Differential Equations: New Applications
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New applications in the Differential Equations category
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Differential Equations: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=136

The Mortgage Payment Problem: Approximating a Discrete Process with a Differential Equation
http://www.maplesoft.com/applications/view.aspx?SID=153511&ref=Feed
In this guest article in the Tips and Techniques series, Dr. Michael Monagan uses mortgage interest to test how well a differential equation models what is essentially a discrete process.
<img src="/view.aspx?si=153511/thumb.jpg" alt="The Mortgage Payment Problem: Approximating a Discrete Process with a Differential Equation" align="left"/>In this guest article in the Tips and Techniques series, Dr. Michael Monagan uses mortgage interest to test how well a differential equation models what is essentially a discrete process.
153511
Thu, 20 Feb 2014 05:00:00 Z
Prof. Michael Monagan
Prof. Michael Monagan

The House Warming Model
http://www.maplesoft.com/applications/view.aspx?SID=153491&ref=Feed
In this guest article in the Tips and Techniques series, Dr. Michael Monagan discusses a model of heatflow in a house, and shows how he uses this model in his class.
<img src="/view.aspx?si=153491/thumb.jpg" alt="The House Warming Model" align="left"/>In this guest article in the Tips and Techniques series, Dr. Michael Monagan discusses a model of heatflow in a house, and shows how he uses this model in his class.
153491
Wed, 22 Jan 2014 05:00:00 Z
Prof. Michael Monagan
Prof. Michael Monagan

Classroom Tips and Techniques: SliderControl of Parameters in an ODE
http://www.maplesoft.com/applications/view.aspx?SID=152112&ref=Feed
Several ways to provide slidercontrol of parameters in a differential equation are considered. In particular, the cases of one and two parameters are illustrated, and for the case of two parameters, a 2dimensional slider is constructed.
<img src="/view.aspx?si=152112/thumb.jpg" alt="Classroom Tips and Techniques: SliderControl of Parameters in an ODE" align="left"/>Several ways to provide slidercontrol of parameters in a differential equation are considered. In particular, the cases of one and two parameters are illustrated, and for the case of two parameters, a 2dimensional slider is constructed.
152112
Mon, 23 Sep 2013 04:00:00 Z
Dr. Robert Lopez
Dr. Robert Lopez

Classroom Tips and Techniques: Solving Algebraic Equations by the Dragilev Method
http://www.maplesoft.com/applications/view.aspx?SID=149514&ref=Feed
The Dragilev method for solving certain systems of algebraic equations is used to parametrize the closed curve formed by the intersection of two given surfaces. This work is an elucidation of several posts to MaplePrimes.
<img src="/view.aspx?si=149514/thumb.jpg" alt="Classroom Tips and Techniques: Solving Algebraic Equations by the Dragilev Method" align="left"/>The Dragilev method for solving certain systems of algebraic equations is used to parametrize the closed curve formed by the intersection of two given surfaces. This work is an elucidation of several posts to MaplePrimes.
149514
Tue, 16 Jul 2013 04:00:00 Z
Dr. Robert Lopez
Dr. Robert Lopez

Classroom Tips and Techniques: Gems 3135 from the Red Book of Maple Magic
http://www.maplesoft.com/applications/view.aspx?SID=147092&ref=Feed
Five additional "gems" from the Red Book of Maple Magic are detailed. Gem 31 shows how the updated Explore command can be applied to the numeric solution of an initialvalue problem containing parameters. Gem 32 shows some list manipulations. Gem 33 clarifies some issues with the contourplot command, while Gem 34 clarifies some issues with the sample option in the plot command. Finally, Gem 36 examines the Equate command, and its alternatives.
<img src="/view.aspx?si=147092/thumb.jpg" alt="Classroom Tips and Techniques: Gems 3135 from the Red Book of Maple Magic" align="left"/>Five additional "gems" from the Red Book of Maple Magic are detailed. Gem 31 shows how the updated Explore command can be applied to the numeric solution of an initialvalue problem containing parameters. Gem 32 shows some list manipulations. Gem 33 clarifies some issues with the contourplot command, while Gem 34 clarifies some issues with the sample option in the plot command. Finally, Gem 36 examines the Equate command, and its alternatives.
147092
Fri, 10 May 2013 04:00:00 Z
Dr. Robert Lopez
Dr. Robert Lopez

