Improved Adomian Decomposition Method Research Articles

Exploring fractional-order new coupled Korteweg-de Vries system via improved Adomian decomposition method.

This paper aims to extend the applications of the projected fractional improved Adomian Decomposition method (fIADM) to the fractional order new coupled Korteweg-de Vries (cKdV) system. This technique is significantly recognized for its effectiveness in addressing nonlinearities and iteratively handling fractional derivatives. The approximate solutions of the fractional-order new cKdV system are obtained by employing the improved ADM in fractional form. These solutions play a crucial role in designing and optimizing systems in engineering applications where accurate modeling of wave phenomena is essential, including fluid dynamics, plasma physics, nonlinear optics, and other mathematical physics domains. The fractional order new cKdV system, integrating fractional calculus, enhances accuracy in modeling wave interactions compared to the classical cKdV system. Comparison with exact solutions demonstrates the high accuracy and ease of application of the projected method. This proposed technique proves influential in resolving fractional coupled systems encountered in various fields, including engineering and physics. Numerical results obtained using Mathematica software further verify and demonstrate its efficacy.

Analysis of functionally graded rotating disks via analytical approximation methods

Analytical solutions for functionally graded (FG) rotating disks are limited due to the functional representation of the cross-sectional geometry, the density, and the modulus of elasticity which lead to a governing differental equation with variable coefficients. Analytical solutions for such disks are limited according to the functional form chosen for the variation of the property. Hence, analytical approximation approaches gain importance for such problems. In this study, two analytical approximation approaches are proposed for the analysis FG rotating disks. The methods used are the Improved Adomian decomposition method (IADM) and the optimal homotopy asymptotic method (OHAM). The resulting solution is an analytical solution in power series form. It is assumed that the disks considered are subjected to constant angular velocity. Material properties such as modulus of elasticity, density and yield strength are assumed to vary along the radial direction according to the power law function. Stresses are obtained for different values of the gradient parameters to investigate the effects of the parameters. Von Mises yield criterion is used to determine elastic limit angular velocities. The results from this study are in very good agreement with previous results available in the literature. In addition, it is observed that OHAM converges to results much faster than IADM.

Elastic Stress Analysis of Hyperbolic Rotating Disks via Optimal Homotopy Asymptotic Method

This study, it is aimed to use an effective approximation method, called Optimal Homotopy Asymptotic Method (OHAM) for elastic stress analysis. This study is carried out with the idea that utilizable of the considered method in many areas and having advantages will add a new perspective to the rotating disc problems that can provide convenience and practicality. The considered disk is an annular disk with both free ends and is subjected to centrifugal force. The thickness variation is assumed as hyperbolic. Deformations occurred on the disk, and radial and tangential stresses are calculated with the proposed method. The approximate solutions are in very good agreement with the exact solution. It is observed that the results converge to exact solutions faster than other approximate methods, like the Improved Adomian Decomposition method, in the literature. The results of this study show that OHAM is a very practical method with fast results for rotating disk problems. The study also shows the advantages of OHAM being directly applicable to differential equations of rotating disks without any transform function. Results by the comparison of approximate results and exact solutions indicated that OHAM can effectively be used in the analysis of rotating variable thickness disks.

Analytical Solution of Nonlinear Nonhomogeneous Space and Time Fractional Physical Models by Improved Adomian Decomposition Method

The paper aims to obtain exact analytical solution of nonlinear nonhomogeneous space-time fractional order partial differential equations in Gas dynamics model, Advection model, Wave model and Klein-Gordon model by improved Adomian decomposition method coupled with fractional Taylor expansion series.The solution of these equations are in series form may have rapid convergence to a closed-form solution. The effectiveness and sharpness of this method is shown by obtaining the exact solution of these equations with suitable initial conditions(ICs). With the help of this method, it is possible to investigate nature of solutions when we vary order of the fractional derivative. Behaviour of the solution of these equations are represented by graphs using MATHEMATICA software.

Numerical Simulation of Cubic-Quartic Optical Solitons with Perturbed Fokas–Lenells Equation Using Improved Adomian Decomposition Algorithm

The current manuscript displays elegant numerical results for cubic-quartic optical solitons associated with the perturbed Fokas–Lenells equations. To do so, we devise a generalized iterative method for the model using the improved Adomian decomposition method (ADM) and further seek validation from certain well-known results in the literature. As proven, the proposed scheme is efficient and possess a high level of accuracy.

