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Mathematical Modeling
Course Description

Approach:

  • The focus is on physical applications

  • The emphasis is on analyzing, understanding and interpreting the solutions


    Steps for each Problem

    • formulate a mathematical model of a physical system

    • analyze the mathematical model

    • interpret the results

    • predict the behavior of the original physical system

Possible Application Areas

  • Dynamics of Mechanical Systems

    • Vibration of a spring-mass system with and without a damper
    • Applications of spring-mass-damper systems to seismographs, moving coil instruments, accelerometers, transducers etc.
    • Wave motion on a string with application to stringed instruments

  • Dynamics of Electrical Circuits

    • Oscillations in RLC circuits
    • Response of RLC circuits to sinusoidal, square, pulse, ramp and burst waveforms
    • Applications of RLC circuits to frequency compensation networks, phase shift networks and filters
    • Electrical analogs of oscillatory mechanical systems

  • Mathematical Modeling of Chemical and Biological Systems

    • Chemical kinetics
    • Enzyme-catalyzed reactions
    • Control and regulation in multienzyme biochemical pathways

  • Introduction to Nonlinear Dynamics and Chaos

    Behavior of nonlinear mechanical, electrical and chemical systems in contrast to linear systems

    • Phase diagrams
    • Period doubling route to chaos
    • Poincare sections
    • Lyapunov Exponents
    • Duffing equation
    • van der Pol equation
    • B-Z reactions.

For each of these systems the students will learn how to derive the governing differential equations, carry out a complete analysis of these equations, and perform computer simulations as needed. The goal is to use differential equations to bring out the physics or chemistry of the problem and at the same time use the physics or chemistry to construct a realistic mathematical model. Every effort is made to promote critical thinking even at the simplest level.

Laboratory Experiments

  • Measurement of the response of a mass-spring system driven by an eccentric rotor

  • Generation and measurement of transverse waves on a string

  • Use of oscilloscopes, function generators, multimeters etc.

  • RLC circuits; response to sinusoidal and other types of forcing

  • Observation of chaotic dynamics in a nonlinear circuit

The students will do these experiments in groups of three and write a technical report on each experiment. In particular, they will be asked to make a detailed comparison of the results of the experiment with their predictions from an appropriate mathematical model.


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