Dynamics of the Battlefield: The Lanchester Model
http://www.maplesoft.com/applications/view.aspx?SID=146801&ref=Feed
<p>Around the time of World War I, July 28, 1914 to November 11, 1918, many mathematicians and engineers, including Frederick W. Lanchester, became fascinated by the dynamics of the battlefield. Various mathematical models were proposed in an effort to explainand to predicthow military forces interacted on the battlefield. During World War I these mathematical investigations were mainly academic, although during World War II the United States government actually applied these models to make important decisions about the Battle of Iwo Jima in which the American forces seized control of the Japanese island of Iwo Jima. Outnumbered and outgunned by the Americans, the Japanese were defeated even before the battle began although the American forces suffered many casualties and injuries.</p>
<img src="/view.aspx?si=146801/army2.JPG" alt="Dynamics of the Battlefield: The Lanchester Model" align="left"/><p>Around the time of World War I, July 28, 1914 to November 11, 1918, many mathematicians and engineers, including Frederick W. Lanchester, became fascinated by the dynamics of the battlefield. Various mathematical models were proposed in an effort to explainand to predicthow military forces interacted on the battlefield. During World War I these mathematical investigations were mainly academic, although during World War II the United States government actually applied these models to make important decisions about the Battle of Iwo Jima in which the American forces seized control of the Japanese island of Iwo Jima. Outnumbered and outgunned by the Americans, the Japanese were defeated even before the battle began although the American forces suffered many casualties and injuries.</p>
146801
Mon, 06 May 2013 04:00:00 Z
Douglas Lewit
Douglas Lewit

Calculation of the Average Duration of an Illness and Computation of the Reproduction Number in the SIR Model
http://www.maplesoft.com/applications/view.aspx?SID=142794&ref=Feed
<p>I prepared this Maple worksheet as part of a presentation to Professor Mubayi's lab group at Northeastern Illinois University. Every member of the research group explores a different aspect of how mathematics is used to study public health. During this presentation, I explore two different SIR models.</p>
<img src="/applications/images/app_image_blank_lg.jpg" alt="Calculation of the Average Duration of an Illness and Computation of the Reproduction Number in the SIR Model" align="left"/><p>I prepared this Maple worksheet as part of a presentation to Professor Mubayi's lab group at Northeastern Illinois University. Every member of the research group explores a different aspect of how mathematics is used to study public health. During this presentation, I explore two different SIR models.</p>
142794
Tue, 29 Jan 2013 05:00:00 Z
Douglas Lewit
Douglas Lewit

Alexander Friedmann's Cosmic Scenarios
http://www.maplesoft.com/applications/view.aspx?SID=142459&ref=Feed
<p>The Russian mathematician and physicist Alexander Friedmann (18881925) is well known among relativists, but his contributions to cosmology are largely misunderstood. Even the Royal Swedish Academy of Sciences misrepresented Friedmann's work in the 2011 Nobel Prize scientific background essay. Friedmann was the first physicist who demonstrated that Albert Einstein's general relativity admits nonstatic solutions, and the universe can expand, oscillate, and be born in a singularity. Friedmann's conclusion was based on his analysis of an elliptic integral; this worksheet employs Maple's utility of handling elliptic integrals to present Friedmann's results graphically. Friedmann's differential equation governing the evolution of the universe based on Einstein's general theory of relativity is also derived using Maple's tensor package. </p>
<img src="/view.aspx?si=142459/friedmannscenario.jpg" alt="Alexander Friedmann's Cosmic Scenarios" align="left"/><p>The Russian mathematician and physicist Alexander Friedmann (18881925) is well known among relativists, but his contributions to cosmology are largely misunderstood. Even the Royal Swedish Academy of Sciences misrepresented Friedmann's work in the 2011 Nobel Prize scientific background essay. Friedmann was the first physicist who demonstrated that Albert Einstein's general relativity admits nonstatic solutions, and the universe can expand, oscillate, and be born in a singularity. Friedmann's conclusion was based on his analysis of an elliptic integral; this worksheet employs Maple's utility of handling elliptic integrals to present Friedmann's results graphically. Friedmann's differential equation governing the evolution of the universe based on Einstein's general theory of relativity is also derived using Maple's tensor package. </p>
142459
Sun, 20 Jan 2013 05:00:00 Z
Dr. Frank Wang
Dr. Frank Wang

