scholarly journals On the Solution of an Imprecisely Defined Nonlinear Time-Fractional Dynamical Model of Marriage

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 689 ◽  
Author(s):  
Rajarama Mohan Jena ◽  
Snehashish Chakraverty ◽  
Dumitru Baleanu
Keyword(s):  
Dynamical System ◽  
Fuzzy Numbers ◽  
Initial Conditions ◽  
Coupled System ◽  
Dynamical Model ◽  
Transform Method ◽  
System Parameters ◽  
Nonlinear Coupled System ◽  
Differential Transform ◽  
Fractional Dynamical System

The present paper investigates the numerical solution of an imprecisely defined nonlinear coupled time-fractional dynamical model of marriage (FDMM). Uncertainties are assumed to exist in the dynamical system parameters, as well as in the initial conditions that are formulated by triangular normalized fuzzy sets. The corresponding fractional dynamical system has first been converted to an interval-based fuzzy nonlinear coupled system with the help of a single-parametric gamma-cut form. Further, the double-parametric form (DPF) of fuzzy numbers has been used to handle the uncertainty. The fractional reduced differential transform method (FRDTM) has been applied to this transformed DPF system for obtaining the approximate solution of the FDMM. Validation of this method was ensured by comparing it with other methods taking the gamma-cut as being equal to one.

A robust technique based solution of time-fractional seventh-order Sawada–Kotera and Lax’s KdV equations

2021 ◽  
pp. 2150265
Author(s):  
Rajarama Mohan Jena ◽  
Snehashish Chakraverty ◽  
Dumitru Baleanu ◽  
Waleed Adel ◽  
Hadi Rezazadeh
Keyword(s):  
Convergence Analysis ◽  
Series Solution ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
Absolute Error ◽  
Transform Method ◽  
Cpu Time ◽  
Kdv Equations ◽  
Differential Transform ◽  
Seventh Order

In this paper, the fractional reduced differential transform method (FRDTM) is used to obtain the series solution of time-fractional seventh-order Sawada–Kotera (SSK) and Lax’s KdV (LKdV) equations under initial conditions (ICs). Here, the fractional derivatives are considered in the Caputo sense. The results obtained are contrasted with other previous techniques for a specific case, [Formula: see text] revealing that the presented solutions agree with the existing solutions. Further, convergence analysis of the present results with an increasing number of terms of the solution and absolute error has also been studied. The behavior of the FRDTM solution and the effects on different values [Formula: see text] are illustrated graphically. Also, CPU-time taken to obtain the solutions of the title problems using FRDTM has been demonstrated.

scholarly journals SYMBOLIC DYNAMICAL MODEL OF AVERAGE QUEUE SIZE OF RANDOM EARLY DETECTION ALGORITHM

2010 ◽  
Vol 20 (05) ◽  
pp. 1415-1437 ◽  
Author(s):  
CHARLOTTE YUK-FAN HO ◽  
BINGO WING-KUEN LING ◽  
HERBERT H. C. IU
Keyword(s):  
Fixed Point ◽  
Early Detection ◽  
Initial Conditions ◽  
Detection Algorithm ◽  
Dynamical Model ◽  
Random Early Detection ◽  
Queue Size ◽  
System Parameters ◽  
Internet Networks

In this paper, a symbolic dynamical model of the average queue size of the random early detection (RED) algorithm is proposed. The conditions for both the system parameters and the initial conditions that the average queue size of the RED algorithm would converge to a fixed point are derived. These results are useful for network engineers to design both the system parameters and the initial conditions so that internet networks can achieve a good performance.

scholarly journals Analytical Solution of Two-Dimensional Sine-Gordon Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Alemayehu Tamirie Deresse ◽  
Yesuf Obsie Mussa ◽  
Ademe Kebede Gizaw
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Test Problems ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Science And Engineering ◽  
Transform Method ◽  
Differential Transform ◽  
Good Agreement

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear sine-Gordon equations subject to appropriate initial conditions. Some lemmas which help us to solve the governing problem using the proposed method are proved. This scheme has the advantage of generating an analytical approximate solution or exact solution in a convergent power series form with conveniently determinable components. The method considers the use of the appropriate initial conditions and finds the solution without any discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the proposed method are demonstrated by four of our test problems, and solution behavior of the test problems is presented using tables and graphs. Further, the numerical results are found to be in a good agreement with the exact solutions and the numerical solutions that are available in literature. We have showed the convergence of the proposed method. Also, the obtained results reveal that the introduced method is promising for solving other types of nonlinear partial differential equations (NLPDEs) in the fields of science and engineering.

