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1
Content available On the semi-analytic technique to deal with nonlinear fractional differential equations
Khirsariya Sagar R., Rao Snehal B.
Journal of Applied Mathematics and Computational Mechanics
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2023
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Vol. 22, nr 1
17--30
EN
In this article, we present a novel hybrid approach, by combining the Sawi transform with the homotopy perturbation method, to achieve the approximate and analytic solutions of nonlinear fractional differential equations (ODE as well as PDE) using the time-fractional Caputo derivative. The proposed algorithm is faster and simple compared to other iterative methods. The Sawi transform is used along with the homotopy perturbation method to accelerate the convergence of the series solution. The results discussed using calculations, graphs and tables are compatible for comparison with other known methods like the residual power series method and the exact solution which are discussed in the literature.
2
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
|
2023
|
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.
3
Content available Existence, uniqueness and parameter perturbation analysis results of a fractional integro-differential boundary problem
Zhang Nan, Zhang Lingling, Ngungu Mercy, Adeniji Adejimi, Addai Emmanuel
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2023
|
Vol. 71, nr 4
art. no. e145938
EN
In the formulation, the existence, uniqueness and stability of solutions and parameter perturbation analysis to Riemann-Liouville fractional differential equations with integro-differential boundary conditions are discussed by the properties of Green’s function and cone theory. First, some theorems have been established from standard fixed point theorems in a proper Banach space to guarantee the existence and uniqueness of positive solution. Moreover, we discuss the Hyers-Ulam stability and parameter perturbation analysis, which examines the stability of solutions in the presence of small changes in the equation main parameters, that is, the derivative order η, the integral order β of the boundary condition, the boundary parameter ξ , and the boundary value τ. As an application, we present a concrete example to demonstrate the accuracy and usefulness of the proposed work. By using numerical simulation, we obtain the figure of unique solution and change trend figure of the unique solution with small disturbances to occur in different kinds of parameters.
4
Content available remote Positive solutions for fractional differential equation at resonance under integral boundary conditions
Wang Youyu, Huang Yue, Li Xianfei
Demonstratio Mathematica
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2022
|
Vol. 55, nr 1
238--253
EN
By using the theory of fixed point index and spectral theory of linear operators, we study the existence of positive solutions for Riemann-Liouville fractional differential equations at resonance. Our approach will provide some new ideas for the study of this kind of problem.
5
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
|
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.
6
Content available remote Existence and uniqueness of mild solutions for a fractional differential equation under Sturm-Liouville boundary conditions when the data function is of Lipschitzian type
Harjani Jackie, López Belen, Sadarangani Kishin
Demonstratio Mathematica
|
2020
|
Vol. 53, nr 1
167--173
EN
In this article, we present a sufficient condition about the length of the interval for the existence and uniqueness of mild solutions to a fractional boundary value problem with Sturm-Liouville boundary conditions when the data function is of Lipschitzian type. Moreover, we present an application of our result to the eigenvalues problem and its connection with a Lyapunov-type inequality.
7
Some fractional differential equations involving generalized hypergeometric functions
Agarwal Praveen, Qi Feng, Chand Mehar, Singh Gurmej
Journal of Applied Analysis
|
2019
|
Vol. 25, nr 1
37--44
EN
In the paper, using the generalized Marichev-Saigo-Maeda fractional operator, the authors establish some fractional differential equations associated with generalized hypergeometric functions and, by employing integral transforms, present some image formulas of the resulting equations.
8
Content available remote Positive solution for nonlinear fractional differential equation with integral boundary value condition
Zatla Y. T., Daudi-Merzagui N.
Fasciculi Mathematici
|
2018
|
Nr 61
163--177
EN
In this paper, we consider a fractional differential equation, with integral boundary conditions, when the nonlinearities are sign changing. Our approach is based on the Krasnoselskii theorem in double cones. We generalize some recent results.
9
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
|
2018
|
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].
10
Content available Positive solutions of a singular fractional boundary value problem with a fractional boundary condition
Lyons J. W., Neugebauer J. T.
Opuscula Mathematica
|
2017
|
Vol. 37, no. 3
421--434
EN
For α ∈ (1,2] the singular fractional boundary value problem [formula] satisfying the boundary conditions [formula] where β ∈ (0,α - 1], μ ∈ (0,α - 1], and [formula] are Riemann-Liouville derivatives of order α, β, and μ respectively, is considered. Here ƒ satisfies a local Carathéodory condition, and ƒ (t, x, y) may be singular at the value 0 in its space variable x. Using regularization and sequential techniques and Krasnosel’skii’s fixed point theorem, it is shown this boundary value problem has a positive solution. An example is given.
11
Content available Fractional boundary value problems on the half line
Frioui A., Guezane-Lakoud A., Khaldi R.
