Solutions of complex equations with adomian decomposition method

In this study, first order linear complex differential equations have been solved with adomian decomposition method.Publisher's Versio

Laguerre wavelet solution of Bratu and Duffing equations

The aim of this study is to solve the Bratu and Duffing equations by using the Laguerre wavelet method. The solution of these nonlinear equations is approximated by Laguerre wavelets which are defined by well known Laguerre polynomials. One of the advantages of the proposed method is that it does not require the approximation of the nonlinear term like other numerical methods. The application of the method converts the nonlinear differential equation to a system of algebraic equations. The method is tested on four examples and the solutions are compared with the analytical and other numerical solutions and it is observed that the proposed method has a better accuracy.Publisher's Versio

Approximation Method for the Heat Equation with Derivative Boundary Conditions

In this paper, modification of Adomian decomposition method is introduced for solving heat equation with derivative boundary conditions. Some examples and the obtained results demonstrate efficiency of the proposed method. Keywords: Modified decomposition method, Heat equation, Derivative boundary conditions

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

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

Numerical Investigation of the Fractional Oscillation Equations under the Context of Variable Order Caputo Fractional Derivative via Fractional Order Bernstein Wavelets

This article describes an approximation technique based on fractional order Bernstein wavelets for the numerical simulations of fractional oscillation equations under variable order, and the fractional order Bernstein wavelets are derived by means of fractional Bernstein polynomials. The oscillation equation describes electrical circuits and exhibits a wide range of nonlinear dynamical behaviors. The proposed variable order model is of current interest in a lot of application areas in engineering and applied sciences. The purpose of this study is to analyze the behavior of the fractional force-free and forced oscillation equations under the variable-order fractional operator. The basic idea behind using the approximation technique is that it converts the proposed model into non-linear algebraic equations with the help of collocation nodes for easy computation. Different cases of the proposed model are examined under the selected variable order parameters for the first time in order to show the precision and performance of the mentioned scheme. The dynamic behavior and results are presented via tables and graphs to ensure the validity of the mentioned scheme. Further, the behavior of the obtained solutions for the variable order is also depicted. From the calculated results, it is observed that the mentioned scheme is extremely simple and efficient for examining the behavior of nonlinear random (constant or variable) order fractional models occurring in engineering and science.Comment: This is a preprint of a paper whose final and definite form is published Open Access in 'Mathematics' at [http://dx.doi.org/10.3390/math11112503

Solusi Numerik Model Epidemi Seir pada Penyebaran Tuberkulosis dengan Metode Dekomposisi Adomian

Tuberkulosis (TB) merupakan penyakit menular yang menyerang paru-paru. Dalam program penanggulangan TB, sangat perlu diperhatikan jumlah pasien dengan hasil pengobatan lengkap, meninggal, gagal, default dan pindah. Pada artikel ini, dibangun sebuah model matematika yang menggambarkan tingkat penyebaran infeksi penyakit TB dengan menggunakan metode Dekomposisi Adomian. Kemudian melakukan simulasi terhadap model penyebaran penyakit tersebut. Tahapan metode penelitian meliputi membangun model epidemi SEIR untuk penyebaran penyakit TB; mencari solusi untuk penyebaran penyakit TB dengan menggunakan metode Dekomposisi Adomian; melakukan simulasi berbantuan perangkat lunak; menganalisis hasil simulasi; serta menyimpulkan perilaku model yang dianalisis tersebut. Dari hasil simulasi, ditemukan bahwa model ini cukup baik untuk menggambarkan penyebaran penyakit TB dengan menggunakan parameter yang bersesuaian

A Comparison of Numerical Methods for Solving the Unforced Van Der Pol’s Equation

Due to the advancements in the field of computational mathematics, numerical methods are most widely being utilized to solve the equations arising in the fields of applied medical sciences, engineering and technology. In this paper, the numerical solutions of an important equation of applied dynamics: namely, the Unforced Van der Pol’s Equation (UFVDP) are obtained by reducing it to a system of two first order differential equations. The objective of this work is to investigate the efficiency of improved Heun’s (IH) method against the classical Runge-Kutta (RK4) and Mid-point (MP) methods for UFVDP equation. For analysis of accuracy, the Poincare-Lindstedt method has been used as a comparison criterion and respective error bounds are obtained. The results show that the popular RK4 method retains its better accuracy than other methods used for comparison. Keywords: Van der Pol, Runge-Kutta, Mid-point, Improved Heun’s, Poincare-Lindstedt