scholarly journals Approximate solutions and Hyers–Ulam stability for a system of the coupled fractional thermostat control model via the generalized differential transform

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sina Etemad ◽  
Brahim Tellab ◽  
Jehad Alzabut ◽  
Shahram Rezapour ◽  
Mohamed Ibrahim Abbas
Keyword(s):  
Fixed Point ◽  
Fixed Point Theory ◽  
Numerical Technique ◽  
Coupled System ◽  
Control Model ◽  
Ulam Stability ◽  
Transform Method ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Differential Transform ◽  
Generalized Differential

AbstractIn this paper, we consider a new coupled system of fractional boundary value problems based on the thermostat control model. With the help of fixed point theory, we investigate the existence criterion of the solution to the given coupled system. This property is proved by using the Krasnoselskii’s fixed point theorem and its uniqueness is proved via the Banach principle for contractions. Further, the Hyers–Ulam stability of solutions is investigated. Then, we find the approximate solution of the coupled fractional thermostat control system by using a numerical technique called the generalized differential transform method. To show the consistency and validity of our theoretical results, we provide two illustrative examples.

scholarly journals Approximate Analytic Solutions of Two-Dimensional Nonlinear Klein–Gordon Equation by Using the Reduced Differential Transform Method

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Wayinhareg Gashaw Belayeh ◽  
Yesuf Obsie Mussa ◽  
Ademe Kebede Gizaw
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Type Systems ◽  
Analytical Approximation ◽  
Differential Transform Method ◽  
Two Dimensional ◽  
Transform Method ◽  
Differential Transform ◽  
Klein Gordon ◽  
Reduced Differential Transform Method

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear Klein–Gordon equations (NLKGEs) with quadratic and cubic nonlinearities subject to appropriate initial conditions. The proposed technique has the advantage of producing an analytical approximation in a convergent power series form with a reduced number of calculable terms. Two test examples from mathematical physics are discussed to illustrate the validity and efficiency of the method. In addition, numerical solutions of the test examples are presented graphically to show the reliability and accuracy of the method. Also, the results indicate that the introduced method is promising for solving other type systems of NLPDEs.

scholarly journals HERD BEHAVIOR AND NONFUNDAMENTAL ASSET PRICE FLUCTUATIONS IN FINANCIAL MARKETS

2006 ◽  
Vol 10 (4) ◽  
pp. 502-528 ◽  
Author(s):  
GIAN-ITALO BISCHI ◽  
MAURO GALLEGATI ◽  
LAURA GARDINI ◽  
ROBERTO LEOMBRUNI ◽  
ANTONIO PALESTRINI
Keyword(s):  
Dynamical System ◽  
Financial Markets ◽  
Path Dependence ◽  
Initial Conditions ◽  
Discrete Dynamical System ◽  
Asset Price ◽  
Mean Field ◽  
High Sensitivity ◽  
Coupled System ◽  
Herd Behavior

In this paper we investigate the effects of herding on asset price dynamics during continuous trading. We focus on the role of interaction among traders, and we investigate the dynamics emerging when we allow for a tendency to mimic the actions of other investors, that is, to engage in herd behavior. The model, built as amean fieldin a binary setting (buy/sell decisions of a risky asset), is expressed by a three-dimensional discrete dynamical system describing the evolution of the asset price, its expected price, and its excess demand. We show that such dynamical system can be reduced to a unidirectionally coupled system. In line with therational herd behaviorliterature [Bikhchandani, S., Sharma, S. (2000), Herd Behavior in Financial Markets: A Review. Working paper, IMF, WP/00/48], situations of multistability are observed, characterized by strongpath dependence; that is, the dynamics of the system are strongly influenced by historical accidents. We describe the different kinds of dynamic behavior observed, and we characterize the bifurcations that mark the transitions between qualitatively different time evolutions. Some situations give rise to high sensitivity with respect to small changes of the parameters and/or initial conditions, including the possibility ofinvest or reject cascades(i.e., sudden uncontrolled increases or crashes of the prices).

scholarly journals Homotopy Series Solutions to Time-Space Fractional Coupled Systems

2017 ◽  
Vol 2017 ◽  
pp. 1-19 ◽  
Author(s):  
Jin Zhang ◽  
Ming Cai ◽  
Bochao Chen ◽  
Hui Wei
Keyword(s):  
Water System ◽  
Coupled System ◽  
Coupled Systems ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Time Space ◽  
Nonlinear Coupled System ◽  
Sumudu Transform ◽  
Shallow Water System ◽  
Burgers System

