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1
Content available Solution of fractional integro-differential equations using least squares and shifted legendre methods
Mahdy Amr M.S., Nagdy Abbas S., Mohamed Doaa Sh.
Journal of Applied Mathematics and Computational Mechanics
|
2024
|
Vol. 23, nr 1
59-70
EN
In recent years, fractional calculus (FC) has filled in a hole in traditional calculus in terms of the effect of memory, which lets us know things about the past and present and guess what will happen in the future. It is very important to have this function, especially when studying biological models and integral equations. This paper introduces developed mathematical strategies for understanding a direct arrangement of fractional integro-differential equations (FIDEs). We have presented the least squares procedure and the Legendre strategy for discussing FIDEs. We have given the form of the Caputo concept fractional order operator and the properties. We have presented the properties of the shifted Legendre polynomials. We have shown the steps of the technique to display the solution. Some test examples are given to exhibit the precision and relevance of the introduced strategies. Mathematical outcomes show that this methodology is a comparison between the exact solution and the methods suggested. To show the theoretical results gained, the simulation of suggested strategies is given in eye-catching figures and tables. Program Mathematica 12 was used to get all of the results from the techniques that were shown.
2
Content available Remarks on the Caputo fractional derivative
Sikora Beata
MINUT Matematyka i Informatyka na Uczelniach Technicznych
|
2023
|
Nr 5
76-84
EN
The purpose of the paper is to familiarise the reader with the concept of the Caputo fractional derivative. The definition and basic properties of the Caputo derivative are given. Formulas for the derivatives of selected functions are derived. Examples of calculating the derivatives of basic functions are presented. The paper also contains a number of self-solving exercises, with answers.
3
Content available An iterative approach for solving fractional order Cauchy reaction-diffusion equations
Kumar Manoj
Journal of Applied Mathematics and Computational Mechanics
|
2023
|
Vol. 22, nr 3
19--32
EN
Reaction-diffusion equations are vitally important due to their role in developing sturdy models in various scientific fields. In the present work, we address an algorithm of the Daftardar-Gejji and Jafari method for solving the nonlinear functional equations of the form ψ = f +L(ψ) + N(ψ). Further, we employ this algorithm to solve Caputo derivative-based time-fractional Cauchy reaction-diffusion equations. We obtain solutions in a series form that converges to a closed form. Furthermore, we perform numerical simulations for the various values of the order of fractional derivatives. The computational procedure of the proposed algorithm is not burdensome. However, it is time-efficient and can easily be implemented using a computer algebra system.
4
Content available Results on the controllability of Caputo’s fractional descriptor systems with constant delays
Beata Sikora
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2023
|
Vol. 71, nr 4
art. no. e146287
EN
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.
5
Content available Time-fractional heat conduction in a finite composite cylinder with heat source
Kukla Stanisław, Siedlecka Urszula
Journal of Applied Mathematics and Computational Mechanics
|
2020
|
Vol. 19, nr 2
85--94
EN
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.
6
Content available Application of fractional order theory of thermoelasticity in an elliptical disk and associated thermal stresses
Thakare S., Panke Y., Hadke K.
International Journal of Applied Mechanics and Engineering
|
2020
|
Vol. 25, no. 3
169--180
EN
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.
7
Content available Analytical solution of a fractional model of fluid flow through narrowing system in terms of Mittag-Leffler function
Choudhary A., Kumar D., Singh J.
International Journal of Applied Mechanics and Engineering
|
2020
|
Vol. 25, no. 1
1--11
EN
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.
8
Content available Heat conduction in a finite medium using the fractional single-phase-lag model
Siedlecka Urszula
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2019
|
Vol. 67, nr 2
401--407
EN
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.
9
Content available The Caputo vs. Caputo-Fabrizio operators in modeling of heat transfer process
Oprzędkiewicz K., Mitkowski W., Gawin E., Dziedzic K.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2018
|
Vol. 66, nr 4
501--507
EN
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.
10
Content available Non integer order, state space model of heat transfer process using Caputo-Fabrizio operator
Oprzędkiewicz K.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2018
|
Vol. 66, nr 3
249--255
EN
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.
11
Content available Harmonic impact on the surface of a half-plane in the framework of Time-Fractional Heat Conduction
Povstenko Y.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
|
2016
|
Vol. 21
85--92
EN
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.
12
Content available Some new existence results and stability concepts for fractional partial random differential equations
Abbas S., Benchohra M., Darwish M. A.
Journal of Mathematics and Applications
|
2016
|
Vol. 39
5--22
EN
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.
13
Content available Fractional heat conduction in multilayer spherical bodies
Kukla S., Siedlecka U.
Journal of Applied Mathematics and Computational Mechanics
|
2016
|
Vol. 15, nr 4
83--92
EN
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.
14
Content available Solutions to fractional diffusion-wave equation in a circular sector
Povstenko Y.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
|
2013
|
Vol. 18
41--54
EN
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.
15
Content available Different kinds of boundary conditions for time-fractional heat conduction equation
Povstenko Y.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
|
2011
|
Vol. 16
61--66
EN
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.
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