cylinder kompozytowy
    przewodzenie ciepła
    Opis:
    In this paper, the effect of the fractional order of the Caputo time-derivative occurring in heat conduction models on the temperature distribution in a finite cylinder consisting of an inner solid cylinder and an outer concentric layer is investigated. The inner cylinder (core) and the cylindrical layer are in perfect thermal contact. The Robin boundary condition on the outer surface and the Neumann conditions on the ends of the cylinder are assumed. An internal heat source is represented in the mathematical model by taking into account in the heat conduction equation of a function which depends on the space and time variable. An analytical solution of the problem is derived in the form of the double series of eigenfunctions. Numerical examples are presented.
    Źródło:
    Journal of Applied Mathematics and Computational Mechanics; 2020, 19, 2; 85-94
    2299-9965
    Pojawia się w:
    Journal of Applied Mathematics and Computational Mechanics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Heat conduction in a finite medium using the fractional single-phase-lag model
    Autorzy:
    Siedlecka, Urszula
    Powiązania:
    https://bibliotekanauki.pl/articles/200748.pdf
    Data publikacji:
    2019
    Wydawca:
    Polska Akademia Nauk. Czytelnia Czasopism PAN
    Tematy:
    single-phase-lagging heat conduction
    Caputo derivative
    fractional heat conduction
    przewodzenie ciepła
    Pochodna Caputo
    Opis:
    In the paper, a solution of the time-fractional single-phase-lagging heat conduction problem in finite regions is presented. The heat conduction equation with the Caputo time-derivative is complemented by the Robin boundary conditions. The Laplace transform with respect to the time variable and an expansion in the eigenfunctions series with respect to the space variable was applied. A method for the numerical inversion of the Laplace transforms was used. Formulation and solution of the problem cover the heat conduction in a finite slab, hollow cylinder and hollow sphere. The effect of the fractional order of the Caputo derivative and the phase-lag parameter on the temperature distribution in a slab has been numerically investigated.
    Źródło:
    Bulletin of the Polish Academy of Sciences. Technical Sciences; 2019, 67, 2; 401-407
    0239-7528
    Pojawia się w:
    Bulletin of the Polish Academy of Sciences. Technical Sciences
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Fractional heat conduction in multilayer spherical bodies
    Autorzy:
    Kukla, S.
    Siedlecka, U.
    Powiązania:
    https://bibliotekanauki.pl/articles/122775.pdf
    Data publikacji:
    2016
    Wydawca:
    Politechnika Częstochowska. Wydawnictwo Politechniki Częstochowskiej
    Tematy:
    heat conduction
    multilayer bodies
    Caputo derivative
    spherical coordinate
    przewodnictwo cieplne
    pochodna Caputo
    współrzędna sferyczna
    Opis:
    In this paper an analytical solution of the time-fractional heat conduction problem in a spherical coordinate system is presented. The considerations deal the two-dimensional problem in multilayer spherical bodies including a hollow sphere, hemisphere and spherical wedge. The mathematical Robin conditions are assumed. The solution is a sum of time-dependent function satisfied homogenous boundary conditions and of a solution of the steady-state problem. Numerical example shows the temperature distributions in the hemisphere for various order of time-derivative.
    Źródło:
    Journal of Applied Mathematics and Computational Mechanics; 2016, 15, 4; 83-92
    2299-9965
    Pojawia się w:
    Journal of Applied Mathematics and Computational Mechanics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    A new numerical technique for solving fractional Bratu’s initial value problems in the Caputo and Caputo-Fabrizio sense
    Autorzy:
    Khalouta, Ali
    Kadem, Abdelouahab
    Powiązania:
    https://bibliotekanauki.pl/articles/1839797.pdf
    Data publikacji:
    2020
    Wydawca:
    Politechnika Częstochowska. Wydawnictwo Politechniki Częstochowskiej
    Tematy:
    fractional Bratu’s initial value problem
    Caputo fractional derivative
    Caputo-Fabrizio fractional derivative
    natural transform method
    Adomian decomposition method
    ułamkowa pochodna Caputo
    ułamkowa pochodna Caputo-Fabrizio
    metoda transformacji naturalnej
    metoda dekompozycji Adomiana
    Opis:
    The purpose of this paper is to propose a new numerical technique called the natural decomposition method (NDM) for solving fractional Bratu’s initial value problems (FBIVP) in the Caputo and Caputo-Fabrizio sense. The NDM is a combined form of the natural transform method and the Adomian decomposition method. The numerical example is provided in order to validate the efficiency and reliability of the proposed method. The obtained results reveal that the proposed method is a very efficient and simple tool for solving fractional differential equations.
