
1030TextSRandomPerturbation
05 Mar 2020  Technique Narratives  Contributor(s):: Reza R Ahangar
After a brief historical view of this problem, we will demonstrate the derivation of first order linear differential equations with random perturbations. Students in their semester research project under a course “special topics” or “independent study” will learn a past...

1010TextSAtmosphericCO2Bifurcation
12 Jan 2020  Technique Narratives  Contributor(s):: Jakob Kotas
Students are introduced to the concept of a bifurcation in a firstorder ordinary differential equation through a modeling scenario involving atmospheric carbon dioxide. Carbon dioxide is taken as a parameter and temperature is a function of time.

2005TextSLinearizeItAll
09 Dec 2019  Technique Narratives  Contributor(s):: Don Hickethier
Linear approximations are often used to simplify nonlinear ordinary differential equations (ODEs) for ease in analysis. The resulting linear approximation produces an ODE where closed form solutions may be obtained. A simple model using Torricelli's Law will compare the exact solution to...

5010SMatrixExponential
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...

1060TextSRegularPerturbation
21 Apr 2019  Technique Narratives  Contributor(s):: Mark 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...

1005TextSNavigatingNumericalMethods
31 Mar 2019  Technique Narratives  Contributor(s):: Corban Harwood
This technique narrative guides a discoverybased 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 velocity...

7011TextSCoupledSystemLaplace
31 Mar 2019  Technique Narratives  Contributor(s):: Mitaxi Pranlal 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...

5005TextSStiffDifferentialEquations
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...

1003STextIntroNumericalMethods
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.

1015TextSDimensionlessVariables
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...

2001TextSNumericalMethodsComparisons
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...

9001TextSSkinBurnModelNumericalMethods
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...

5030STextLinNonHomoSystemSol
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.

7006STextLaplaceTransformBirth
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.

5012STextLinearSystemConjecture
17 Sep 2015  Technique Narratives  Contributor(s):: Brian Winkel

7005STextLaplaceTransformOverview
06 Jun 2015  Technique Narratives  Contributor(s):: Brian Winkel
The Laplace Transform is a mathematical construct that has proven very useful in both solving and understanding differential equations. We introduce it and show its power here. This is done in a Mathematica notebook with pdf provided.

8002STextTrigSumRepresentation
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.

3090STextChebyshevPolynomialSolution
03 Jun 2015  Technique Narratives  Contributor(s):: FR Gabriel Costa

1009STextBifurcation
31 May 2015  Technique Narratives  Contributor(s):: Edward Swim

1002STextIntegratingFactor
30 May 2015  Technique Narratives  Contributor(s):: Brian Winkel