What's New:

Technique Narratives feed

  1. 5-010-S-MatrixExponential

    12 Sep 2019 | Technique Narratives | Contributor(s):: Kurt Bryan

    The matrix exponential is a powerful computational and conceptual tool for analyzing systems of linear, constant coefficient, ordinary differential equations (ODE's). This narrative offers a quick introduction to the technique, with examples and exercises. It also includes an introduction to...

  2. 1-060-Text-S-RegularPerturbation

    21 Apr 2019 | Technique Narratives | Contributor(s):: Mark A. Lau

    This Technique Narrative presents an introduction to a set of analytical approximations referred to as regular perturbation.This is a particular mathematical tool within the broader set of perturbation methods. In general, perturbation methods are analytical techniques for obtaining...

  3. 1-005-Text-S-NavigatingNumericalMethods

    31 Mar 2019 | Technique Narratives | Contributor(s):: Corban Harwood

    This technique narrative guides a discovery-based approach to learning the basics of numerical methods for first order differential equations, by following the graphical and analytical perspectives of the forward Euler method and second order Taylor method. These methods are motivated by...

  4. 7-011-Text-S-CoupledSystemLaplace

    31 Mar 2019 | Technique Narratives | Contributor(s):: Mitaxi Mehta

    Differential equations and Laplace transforms are an integral part of control problems in engineering systems. However a clear explanation of the relationship of Laplace transforms with the differential equation formalism is difficult to find for coupled differential equations. Here we describe...

  5. 5-005-Text-S-StiffDifferentialEquations

    05 Mar 2019 | Technique Narratives | Contributor(s):: Kurt Bryan

    This material introduces the topic of ``stiffness'' for a system of ordinary differential equations (ODE's), through a series of examples.Stiffness is a property that a system of ODE's may possess that make it difficult to solve numerically with standard methods, and it is a...

  6. 1-003-S-Text-IntroNumericalMethods

    10 Jan 2019 | Technique Narratives | Contributor(s):: Brian Winkel

    We ask students to develop two numerical methods for solving first order differential equations  geometrically and to compute numeric solutions and compare them to the analytic solutions for a number of different step sizes. 

  7. 1-015-Text-S-DimensionlessVariables

    25 Aug 2018 | Technique Narratives | Contributor(s):: Kurt Bryan

    This material introduces the idea of ``rescaling'' for ordinary differential equations (ODE's) by the introduction of dimensionless variables. In practice this is an extremely common and useful prelude to the analysis and solution of ODE's, and yet is not often taught in an...

  8. 2-001-Text-S-NumericalMethodsComparisons

    18 Aug 2018 | Technique Narratives | Contributor(s):: Swarn Singh

    It is not always possible to solve a differential equation analytically.This material makes an effort to teach the basics of numerical methods for first order differential equations by following graphical and numerical approaches. Here we also discuss the order of accuracy of the methods and...

  9. 9-001-Text-S-SkinBurnModelNumericalMethods

    18 Aug 2018 | Technique Narratives | Contributor(s):: Suruchi Singh

    The heat equation is an important partial differential equation (PDE) which describes the distribution of heat in a given region over time. Here we learn to solve a heat equation numerically. It is difficult to study the behavior of temperature in problems with interfaces analytically so...

  10. 5-030-S-Text-LinNonHomoSystemSol

    26 Aug 2016 | Technique Narratives | Contributor(s):: Brian Winkel

    We offer strategies for solving linear systems of nonhomogeneous differential equations of the form  X'(t) = A X(t) + G(t) using a conjectured solution strategy for a system of constant coefficient, linear, nonhomogeneous, differential equations.

  11. 7-006-S-Text-LaplaceTransformBirth

    24 Feb 2016 | Technique Narratives | Contributor(s):: Sania Qureshi

    We present a way of introducing the Laplace Transform as the continuous analogue of a power series expression of a function.

  12. 5-012-S-Text-LinearSystemConjecture

    17 Sep 2015 | Technique Narratives

  13. 7-005-S-Text-LaplaceTransformOverview

    06 Jun 2015 | Technique Narratives | Contributor(s):: Brian Winkel

  14. 8-002-S-Text-TrigSumRepresentation

    04 Jun 2015 | Technique Narratives | Contributor(s):: Brian Winkel

    Students discover how to represent functions as sums of trigonometric functions and the value of such representations in many fields.

  15. 3-090-S-Text-ChebyshevPolynomialSolution

    03 Jun 2015 | Technique Narratives | Contributor(s):: FR Gabriel Costa

  16. 1-009-S-Text-Bifurcation

    31 May 2015 | Technique Narratives | Contributor(s):: Edward Swim

  17. 1-002-S-Text-IntegratingFactor

    30 May 2015 | Technique Narratives | Contributor(s):: Brian Winkel

  18. 1-001-S-Text-SeparationOfVariables

    30 May 2015 | Technique Narratives | Contributor(s):: Brian Winkel