    Źródło:
    Journal of Applied Mathematics and Computational Mechanics; 2020, 19, 1; 43-56
    2299-9965
    Pojawia się w:
    Journal of Applied Mathematics and Computational Mechanics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    A new numerical technique for solving fractional Bratu’s initial value problems in the Caputo and Caputo-Fabrizio sense
    Autorzy:
    Khalouta, Ali
    Kadem, Abdelouahab
    Powiązania:
    https://bibliotekanauki.pl/articles/122619.pdf
    Data publikacji:
    2020
    Wydawca:
    Politechnika Częstochowska. Wydawnictwo Politechniki Częstochowskiej
    Tematy:
    fractional Bratu’s initial value problem
    Caputo fractional derivative
    Caputo-Fabrizio fractional derivative
    natural transform method
    Adomian decomposition method
    ułamkowa pochodna Caputo
    ułamkowa pochodna Caputo-Fabrizio
    metoda transformacji naturalnej
    metoda dekompozycji Adomiana
    Opis:
    The purpose of this paper is to propose a new numerical technique called the natural decomposition method (NDM) for solving fractional Bratu’s initial value problems (FBIVP) in the Caputo and Caputo-Fabrizio sense. The NDM is a combined form of the natural transform method and the Adomian decomposition method. The numerical example is provided in order to validate the efficiency and reliability of the proposed method. The obtained results reveal that the proposed method is a very efficient and simple tool for solving fractional differential equations.
    Źródło:
    Journal of Applied Mathematics and Computational Mechanics; 2020, 19, 1; 43-56
    2299-9965
    Pojawia się w:
    Journal of Applied Mathematics and Computational Mechanics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Different kinds of boundary conditions for time-fractional heat conduction equation
    Autorzy:
    Povstenko, Y.
    Powiązania:
    https://bibliotekanauki.pl/articles/121936.pdf
    Data publikacji:
    2011
    Wydawca:
    Uniwersytet Humanistyczno-Przyrodniczy im. Jana Długosza w Częstochowie. Wydawnictwo Uczelniane
    Tematy:
    boundary conditions
    heat conduction
    time-fractional equation
    Caputo derivative
    przewodzenie ciepła
    warunki brzegowe
    pochodna Caputo
    Opis:
    The time-fractional heat conduction equation with the Caputo derivative of the order 0 ˂ α ˂ 2 is considered in a bounded domain. For this equation different types of boundary conditions can be given. The Dirichlet boundary condition prescribes the temperature over the surface of the body. In the case of mathematical Neumann boundary condition the boundary values of the normal derivative are set, the physical Neumann boundary condition specifies the boundary values of the heat flux. In the case of the classical heat conduction equation (α = 1), these two types of boundary conditions are identical, but for fractional heat conduction they are essentially different. The mathematical Robin boundary condition is a specification of a linear combination of the values of temperature and the values of its normal derivative at the boundary of the domain, while the physical Robin boundary condition prescribes a linear combination of the values of temperature and the values of the heat flux at the surface of a body.
    Źródło:
    Scientific Issues of Jan Długosz University in Częstochowa. Mathematics; 2011, 16; 61-66
    2450-9302
    Pojawia się w:
    Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Harmonic impact on the surface of a half-plane in the framework of Time-Fractional Heat Conduction
    Autorzy:
    Povstenko, Y.
    Powiązania:
    https://bibliotekanauki.pl/articles/121966.pdf
    Data publikacji:
    2016
    Wydawca:
    Uniwersytet Humanistyczno-Przyrodniczy im. Jana Długosza w Częstochowie. Wydawnictwo Uczelniane
    Tematy:
    równanie przewodzenia ciepła
    technika transformacji całkowitej
    pochodna Caputo
    heat conduction equation
    total transformation technique
    derivative Caputo
    Opis:
    The time-fractional heat conduction equation with the Caputo derivative is considered in a half-plane. The boundary value of temperature varies harmonically in time. The integral transform technique is used; the solution is obtained in terms of integral with integrand being the Mittag-Leffler functions. The particular case of solution corresponding to the classical heat conduction equation is discussed in details.
    Źródło:
    Scientific Issues of Jan Długosz University in Częstochowa. Mathematics; 2016, 21; 85-92
    2450-9302
    Pojawia się w:
    Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Application of fractional order theory of thermoelasticity in an elliptical disk and associated thermal stresses
    Autorzy:
    Thakare, S.
    Panke, Y.