Green's functions
http://www.maplesoft.com/applications/view.aspx?SID=136531&ref=Feed
<p>This is a derivation and specific construction and application of Green's functions as an "Inverse" to differential operators.</p>
<img src="/applications/images/app_image_blank_lg.jpg" alt="Green's functions" align="left"/><p>This is a derivation and specific construction and application of Green's functions as an "Inverse" to differential operators.</p>
136531
Wed, 15 Aug 2012 04:00:00 Z
Dr. Jack Wagner
Dr. Jack Wagner

Classroom Tips and Techniques: SliderControl of Parameters in Numeric Solutions of ODEs
http://www.maplesoft.com/applications/view.aspx?SID=135062&ref=Feed
In the article "Sliders for ParameterDependent Curves", and again in the article "Caustics for a Plane Curve", the use of sliders to control parameters was explored. This month's article explores the use of sliders to control parameters in a differential equation that must be solved numerically.
<img src="/view.aspx?si=135062/thumb.jpg" alt="Classroom Tips and Techniques: SliderControl of Parameters in Numeric Solutions of ODEs" align="left"/>In the article "Sliders for ParameterDependent Curves", and again in the article "Caustics for a Plane Curve", the use of sliders to control parameters was explored. This month's article explores the use of sliders to control parameters in a differential equation that must be solved numerically.
135062
Tue, 12 Jun 2012 04:00:00 Z
Dr. Robert Lopez
Dr. Robert Lopez

Classroom Tips and Techniques: Fourier Series and an Orthogonal Expansions Package
http://www.maplesoft.com/applications/view.aspx?SID=134198&ref=Feed
The OrthogonalExpansions package contributed to the Maple Application Center by Dr. Sergey Moiseev is considered as a tool for generating a Fourier series and its partial sums. This package provides commands for expansions in 17 other bases of orthogonal functions. In addition to looking at the Fourier series option, this article also considers the Bessel series expansion.
<img src="/view.aspx?si=134198/thumb.jpg" alt="Classroom Tips and Techniques: Fourier Series and an Orthogonal Expansions Package" align="left"/>The OrthogonalExpansions package contributed to the Maple Application Center by Dr. Sergey Moiseev is considered as a tool for generating a Fourier series and its partial sums. This package provides commands for expansions in 17 other bases of orthogonal functions. In addition to looking at the Fourier series option, this article also considers the Bessel series expansion.
134198
Mon, 14 May 2012 04:00:00 Z
Dr. Robert Lopez
Dr. Robert Lopez

Differential Equations in Maple 16
http://www.maplesoft.com/applications/view.aspx?SID=132225&ref=Feed
Maple 16 continues to push the frontiers in differential equation solving and extends its lead in computing closedform solutions to differential equations, adding in even more classes of problems that can be handled. The numeric ODE, DAE, and PDE solvers also continue to evolve. Maple 16 shows significant performance improvements for these solvers, as well as enhanced event handling. This application illustrates many of these improvements.
<img src="/view.aspx?si=132225/thumb2.jpg" alt="Differential Equations in Maple 16" align="left"/>Maple 16 continues to push the frontiers in differential equation solving and extends its lead in computing closedform solutions to differential equations, adding in even more classes of problems that can be handled. The numeric ODE, DAE, and PDE solvers also continue to evolve. Maple 16 shows significant performance improvements for these solvers, as well as enhanced event handling. This application illustrates many of these improvements.
132225
Tue, 27 Mar 2012 04:00:00 Z
Maplesoft
Maplesoft

