Analytical Solution of a Dynamic System Containing Fractional Derivative of Order One-Half by Adomian Decomposition Method

2005 ◽  
Vol 72 (2) ◽  
pp. 290-295 ◽  
Author(s):  
S. Saha Ray ◽  
B. P. Poddar ◽  
R. K. Bera
Keyword(s):  
Fractional Derivative ◽  
Dynamic System ◽  
Decomposition Method ◽  
Materials Science ◽  
Adomian Decomposition Method ◽  
Expansion Method ◽  
Physical Problems ◽  
Relaxation Functions ◽  
Damping Behavior ◽  
Adomian Decomposition

The fractional derivative has been occurring in many physical problems, such as frequency-dependent damping behavior of materials, motion of a large thin plate in a Newtonian fluid, creep and relaxation functions for viscoelastic materials, the PIλDμ controller for the control of dynamical systems, etc. Phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry, and materials science are also described by differential equations of fractional order. The solution of the differential equation containing a fractional derivative is much involved. Instead of an application of the existing methods, an attempt has been made in the present analysis to obtain the solution of an equation in a dynamic system whose damping behavior is described by a fractional derivative of order 1/2 by the relatively new Adomian decomposition method. The results obtained by this method are then graphically represented and compared with those available in the work of Suarez and Shokooh [Suarez, L. E., and Shokooh, A., 1997, “An Eigenvector Expansion Method for the Solution of Motion Containing Fraction Derivatives,” ASME J. Appl. Mech., 64, pp. 629–635]. A good agreement of the results is observed.

scholarly journals Analytical Solutions of Boundary Values Problem of 2D and 3D Poisson and Biharmonic Equations by Homotopy Decomposition Method

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Abdon Atangana ◽  
Adem Kılıçman
Keyword(s):  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Variational Iteration Method ◽  
Biharmonic Equations ◽  
Poisson Equations ◽  
Physical Problems ◽  
Adomian Decomposition ◽  
Homotopy Decomposition ◽  
2D And 3D ◽  
Corrected Function

The homotopy decomposition method, a relatively new analytical method, is used to solve the 2D and 3D Poisson equations and biharmonic equations. The method is chosen because it does not require the linearization or assumptions of weak nonlinearity, the solutions are generated in the form of general solution, and it is more realistic compared to the method of simplifying the physical problems. The method does not require any corrected function or any Lagrange multiplier and it avoids repeated terms in the series solutions compared to the existing decomposition method including the variational iteration method, the Adomian decomposition method, and Homotopy perturbation method. The approximated solutions obtained converge to the exact solution as tends to infinity.

scholarly journals Analytic Fuzzy Formulation of a Time-Fractional Fornberg–Whitham Model with Power and Mittag–Leffler Kernels

2021 ◽  
Vol 5 (3) ◽  
pp. 113 ◽  
Author(s):  
Saima Rashid ◽  
Rehana Ashraf ◽  
Ahmet Ocak Akdemir ◽  
Manar A. Alqudah ◽  
Thabet Abdeljawad ◽  
...  
Keyword(s):  
Fractional Derivative ◽  
Decomposition Method ◽  
Numerical Solutions ◽  
Integral Transform ◽  
Initial Conditions ◽  
Adomian Decomposition Method ◽  
Algorithmic Approach ◽  
Closed Form Solutions ◽  
Adomian Decomposition ◽  
Fractional Orders

This manuscript assesses a semi-analytical method in connection with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian decomposition method (briefly, EADM). Moreover, we use the aforesaid strategy to address the time-fractional Fornberg–Whitham equation (FWE) under gH-differentiability by employing different initial conditions (IC). Several algebraic aspects of the fuzzy Caputo fractional derivative (CFD) and fuzzy Atangana–Baleanu (AB) fractional derivative operator in the Caputo sense, with respect to the Elzaki transform, are presented to validate their utilities. Apart from that, a general algorithm for fuzzy Caputo and AB fractional derivatives in the Caputo sense is proposed. Some illustrative cases are demonstrated to understand the algorithmic approach of FWE. Taking into consideration the uncertainty parameter ζ∈[0,1] and various fractional orders, the convergence and error analysis are reported by graphical representations of FWE that have close harmony with the closed form solutions. It is worth mentioning that the projected approach to fuzziness is to verify the supremacy and reliability of configuring numerical solutions to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures.