    Hadke, K.
    Powiązania:
    https://bibliotekanauki.pl/articles/1839826.pdf
    Data publikacji:
    2020
    Wydawca:
    Uniwersytet Zielonogórski. Oficyna Wydawnicza
    Tematy:
    naprężenia termiczne
    pochodna Caputo
    termosprężystość
    elliptical disk
    fractional order theory
    Caputo derivative
    integral transform
    thermal stresses
    thermal deflection
    Opis:
    In this article, a time fractional-order theory of thermoelasticity is applied to an isotropic homogeneous elliptical disk. The lower and upper surfaces of the disk are maintained at zero temperature, whereas the sectional heat supply is applied on the outer curved surface. Thermal deflection and associated thermal stresses are obtained in terms of Mathieu function of the first kind of order 2n. Numerical evaluation is carried out for the temperature distribution, Thermal deflection and thermal stresses and results of the resulting quantities are depicted graphically.
    Źródło:
    International Journal of Applied Mechanics and Engineering; 2020, 25, 3; 169-180
    1734-4492
    2353-9003
    Pojawia się w:
    International Journal of Applied Mechanics and Engineering
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Results on the controllability of Caputo’s fractional descriptor systems with constant delays
    Autorzy:
    Beata, Sikora
    Powiązania:
    https://bibliotekanauki.pl/articles/27311455.pdf
    Data publikacji:
    2023
    Wydawca:
    Polska Akademia Nauk. Czasopisma i Monografie PAN
    Tematy:
    descriptor systems
    fractional-order systems
    Caputo derivative
    Drazin inverse
    controllability
    pochodna Caputo
    odwrotność Drazina
    sterowalność
    system deskryptorowy
    system ułamkowego rzędu
    Opis:
    The paper investigates the controllability of fractional descriptor linear systems with constant delays in control. The Caputo fractional derivative is considered. Using the Drazin inverse and the Laplace transform, a formula for solving of the matrix state equation is obtained. New criteria of relative controllability for Caputo’s fractional descriptor systems are formulated and proved. Both constrained and unconstrained controls are considered. To emphasize the importance of the theoretical studies, an application to electrical circuits is presented as a practical example.
    Źródło:
    Bulletin of the Polish Academy of Sciences. Technical Sciences; 2023, 71, 4; art. no. e146287
    0239-7528
    Pojawia się w:
    Bulletin of the Polish Academy of Sciences. Technical Sciences
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The Caputo vs. Caputo-Fabrizio operators in modeling of heat transfer process
    Autorzy:
    Oprzędkiewicz, K.
    Mitkowski, W.
    Gawin, E.
    Dziedzic, K.
    Powiązania:
    https://bibliotekanauki.pl/articles/200001.pdf
    Data publikacji:
    2018
    Wydawca:
    Polska Akademia Nauk. Czytelnia Czasopism PAN
    Tematy:
    fractional order systems
    Caputo operator
    Caputo-Fabrizio operator
    non singular kernel operator
    heat transfer
    simplex method
    transfer ciepła
    metoda simplex
    system ułamkowy
    pochodna Caputo
    Opis:
    In the paper two non-integer order, state space models of heat transfer process are compared. The first uses a known Caputo operator and the second – a new operator proposed by Caputo and Fabrizio in 2015. Both discussed models are modifications of a known, integer order, state space, semigroup model of heat transfer process. Parameters of both models were identified by means of optimization of MSE cost function with the use of simplex method, available in MATLAB. Both proposed models have been compared in the aspect of accuracy and convergence. Analytical and numerical results show that the Caputo-Fabrizio model is faster convergent and easier to implement than the Caputo model. However, its accuracy in the sense of MSE cost function is worse.
    Źródło:
    Bulletin of the Polish Academy of Sciences. Technical Sciences; 2018, 66, 4; 501-507
    0239-7528
    Pojawia się w:
    Bulletin of the Polish Academy of Sciences. Technical Sciences
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Analytical solution of a fractional model of fluid flow through narrowing system in terms of Mittag-Leffler function
    Autorzy:
    Choudhary, A.
    Kumar, D.
    Singh, J.
