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1
Content available An iterative approach for solving fractional order Cauchy reaction-diffusion equations
Kumar Manoj
Journal of Applied Mathematics and Computational Mechanics
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2023
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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.
2
Content available Fractional heat conduction in a rectangular plate with bending moments
Warbhe Shrikant
Journal of Applied Mathematics and Computational Mechanics
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2020
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Vol. 19, nr 4
115--126
EN
In this research work, we consider a thin, simply supported rectangular plate defined as 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ c and determine the thermal stresses by using a thermal bending moment with the help of a time dependent fractional derivative. The constant temperature is prescribed on the surface y = 0 and other surfaces are maintained at zero temperature. A powerful technique of integral transform is used to find the analytical solution of initial-boundary value problem of a thin rectangular plate. The numerical result of temperature distribution, thermal deflection and thermal stress component are computed and represented graphically for a copper plate.
3
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
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2020
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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.
4
Content available Analysis of time-fractional heat transfer and its thermal deflection in a circular plate by a moving heat source
Thakare S., Warbhe M.
International Journal of Applied Mechanics and Engineering
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2020
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Vol. 25, no. 3
158--168
EN
Mathematical modeling of a thin circular plate has been made by considering a nonlocal Caputo type time fractional heat conduction equation of order […], by the action of a moving heat source. Physically convective heat exchange boundary conditions are applied at lower, upper and outer curved surface of the plate. Temperature distribution and thermal deflection has been investigated by a quasi-static approach in the context of fractional order heat conduction. The integral transformation technique is used to analyze the analytical solution to the problem. Numerical computation including the effect of the fractional order parameter has been done for temperature and deflection and illustrated graphically for an aluminum material.
5
Content available On LQ optimization problem subject to fractional order irregular singular systems
Muhafzan, Nazra Admi, Yulianti Lyra, Zulakmal, Revina Refi
Archives of Control Sciences
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2020
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Vol. 30, No. 4
745--756
EN
In this paper we discuss the linear quadratic (LQ) optimization problem subject to fractional order irregular singular systems. The aim of this paper is to find the control-state pairs satisfying the dynamic constraint of the form a fractional order irregular singular systems such that the LQ objective functional is minimized. The method of solving is to convert such LQ optimization into the standard fractional LQ optimization problem. Under some particularly conditions we find the solution of the problem under consideration.
6
Content available Accuracy estimation of the approximated Atangana-Baleanu operator
Oprzędkiewicz Krzysztof, Mitkowski Wojciech
Journal of Applied Mathematics and Computational Mechanics
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2019
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Vol. 18, nr 4
53--62
EN
In the paper, the accuracy analysis of the approximation of the Atangana-Baleanu (AB) operator is presented. The AB operator is the nonsingular kernel operator proposed by Atangana and Baleanu. It is obtained by replacing the exponential function in the Caputo-Fabrizio operator by the Mittag-Leffler function. The Laplace transform of the AB operator requires approximating the factor sa. This is done using the well-known Oustaloup Recursive Approximaion (ORA) approximation. The step and frequency responses of the approximation are compared to the analytical responses. As the cost function, the FIT function available in MATLAB was applied. Results of simulations show that the use of ORA allows us to obtain the accurate approximant of the AB operator.
7
Certain inequalities of meromorphic univalent functions associated with the Mittag-Leffler function
Aouf Mohamed K., Mostafa Adela O.
Journal of Applied Analysis
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2019
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Vol. 25, nr 2
173--178
EN
The purpose of this paper is to prove differential inequalities for meromorphic univalent functions by using a new operator associated with the Mittag-Leffler function.
8
Content available Modeling of IR proximity sensors with the use of interval mittag-leffler function
Oprzędkiewicz K., Garbacz M.
Journal of Automation Mobile Robotics and Intelligent Systems
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2017
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Vol. 11, No. 3
3--6
EN
In the paper new, interval models of IR proximity sensors are presented. The dependence between distance and signal from sensor is described with the use of exponential function and two parameter Mittag-Leffler function with interval parameters. Identification method for was also proposed. Results of experiments show, that two parameter Mittag-Leffler function most accurate describes a behaviour of proximity IR sensor, than exponential function.
9
Content available The fundamental solutions to the central symmetric time-fractional heat conduction equation with heat absorption
Povstenko Y., Klekot J.
Journal of Applied Mathematics and Computational Mechanics
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2017
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Vol. 16, nr 2
101--112
EN
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.
10
Content available The controllability of nonlinear implicit fractional delay dynamical systems
Joice Nirmala R., Balachandran K.
International Journal of Applied Mathematics and Computer Science
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2017
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Vol. 27, no. 3
501--513
EN
This paper is concerned with the controllability of nonlinear fractional delay dynamical systems with implicit fractional derivatives for multiple delays and distributed delays in control variables. Sufficient conditions are obtained by using the Darbo fixed point theorem. Further, examples are given to illustrate the theory.
11
Content available The Cauchy problem for the time-fractional advection diffusion equation in a layer
Povstenko Y., Klekot J.
Technical Sciences / University of Warmia and Mazury in Olsztyn
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2016
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nr 19(3)
231--244
EN
The time-fractional advection-diffusion equation with the Caputo time derivative is studied in a layer. The fundamental solution to the Cauchy problem is obtained using the integral transform technique. The logarithmicsingularity term is separated from the solution. Expressions amenable for numerical treatment are obtained. The numerical results are illustrated graphically.
12
Content available The Dirichlet problem for the time-fractional advection-diffusion equation in a half-space
Povstenko Y., Klekot J.
Journal of Applied Mathematics and Computational Mechanics
|
2015
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Vol. 14, nr 2
73--83
EN
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.
13
Content available Fundamental solution to the Cauchy problem for the time-fractional advection-diffusion equation
Povstenko Y., Klekot J.
Journal of Applied Mathematics and Computational Mechanics
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2014
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Vol. 13, nr 1
95--102
EN
The one-dimensional time-fractional advection-diffusion equation with the Caputo time derivative is considered. The fundamental solution to the Cauchy problem is obtained using the integral transform technique. The numerical results are illustrated graphically.
14
Content available Solutions to fractional diffusion-wave equation in a circular sector
Povstenko Y.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2013
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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 On the controllability of fractional dynamical systems
Balachandran K., Kokila J.
International Journal of Applied Mathematics and Computer Science
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2012
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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.
16
Content available remote Application of Adomian Decomposition Method and Variational Iteration Method for the solution of a fractional (arbitrary) order differential equation
Ray S. S., Bera R. K.
International Journal of Applied Mechanics and Engineering
|
2009
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Vol. 14, no 4
1169-1173
EN
In this paper, the Adomian decomposition method (ADM) and variational iteration method (VIM) are implemented to obtain an approximate solution to a fractional differential equation with an arbitrary order […]. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions to different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. The approximate solution obtained using the VIM is exactly the same and in good agreement as that obtained by using the ADM.
17
Content available remote Generalization of a class of polynomials
Shukla A., Prajapati J.
Demonstratio Mathematica
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2007
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Vol. 40, nr 4
819-826
EN
An attempt is made to investigate a class of polynomials defined in form of Rodrigues type formula and Mittag-Leffler Function. Some generating relations and finite summation formulae have also been obtained.
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