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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 Solution of hybrid time fractional diffusion problem via weighted inner product
Cetinkaya Suleyman, Demir Ali
Journal of Applied Mathematics and Computational Mechanics
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2021
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Vol. 20, nr 2
17--27
EN
In this research, we discuss the construction of the analytic solution of homogenous initial boundary value problem including partial differential equations of fractional order. Since the homogenous initial boundary value problem involves the Hybrid fractional order derivative with various coefficients functions, it has classical initial and boundary conditions. By means of separation of the variables method and the inner product defined on L 2 [0,l], the solution is constructed in the form of a Fourier series including the bivariate Mittag-Leffler function. An illustrative example presents the applicability and influence of the separation of variables method on time fractional diffusion problems. Moreover, as the fractional order α tends to 1, the solution of the fractional diffusion problem tends to the solution of the diffusion problem which proves the accuracy of the solution.
3
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.
4
Content available remote Numerical approach to the controllability of fractional order impulsive differential equations
Kumar Avadhesh, Vats Ramesh K., Kumar Ankit, Chalishajar Dimplekumar N.
Demonstratio Mathematica
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2020
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Vol. 53, nr 1
193--207
EN
In this manuscript, a numerical approach for the stronger concept of exact controllability (total controllability) is provided. The proposed control problem is a nonlinear fractional differential equation of order α ∈ (1, 2] with non-instantaneous impulses in finite-dimensional spaces. Furthermore, the numerical controllability of an integro-differential equation is briefly discussed. The tool for studying includes the Laplace transform, the Mittag-Leffler matrix function and the iterative scheme. Finally, a few numerical illustrations are provided through MATLAB graphs.
5
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.
6
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.
7
Content available Matrix Mittag‑Leffler function in fractional systems and its computation
Matychyn I., Onyshchenko V.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2018
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Vol. 66, nr 4
495--500
EN
Matrix Mittag‑Leffler functions play a key role in numerous applications related to systems with fractional dynamics. That is why the methods for computing the matrix Mittag‑Leffler function are so important. The matrix Mittag‑Leffler function is a generalization of matrix exponential function. This implies that some of numerous existing methods for computing the matrix exponential can be adapted for matrix Mittag‑Leffler functions as well. Unfortunately, the technique of scaling and squaring, widely used in computing of the matrix exponential, cannot be applied to matrix Mittag‑Leffler functions, as the latter do not possess the semigroup property. Here we describe a method of computing the matrix Mittag‑Leffler function based on the Jordan canonical form representation. This method is implemented with Matlab code [1].
8
Content available Global convergence analysis of impulsive fractional order difference systems
He D., Xu L.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2018
|
Vol. 66, nr 5
599--604
EN
This paper is designed to deal with the convergence and stability analysis of impulsive Caputo fractional order difference systems. Using the Lyapunov functions, the Z-transforms of Caputo difference operators, and the properties of discrete Mittag-Leffler functions, some effective criteria are derived to guarantee the global convergence and the exponential stability of the addressed systems.
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 Dirichlet problem for the time-fractional advection-diffusion equation in a half-space
Povstenko Y., Klekot J.
Journal of Applied Mathematics and Computational Mechanics
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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.
12
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.
13
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.
14
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
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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.
15
Content available remote Generalization of a class of polynomials
Shukla A., Prajapati J.
Demonstratio Mathematica
|
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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