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AN APPLICATION OF SINGULAR PERTURBATION THEORY TO THE STUDY OF THE LONGITUDINAL MOTION OF A DISCRETIZED VISCOELASTIC ROD

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2020, Master of Science, University of Akron, Applied Mathematics.
In this thesis we study the longitudinal motion of a light viscoelastic rod carrying a heavy particle. We will fix the leftmost side of our rod and then discretize our rod into K light blocks with one heavy block on the rightmost end where ε is the ratio of the mass of a light block to the mass of the heavy block and ε << 1. We then focus on the simplest case, K = 1. We will then show that in this case we cannot use regular asymptotic expansions to approximate the solutions to the resulting system of differential equations. We instead use a singular perturbation approach from [1] to analyze our problem. We then show that our initial value problem can be transformed into an integral equation and that this integral equation is satisfied by an approximate solution on a compact domain. We then conclude that this approximate solution is an approximate solution to the original initial value problem and we will get estimates on the solutions that are uniform in time and in ε. We then study the outer problem numerically for a non-monotonic function. We will linearize our system about its fixed points and study, using the Stable Manifold Theorem, the stability of the fixed points.
J. Patrick Wilber (Advisor)
Truyen Nguyen (Committee Member)
Lingxing Yao (Committee Member)
67 p.

Recommended Citations

Citations

  • Kane, J. P. (2020). AN APPLICATION OF SINGULAR PERTURBATION THEORY TO THE STUDY OF THE LONGITUDINAL MOTION OF A DISCRETIZED VISCOELASTIC ROD [Master's thesis, University of Akron]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=akron1590589044202922

    APA Style (7th edition)

  • Kane, Joshua. AN APPLICATION OF SINGULAR PERTURBATION THEORY TO THE STUDY OF THE LONGITUDINAL MOTION OF A DISCRETIZED VISCOELASTIC ROD. 2020. University of Akron, Master's thesis. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=akron1590589044202922.

    MLA Style (8th edition)

  • Kane, Joshua. "AN APPLICATION OF SINGULAR PERTURBATION THEORY TO THE STUDY OF THE LONGITUDINAL MOTION OF A DISCRETIZED VISCOELASTIC ROD." Master's thesis, University of Akron, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=akron1590589044202922

    Chicago Manual of Style (17th edition)