    Powiązania:
    https://bibliotekanauki.pl/articles/264897.pdf
    Data publikacji:
    2020
    Wydawca:
    Uniwersytet Zielonogórski. Oficyna Wydawnicza
    Tematy:
    transformata Hankela
    funkcja Bessela
    pochodna Caputo
    narrowing system
    generalized Caputo fractional derivative
    Sumudu transform
    finite Hankel transform
    Bessel function
    Mittag-Leffler function
    Opis:
    In this work, we discuss a fractional model of a flow equation in a simple pipeline. Pipeline narrowing is a crucial aspect in drinking water distribution processes, sewage system and in oil-well schemes. The solution of the mathematical model is determined with the aid of the Sumudu transform and finite Hankel transform. The results derived in the current study are in compact and graceful forms in terms of the Mittag-Leffler type function, which are convenient for numerical and theoretical evaluation.
    Źródło:
    International Journal of Applied Mechanics and Engineering; 2020, 25, 1; 1-11
    1734-4492
    2353-9003
    Pojawia się w:
    International Journal of Applied Mechanics and Engineering
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Some new existence results and stability concepts for fractional partial random differential equations
    Autorzy:
    Abbas, S.
    Benchohra, M.
    Darwish, M. A.
    Powiązania:
    https://bibliotekanauki.pl/articles/357770.pdf
    Data publikacji:
    2016
    Wydawca:
    Politechnika Rzeszowska im. Ignacego Łukasiewicza. Oficyna Wydawnicza
    Tematy:
    random differential equations
    left-sided mixed Riemann-Liouville integral
    Caputo fractional-order derivative
    Banach space
    Darboux problem
    Ulam stability
    równanie różniczkowe
    pochodna Caputo
    przestrzeń Banacha
    problem Darbouxa
    Opis:
    In the present paper we provide some existence results and Ulam’s type stability concepts for the Darboux problem of partial fractional random differential equations in Banach spaces, by applying the measure of noncompactness and a random fixed point theorem with stochastic domain.
    Źródło:
    Journal of Mathematics and Applications; 2016, 39; 5-22
    1733-6775
    2300-9926
    Pojawia się w:
    Journal of Mathematics and Applications
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The Generalized Differential Transform Method for solution of a free vibration linear differential equation with fractional derivative damping
    Autorzy:
    Das, Deepanjan
    Powiązania:
    https://bibliotekanauki.pl/articles/122961.pdf
    Data publikacji:
    2019
    Wydawca:
    Politechnika Częstochowska. Wydawnictwo Politechniki Częstochowskiej
    Tematy:
    fractional differential equations
    Caputo fractional derivative
    generalized
    differential transform method
    analytic solution
    ułamkowe równania różniczkowe
    ułamkowa pochodna Caputo
    metoda transformacji różnicowej
    rozwiązanie analityczne
    ułamkowe równanie różniczkowe
    Opis:
    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.
    Źródło:
    Journal of Applied Mathematics and Computational Mechanics; 2019, 18, 2; 19-29
    2299-9965
    Pojawia się w:
    Journal of Applied Mathematics and Computational Mechanics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Some closed form series solutions for the time-fractional diffusion-wave equation in polar coordinates with a generalized Caputo fractional derivative
    Autorzy:
    Elkott, Ibrahim
    Abdel-Latif, Mohamed S.
    El-Kalla, Ibrahim L.
    Abdel Kader, Abass H.
    Powiązania:
    https://bibliotekanauki.pl/articles/24201502.pdf
    Data publikacji:
    2023
    Wydawca:
    Politechnika Częstochowska. Wydawnictwo Politechniki Częstochowskiej
    Tematy:
    generalized time-fractional Caputo derivative
    generalized Laplace transform
    Hankel transform
    diffusion-wave equation
    uogólniona ułamkowa pochodna Caputo w czasie
    uogólniona transformata Laplace'a
    transformata Hankela
    równanie fali dyfuzyjnej
    Opis:
    In this paper, we obtain some closed form series solutions for the time fractional diffusion-wave equation (TFDWE) with the generalized time-fractional Caputo derivative (GTFCD) associated with a source term in polar coordinates. These solutions are found using generalized Laplace and Hankel transforms. We obtained the closed form series solutions in the form of the Polygamma function. The effect of the fractional order derivative on the diffusion-wave variable is illustrated graphically.
    Źródło:
    Journal of Applied Mathematics and Computational Mechanics; 2023, 22, 2; 5--14
    2299-9965
    Pojawia się w:
    Journal of Applied Mathematics and Computational Mechanics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    An analytical solution to the problem of time-fractional heat conduction in a composite sphere
    Autorzy:
    Kukla, S.
    Siedlecka, U.
