scholarly journals On Ulam Stability of a Generalized Delayed Differential Equation of Fractional Order

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Janusz Brzdęk ◽  
Nasrin Eghbali ◽  
Vida Kalvandi
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Exact Solutions ◽  
Fractional Order ◽  
Nonlinear Operator ◽  
Main Tool ◽  
Ulam Stability ◽  
Speed Of Convergence ◽  
Delayed Differential Equation ◽  
Pointwise Limit

AbstractWe investigate Ulam stability of a general delayed differential equation of a fractional order. We provide formulas showing how to generate the exact solutions of the equation using functions that satisfy it only approximately. Namely, the approximate solution $$\phi $$ ϕ generates the exact solution as a pointwise limit of the sequence $$\varLambda ^n\phi $$ Λ n ϕ with some integral (possibly, nonlinear) operator $$\varLambda $$ Λ . We estimate the speed of convergence and the distance between those approximate and exact solutions. Moreover, we provide some exemplary calculations, involving the Chebyshev and Bielecki norms and some semigauges, that could help to obtain reasonable outcomes for such estimations in some particular cases. The main tool is the Diaz–Margolis fixed point alternative.

New exact solutions for the Wick-type stochastic Zakharov–Kuznetsov equation for modelling waves on shallow water surfaces

2017 ◽  
Vol 25 (2) ◽  
Author(s):  
S. Saha Ray ◽  
S. Singh
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solution ◽  
Exact Solutions ◽  
White Noise ◽  
Shallow Water ◽  
Hermite Transform ◽  
New Exact Solutions ◽  
Kudryashov Method ◽  
Stochastic Solution

AbstractIn this article, an exact solution of the Wick-type stochastic Zakharov–Kuznetsov equation has been obtained by using the Kudryashov method. We have used the Hermite transform for transforming the Wick-type stochastic Zakharov–Kuznetsov equation into a deterministic partial differential equation. Also we have applied the inverse Hermite transform for obtaining a set of stochastic solution in the white noise space.

scholarly journals Unified Treatment of a Class of Spherically Symmetric Potentials: Quasi-Exact Solution

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
H. Panahi ◽  
M. Baradaran
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Representation Theory ◽  
Exact Solutions ◽  
Lie Algebra ◽  
Algebraic Approach ◽  
Spherically Symmetric ◽  
Exact Solvability ◽  
Enveloping Algebra ◽  
Basic Differential Equation

We investigate the Schrödinger equation for a class of spherically symmetric potentials in a simple and unified manner using the Lie algebraic approach within the framework of quasi-exact solvability. We illustrate that all models give rise to the same basic differential equation, which is expressible as an element of the universal enveloping algebra ofsl(2). Then, we obtain the general exact solutions of the problem by employing the representation theory ofsl(2)Lie algebra.

scholarly journals Scalar Singularly Perturbed Cauchy Problem for a Differential Equation of Fractional Order

2019 ◽  
Vol 7 (1) ◽  
Author(s):  
Burkhan Kalimbetov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cauchy Problem ◽  
Exact Solution ◽  
Small Parameter ◽  
Fractional Order ◽  
Regularization Method ◽  
Original Problem ◽  
Convergent Series ◽  
Singularly Perturbed

In this paper we consider initial problem for an ordinary differential equation of fractional order with a small parameter for the derivative. S.A. Lomov regularization method is used to construct an asymptotic approximate solution of the problem with accuracy up to any power of a small parameter. Using the computer mathematics system (CMS) Maple, a symbolic solution of the original problem is obtained, and solution schedules are constructed, depending on the initial data and various values of the small parameter. It is shown that the asymptotic solution presented in the form of a specific convergent series and the solution represented by the CMS Maple coincides with the exact solution of the original problem. 

scholarly journals EXISTENCE RESULTS AND STABILITY CRITERIA FOR ABC-FUZZY-VOLTERRA INTEGRO-DIFFERENTIAL EQUATION

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040048 ◽  
Author(s):  
HASIB KHAN ◽  
J. F. GOMEZ-AGUILAR ◽  
THABET ABDELJAWAD ◽  
AZIZ KHAN
Keyword(s):  
Differential Equation ◽  
Stability Analysis ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Ulam Stability ◽  
Stability Criteria ◽  
Science And Engineering

In the modeling of dynamical problems the fractional order integro-differential equations (IDEs) are very common in science and engineering. The scientists are developing different aspects of these models. The existence of solutions, stability analysis and numerical simulations are the most commonly studied aspects. There is no paper in literature describing the Hyers–Ulam stability (HU-stability) for fuzzy-fractional order models. Therefore, keeping the importance of the study, we consider the existence, uniqueness and HU-stability of a fractional order fuzzy-Volterra IDE.