Spherical Pendulum with Animation
http://www.maplesoft.com/applications/view.aspx?SID=132143&ref=Feed
<p>Some years ago I have written a Maple document ( already on Maple's online) on the subject of animating a simple pendulum for large angles of oscillation. This gave me the chance to test Maple command JacobiSN(time, k). I was very much pleased to see Maple do a wonderful job in getting these Jacobi's elliptic functions without a glitch.<br />Today I am back to these same functions for a similar purpose though much more sophisticated than the previous one.<br />The idea is:<br />1 to get the differential equations of motion for the Spherical Pendulum (SP),<br />2 to solve them,<br />3 to use Maple for finding the inverse of these Elliptic Integrals i.e. finding the displacement z as function of time,<br />4 to get a set of coordinates [x, y, z] for the positions of the bob at different times for plotting,<br />5 finally to work out the necessary steps for the purpose of animation.<br />It turns out that even with only 3 oscillations where each is defined with only 20 positions of the bob for a total of 60 points on the graph, the animation is so overwhelming that Maple reports:<br /> " the length of the output exceeds 1 million".<br />Not withstanding this warning, Maple did a perfect job by getting the animation to my satisfaction. <br />Note that with only 60 positions of the bob, the present article length is equal to 11.3 MB! To be able to upload it, I have to save it without running the last command related to the animation. Doing so I reduced it to a mere 570 KB.<br /><br />It was tiring to get through a jumble of formulas, calculations and programming so I wonder why I have to go through all this trouble to get this animation and yet one can get the same thing with much better animation from the internet. I think the reason is the challenge to be able to do things that others have done before and secondly the idea of creating something form nothing then to see it working as expected, gives (at least to me) a great deal of pleasure and satisfaction.<br />This is beside the fact that, to my knowledge, no such animation for (SP) has been published on Maple online with detailed calculations & programming as I did.<br /><br /></p>
<img src="/view.aspx?si=132143/433082\Spherical_Pendulum_p.jpg" alt="Spherical Pendulum with Animation" align="left"/><p>Some years ago I have written a Maple document ( already on Maple's online) on the subject of animating a simple pendulum for large angles of oscillation. This gave me the chance to test Maple command JacobiSN(time, k). I was very much pleased to see Maple do a wonderful job in getting these Jacobi's elliptic functions without a glitch.<br />Today I am back to these same functions for a similar purpose though much more sophisticated than the previous one.<br />The idea is:<br />1 to get the differential equations of motion for the Spherical Pendulum (SP),<br />2 to solve them,<br />3 to use Maple for finding the inverse of these Elliptic Integrals i.e. finding the displacement z as function of time,<br />4 to get a set of coordinates [x, y, z] for the positions of the bob at different times for plotting,<br />5 finally to work out the necessary steps for the purpose of animation.<br />It turns out that even with only 3 oscillations where each is defined with only 20 positions of the bob for a total of 60 points on the graph, the animation is so overwhelming that Maple reports:<br /> " the length of the output exceeds 1 million".<br />Not withstanding this warning, Maple did a perfect job by getting the animation to my satisfaction. <br />Note that with only 60 positions of the bob, the present article length is equal to 11.3 MB! To be able to upload it, I have to save it without running the last command related to the animation. Doing so I reduced it to a mere 570 KB.<br /><br />It was tiring to get through a jumble of formulas, calculations and programming so I wonder why I have to go through all this trouble to get this animation and yet one can get the same thing with much better animation from the internet. I think the reason is the challenge to be able to do things that others have done before and secondly the idea of creating something form nothing then to see it working as expected, gives (at least to me) a great deal of pleasure and satisfaction.<br />This is beside the fact that, to my knowledge, no such animation for (SP) has been published on Maple online with detailed calculations & programming as I did.<br /><br /></p>
132143
Mon, 26 Mar 2012 04:00:00 Z
Dr. Ahmed Baroudy
Dr. Ahmed Baroudy

Parameterizing Motion along a Curve
http://www.maplesoft.com/applications/view.aspx?SID=130465&ref=Feed
<p>We use the EulerLagrange equation to parameterize the motion of a bead on a parabola, a helix, and a piecewise defined combination of the two.</p>
<img src="/applications/images/app_image_blank_lg.jpg" alt="Parameterizing Motion along a Curve" align="left"/><p>We use the EulerLagrange equation to parameterize the motion of a bead on a parabola, a helix, and a piecewise defined combination of the two.</p>
130465
Wed, 08 Feb 2012 05:00:00 Z
Shawn Hedman
Shawn Hedman

