scholarly journals On the Singular Perturbations for Fractional Differential Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Abdon Atangana
Keyword(s):  
Differential Equation ◽  
Singular Perturbation ◽  
Variational Iteration Method ◽  
Regular Perturbation ◽  
Homotopy Decomposition ◽  
New Development ◽  
The Laplace Transform ◽  
Nonhomogeneous Differential Equation ◽  
Fractional Differential ◽  
Linear Differential

The goal of this paper is to examine the possible extension of the singular perturbation differential equation to the concept of fractional order derivative. To achieve this, we presented a review of the concept of fractional calculus. We make use of the Laplace transform operator to derive exact solution of singular perturbation fractional linear differential equations. We make use of the methodology of three analytical methods to present exact and approximate solution of the singular perturbation fractional, nonlinear, nonhomogeneous differential equation. These methods are including the regular perturbation method, the new development of the variational iteration method, and the homotopy decomposition method.

scholarly journals Comparison Methods for Solving Non-Linear Sturm–Liouville Eigenvalues Problems

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1179 ◽  
Author(s):  
Kamel Al-Khaled ◽  
Ashwaq Hazaimeh
Keyword(s):  
Variational Iteration Method ◽  
The Other ◽  
Sinc Method ◽  
Linear System Of Equations ◽  
Non Linear ◽  
The Laplace Transform ◽  
Sturm Liouville ◽  
Linear Differential ◽  
Non Linear System ◽  
Adomian’S Polynomials

In this paper, we present a comparative study between Sinc–Galerkin method and a modified version of the variational iteration method (VIM) to solve non-linear Sturm–Liouville eigenvalue problem. In the Sinc method, the problem under consideration was converted from a non-linear differential equation to a non-linear system of equations, that we were able to solve it via the use of some iterative techniques, like Newton’s method. The other method under consideration is the VIM, where the VIM has been modified through the use of the Laplace transform, and another effective modification has also been made to the VIM by replacing the non-linear term in the integral equation resulting from the use of the well-known VIM with the Adomian’s polynomials. In order to explain the advantages of each method over the other, several issues have been studied, including one that has an application in the field of spectral theory. The results in solutions to these problems, which were included in tables, showed that the improved VIM is better than the Sinc method, while the Sinc method addresses some advantages over the VIM when dealing with singular problems.

scholarly journals Exact Solution of Ambartsumian Delay Differential Equation and Comparison with Daftardar-Gejji and Jafari Approximate Method

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 331 ◽  
Author(s):  
Huda Bakodah ◽  
Abdelhalim Ebaid
Keyword(s):  
Differential Equation ◽  
Closed Form ◽  
Closed Form Solution ◽  
Approximate Solutions ◽  
Form Solution ◽  
Proportional Delay ◽  
Delay Term ◽  
The Laplace Transform ◽  
Linear Differential ◽  
Delay Differential

The Ambartsumian equation, a linear differential equation involving a proportional delay term, is used in the theory of surface brightness in the Milky Way. In this paper, the Laplace-transform was first applied to this equation, and then the decomposition method was implemented to establish a closed-form solution. The present closed-form solution is reported for the first time for the Ambartsumian equation. Numerically, the calculations have demonstrated a rapid rate of convergence of the obtained approximate solutions, which are displayed in several graphs. It has also been shown that only a few terms of the new approximate solution were sufficient to achieve extremely accurate numerical results. Furthermore, comparisons of the present results with the existing methods in the literature were introduced.

scholarly journals A fractional differential equation with multi-point strip boundary condition involving the Caputo fractional derivative and its Hyers–Ulam stability

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mehboob Alam ◽  
Akbar Zada ◽  
Ioan-Lucian Popa ◽  
Alireza Kheiryan ◽  
Shahram Rezapour ◽  
...  
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Caputo Fractional Derivative ◽  
Integral Boundary Conditions ◽  
Ulam Stability ◽  
Integral Boundary ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Fractional Differential

