Adomian Decomposition Method for Solving Higher Order Boundary Value Problems

In this paper, we present efficient numerical algorithms for the approximate solution of linear and non-linear higher order boundary value problems. Algorithms are, based on Adomian decomposition. Also, the Laplace Transformation with Adomian decomposition technique is proposed to solve the problems when Adomian series diverges. Three examples are given to illustrate the performance of each technique. Keyword: Higher order Singular boundary value problems, Adomian decomposition techniques, Laplace transformations

An Examination Of The Effectiveness Of The Adomian Decomposition Method In Fluid Dynamic Applications

Since its introduction in the 1980\u27s, 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\u27s delta-perturbation method

Analytical solutions of orientation aggregation models, multiple solutions and path following with the Adomian decomposition method

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.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

Analytic approximate solutions of Volterra’s population and some scientific models by power series method

In this paper, we have implement an analytic approximate method based on power series method (PSM) to obtain asolutions for Volterra’s population model of population growth of a species in a closed system. The numerical solution isobtained by combining the PSM and Pad´e technique. The Pad´e approximation that often show superior performance overseries approximation are effectively used in the analysis to capture essential behavior of the population u(t) of identicalindividuals. The results demonstrate that the method has many merits such as being derivative-free, overcome the difficultyarising in calculating Adomian polynomials to handle the nonlinear terms in Adomian Decomposition Method (ADM).It does not require to calculate Lagrange multiplier as in Variational Iteration Method (VIM) and no needs to construct ahomotopy and solve the corresponding algebraic equations as in Homotopy Perturbation Method (HPM). Moreover, weused this method to solve some scientific models, namely, the hybrid selection model, the Riccati model and the logisticmodel to provide the analytic solutions. The obtained analytic approximate solutions of applying the PSM is in fullagreement with the results obtained with those methods available in the literature. The software used for the calculationsin this study was MATHEMATICAr 8.0

Numerical study of chemical reaction effects in magnetohydrodynamic Oldroyd B oblique stagnation flow with a non-Fourier heat flux model

Reactive magnetohydrodynamic (MHD) flows arise in many areas of nuclear reactor transport. Working fluids in such systems may be either Newtonian or non-Newtonian. Motivated by these applications, in the current study, a mathematical model is developed for electrically-conducting viscoelastic oblique flow impinging on stretching wall under transverse magnetic field. A non-Fourier Cattaneo-Christov model is employed to simulate thermal relaxation effects which cannot be simulated with the classical Fourier heat conduction approach. The Oldroyd-B non-Newtonian model is employed which allows relaxation and retardation effects to be included. A convective boundary condition is imposed at the wall invoking Biot number effects. The fluid is assumed to be chemically reactive and both homogeneous-heterogeneous reactions are studied. The conservation equations for mass, momentum, energy and species (concentration) are altered with applicable similarity variables and the emerging strongly coupled, nonlinear non-dimensional boundary value problem is solved with robust well-tested Runge-Kutta-Fehlberg numerical quadrature and a shooting technique with tolerance level of 10−4. Validation with the Adomian decomposition method (ADM) is included. The influence of selected thermal (Biot number, Prandtl number), viscoelastic hydrodynamic (Deborah relaxation number), Schmidt number, magnetic parameter and chemical reaction parameters, on velocity, temperature and concentration distributions are plotted for fixed values of geometric (stretching rate, obliqueness) and thermal relaxation parameter. Wall heat transfer rate (local heat flux) and wall species transfer rate (local mass flux) are also computed and it is observed that local mass flux increases with strength of heterogeneous reactions whereas it decreases with strength of homogeneous reactions. The results provide interesting insights into certain nuclear reactor transport phenomena and furthermore a benchmark for more general CFD simulations

A reliable iterative method for Cauchy problems

In the present paper, the new iterative method proposed by Daftardar-Gejji and Jafari (NIM or DJM) [V. Daftardar-Gejji, H. Jafari, An iterative method for solving non linear functional equations, J. Math. Anal. Appl. 316 (2006) 753-763] is used to solve the Cauchy problems. In this iterative method the solution is obtained in the series form that converge to the exact solution with easily computed components. The results demonstrate that the method has many merits such as being derivative-free, overcome the difficulty arising in calculating calculating Adomian polynomials to handle the nonlinear terms in Adomian Decomposition Method (ADM). It does not require to calculate Lagrange multiplier in Variational Iteration Method (VIM) and no needs to construct a homotopy and solve the corresponding algebraic equations in Homotopy Perturbation Method (HPM) and can be easily comprehended with only a basic knowledge of Calculus. The results show that the present method is very effective, simple and provide the analytic solutions. The software used for the calculations in this study was MATHEMATICA r 8.0.Keywords: New iterative method; Cauchy problems; Inviscid Burger’s equation; transport equatio

Numerical solution of Boundary Value Problems by Piecewise Analysis Method

In this paper, we use an efficient numerical algorithm for solving two point fourth-order linear  and nonlinear boundary value problems, which is based on the homotopy analysis method (HAM), namely, the piecewise – homotopy analysis method ( P-HAM).The method contains an auxiliary parameter that provides a powerful tool to analysis strongly linear and nonlinear ( without linearization ) problems directly. Numerical examples are presented to compare the results obtained with some existing results found in literatures. Results obtained by the RHAM performed better in terms of accuracy achieved. Keywords:            Piecewise-homotopy analysis, perturbation, Adomain decomposition method, Variational Iteration, Boundary Value Problems

Homotopy Perturbation Method for the Strongly Nonlinear Darcy-Forscheimer Model

We derive approximate analytical solution to the strongly nonlinear Darcy-Forscheimer model through the homotopy perturbation method. The approximate solutions compared well with numerical results computed via the bvp4c routine of Matlab as well as existing results in the literature. Keywords: Semi-analytical Solution, Homotopy Perturbation, Variational Iteration, Darcy-Forscheimer Model, bvp4

On Nonperturbative Techniques for Thermal Radiation Effect on Natural Convection past a Vertical Plate Embedded in a Saturated Porous Medium

In this article, the heat transfer characteristics of natural convection about a vertical permeable flat surface embedded in a saturated porous medium are studied by taking into account the thermal radiation effect. The plate is assumed to have a power-law temperature distribution. Similarity variables are employed in order to transform the governing partial differential equations into a nonlinear ordinary differential equation. Both Adomian decomposition method (ADM) and He's variational iteration method (VIM) coupled with Padé approximation technique are implemented to solve the reduced system. Comparisons with previously published works are performed, and excellent agreement between the results is obtained