Academic literature on the topic 'Adomian Decomposition Method'

Since its introduction in the 1980's, the Adomian Decomposition Method (ADM) has proven to be an efficient and reliable method for solving many types of problems. Originally developed to solve nonlinear functional equations, the ADM has since been used for a wide range of equation types (like boundary value problems, integral equations, equations arising in flow of incompressible and compressible fluids etc...). This work is devoted to an evaluation of the effectiveness of this method when used for fluid dynamic applications. In particular, the ADM has been applied to the Blasius equation, the Falkner-Skan equation, and the Orr-Sommerfeld equation. This study is divided into five Chapters and an Appendix. The first chapter is devoted to an introduction of the Adomian Decomposition method (ADM) with simple illustrations. The Second Chapter is devoted to the application of the ADM to generalized Blasius Equation and our result is compared to other published results when the parameter values are appropriately set. Chapter 3 presents the solution generated for the Falkner-Skan equation. Finally, the Orr-Sommerfeld equation is dealt with in the fourth Chapter. Chapter 5 is devoted to the findings and recommendations based on this study. The Appendix contains details of the solutions considered as well as an alternate solution for the generalized Blasius Equation using Bender's delta-perturbation method.<br>Ph.D.<br>Department of Mathematics<br>Sciences<br>Mathematics PhD

In this work we apply the Adomian decomposition method to an orientation aggregation problem modelling the time distribution of filaments. We find analytical solutions under certain specific criteria and programmatically implement the Adomian method to two variants of the orientation aggregation model. We extend the utility of the Adomian decomposition method beyond its original capability to enable it to converge to more than one solution of a nonlinear problem and further to be used as a corrector in path following bifurcation problems.

This master's thesis deals with solving fractional-order ordinary differential equations by the Adomian decomposition method. A part of the work is therefore devoted to the theory of equations containing differential operators of non-integer order, especially the Caputo operator. The next part is devoted to the Adomian decomposition method itself, its properties and implementation in the case of Chen system. The work also deals with bifurcation analysis of this system, both for integer and non-integer case. One of the objectives is to clarify the discrepancy in the literature concerning the fractional-order Chen system, where experiments based on the use of the Adomian decomposition method give different results for certain input parameters compared with numerical methods. The clarification of this discrepancy is based on recent theoretical knowledge in the field of fractional-order differential equations and their systems. The conclusions are supported by numerical experiments, own code implementing the Adomian decomposition method on the Chen system was used.

Nesta contribuição apresentamos uma solução para a equação de transferência radiativa condutiva em geometria cilíndrica. Esta solução é aplicada para simular a radiação e campo de temperatura juntamente com o transporte de energia radiativa e condutiva proveniente do escape liberado em lançamentos de foguetes. Para este fim, discutimos uma abordagem semianalítica reduzindo a equação original, que é contínua nas variáveis angulares, numa equação semelhante ao problema SN da transferência radiativa condutiva. A solução é construída usando um método de composição por transformada de Laplace e o método da decomposição de Adomian. O esquema recursivo ´e apresentado para o sistema de equações de ordenadas duplamente discretas juntamente com as dependências dos parâmetros e suas influências sobre a convergência heurística da solução. A solução obtida, em seguida, permite construir o campo próximo relevante para caracterizar o termo fonte para problemas de dispersão ao ajustar os parâmetros do modelo, tais como, emissividade, refletividade, albedo e outros, em comparação com a observação, que são relevantes para os processos de dispersão de campo distante e podem ser manipulados de forma independente do presente problema. Além do método de solução, também relatamos sobre algumas soluções e simulações numéricas.<br>In this contribution we present a solution for the radiative conductive transfer equation in cylinder geometry. This solution is applied to simulate the radiation and temperature field together with conductive and radiative energy transport originated from the exhaust released in rocket launches. To this end we discuss a semi-analytical approach reducing the original equation, which is continuous in the angular variables, into an equation similar to the SN radiative conductive transfer problem. The solution is constructed using a composite method by Laplace transform and Adomian decomposition method. The recursive scheme is presented for the doubly discrete ordinate equations system together with parameter dependencies and their influence on heuristic convergence of the solution. The obtained solution allows then to construct the relevant near field to characterize the source term for dispersion problems when adjusting the model parameters such as emissivity, reflectivity, albedo and others in comparison to the observation, that are relevant for far field dispersion processes and may be handled independently from the present problem. In addition to the solution method we also report some solutions and numerical simulations.