Classroom Tips and Techniques: An Undamped Coupled Oscillator
http://www.maplesoft.com/applications/view.aspx?SID=129521&ref=Feed
<p>Even for just three degrees of freedom, an undamped coupled oscillator modeled by the ODE system <em>M</em> ü + <em>K</em> u = 0 is difficult to solve analytically because, ultimately, a cubic characteristic equation has to be solve exactly. Instead, we simultaneously diagonalize <em>M</em> and <em>K</em>, the mass and stiffness matrices, thereby uncoupling the equations, and obtaining an explicit solution.</p>
<img src="/view.aspx?si=129521/thumb.jpg" alt="Classroom Tips and Techniques: An Undamped Coupled Oscillator" align="left"/><p>Even for just three degrees of freedom, an undamped coupled oscillator modeled by the ODE system <em>M</em> ü + <em>K</em> u = 0 is difficult to solve analytically because, ultimately, a cubic characteristic equation has to be solve exactly. Instead, we simultaneously diagonalize <em>M</em> and <em>K</em>, the mass and stiffness matrices, thereby uncoupling the equations, and obtaining an explicit solution.</p>
129521
Tue, 10 Jan 2012 05:00:00 Z
Dr. Robert Lopez
Dr. Robert Lopez

Coriolis Effect
http://www.maplesoft.com/applications/view.aspx?SID=1437&ref=Feed
The Coriolis effect is a force that modifies the trajectory of falling object on Earth. It is due to the rotation of the referential and, thereby, it is not a real force. The mathematical expression of this effect is obtained from the crossproduct of Earth's angular velocity (omega) with the object's linear velocity (v). The exact equation is F = 2m(v x omega). This worksheet demonstrates the action of the Coriolis effect on a projectile launched from our planet. It includes a graphic of the projectile's path as well as a procedure that determines how far the projectile will travel.
<img src="/view.aspx?si=1437/coriolis_sm.jpg" alt="Coriolis Effect" align="left"/>The Coriolis effect is a force that modifies the trajectory of falling object on Earth. It is due to the rotation of the referential and, thereby, it is not a real force. The mathematical expression of this effect is obtained from the crossproduct of Earth's angular velocity (omega) with the object's linear velocity (v). The exact equation is F = 2m(v x omega). This worksheet demonstrates the action of the Coriolis effect on a projectile launched from our planet. It includes a graphic of the projectile's path as well as a procedure that determines how far the projectile will travel.
1437
Fri, 09 Dec 2011 05:00:00 Z
Pascal ThÃ©riault Lauzier
Pascal ThÃ©riault Lauzier

Classroom Tips and Techniques: Simultaneous Diagonalization and the Generalized Eigenvalue Problem
http://www.maplesoft.com/applications/view.aspx?SID=128444&ref=Feed
<p>This article explores the connections between the generalized eigenvalue problem and the problem of simultaneously diagonalizing a pair of <em>n × n</em> matrices.</p>
<p>Given the <em>n × n</em> matrices <em>A</em> and <em>B</em>, the <em>generalized eigenvalue problem</em> seeks the eigenpairs <em>(lambda<sub>k</sub>, x<sub>k</sub>)</em>, solutions of the equation <em>Ax = lambda Bx</em>, or <em>(A  lambda B) x = 0</em>. If <em>B</em> is nonsingular, the eigenpairs of <em>B<sup>1</sup> A</em> are solutions. If a matrix <em>S</em> exists for which<em> S<sup>T</sup> A S = Lambda</em>, and <em>S<sup>T</sup> B S = I</em>, where <em>Lambda</em> is a diagonal matrix and <em>I</em> is the <em>n × n</em> identity, then <em>A</em> and <em>B</em> are said to be <em>diagonalized simultaneously</em>, in which case the diagonal entries of <em>Lambda</em> are the generalized eigenvalues for <em>A</em> and <em>B</em>. Such a matrix <em>S</em> exists if <em>A</em> is symmetric and <em>B</em> is positive definite. (Our definition of positive definite includes symmetry.)</p>
<img src="/view.aspx?si=128444/thumb.jpg" alt="Classroom Tips and Techniques: Simultaneous Diagonalization and the Generalized Eigenvalue Problem" align="left"/><p>This article explores the connections between the generalized eigenvalue problem and the problem of simultaneously diagonalizing a pair of <em>n × n</em> matrices.</p>
<p>Given the <em>n × n</em> matrices <em>A</em> and <em>B</em>, the <em>generalized eigenvalue problem</em> seeks the eigenpairs <em>(lambda<sub>k</sub>, x<sub>k</sub>)</em>, solutions of the equation <em>Ax = lambda Bx</em>, or <em>(A  lambda B) x = 0</em>. If <em>B</em> is nonsingular, the eigenpairs of <em>B<sup>1</sup> A</em> are solutions. If a matrix <em>S</em> exists for which<em> S<sup>T</sup> A S = Lambda</em>, and <em>S<sup>T</sup> B S = I</em>, where <em>Lambda</em> is a diagonal matrix and <em>I</em> is the <em>n × n</em> identity, then <em>A</em> and <em>B</em> are said to be <em>diagonalized simultaneously</em>, in which case the diagonal entries of <em>Lambda</em> are the generalized eigenvalues for <em>A</em> and <em>B</em>. Such a matrix <em>S</em> exists if <em>A</em> is symmetric and <em>B</em> is positive definite. (Our definition of positive definite includes symmetry.)</p>
128444
Tue, 06 Dec 2011 05:00:00 Z
Dr. Robert Lopez
Dr. Robert Lopez