A New Improved Adomian Decomposition Method for Solving Emden{Fowler Type Equations of Higher{Order

In this paper, we present a suggested modification for Adomain decomposition method to solve Emden{Fowler Types Equations of higher-order ordinary differential equations. The proposed method can be applied to linear and non-linear problems. By using some illustrative examples, we tested the reliability and effectiveness of the proposed method and we found that the obtained results approximate the exact solution. Thus, we can conclude that this proposed method is efficient and reliable .

Corrigendum to “Numerical Solution of Dispersive Optical Solitons with Schrödinger-Hirota Equation by Improved Adomian Decomposition Method”

Corrigendum to “Numerical Solution of Dispersive Optical Solitons with Schrödinger-Hirota Equation by Improved Adomian Decomposition Method”

Adomian Decomposition Method with Modified Bernstein Polynomials for Solving Nonlinear Fredholm and Volterra Integral Equations

Many different problems in mathematics, physics, engineering can be expressed in the form of integral equations. Among these are diffraction problems, population growth, heat transfer, particle transport problems, electrical engineering, elasticity, control, elastic waves, diffusion problems, quantum mechanics, heat radiation, electrostatics and contact problems. Therefore, the solutions which are obtained by the mathematical methods play an important role in these fields. The most two basic types of integral equations are called Fredholm (FIEs) and Volterra (VIEs). In many instances, the ordinary and partial differential equations can be converted into Fredhom and Volterra integral equations that are solved more effectively. We aim through this research to present an improved Adomian decomposition method based on modified Bernstein polynomials (ADM-MBP) to solve nonlinear integral equations of the second kind. We introduced efficient method, constructed on modified Bernstein polynomials. The formulation is developed to solve nonlinear Fredholm and Volterra integral equations of second kind. This method is tested for some examples from nonlinear integral equations. Maple software was used to obtain the solutions of these examples. The results demonstrate reliability of the proposed method. Generally, the proposed method is very convenient to apply to find the solutions of Fredholm and Volterra integral equations of second kind.

Deformation Analysis of Variable Thickness Rotating Disks Using an Improved Adomian Decomposition Technique

In this study, axisymmetric elastic deformation analysis of rotating disks with variable thickness is conducted using an improved Adomian decomposition method (IADM). Variation of thickness is assumed as hyperbolic and different variations are employed for each case considered. Several analytical approximate solutions in different orders are obtained for radial stress, tangential stress, radial displacement and are compared with exact solutions. Results show that IADM can effectively be used in the deformation analysis of rotating variable thickness disks providing the solution as an analytical continuous function in the solution domain.

Numerical Analysis, Circuit Simulation, and Control Synchronization of Fractional-Order Unified Chaotic System

The traditional method of solving fractional chaotic system has the problem of low precision and is computationally cumbersome. In this paper, different fractional-order calculus solutions, the Adams prediction–correction method, the Adomian decomposition method and the improved Adomian decomposition method, are applied to the numerical analysis of the fractional-order unified chaotic system. The result shows that different methods have higher precision, smaller computational complexity, and shorter running time, in which the improved Adomian decomposition method works best. Then, based on the fractional-order chaotic circuit design theory, the circuit diagram of fractional-order unified chaotic system is designed. The result shows that the circuit simulation diagram of fractional-order unified chaotic system is basically consistent with the phase space diagram obtained from the numerical solution of the system, which verifies the existence of the fractional-order unified chaotic system of 0.9-order. Finally, the active control method is used to control and synchronize in the fractional-order unified chaotic system, and the experiment result shows that the method can achieve synchronization in a shorter time and has a better control performance.

Numerical Solution of Dispersive Optical Solitons with Schrödinger-Hirota Equation by Improved Adomian Decomposition Method

In this paper, we present new numerical results for the dispersive optical soliton solutions of the nonlinear Schrödinger-Hirota equation. The spatio-temporal dispersion term is included, in addition to group velocity dispersion Kerr law of nonlinearity are studied. A general recursive numerical scheme for the equation is devised via the Improved Adomian Decomposition Method (IADM) and further sought for some analytical results for validation. The scheme is shown to be efficient and possessed high level of accuracy as demonstrated.