The thesis deals with qualitative properties of solutions of singular initial value problems for ordinary differential and integrodifferential equations which occur in the theory of linear and nonlinear electrical circuits and the theory of therminionic currents. The research is concentrated especially on questions of existence and uniqueness of solutions, asymptotic estimates of solutions and modications of Adomian decomposition method for singular initial problems. Solution algoritms are derived for scalar differential equations of Lane-Emden type using Taylor series and modication of the Adomian decomposition method. For certain classes of nonlinear of integrodifferential equations asymptotic expansions of solutions are constructed in a neighbourhood of a singular point. By means of the combination of Wazewski's topological method and Schauder xed-point theorem there are proved asymptotic estimates of solutions in a region which is homeomorphic to a cone having vertex coinciding with the initial point. Using Banach xed-point theorem the uniqueness of a solution of the singular initial value problem is proved for systems of integrodifferential equations of Volterra and Fredholm type including implicit systems. Moreover, conditions of continuous dependence of a solution on a parameter are determined. Obtained results are presented in illustrative examples.

博士<br>國立成功大學<br>機械工程學系碩博士班<br>98<br>The Laplace Adomian decomposition method (LADM) combines the numerical Laplace transform algorithm and the Admoian decomposition method (ADM). The truncated series solution solved by the LADM diverges rapidly as the applicable domain increases. However, the Pad? approximant extends the domain of the truncated series solution to obtain better accuracy and convergence. In this paper, a hybrid method of the LADM combined with the Pad? approximant, named the hybrid Laplace Adomian decomposition method is proposed to solve the nonlinear physical systems to demonstrate efficient and reliable results. The linearization and small parameter assumptions are unnecessary for solving the nonlinear system problems by the hybrid Laplace Adomian decomposition method. The LADM─Pad? approximant solution is easy to obtain to demonstrate a real nonlinear physical phenomenon, and the transformation of the boundary value conditions into an initial value problem is also unnecessary when solving a boundary value problem. Furthermore, the LADM─Pad? approximant solution is able to demonstrate a nonlinear physical system by an algebra form. So the calculation is no like the numerical method that every value needs to be known every time. The hybrid Laplace Adomian decomposition method has been successfully applied to solve various nonlinear problems such as, nonlinear pendulum systems, nonlinear oscillation systems, nonlinear control systems, and nonlinear fluid dynamic systems. The LADM-Pad? approximant solutions demonstrate efficient and reliable results and have been shown a good accuracy and convergence in comparison with the exact solutions and other numerical method solutions. Moreover, the LADM─Pad? approximant solutions have been demonstrated not only the superiority of the accuracy and convergence over both the ADM and LADM solutions, but also extended the applicable domain to overcome their drawbacks.

博士<br>國立成功大學<br>機械工程學系碩博士班<br>97<br>The paper solves the eigenvalue problems and deals with the free vibration problems by using the Adomian decomposition method (ADM) and Adomian modified decomposition method (AMDM). First, using the ADM, the eigenvalues and normalized eigenfunctions for the Strum-Liouville eigenvalue problem are solved, and the governing differential equation becomes a recursive algebraic equation and boundary conditions become simple algebraic frequency equations which are suitable for symbolic computation. Moreover, after some simple algebraic operations on these frequency equations any th natural frequency, the closed form series solution of any th mode shape can be obtained. Second, the free vibration problems of Euler-Bernoulli beam under various supporting conditions are discussed. Third, using the AMDM, the free vibration problems of an elastically restrained non-uniform Euler-Bernoulli beam with tip mass of rotatory inertia and eccentricity resting on an elastic foundation and subjected to an axial load are proposed. Some numerical results are given to illustrate the influence of the physical parameters on the natural frequencies of the dynamic system. Finally, this paper deals with free vibration problems of non-uniform Timoshenko beams. In this paper, the computed results agree well with those analytical and numerical results given in the literature. These results indicate that the present analysis is accurate, and provides a unified and systematic procedure which is simple and more straightforward than the other analyses.

碩士<br>國立成功大學<br>機械工程學系<br>102<br>When we try to solve physical problems ,we usually build math models to approach our problems. Nonlinear terms are very common both in physical problems and math models, but they will complicate the solving process. In this paper , we use LADM method to solve nonlinear heat transfer problems. We have two cases , nonlinear fin system and nonlinear continuously moving plates system. Some parameters like Convection, radiation, slope of the thermal conductivity-temperature curve , slope of the surface emissivity-temperature curve are discussed. We found when dimensionless numbers as following increase：Nc、Nr 、B(which presents the conventional intensity to conductional intensity、radiative intensity to conductional intensity、surface emission , respectively)will speed up heat transfer in fin or plate. Dimensionless number A(which presents slope of the thermal conductivity-temperature curve)increases will make heat transfer more faster in fin or plate. Dimensionless number Pe(which presents peclet number) increases (if we have a constant fin length or plate length and constant thermal diffusivity )will make final temperature higher. Assuming a power law variation (decided by parameter m ) of the convection coefficient . Nonlinear terms can also be increased by parameter m . In conclusion, LADM is an effective way to solve nonlinear system. Following this paper , we can know more in material or fluid selecting.