    Powiązania:
    https://bibliotekanauki.pl/articles/202020.pdf
    Data publikacji:
    2017
    Wydawca:
    Polska Akademia Nauk. Czytelnia Czasopism PAN
    Tematy:
    heat conduction
    multilayered sphere
    Caputo fractional derivative
    przewodzenie ciepła
    powłoka wielowarstwowa
    pochodna ułamkowego rzędu Caputo
    Opis:
    An analytical solution to the problem of time-fractional heat conduction in a sphere consisting of an inner solid sphere and concentric spherical layers is presented. In the heat conduction equation, the Caputo time-derivative of fractional order and the Robin boundary condition at the outer surface of the sphere are assumed. The spherical layers are characterized by different material properties and perfect thermal contact is assumed between the layers. The analytical solution to the problem of heat conduction in the sphere for time-dependent surrounding temperature and for time-space-dependent volumetric heat source is derived. Numerical examples are presented to show the effect of the harmonically varying intensity of the heat source and the harmonically varying surrounding temperature on the temperature in the sphere for different orders of the Caputo time-derivative.
    Źródło:
    Bulletin of the Polish Academy of Sciences. Technical Sciences; 2017, 65, 2; 179-186
    0239-7528
    Pojawia się w:
    Bulletin of the Polish Academy of Sciences. Technical Sciences
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Non integer order, state space model of heat transfer process using Caputo-Fabrizio operator
    Autorzy:
    Oprzędkiewicz, K.
    Powiązania:
    https://bibliotekanauki.pl/articles/201464.pdf
    Data publikacji:
    2018
    Wydawca:
    Polska Akademia Nauk. Czytelnia Czasopism PAN
    Tematy:
    non-integer order systems
    heat transfer equation
    infinite dimensional systems
    fractional order state equation
    Riesz operator
    Caputo operator
    Caputo-Fabrizio operator
    systemy niecałkowitego rzędu
    układ niecałkowitego rzędu
    twierdzenie Riesza
    pochodna Caputo
    Opis:
    The paper is intended to show a new state space, non integer order model of an one-dimensional heat transfer process. The proposed model derives directly from time continuous, state space semigroup model. The fractional order derivative with respect to time is by a new operator proposed by Caputo and Fabrizio, the non integer order spatial derivative is expressed by Riesz operator. The Caputo-Fabrizio operator can be directly implementated using MATLAB, because it does not require us to apply any approximation. Analytical formulae of step response are given, the system decomposition was discussed also. Main results from the paper show that the use of Caputo Fabrizio operator allows us to obtain the simple in implementation and analysis model of the considered heat transfer process. The accuracy of the proposed model in the sense of a MSE cost function is satisfying.
    Źródło:
    Bulletin of the Polish Academy of Sciences. Technical Sciences; 2018, 66, 3; 249-255
    0239-7528
    Pojawia się w:
    Bulletin of the Polish Academy of Sciences. Technical Sciences
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Application of LDG scheme to solve semi-differential equations
    Autorzy:
    Izadi, Mohammad
    Powiązania:
    https://bibliotekanauki.pl/articles/122346.pdf
    Data publikacji:
    2019
    Wydawca:
    Politechnika Częstochowska. Wydawnictwo Politechniki Częstochowskiej
    Tematy:
    Caputo fractional derivative
    local discontinuous Galerkin method
    semi-differential equations
    pochodna ułamkowa Caputo
    nieciągła metoda Galerkina
    schemat LDG
    równania różnicowe
    Opis:
    In the current work, we investigate a technique based on discontinuous Galerkin method for the numerical approximation of semi-differential equations with Caputo’s fractional derivative. In this approach, using the natural upwind fluxes enables us to solve the model problem element by element locally in each subintervals and there is no need to solve a full global matrix. Numerical experiments are given to verify the efficiency and accuracy of the proposed method. Numerical solutions are compared with the exact solutions as well as the numerical solutions obtained by other available well-established computational procedures. The results show that the LDG method is more accurate for solving this class of differential equation with relatively low degrees of polynomials and number of elements.
    Źródło:
    Journal of Applied Mathematics and Computational Mechanics; 2019, 18, 4; 27-39
    2299-9965
    Pojawia się w:
    Journal of Applied Mathematics and Computational Mechanics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Existence and convergence results for Caputo fractional Volterra integro-differential equations
    Autorzy:
    Hamoud, A. A.
    Bani Issa, M. Sh.
    Ghadle, K. P.
    Abdulghani, M.
