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1-030-Text-S-RandomPerturbation
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...
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1-010-Text-S-AtmosphericCO2Bifurcation
12 Jan 2020 | Technique Narratives | Contributor(s):: Jakob Kotas
Students are introduced to the concept of a bifurcation in a first-order 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.
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2-005-Text-S-LinearizeItAll
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...
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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...
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1-060-Text-S-RegularPerturbation
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...
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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 velocity...
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7-011-Text-S-CoupledSystemLaplace
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...
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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...
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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.
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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...
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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...
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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...
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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.
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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.
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5-012-S-Text-LinearSystemConjecture
17 Sep 2015 | Technique Narratives | Contributor(s):: Brian Winkel
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7-005-S-Text-LaplaceTransformOverview
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.
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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.
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3-090-S-Text-ChebyshevPolynomialSolution
03 Jun 2015 | Technique Narratives | Contributor(s):: FR Gabriel Costa
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1-009-S-Text-Bifurcation
31 May 2015 | Technique Narratives | Contributor(s):: Edward Swim
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1-002-S-Text-IntegratingFactor
30 May 2015 | Technique Narratives | Contributor(s):: Brian Winkel