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Analytical solutions to nonlinear differential equations arising in physical problems

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Date Issued:
2014
Abstract/Description:
Nonlinear partial differential equations are difficult to solve, with many of the approximate solutions in the literature being numerical in nature. In this work, we apply the Homotopy Analysis Method to give approximate analytical solutions to nonlinear ordinary and partial differential equations. The main goal is to apply different linear operators, which can be chosen, to solve nonlinear problems. In the first three chapters, we study ordinary differential equations (ODEs) with one or two linear operators. As we progress, we apply the method to partial differential equations (PDEs) and use several linear operators. The results are all purely analytical, meaning these are approximate solutions that we can evaluate at points and take their derivatives.Another main focus is error analysis, where we test how good our approximations are. The method will always produce approximations, but we use residual errors on the domain of the problem to find a measure of error.In the last two chapters, we apply similarity transforms to PDEs to transform them into ODEs. We then use the Homotopy Analysis Method on one, but are able to find exact solutions to both equations.
Title: Analytical solutions to nonlinear differential equations arising in physical problems.
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Name(s): Baxter, Mathew, Author
Vajravelu, Kuppalapalle, Committee Chair
Li, Xin, Committee Member
Mohapatra, Ram, Committee Member
Shuai, Zhisheng, Committee Member
Kassab, Alain, Committee Member
University of Central Florida, Degree Grantor
Type of Resource: text
Date Issued: 2014
Publisher: University of Central Florida
Language(s): English
Abstract/Description: Nonlinear partial differential equations are difficult to solve, with many of the approximate solutions in the literature being numerical in nature. In this work, we apply the Homotopy Analysis Method to give approximate analytical solutions to nonlinear ordinary and partial differential equations. The main goal is to apply different linear operators, which can be chosen, to solve nonlinear problems. In the first three chapters, we study ordinary differential equations (ODEs) with one or two linear operators. As we progress, we apply the method to partial differential equations (PDEs) and use several linear operators. The results are all purely analytical, meaning these are approximate solutions that we can evaluate at points and take their derivatives.Another main focus is error analysis, where we test how good our approximations are. The method will always produce approximations, but we use residual errors on the domain of the problem to find a measure of error.In the last two chapters, we apply similarity transforms to PDEs to transform them into ODEs. We then use the Homotopy Analysis Method on one, but are able to find exact solutions to both equations.
Identifier: CFE0005303 (IID), ucf:50527 (fedora)
Note(s): 2014-08-01
Ph.D.
Sciences, Mathematics
Doctoral
This record was generated from author submitted information.
Subject(s): Differential Equations -- Homotopy Analysis Method
Persistent Link to This Record: http://purl.flvc.org/ucf/fd/CFE0005303
Restrictions on Access: campus 2019-08-15
Host Institution: UCF

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