Analytical Solution of Unsteady Boundary Layer Flow of a Nanofluid Past a Stretching Inclined Sheet With Effects of Magnetic Field

Flow of a nanofluid in a boundary layer in an inclined moving sheet at angle is considered analytically. The Mathematical formulation consists of the Magnetic parameter, thermophoresis, and Brownian motion. Previously published work considered convective boundary condition. The present study considered an inclined stretching sheet at angle in one dimension and considered thermal conditions of non convective heating and heat flux. Solutions to momentum, temperature and concentration distribution depends on seven parameters, Magnetic parameter M, Lewis number Le, Prandtl number Pr, thermophoresis parameter Nt, the Brownian motion parameter Nb, unsteady parameter c and Grashof numbers Gr and Gc. The non linear coupled Differential equations were solved using the improved Adomian decomposition method and a good agreement was established with the numerical method (Shooting technique). Analytical result is also presented graphically to illustrate the effect of the earlier listed parameters on Momentum, temperature and nanofraction boundary layers. Momentum boundary layer increases with increase in Grashof numbers, angle of inclination and unsteady parameter..Keywords— Adomian Decompostion Method (ADM), Nanofluid, Inclined sheet, magnetic field effects, Numerical Method (NM).

Bright optical solitons of Chen-Lee-Liu equation with improved Adomian decomposition method

This research work presents some numerical the results and analyses of three types of bright soliton solutions for the Chen-Lee-Liu (CLL) equation obtained from the improved Adomian decomposition method. The error analyses of the algorithm ae also discussed.

Accelerating the convergence of Adomian decomposition method (ADM)

A successful and easy to use modification of the classical Adomian decomposition method (ADM) was implemented by Duan and Rach [1] by simply adding a predetermined parameter into the ADM formulation. We provide a mathematical rigor in this paper to approve that the formulation of Duan and Rach indeed improves the classical Adomian method by both preventing its divergence and speeding up its convergence. Instead of prescribing it, the best suitable value of the added parameter is determined within the global squared residual approximation to ensure the quickest convergence of the recursive scheme. Within the presence of a least change interval of the approximate series, an optimum value of the added parameter leading to the accelerated convergence is shown to exist. Moreover, via the improved Adomian Decomposition Method, the convergence region of the series approximation is found to be enlarged to a bigger physical domain. The most important outcome of the present approach is that the results generated by the ADM series are no longer have to be validated via other numerical means.

Soliton Solutions and Numerical Treatment of the Nonlinear Schrodinger’s Equation Using Modified Adomian Decomposition Method

In this paper, the improved Adomian decomposition method (ADM) is applied to the nonlinear Schr&oumldinger’s equation (NLSE), one of the most important partial differential equations in quantum mechanics that governs the propagation of solitons through optical fibers. The performance and the accuracy of our improved method are supported by investigating several numerical examples that include initial conditions. The obtained results are compared with the exact solutions. It is shown that the method does not need linearization, weak or perturbation theory to obtain the solutions.

The Solution of Two-Phase Inverse Stefan Problem Based on a Hybrid Method with Optimization

The two-phase Stefan problem is widely used in industrial field. This paper focuses on solving the two-phase inverse Stefan problem when the interface moving is unknown, which is more realistic from the practical point of view. With the help of optimization method, the paper presents a hybrid method which combines the homotopy perturbation method with the improved Adomian decomposition method to solve this problem. Simulation experiment demonstrates the validity of this method. Optimization method plays a very important role in this paper, so we propose a modified spectral DY conjugate gradient method. And the convergence of this method is given. Simulation experiment illustrates the effectiveness of this modified spectral DY conjugate gradient method.

A new improved Adomian decomposition method and its application to fractional differential equations

In this paper, a new improved Adomian decomposition method is proposed, which introduces a convergence-control parameter into the standard Adomian decomposition method and establishes a new iterative formula. The examples prove that the presented method is reliable, efficient, easy to implement from a computational viewpoint and can be employed to derive successfully analytical approximate solutions of fractional differential equations.

Improved Adomian decomposition method (solving nonlinear non-homogenous initial value problem)

The Adomian decomposition method (ADM) cannot deal generally with the non-homogenous differential equations. The method still needs improvements. This paper introduces a qualitative improvement in the method. The improved method is called the improved Adomian decomposition method (IADM). A comparison between ADM and IADM shows that the ADM gives results that is not accepted in some examples and cannot solve the other examples while IADM solves the problems effectively and efficiently.

Improved Adomian decomposition method

In this paper a new treatment for the Adomian decomposition method (ADM) is introduced. The new treatment is called the improved Adomian decomposition method (IADM) which improves the results obtained from the known Adomian decomposition method. The improved Adomian decomposition method is applied for the analytic treatment of nonlinear initial value problems. The improved method accelerates the convergence of the series solution, and provides the exact power series solution. It solves the drawbacks in the standard Adomian decomposition method.