    Powiązania:
    https://bibliotekanauki.pl/articles/2052432.pdf
    Data publikacji:
    2018
    Wydawca:
    Politechnika Rzeszowska im. Ignacego Łukasiewicza. Oficyna Wydawnicza
    Tematy:
    homotopy analysis method
    Caputo fractional derivative
    Volterra integro-differential equation
    approximate solution
    homotopijna metoda analizy
    pochodna ułamkowa Caputo
    równanie różniczkowe Volterry
    rozwiązanie przybliżone
    Opis:
    In this article, homotopy analysis method is successfully applied to find the approximate solution of Caputo fractional Volterra integro-differential equation. The reliability of the method and reduction in the size of the computational work give this method a wider applicability. Also, the behavior of the solution can be formally determined by analytical approximate. Moreover, we proved the existence and convergence of the solution. Finally, an example is included to demonstrate the validity and applicability of the proposed technique.
    Źródło:
    Journal of Mathematics and Applications; 2018, 41; 109-121
    1733-6775
    2300-9926
    Pojawia się w:
    Journal of Mathematics and Applications
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Analysis of solutions of the 1D fractional Cattaneo heat transfer equation
    Autorzy:
    Siedlecka, Urszula
    Ciesielski, Mariusz
    Powiązania:
    https://bibliotekanauki.pl/articles/2175501.pdf
    Data publikacji:
    2021
    Wydawca:
    Politechnika Częstochowska. Wydawnictwo Politechniki Częstochowskiej
    Tematy:
    heat transfer
    Cattaneo equation
    fractional Caputo derivative
    Laplace transform
    Fourier transform
    wymiana ciepła
    równanie Cattaneo
    pochodna ułamkowa Caputo
    transformata Laplace'a
    transformata Fouriera
    Opis:
    In this paper, a solution of the single-phase lag heat conduction problem is presented. The research concerns the generalized 1D Cattaneo equation in a whole-space domain, where a second order time derivative is replaced by the fractional Caputo derivative. The Fourier-Laplace transform technique is used to determine a solution of the considered problem. The numerical inversion of the Laplace transforms is applied. The effect of the order of the fractional derivative on the temperature distribution is investigated.
    Źródło:
    Journal of Applied Mathematics and Computational Mechanics; 2021, 20, 4; 87--98
    2299-9965
    Pojawia się w:
    Journal of Applied Mathematics and Computational Mechanics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Extremal solutions to a coupled system of nonlinear fractional differential equations with Caputo fractional derivatives
    Autorzy:
    Derbazi, Choukri
    Baitiche, Zidane
    Benchohra, Mouffak
    Graef, John R.
    Powiązania:
    https://bibliotekanauki.pl/articles/2052385.pdf
    Data publikacji:
    2021
    Wydawca:
    Politechnika Rzeszowska im. Ignacego Łukasiewicza. Oficyna Wydawnicza
    Tematy:
    Caputo fractional derivative
    coupled system
    extremal solutions
    monotone iterative technique
    upper and lower solutions
    pochodna ułamkowa Caputo
    system sprzężony
    ekstremalne rozwiązania
    monotoniczna technika iteracyjna
    Opis:
    Using the well-known monotone iterative technique together with the method of upper and lower solutions, the authors investigate the existence of extremal solutions to a class of coupled systems of nonlinear fractional differential equations involving the $\psi$-Caputo derivative with initial conditions. As applications of this work, two illustrative examples are presented.
    Źródło:
    Journal of Mathematics and Applications; 2021, 44; 19-34
    1733-6775
    2300-9926
    Pojawia się w:
    Journal of Mathematics and Applications
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Controllability criteria for time-delay fractional systems with a retarded state
    Autorzy:
    Sikora, B.
    Powiązania:
    https://bibliotekanauki.pl/articles/330421.pdf
    Data publikacji:
    2016
    Wydawca:
    Uniwersytet Zielonogórski. Oficyna Wydawnicza
    Tematy:
    fractional dynamical systems
    controllability criteria
    delays in the state
    constraints
    pseudotransition matrix
    Caputo derivative
    układ dynamiczny ułamkowy
    sterowalność
    pochodna ułamkowego rzędu Caputo
    Opis:
    The paper is concerned with time-delay linear fractional systems with multiple delays in the state. A formula for the solution of the discussed systems is presented and derived using the Laplace transform. Definitions of relative controllability with and without constraints for linear fractional systems with delays in the state are formulated. Relative controllability, both with and without constraints imposed on control values, is discussed. Various types of necessary and sufficient conditions for relative controllability and relative U-controllability are established and proved. Numerical examples illustrate the obtained theoretical results.