Opuscula Mathematica
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2017
|
Vol. 37, no. 2
265--280
EN
In this paper, we focus on the solvability of a fractional boundary value problem at resonance on an unbounded interval. By constructing suitable operators, we establish an existence theorem upon the coincidence degree theory of Mawhin. The obtained results are illustrated by an example.
12
Content available Existence of three solutions for impulsive nonlinear fractional boundary value problems
Heidarkhani S., Ferrara M., Caristi G., Salari A.
Opuscula Mathematica
|
2017
|
Vol. 37, no. 2
281--301
EN
In this work we present new criteria on the existence of three solutions for a class of impulsive nonlinear fractional boundary-value problems depending on two parameters. We use variational methods for smooth functionals defined on reflexive Banach spaces in order to achieve our results.
13
Content available New perspectives of analog and digital simulations of fractional order systems
Charef A., Charef M., Djouambi A., Voda A.
Archives of Control Sciences
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2017
|
Vol. 27, no. 1
91--118
EN
In the recent decades, fractional order systems have been found to be useful in many areas of physics and engineering. Hence, their efficient and accurate analog and digital simulations and numerical calculations have become very important especially in the fields of fractional control, fractional signal processing and fractional system identification. In this article, new analog and digital simulations and numerical calculations perspectives of fractional systems are considered. The main feature of this work is the introduction of an adjustable fractional order structure of the fractional integrator to facilitate and improve the simulations of the fractional order systems as well as the numerical resolution of the linear fractional order differential equations. First, the basic ideas of the proposed adjustable fractional order structure of the fractional integrator are presented. Then, the analog and digital simulations techniques of the fractional order systems and the numerical resolution of the linear fractional order differential equation are exposed. Illustrative examples of each step of this work are presented to show the effectiveness and the efficiency of the proposed fractional order systems analog and digital simulations and implementations techniques
14
Content available remote Analytical Solutions of Classical and Fractional KP-Burger Equation and Coupled KdV Equation
Ghosh U., Sarkar S., Das S.
Computational Methods in Science and Technology
|
2016
|
Vol. 22, No. 3
143--152
EN
Development of new analytical and numerical methods and their applications for solving non-linear partial differential equations (both classical and fractional) is a rising field of Applied Mathematical research because of its applications in Physical, Biological and Social Sciences. In this paper we have used a generalized Tanh method to find the exact solution of KP-Burger equation and coupled KdV equation. The fractional Sub-equation method has been used to find the solution of fractional KP-Burger equation and fractional coupled KdV equations. The exact solution obtained by the fractional sub-equation method reduces to classical solution when the order of fractional derivative tends to one. Finally numerical simulation has been done. The numerical simulation justifies that the solutions of two fractional differential equations reduce to shock solution for KP-Burger equation and soliton solution for coupled KdV equations when the order of derivative tends to one.
15
Content available Analytical solution of the time fractional Fokker-Planck equation
Sutradhar T., Datta B. K., Bera R. K.
International Journal of Applied Mechanics and Engineering
|
2014
|
Vol. 19, no. 2
435--440
EN
A nonperturbative approximate analytic solution is derived for the time fractional Fokker-Planck (F-P) equation by using Adomian’s Decomposition Method (ADM). The solution is expressed in terms of Mittag-Leffler function. The present method performs extremely well in terms of accuracy, efficiency and simplicity.
16
Content available Numerical Solution of Fractional Neutron Point Kinetics Model in Nuclear Reactor
Nowak T. K., Duzinkiewicz K., Piotrowski R.
Archives of Control Sciences
|
2014
|
Vol. 24, no. 2
129--154
EN
This paper presents results concerning solutions of the fractional neutron point kinetics model for a nuclear reactor. Proposed model consists of a bilinear system of fractional and ordinary differential equations. Three methods to solve the model are presented and compared. The first one entails application of discrete Grünwald-Letnikov definition of the fractional derivative in the model. Second involves building an analog scheme in the FOMCON Toolbox in MATLAB environment. Third is the method proposed by Edwards. The impact of selected parameters on the model’s response was examined. The results for typical input were discussed and compared.
17
Content available remote Analytic solution of time fractional nonlinear dynamic system by modified decomposition method
Sutradhar T., Datta B. K., Bera R. K.
International Journal of Applied Mechanics and Engineering
|
2012
|
Vol. 17, no. 4
1327-1337
EN
Analytical and numerical results are reported for an analytical approximate solution of a nonlinear dynamic system containing fractional derivative by a modified decomposition method. Comparison with the exact and numerical solution shows that the present method performs extremely well in terms of accuracy, efficiency and simplicity.
18
Existence and uniqueness results of some fractional boundary value problem
Seddiki H., Mazouzi S.
Journal of Applied Analysis
|
2011
|
Vol. 17, nr 1
91-103
EN
We establish in this paper some existence results of a solution to a boundary value problem of fractional differential equation. We obtain two results, the first one by the Banach fixed point theorem and the second by a nonlinear alternative of Leray-Schauder type.
19
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
|
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.
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