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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
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2023
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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
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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
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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 Effects of the porous boundary and inclined magnetic field on MHD flow in a rectangular duct
Chutia Muhim
Journal of Applied Mathematics and Computational Mechanics
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2020
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Vol. 19, nr 4
33--44
EN
In this work, a steady two dimensional MHD flow of a viscous incompressible fluid through a rectangular duct under the action of an inclined magnetic field with a porous boundary has been investigated. The coupled partial differential equations are transformed into a system of algebraic equations using the finite difference method and are then solved simultaneously using the Gauss Seidal iteration method by programming in Matlab software. Numerical solutions for velocity, induced magnetic field and current density lines are obtained and analyzed for different values of dimensionless parameters namely suction/injection parameter (S), Hartmann number (M) and inclination angle (θ) and are presented graphically.
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
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2020
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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
Content available The Generalized Differential Transform Method for solution of a free vibration linear differential equation with fractional derivative damping
Das Deepanjan
Journal of Applied Mathematics and Computational Mechanics
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2019
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Vol. 18, nr 2
19--29
EN
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.
8
Some fractional differential equations involving generalized hypergeometric functions
Agarwal Praveen, Qi Feng, Chand Mehar, Singh Gurmej
Journal of Applied Analysis
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2019
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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.
9
Existence and uniqueness results of some fractional boundary value problem
Seddiki H., Mazouzi S.
Journal of Applied Analysis
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2011
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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.
10
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.
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