    Źródło:
    International Journal of Applied Mathematics and Computer Science; 2016, 26, 3; 521-531
    1641-876X
    2083-8492
    Pojawia się w:
    International Journal of Applied Mathematics and Computer Science
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Nonlinear fractional differential equations with non-instantaneous impulses in Banach spaces
    Autorzy:
    Benchohra, M.
    Slimane, M.
    Powiązania:
    https://bibliotekanauki.pl/articles/2052426.pdf
    Data publikacji:
    2018
    Wydawca:
    Politechnika Rzeszowska im. Ignacego Łukasiewicza. Oficyna Wydawnicza
    Tematy:
    initial value problem
    impulses
    Caputo fractional derivative
    measure of noncompactness
    fixed point
    Banach space
    wartość początkowa
    impulsy
    pochodna ułamkowa Caputo
    miara braku zgodności
    punkt stały
    przestrzeń Banacha
    Opis:
    This paper is devoted to study the existence of solutions for a class of initial value problems for non-instantaneous impulsive fractional differential equations involving the Caputo fractional derivative in a Banach space. The arguments are based upon Monch's fixed point theorem and the technique of measures of noncompactness.
    Źródło:
    Journal of Mathematics and Applications; 2018, 41; 39-51
    1733-6775
    2300-9926
    Pojawia się w:
    Journal of Mathematics and Applications
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Analytical and numerical study for a fractional boundary value problem with a conformable fractional derivative of Caputo and its fractional integral
    Autorzy:
    Bekkouche, M. Moumen
    Guebbai, H.
    Powiązania:
    https://bibliotekanauki.pl/articles/1839786.pdf
    Data publikacji:
    2020
    Wydawca:
    Politechnika Częstochowska. Wydawnictwo Politechniki Częstochowskiej
    Tematy:
    fractional boundary value problem
    Volterra-Fredholm integral equation
    fractional derivative
    fractional integral
    równanie całkowe Volterry-Fredholma
    ułamkowa wartość brzegowa
    pochodna ułamkowa Caputo
    Opis:
    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.
    Źródło:
    Journal of Applied Mathematics and Computational Mechanics; 2020, 19, 2; 31-42
    2299-9965
    Pojawia się w:
    Journal of Applied Mathematics and Computational Mechanics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The Dirichlet problem for the time-fractional advection-diffusion equation in a half-space
    Autorzy:
    Povstenko, Y.
    Klekot, J.
    Powiązania:
    https://bibliotekanauki.pl/articles/122941.pdf
    Data publikacji:
    2015
    Wydawca:
    Politechnika Częstochowska. Wydawnictwo Politechniki Częstochowskiej
    Tematy:
    Caputo fractional derivative
    advection-diffusion equation
    Laplace integral transform
    Fourier sine transform
    Mittag-Leffler function
    pochodna rzędu ułamkowego Caputo
    funkcja Mittag-Lefflera
    Opis:
    The one-dimensional time-fractional advection-diffusion equation with the Caputo time derivative is considered in a half-space. The fundamental solution to the Dirichlet problem and the solution of the problem with constant boundary condition are obtained using the integral transform technique. The numerical results are illustrated graphically.
    Źródło:
    Journal of Applied Mathematics and Computational Mechanics; 2015, 14, 2; 73-83
    2299-9965
    Pojawia się w:
    Journal of Applied Mathematics and Computational Mechanics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Existence and Ulam-Hyers stability of the implicit fractional boundary value problem with ψ-Caputo fractional derivative
    Autorzy:
    Wahash, Hanan A.
    Abdo, Mohammed S
    Panchal, Satish K.
    Powiązania:
    https://bibliotekanauki.pl/articles/122800.pdf
    Data publikacji:
    2020
    Wydawca:
    Politechnika Częstochowska. Wydawnictwo Politechniki Częstochowskiej
    Tematy:
    fractional differential equations
    ψ-fractional integral and derivative
    existence and Ulam-Hyers stability
    fixed point theorem
    równania różniczkowe ułamkowe
    równania różniczkowe cząstkowe
    pochodna ułamkowa
    twierdzenie o punkcie stałym
    pochodna ułamkowa Caputo
    Opis:
    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.
    Źródło:
    Journal of Applied Mathematics and Computational Mechanics; 2020, 19, 1; 89-101
    2299-9965
    Pojawia się w:
    Journal of Applied Mathematics and Computational Mechanics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Existence and Ulam-Hyers stability of the implicit fractional boundary value problem with ψ-Caputo fractional derivative
    Autorzy:
    Wahash, Hanan A.
    Abdo, Mohammed S
    Panchal, Satish K.
