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### Technique Narratives feed

1. 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...

2. 05 Mar 2020 | Technique Narratives | Contributor(s):: Reza 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...

3. 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.

4. 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...

5. 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...

6. 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...

7. 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...

8. 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...

9. 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...

10. 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.

11. 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...

12. 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...

13. 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...

14. 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.

15. 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.

16. 17 Sep 2015 | Technique Narratives

17. 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.

18. 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.

19. 03 Jun 2015 | Technique Narratives | Contributor(s):: FR Gabriel Costa

20. 31 May 2015 | Technique Narratives | Contributor(s):: Edward Swim