Differential Equations: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=136
en-us2016 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemWed, 26 Oct 2016 07:33:04 GMTWed, 26 Oct 2016 07:33:04 GMTNew applications in the Differential Equations categoryhttp://www.mapleprimes.com/images/mapleapps.gifDifferential Equations: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=136
Digitizing mathematics: ODEs, Special Functions and Solutions to Einstein's Equations
http://www.maplesoft.com/applications/view.aspx?SID=154174&ref=Feed
The material below was presented in the <A HREF="https://www.fields.utoronto.ca/programs/scientific/15-16/semantic/">Semantic Representation of Mathematical Knowledge Workshop</A>, February 3-5, 2016 at the Fields Institute, University of Toronto. It shows the approach used for “digitizing mathematical knowledge" regarding Differential Equations, Special Functions and Solutions to Einstein's equations. While for these areas using databases of information helps (for example textbooks frequently contain these sort of databases), these are areas that, at the same time, are very suitable for using algorithmic mathematical approaches, that result in much richer mathematics than what can be hard-coded into a database. The material also focuses on an interesting cherry-picked collection of Maple functionality, that I think is beautiful, not well know, and seldom focused inter-related as here.<BR><BR>
This application is also featured in a <A HREF="http://www.mapleprimes.com/posts/206715-Digitizing-Mathematics-ODEs-Special">MaplePrimes blog post</A>.<img src="/applications/images/app_image_blank_lg.jpg" alt="Digitizing mathematics: ODEs, Special Functions and Solutions to Einstein's Equations" align="left"/>The material below was presented in the <A HREF="https://www.fields.utoronto.ca/programs/scientific/15-16/semantic/">Semantic Representation of Mathematical Knowledge Workshop</A>, February 3-5, 2016 at the Fields Institute, University of Toronto. It shows the approach used for “digitizing mathematical knowledge" regarding Differential Equations, Special Functions and Solutions to Einstein's equations. While for these areas using databases of information helps (for example textbooks frequently contain these sort of databases), these are areas that, at the same time, are very suitable for using algorithmic mathematical approaches, that result in much richer mathematics than what can be hard-coded into a database. The material also focuses on an interesting cherry-picked collection of Maple functionality, that I think is beautiful, not well know, and seldom focused inter-related as here.<BR><BR>
This application is also featured in a <A HREF="http://www.mapleprimes.com/posts/206715-Digitizing-Mathematics-ODEs-Special">MaplePrimes blog post</A>.154174Fri, 07 Oct 2016 04:00:00 ZDr. Edgardo Cheb-TerrabDr. Edgardo Cheb-TerrabODEs, PDE solutions: when are they "general"?
http://www.maplesoft.com/applications/view.aspx?SID=154165&ref=Feed
This presentation discusses the concept of “general solution” of a Partial Differential Equation, or a system of them, possibly including ODEs and/or algebraic equations, and shows how to tell whether a solution returned by Maple’s <A HREF="/support/help/Maple/view.aspx?path=pdsolve">pdsolve</A> is or not a general (as opposed to particular) solution.<BR><BR>
This application is also the subject of a <A HREF="http://www.mapleprimes.com/posts/204437-PDE-Solutions-When-Are-They-general">blog post on MaplePrimes</A>.<img src="/applications/images/app_image_blank_lg.jpg" alt="ODEs, PDE solutions: when are they "general"?" align="left"/>This presentation discusses the concept of “general solution” of a Partial Differential Equation, or a system of them, possibly including ODEs and/or algebraic equations, and shows how to tell whether a solution returned by Maple’s <A HREF="/support/help/Maple/view.aspx?path=pdsolve">pdsolve</A> is or not a general (as opposed to particular) solution.<BR><BR>
This application is also the subject of a <A HREF="http://www.mapleprimes.com/posts/204437-PDE-Solutions-When-Are-They-general">blog post on MaplePrimes</A>.154165Fri, 30 Sep 2016 04:00:00 ZDr. Edgardo Cheb-TerrabDr. Edgardo Cheb-TerrabFactorizing with non-commutative variables
http://www.maplesoft.com/applications/view.aspx?SID=154166&ref=Feed
New capabilities for factorizing expressions involving noncommutative variables are presented and illustrated with a set of examples.<BR><BR>
This application is also the subject of a <A HREF="http://www.mapleprimes.com/posts/201368-New-Factorizing-With-Noncommutative-Variables">blog post on MaplePrimes</A>.<img src="/applications/images/app_image_blank_lg.jpg" alt="Factorizing with non-commutative variables" align="left"/>New capabilities for factorizing expressions involving noncommutative variables are presented and illustrated with a set of examples.<BR><BR>