    Powiązania:
    https://bibliotekanauki.pl/articles/1839794.pdf
    Data publikacji:
    2020
    Wydawca:
    Politechnika Częstochowska. Wydawnictwo Politechniki Częstochowskiej
    Tematy:
    fractional differential equations
    ψ-fractional integral and derivative
    existence and Ulam-Hyers stability
    fixed point theorem
    równania różniczkowe ułamkowe
    równania różniczkowe cząstkowe
    pochodna ułamkowa
    twierdzenie o punkcie stałym
    pochodna ułamkowa Caputo
    Opis:
    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.
    Źródło:
    Journal of Applied Mathematics and Computational Mechanics; 2020, 19, 1; 89-101
    2299-9965
    Pojawia się w:
    Journal of Applied Mathematics and Computational Mechanics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The fundamental solutions to the central symmetric time-fractional heat conduction equation with heat absorption
    Autorzy:
    Povstenko, Y.
    Klekot, J.
    Powiązania:
    https://bibliotekanauki.pl/articles/122778.pdf
    Data publikacji:
    2017
    Wydawca:
    Politechnika Częstochowska. Wydawnictwo Politechniki Częstochowskiej
    Tematy:
    non-Fourier heat conduction
    Caputo fractional derivative
    heat absorption
    Laplace integral transform
    Fourier transform
    Mittag-Leffler function
    przewodzenie ciepła
    pochodna ułamkowa Caputo
    absorpcja ciepła
    transformata Laplace'a
    transformata Fouriera
    funkcja Mittag-Lefflera
    Opis:
    The time-fractional heat conduction equation with heat absorption proportional to temperature is considered in the case of central symmetry. The fundamental solutions to the Cauchy problem and to the source problem are obtained using the integral transform technique. The numerical results are presented graphically.
    Źródło:
    Journal of Applied Mathematics and Computational Mechanics; 2017, 16, 2; 101-112
    2299-9965
    Pojawia się w:
    Journal of Applied Mathematics and Computational Mechanics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    A new modification of the reduced differential transform method for nonlinear fractional partial differential equations
    Autorzy:
    Khalouta, Ali
    Kadem, Abdelouahab
    Powiązania:
    https://bibliotekanauki.pl/articles/1839758.pdf
    Data publikacji:
    2020
    Wydawca:
    Politechnika Częstochowska. Wydawnictwo Politechniki Częstochowskiej
    Tematy:
    nonlinear fractional partial differential equations
    Caputo fractional derivative
    Shehu transform method
    reduced differential transform method
    approximate analytical solution
    nieliniowe równania różniczkowe cząstkowe ułamkowe
    pochodna ułamkowa Caputo
    metoda transformacji Shehu
    metoda transformacji różnicowej
    Opis:
    The objective of this study is to present a new modification of the reduced differential transform method (MRDTM) to find an approximate analytical solution of a certain class of nonlinear fractional partial differential equations in particular, nonlinear time-fractional wave-like equations with variable coefficients. This method is a combination of two different methods: the Shehu transform method and the reduced differential transform method. The advantage of the MRDTM is to find the solution without discretization, linearization or restrictive assumptions. Three different examples are presented to demonstrate the applicability and effectiveness of the MRDTM. The numerical results show that the proposed modification is very effective and simple for solving nonlinear fractional partial differential equations.
    Źródło:
    Journal of Applied Mathematics and Computational Mechanics; 2020, 19, 3; 45-58
    2299-9965
    Pojawia się w:
    Journal of Applied Mathematics and Computational Mechanics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Numerical solution of a fractional coupled system with the Caputo-Fabrizio fractional derivative
    Autorzy:
    Mansouri, Ikram
    Bekkouche, Mohammed Moumen
    Ahmed, Abdelaziz Azeb
    Powiązania:
    https://bibliotekanauki.pl/articles/2202061.pdf
    Data publikacji:
    2023
    Wydawca:
    Politechnika Częstochowska. Wydawnictwo Politechniki Częstochowskiej
    Tematy:
    Caputo-Fabrizio fractional derivative
    fractional integral
    coupled system
    fractional differential equation
    fixed point
    Adomian decomposition method
    pochodna ułamkowa Caputo-Fabrizio
    całkowanie ułamkowe
    system sprzężony
    ułamkowe równanie różniczkowe
    punkt stały
    metoda dekompozycji Adomiana
    Opis:
    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.
    Źródło:
    Journal of Applied Mathematics and Computational Mechanics; 2023, 22, 1; 46--56
    2299-9965
    Pojawia się w:
    Journal of Applied Mathematics and Computational Mechanics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
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