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Renormalization Group Methods in Applied Mathematical Problems

Merdan, Huseyin (2004) Renormalization Group Methods in Applied Mathematical Problems. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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This work presents the application of the methods known as renormalization group (RG) and scaling in the physics literature to applied mathematics problems after a brief review ofthe methodology.The first part of the thesis involves an application to a class of nonlinear parabolic differential equations. We consider equationsof the form $u_{t}=frac{1}{2}u_{xx}+varepsilonN(x,u,u_{x},u_{xx})$ where $varepsilon$ is a small positive number and $N$ is dimensionally consistent without additional dimensional constants. First, RG methods are described fordetermining the key exponents related to the decay of solutions to these equations. The determination of decay exponents is viewed asan asymptotically self-similar process that facilitates an RG approach. These methods are extended to higher order in the smallcoefficient of the nonlinearity. The RG calculations lead to the result that for large space and time, the solution is characterized by $u(x,t)sim t^{-frac{1}{2}-alpha }u^{ast}(xt^{-1/2},1)$ where the exponent $alpha$ is a simple function of the exponents of the terms in $N$. Finally, the RG results are verified in some cases by rigorous proofs and other calculations.In the second part, the application of renormalization technique to systems of equations describing interface problems are presented. The temporal evaluation of an interface separating twophases is analyzed for large time. We study the standard sharp interface problem in the quasi-static regime. The characteristic length, $R(t)$, of a self-similar system that is the timedependent length scale characterizing the pattern growth is calculated by implementing a renormalization procedure. It behaves as t^β where β has values in the continuous spectrum [1/3,1/2] when the dynamical undercooling is non-zero, and β in [1/3,∞] when the undercooling is set atzero. The single value of β=1 is extracted from this continuous spectrum as a consequence of boundary conditions that impose a plane wave. It is also shown that in almost all of thesecases, the capillarity length (arising from surface tension) is irrelevant for the large time behavior even though it has a crucial role at the early stage evolution of an interface.end{abstract}


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Merdan, Huseyinhumst4@pitt.eduHUMST4
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairCaginalp, Gunduzcaginalp@pitt.eduCAGINALP
Committee MemberWang, Dehuadwang@math.pitt.eduDHWANG
Committee MemberBurbea, Jacobburbea@pitt.eduBURBEA
Committee MemberSofa, William Awsoffa@pitt.eduWSOFFA
Committee MemberTroy, William Ctroy@math.pitt.eduTROY
Date: 24 September 2004
Date Type: Completion
Defense Date: 28 May 2004
Approval Date: 24 September 2004
Submission Date: 31 July 2004
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: Asymptotic self-similarity; Capilarity length; Interface dynamics; Nonlinear parabolic equations; Quasi-static regime; Renormalization group
Other ID:, etd-07312004-200513
Date Deposited: 10 Nov 2011 19:55
Last Modified: 15 Nov 2016 13:47


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