On biological population model of fractional order

2016 ◽  
Vol 09 (05) ◽  
pp. 1650070 ◽  
Author(s):  
Syed Tauseef Mohyud-Din ◽  
Ayyaz Ali ◽  
Bandar Bin-Mohsin
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solutions ◽  
Fractional Order ◽  
Population Model ◽  
Numerical Data ◽  
Population Changes ◽  
Partial Differential ◽  
Biological Population ◽  
Fractional Pdes

In this paper, we extensively studied a mathematical model of biology. It helps us to understand the dynamical procedure of population changes in biological population model and provides valuable predictions. In this model, we establish a variety of exact solutions. To study the exact solutions, we used a fractional complex transform to convert the particular partial differential equation of fractional order into corresponding partial differential equation and modified exp-function method is implemented to investigate the nonlinear equation. Graphical demonstrations along with the numerical data reinforce the efficacy of the used procedure. The specified idea is very effective, unfailing, well-organized and pragmatic for fractional PDEs and could be protracted to further physical happenings.

Modified Auxiliary Differential Equation Method and Exact Solutions Generalized Schrödinger

2014 ◽  
Vol 513-517 ◽  
pp. 4470-4473 ◽  
Author(s):  
Lin Tian ◽  
Yu Ping Qin
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Exact Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Equation Method ◽  
Hyperbolic Function ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Solution ◽  
Auxiliary Differential Equation Method ◽  
Differential Equation Method

This paper describes a method on which modify auxiliary differential equation method by using this method for solving nonlinear partial differential equations and with aid of Maple Software ,we get the exact solution of the generalized schrödinger, including hyperbolic function solutions, trigonometric solution.

scholarly journals On finite series solutions of conformable time-fractional Cahn-Allen equation

2020 ◽  
Vol 9 (1) ◽  
pp. 194-200 ◽  
Author(s):  
Asim Zafar ◽  
Hadi Rezazadeh ◽  
Khalid K. Ali
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Exact Solutions ◽  
Fractional Order ◽  
Symbolic Computation ◽  
Nonlinear Ordinary Differential Equation ◽  
Hyperbolic Function ◽  
Series Solutions ◽  
Finite Series ◽  
New Exact Solutions

AbstractThe aim of this article is to derive new exact solutions of conformable time-fractional Cahn-Allen equation. We have achieved this aim by hyperbolic function and expa function methods with the aid of symbolic computation using Mathematica. This idea seems to be very easy to employ with reliable results. The time fractional Cahn-Allen equation is reduced to respective nonlinear ordinary differential equation of fractional order. Also, we have depicted graphically the constructed solutions.

scholarly journals "Approximate solution for the two-dimensional Volterra-Integro differential equation of fractional order by using Generalized Deferential Transform method "

2018 ◽  
Vol 5 (2) ◽  
pp. 1-9
Author(s):  
Mohammed G. S. AL-Safi ◽  
Wurood R. Abd AL- Hussein
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Fractional Order ◽  
Two Dimensional ◽  
Transform Method ◽  
Integro Differential Equation ◽  
Numerical Example ◽  
Differential Transform ◽  
Generalized Differential

"In this work, an efficient generalized differential transform method (GDTM) is proposed for solving the twodimensional Volterra-Integro differential equation (2-DVIDE) of fractional order. The results of the proposed method are compared with exact solution, a numerical example is considered for testing the accuracy and validity of this method."

scholarly journals Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations

2022 ◽  
Vol 6 (1) ◽  
pp. 24
Author(s):  
Muhammad Shakeel ◽  
Nehad Ali Shah ◽  
Jae Dong Chung
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solutions ◽  
Closed Form ◽  
Fractional Order ◽  
Soliton Solutions ◽  
Wave Transformation ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Closed Form Solutions

In this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fractional order partial differential equation into an integer order ordinary differential equation. Using the novel (G’/G2)-expansion method, several exact solutions with extra free parameters are found in the form of hyperbolic, trigonometric, and rational function solutions. When parameters are given appropriate values along with distinct values of fractional order α travelling wave solutions such as singular periodic waves, singular kink wave soliton solutions are formed which are forms of soliton solutions. Also, the solutions obtained by the proposed method depend on the value of the arbitrary parameters H. Previous results are re-derived when parameters are given special values. Furthermore, for numerical presentations in the form of 3D and 2D graphics, the commercial software Mathematica 10 is incorporated. The method is accurately depicted, and it provides extra general exact solutions.

scholarly journals Hyers-Ulam Stability of Nonhomogeneous Linear Differential Equations of Second Order

2009 ◽  
Vol 2009 ◽  
pp. 1-7 ◽  
Author(s):  
Yongjin Li ◽  
Yan Shen
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Differential Equations ◽  
Second Order ◽  
Linear Differential Equations ◽  
Ulam Stability ◽  
Linear Differential ◽  
The Stability

The aim of this paper is to prove the stability in the sense of Hyers-Ulam of differential equation of second ordery′′+p(x)y′+q(x)y+r(x)=0. That is, iffis an approximate solution of the equationy′′+p(x)y′+q(x)y+r(x)=0, then there exists an exact solution of the equation near tof.

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