scholarly journals Application of Natural Transform Method to Fractional Pantograph Delay Differential Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
M. Valizadeh ◽  
Y. Mahmoudi ◽  
F. Dastmalchi Saei
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Perturbation Method ◽  
Delay Differential Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
New Method ◽  
Transform Method ◽  
Adomian Decomposition ◽  
Delay Differential

In this paper, a new method based on combination of the natural transform method (NTM), Adomian decomposition method (ADM), and coefficient perturbation method (CPM) which is called “perturbed decomposition natural transform method” (PDNTM) is implemented for solving fractional pantograph delay differential equations with nonconstant coefficients. The fractional derivative is regarded in Caputo sense. Numerical evaluations are included to demonstrate the validity and applicability of this technique.

scholarly journals A Novel Treatment of Fuzzy Fractional Swift–Hohenberg Equation for a Hybrid Transform within the Fractional Derivative Operator

2021 ◽  
Vol 5 (4) ◽  
pp. 209
Author(s):  
Saima Rashid ◽  
Rehana Ashraf ◽  
Fatimah S. Bayones
Keyword(s):  
Fractional Derivative ◽  
Decomposition Method ◽  
Numerical Solutions ◽  
Integral Transform ◽  
Initial Conditions ◽  
Adomian Decomposition Method ◽  
Initial Value ◽  
Adomian Decomposition ◽  
In Series ◽  
Fractional Orders

This article investigates the semi-analytical method coupled with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian decomposition method (briefly, EADM). In addition, we apply this method to the time-fractional Swift–Hohenberg equation (SHe) with various initial conditions (IC) under gH-differentiability. Some aspects of the fuzzy Caputo fractional derivative (CFD) with the Elzaki transform are presented. Moreover, we established the general formulation and approximate findings by testing examples in series form of the models under investigation with success. With the aid of the projected method, we establish the approximate analytical results of SHe with graphical representations of initial value problems by inserting the uncertainty parameter 0≤℘≤1 with different fractional orders. It is expected that fuzzy EADM will be powerful and accurate in configuring numerical solutions to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures.

scholarly journals Mathematical Modelling in Engineering with Integral Transforms via Modified Adomian Decomposition Method

2021 ◽  
Vol 8 (3) ◽  
pp. 409-417
Author(s):  
Minakshi Mohanty ◽  
Saumya Ranjan Jena ◽  
Satya Kumar Misra
Keyword(s):  
Differential Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Integral Transforms ◽  
Physical Problems ◽  
Modified Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Different Types ◽  
Linear Differential ◽  
Homogeneous Linear

In this work three integral transforms through modified Adomian decomposition method (ADM) are proposed to obtain the approximate analytical solution of different types of mathematical models arising in physical problems. These transformations are applied for both homogeneous and non-homogeneous linear differential equations. The efficiency and accuracy of the proposed methods are implemented through higher order non-homogeneous ordinary differential equations. Numerical tests are reported for applicability of the current scheme based on different transformations and compared with exact solutions.

scholarly journals Abundant Wave Accurate Analytical Solutions of the Fractional Nonlinear Hirota–Satsuma–Shallow Water Wave Equation

Fluids ◽  
2021 ◽  
Vol 6 (7) ◽  
pp. 235
Author(s):  
Chen Yue ◽  
Dianchen Lu ◽  
Mostafa M. A. Khater
Keyword(s):  
Wave Equation ◽  
Fractional Derivative ◽  
Shallow Water ◽  
Analytical Solutions ◽  
Water Wave ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Conformable Fractional Derivative ◽  
Adomian Decomposition ◽  
Fractional System

This research paper targets the fractional Hirota’s analytical solutions–Satsuma (HS) equations. The conformable fractional derivative is employed to convert the fractional system into a system with an integer–order. The extended simplest equation (ESE) and modified Kudryashov (MKud) methods are used to construct novel solutions of the considered model. The solutions’ accuracy is investigated by handling the computational solutions with the Adomian decomposition method. The solutions are explained in some different sketches to demonstrate more novel properties of the considered model.

scholarly journals Solution to the Linear Fractional Differential Equation Using Adomian Decomposition Method

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Jin-Fa Cheng ◽  
Yu-Ming Chu
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Fractional Derivative ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Initial Value ◽  
Adomian Decomposition ◽  
Constant Coefficients ◽  
Fractional Differential ◽  
Linear Fractional Differential Equations

We obtain the analytical general solution of the linear fractional differential equations with constant coefficients by Adomian decomposition method under nonhomogeneous initial value condition, which is in the sense of the Caputo fractional derivative.

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