A Fourier Spectral Method to Solve Linear and Non-Linear Differential Equations and its Applications

A Fourier Spectral Method to Solve Linear and Non-Linear Differential Equations and its Applications

Bibliographic Details
Main Author: Akkineni, Dharma Teja
Language:English
Published: University of Akron / OhioLINK 2014
Subjects:
Electrical Engineering
Online Access:http://rave.ohiolink.edu/etdc/view?acc_num=akron1418994964
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spelling ndltd-OhioLink-oai-etd.ohiolink.edu-akron14189949642021-08-03T06:28:37Z A Fourier Spectral Method to Solve Linear and Non-Linear Differential Equations and its Applications Akkineni, Dharma Teja Electrical Engineering <p>     This thesis is dedicated to develop a Fourier spectral method based on discrete Fourier transform for finding the solutions of partial differential equations. Spectral methods with a variety of basis functions such as Galerkin method, Tau method, and collocation method are used to solve linear and non-linear complex differential equations numerically in various fields. Unlike the existing methods where the basis functions are orthogonal in the domain, we have developed procedures without this requirement. In other words the numerical methods developed here are based on non-harmonic Fourier series.</p><p>     We considered several classes of problems such as Blasius boundary layer problem in flow of a liquid on a semi-infinite plate, the Euler-Lagrange equation arising in the Landau theory of phase transitions and the Fokker-Plank equation in Fokker-Plank Kolmogorov approach for modeling geo materials in one dimension. The non-harmonic Fourier spectral method is tested on different kinds of sample differential equations to which analytical solutions are known. The solutions are compared with analytical solutions for accuracy and precision. The Newton-Kantorovich iterative approach is used for non-linear problems. The solution to Blasius boundary layer problem is calculated with an error of order 10<sup>-3</sup> with 32 grid points and an error of order 10<sup>-6</sup> with 100 grid points over the domain. The 2-D Euler-Lagrange equation is solved in only 4 iterations with 32 grid points over the domain while keeping the error in the order of 10<sup>-6</sup>. A hybrid method involving finite difference method and spectral method is used to solve the time varying Fokker-Plank equations.</p><p>     The developed methods can be applied to other class of problems. The class of problems considered in this thesis are on regular boundaries. The developed method can be extended to problems with irregular boundaries using appropriate domain mapping techniques.</p> 2014-12-19 English text University of Akron / OhioLINK http://rave.ohiolink.edu/etdc/view?acc_num=akron1418994964 http://rave.ohiolink.edu/etdc/view?acc_num=akron1418994964 unrestricted This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws.
collection NDLTD
language English
sources NDLTD
topic Electrical Engineering
spellingShingle Electrical Engineering
Akkineni, Dharma Teja
A Fourier Spectral Method to Solve Linear and Non-Linear Differential Equations and its Applications
author Akkineni, Dharma Teja
author_facet Akkineni, Dharma Teja
author_sort Akkineni, Dharma Teja
title A Fourier Spectral Method to Solve Linear and Non-Linear Differential Equations and its Applications
title_short A Fourier Spectral Method to Solve Linear and Non-Linear Differential Equations and its Applications
title_full A Fourier Spectral Method to Solve Linear and Non-Linear Differential Equations and its Applications
title_fullStr A Fourier Spectral Method to Solve Linear and Non-Linear Differential Equations and its Applications
title_full_unstemmed A Fourier Spectral Method to Solve Linear and Non-Linear Differential Equations and its Applications
title_sort fourier spectral method to solve linear and non-linear differential equations and its applications
publisher University of Akron / OhioLINK
publishDate 2014
url http://rave.ohiolink.edu/etdc/view?acc_num=akron1418994964
work_keys_str_mv AT akkinenidharmateja afourierspectralmethodtosolvelinearandnonlineardifferentialequationsanditsapplications
AT akkinenidharmateja fourierspectralmethodtosolvelinearandnonlineardifferentialequationsanditsapplications
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