Math 322 - Mathematical Analysis for Engineers


Sturm-Liouville Eigenfunctions

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This MATLAB GUI explores the orthogonality properties of several families of Sturm-Liouville eigenfunctions and emphasizes the geometric significance of the orthogonality relationship between two (different) eigenfunctions in the same family.

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Vibrating String

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This MATLAB GUI illustrates the use of Fourier series to simulate the dynamics of a vibrating string. The string is clamped at its end points and its deflection from the horizontal, u, evolves according to the wave equation, utt - uxx = 0.

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Circular Elastic Membrane

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This MATLAB GUI illustrates how the vibrating modes of a circular membrane evolve in time and interact with one another. The membrane is clamped at its boundary and its deflection from the horizontal, u, evolves according to the two-dimensional wave equation, utt = ∇2u.

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Rectangular Elastic Membrane

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This MATLAB GUI illustrates how the vibrating modes of a rectangular membrane evolve in time and interact with one another. The membrane is clamped at its boundary and its deflection from the horizontal, u, evolves according to the two-dimensional wave equation, utt = c2(uxx + uyy).

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One-dimensional Heat Equation

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This MATLAB GUI illustrates the use of Fourier series to simulate the diffusion of heat in a domain of finite size. The quantity u evolves according to the heat equation, ut - uxx = 0, and may satisfy Dirichlet, Neumann, or mixed boundary conditions.

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Heat Equation on the Whole Line

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This MATLAB GUI plots the solution to the one-dimensional heat equation, ut = c2uxx, as a function of time and for "top hat" initial conditions.

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Mass-Spring System

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This MATLAB GUI simulates the solution to the ordinary differential equation m y'' + c y' + k y = F(t), describing the response of a one-dimensional mass spring system with forcing function F(t) given by (i) a unit square wave or (ii) a Dirac delta function (e.g. "hammerblow"). Without loss of generality, m is set to 1.

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