AbstractIn this work, we investigate the existence, uniqueness, and stability of fractional differential equation with multi-point integral boundary conditions involving the Caputo fractional derivative. By utilizing the Laplace transform technique, the existence of solution is accomplished. By applying the Bielecki-norm and the classical fixed point theorem, the Ulam stability results of the studied system are presented. An illustrative example is provided at the last part to validate all our obtained theoretical results.

scholarly journals On the Fractional Nagumo Equation with Nonlinear Diffusion and Convection

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Abdon Atangana ◽  
Suares Clovis Oukouomi Noutchie
Keyword(s):  
Approximate Solution ◽  
Nonlinear Diffusion ◽  
Decomposition Method ◽  
Integral Transform ◽  
Analytical Techniques ◽  
Variational Iteration Method ◽  
Homotopy Decomposition ◽  
New Development ◽  
Nagumo Equation ◽  
Diffusion And Convection

We presented the Nagumo equation using the concept of fractional calculus. With the help of two analytical techniques including the homotopy decomposition method (HDM) and the new development of variational iteration method (NDVIM), we derived an approximate solution. Both methods use a basic idea of integral transform and are very simple to be used.

Analysis on the time and frequency domain for the RC electric circuit of fractional order

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Manule Guía ◽  
Francisco Gómez ◽  
Juan Rosales
Keyword(s):  
Differential Equation ◽  
Frequency Domain ◽  
Time Domain ◽  
Cutoff Frequency ◽  
Electric Circuit ◽  
Electrical Circuit ◽  
High Frequencies ◽  
The Laplace Transform ◽  
Fractional Differential ◽  
The Time Domain

AbstractThis paper provides an analysis in the time and frequency domain of an RC electrical circuit described by a fractional differential equation of the order 0 < α≤ 1. We use the Laplace transform of the fractional derivative in the Caputo sense. In the time domain we emphasize on the delay, rise and settling times, while in the frequency domain the interest is in the cutoff frequency, the bandwidth and the asymptotes in low and high frequencies. All these quantities depend on the order of differential equation.

scholarly journals Positive Solutions for a Weakly Singular Hadamard-Type Fractional Differential Equation with Changing-Sign Nonlinearity

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Xinguang Zhang ◽  
Lixin Yu ◽  
Jiqiang Jiang ◽  
Yonghong Wu ◽  
Yujun Cui
Keyword(s):  
Differential Equation ◽  
Singular Perturbation ◽  
Positive Solutions ◽  
Periodic Boundary ◽  
Existence Of Solutions ◽  
Numerical Techniques ◽  
Periodic Boundary Value Problems ◽  
Weakly Singular ◽  
Existence Of Positive Solutions ◽  
Fractional Differential

In this paper, we focus on the existence of positive solutions for a class of weakly singular Hadamard-type fractional mixed periodic boundary value problems with a changing-sign singular perturbation. By using nonlinear analysis methods combining with some numerical techniques, we further discuss the effect of the perturbed term for the existence of solutions of the problem under the positive, negative, and changing-sign cases. The interesting points are that the nonlinearity can be singular at the second and third variables and be changing-sign.

scholarly journals Solusi Persamaan Diferensial Fraksional Riccati Menggunakan Adomian Decomposition Method dan Variational Iteration Method

Matematika ◽  
2019 ◽  
Vol 18 (1) ◽  
Author(s):  
Muhamad Deni Johansyah ◽  
Herlina Napitupulu ◽  
Erwin Harahap ◽  
Ira Sumiati ◽  
Asep K. Supriatna
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Iteration Method ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
Adomian Decomposition ◽  
Fractional Differential