An Epidemic Model (for Influenza or Zombies)
http://www.maplesoft.com/applications/view.aspx?SID=127836&ref=Feed
<p>Systems of differential equations can be used to model an epidemic of influenza or of zombies. This is an interactive Maple document suitable for use in courses on mathematical biology or differential equations or calculus courses that include differential equations. No knowledge of Maple is required.</p>
<img src="/view.aspx?si=127836/Cholera.jpg" alt="An Epidemic Model (for Influenza or Zombies)" align="left"/><p>Systems of differential equations can be used to model an epidemic of influenza or of zombies. This is an interactive Maple document suitable for use in courses on mathematical biology or differential equations or calculus courses that include differential equations. No knowledge of Maple is required.</p>
127836
Thu, 17 Nov 2011 05:00:00 Z
Dr. Robert Israel
Dr. Robert Israel

The Orbit of Kepler 16b
http://www.maplesoft.com/applications/view.aspx?SID=126766&ref=Feed
<p>NASA's Kepler space telescope recently made the news by finding a planet that orbits a doublestar system, a situation that brought to mind the fictional planet Tatooine of the movie Star Wars. On such a planet, if it had a solid surface, you could see a double sunset. <br /><br />This worksheet explores the orbital mechanics of such a system.</p>
<img src="/view.aspx?si=126766/kepler16b.png" alt="The Orbit of Kepler 16b" align="left"/><p>NASA's Kepler space telescope recently made the news by finding a planet that orbits a doublestar system, a situation that brought to mind the fictional planet Tatooine of the movie Star Wars. On such a planet, if it had a solid surface, you could see a double sunset. <br /><br />This worksheet explores the orbital mechanics of such a system.</p>
126766
Tue, 18 Oct 2011 04:00:00 Z
Dr. Robert Israel
Dr. Robert Israel

Classroom Tips and Techniques: Gems 1620 from the Red Book of Maple Magic
http://www.maplesoft.com/applications/view.aspx?SID=125886&ref=Feed
From the Red Book of Maple Magic, Gems 1620: Vectors with assumptions in VectorCalculus, aliasing commands to symbols, setting iterated integrals from the Expression palette, writing a slider value to a label, and writing text to a math container.
<img src="/view.aspx?si=125886/thumb.jpg" alt="Classroom Tips and Techniques: Gems 1620 from the Red Book of Maple Magic" align="left"/>From the Red Book of Maple Magic, Gems 1620: Vectors with assumptions in VectorCalculus, aliasing commands to symbols, setting iterated integrals from the Expression palette, writing a slider value to a label, and writing text to a math container.
125886
Fri, 23 Sep 2011 04:00:00 Z
Dr. Robert Lopez
Dr. Robert Lopez