We apply the homotopy perturbation Sumudu transform method (HPSTM) to the time-space fractional coupled systems in the sense of Riemann-Liouville fractional integral and Caputo derivative. The HPSTM is a combination of Sumudu transform and homotopy perturbation method, which can be easily handled with nonlinear coupled system. We apply the method to the coupled Burgers system, the coupled KdV system, the generalized Hirota-Satsuma coupled KdV system, the coupled WBK system, and the coupled shallow water system. The simplicity and validity of the method can be shown by the applications and the numerical results.

scholarly journals An Analytical Solution of a Time-fractional Diffusion Equation Withexternal Force and Absorbent Term by Generalized Two-dimensional differential Transform Method

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Diffusion Equation ◽  
Initial Conditions ◽  
Fractional Diffusion ◽  
Differential Transform Method ◽  
Fractional Diffusion Equation ◽  
Transform Method ◽  
Differential Transform ◽  
Time Fractional Diffusion Equation ◽  
Generalized Differential ◽  
External Forces

In this paper, we have used generalized differential transform method in obtaining a general recurrencerelation for determining the solutions of time fractional diffusion equation with external force and absorbent term.Diffusion equations play an improtant part in energy transfer problems. Inclusion of fractional derivatives bring thenon-locality aspect into the physical system containing this equation. The obtained relation will help us to solvesuch equations with various external forces and initial conditions. Three illustrative examples have been discussed.

Generalized Differential Transform Method for Solving RLC Electric Circuit of Non-Integer Order

2018 ◽  
Vol 7 (2) ◽  
pp. 127-135 ◽  
Author(s):  
N. Magesh ◽  
A. Saravanan
Keyword(s):  
Exact Solution ◽  
Initial Conditions ◽  
Electric Circuit ◽  
Convergent Series ◽  
Second Order ◽  
Differential Transform Method ◽  
Transform Method ◽  
Rlc Circuit ◽  
Differential Transform ◽  
Generalized Differential

AbstractSystematic construction of fractional ordinary differential equations [FODEs] has gained much attention nowadays research because dimensional homogeneity plays a major role in mathematical modeling. In order to keep up the dimension of the physical quantities, we need some auxiliary parameters. When we utilize auxiliary parameters, the FODE turns out to be more intricate. One of such kind of model is non-homogeneous fractional second order RLC circuit. To solve this kind of complicated FODEs, we need proficient modern analytical method. In this paper, we use two different methods, one is modern and the other is traditional, namely generalized differential transform Method (GDTM) and Laplace transform method (LTM) to obtain the analytical solution of non-homogeneous fractional second order RLC circuit. We present the solution in terms of convergent series. Though GDTM and LTM are capable to produce the exact solution of fractional RLC circuit, great strength of GDTM over LTM is that differential transform of initial conditions occupy the coefficients of first two terms in series solution so that we arrive exact solution with few iterations and also, it does not allow the noise terms while computing the coefficients. Due to this, GDTM takes less time to converge than LTM and it has been demonstrated. Furthermost, we discuss the characteristics of non-homogeneous fractional second order RLC circuit through numerical illustrations.

scholarly journals Differential Transform Method with Complex Transforms to Some Nonlinear Fractional Problems in Mathematical Physics

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Syed Tauseef Mohyud-Din ◽  
Farah Jabeen Awan ◽  
Jamshad Ahmad ◽  
Saleh M. Hassan
Keyword(s):  
Initial Conditions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Mathematical Tool ◽  
Transform Method ◽  
Alternative Approach ◽  
Differential Transform ◽  
Fractional Differential ◽  
Analytical Series ◽  
Adomian’S Polynomials

This paper witnesses the coupling of an analytical series expansion method which is called reduced differential transform with fractional complex transform. The proposed technique is applied on three mathematical models, namely, fractional Kaup-Kupershmidt equation, generalized fractional Drinfeld-Sokolov equations, and system of coupled fractional Sine-Gordon equations subject to the appropriate initial conditions which arise frequently in mathematical physics. The derivatives are defined in Jumarie’s sense. The accuracy, efficiency, and convergence of the proposed technique are demonstrated through the numerical examples. It is observed that the presented coupling is an alternative approach to overcome the demerit of complex calculation of fractional differential equations. The proposed technique is independent of complexities arising in the calculation of Lagrange multipliers, Adomian’s polynomials, linearization, discretization, perturbation, and unrealistic assumptions and hence gives the solution in the form of convergent power series with elegantly computed components. All the examples show that the proposed combination is a powerful mathematical tool to solve other nonlinear equations also.

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