Abstrak. Pada umumnya orde dari persamaan diferensial adalah bilangan asli, namun orde pada persamaan diferensial dapat dibentuk menjadi orde pecahan yang disebut persamaan diferensial fraksional. Paper ini membahas persamaan diferensial fraksional Riccati dengan orde diantara nol dan satu, dan koefisien konstan. Metode numerik yang digunakan untuk mendapatkan solusi dari persamaan diferensial fraksional Riccati adalah Adomian Decomposition Method (ADM) dan Variational Iteration Method (VIM). Tujuan dari paper ini adalah untuk memperluas penerapan ADM dan VIM dalam menyelesaikan persamaan diferensial fraksional Riccati nonlinear dengan turunan Caputo. Perbandingan solusi yang diperoleh menunjukkan bahwa VIM adalah metode yang lebih sederhana untuk mencari solusi persamaan diferensial fraksional Riccati nonlinier dengan orde antara nol dan satu, kemudian hasil yang diperoleh disajikan dalam bentuk grafik.Kata kunci: diferensial, fraksional, riccati, adomian dekomposisiThe solution of Riccati Fractional Differential Equation using Adomian Decomposition methodAbstract. Generally, the order of differential equations is a natural numbers, but this order can be formed into fractional, called as fractional differential equations.  In this paper, the Riccati fractional differential equations with order between zero and one, and constant coefficient is discussed.  The numerical methods used to obtain solutions from Riccati fractional differential equations are the Adomian Decomposition Method (ADM) and Variational Iteration Method (VIM).  The aim of this paper is to expand the application of ADM and VIM in solving nonlinear Riccati fractional differential equations with Caputo derivatives.  The comparison of the obtained solutions shows that VIM is simpler method for finding solutions to Riccati nonlinear fractional differential equations with order between zero and one. The obtained results are presented graphically.Keywords: riccati, fractional, differential, adomian, decomposition

Some solutions of the nonhomogeneous Bagley–Torvik equation

2019 ◽  
Vol 10 (01) ◽  
pp. 1941002 ◽  
Author(s):  
Temirkhan S. Aleroev ◽  
Sergey Erokhin
Keyword(s):  
Differential Equation ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Numerical Solutions ◽  
Experimental Information ◽  
Polymer Concrete ◽  
The Laplace Transform ◽  
Nonhomogeneous Differential Equation ◽  
The Right ◽  
Derivative Order

In this study, nonhomogeneous differential equation of the second order is considered, which contains fractional derivative (Bagley–Torvik equation), where the derivative order ranges within 1 and 2. This equation is applied in mechanics of oscillation processes. To study the equation, we use the Laplace transform, which allows us to obtain an image of the solution in an explicit form. Two typical kinds of functions of the right-hand side of the equation are considered. Numerical solutions are constructed for each of them. The solutions obtained are compared with experimental information on polymer concrete samples. The comparison allows for the conclusion about the adequacy of the numerical and analytical solutions to the nonhomogeneous Bagley–Torvik equation.

scholarly journals Numerical approach to the controllability of fractional order impulsive differential equations

2020 ◽  
Vol 53 (1) ◽  
pp. 193-207
Author(s):  
Avadhesh Kumar ◽  
Ramesh K. Vats ◽  
Ankit Kumar ◽  
Dimplekumar N. Chalishajar
Keyword(s):  
Differential Equation ◽  
Iterative Scheme ◽  
Impulsive Differential Equations ◽  
Exact Controllability ◽  
Numerical Approach ◽  
Nonlinear Fractional Differential Equation ◽  
Finite Dimensional ◽  
The Laplace Transform ◽  
Fractional Differential ◽  
Total Controllability

AbstractIn 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 \alpha \in (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.

scholarly journals On Solutions of the Initial Value Problem for the Three-Term Fractional Differential Equation with Caputo Derivatives

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1355
Author(s):  
Stanisław Kukla ◽  
Urszula Siedlecka
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Double Series ◽  
Pochhammer Symbol ◽  
Generalized Hypergeometric Functions ◽  
Caputo Derivatives ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Fractional Differential ◽  
Finite Sum

In this paper, two forms of an exact solution and an analytical–numerical solution of the three-term fractional differential equation with the Caputo derivatives are presented. The Prabhakar function and an asymptotic expansion are utilized to present the double series solution. Using properties of the Pochhammer symbol, a solution is obtained in the form of an infinite series of generalized hypergeometric functions. As an alternative for the series solutions of the fractional commensurate equation, a solution received by an analytical–numerical method based on the Laplace transform technique is proposed. This solution is obtained in the form of a finite sum of the Mittag-Leffler type functions. Numerical examples and a discussion are presented.

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