This application is also the subject of a <A HREF="http://www.mapleprimes.com/posts/201368-New-Factorizing-With-Noncommutative-Variables">blog post on MaplePrimes</A>.154166Fri, 30 Sep 2016 04:00:00 ZDr. Edgardo Cheb-TerrabDr. Edgardo Cheb-TerrabNew developments on exact solutions for PDEs with Boundary Conditions
http://www.maplesoft.com/applications/view.aspx?SID=154169&ref=Feed
A collection of two presentations that discuss and illustrate newly implemented methods for computing exact solutions to Partial Differential Equations subject to Boundary conditions.<BR><BR>
These applications are also discussed in two MaplePrimes blog posts:
<UL>
<LI><A HREF="http://www.mapleprimes.com/posts/204436-New-Developments-On-Exact-Solutions">New developments on exact solutions for PDEs with Boundary Conditions</A>
<LI><A HREF="http://www.mapleprimes.com/posts/201226-PDEs-And-Boundary-Conditions--New-Developments">PDEs and Boundary Conditions - new developments</A>
</UL><img src="/applications/images/app_image_blank_lg.jpg" alt="New developments on exact solutions for PDEs with Boundary Conditions" align="left"/>A collection of two presentations that discuss and illustrate newly implemented methods for computing exact solutions to Partial Differential Equations subject to Boundary conditions.<BR><BR>
These applications are also discussed in two MaplePrimes blog posts:
<UL>
<LI><A HREF="http://www.mapleprimes.com/posts/204436-New-Developments-On-Exact-Solutions">New developments on exact solutions for PDEs with Boundary Conditions</A>
<LI><A HREF="http://www.mapleprimes.com/posts/201226-PDEs-And-Boundary-Conditions--New-Developments">PDEs and Boundary Conditions - new developments</A>
</UL>154169Fri, 30 Sep 2016 04:00:00 ZDr. Edgardo Cheb-TerrabDr. Edgardo Cheb-TerrabODEs, PDEs and Special Functions
http://www.maplesoft.com/applications/view.aspx?SID=154164&ref=Feed
This presentation illustrates the Maple capabilities for studying and solving ODEs and PDEs, implemented within the <A HREF="/support/help/Maple/view.aspx?path=DEtools">DEtools</A> and <A HREF="/support/help/Maple/view.aspx?path=PDEtools">PDEtools</A> packages, as well as getting information about and working with Special functions of the mathematical language, implemented within the <A HREF="/support/help/Maple/view.aspx?path=FunctionAdvisor">FunctionAdvisor</A>, the conversion network for mathematical functions and the <A HREF="/support/help/Maple/view.aspx?path=MathematicalFunctions">MathematicalFunctions</A> package.<BR><BR>
This application is also the subject of a <A HREF="http://www.mapleprimes.com/posts/149877-ODEs-PDEs-And-Special-Functions">blog post on MaplePrimes</A>.<img src="/applications/images/app_image_blank_lg.jpg" alt="ODEs, PDEs and Special Functions" align="left"/>This presentation illustrates the Maple capabilities for studying and solving ODEs and PDEs, implemented within the <A HREF="/support/help/Maple/view.aspx?path=DEtools">DEtools</A> and <A HREF="/support/help/Maple/view.aspx?path=PDEtools">PDEtools</A> packages, as well as getting information about and working with Special functions of the mathematical language, implemented within the <A HREF="/support/help/Maple/view.aspx?path=FunctionAdvisor">FunctionAdvisor</A>, the conversion network for mathematical functions and the <A HREF="/support/help/Maple/view.aspx?path=MathematicalFunctions">MathematicalFunctions</A> package.<BR><BR>
This application is also the subject of a <A HREF="http://www.mapleprimes.com/posts/149877-ODEs-PDEs-And-Special-Functions">blog post on MaplePrimes</A>.154164Fri, 30 Sep 2016 04:00:00 ZDr. Edgardo Cheb-TerrabDr. Edgardo Cheb-TerrabEl Niño Temperature Anomalies Modeled by a Delay Differential Equation
http://www.maplesoft.com/applications/view.aspx?SID=154142&ref=Feed
Delay differential equations are differential equations in which the derivative of the unknown function at a certain time depends on past values of the function and/or its derivatives. Max J. Suarez and Paul S. Schopf used such an equation to model the El Niño phenomenon. This worksheet demonstrate how Maple's dsolve command can be used to solve a delay differential equation numerically.<img src="/view.aspx?si=154142/waves.png" alt="El Niño Temperature Anomalies Modeled by a Delay Differential Equation" align="left"/>Delay differential equations are differential equations in which the derivative of the unknown function at a certain time depends on past values of the function and/or its derivatives. Max J. Suarez and Paul S. Schopf used such an equation to model the El Niño phenomenon. This worksheet demonstrate how Maple's dsolve command can be used to solve a delay differential equation numerically.154142Mon, 29 Aug 2016 04:00:00 ZDr. Frank WangDr. Frank WangInterpretación geométrica del proceso de solución de una ecuación trigonométrica
http://www.maplesoft.com/applications/view.aspx?SID=154110&ref=Feed
Esta aplicación tiene como objetivo ayudar al estudiante a comprender el significado geométrico de resolver la ecuación trigonométrica sen(theta) = c en un intervalo de longitud 2Pi.
La barra deslizante de la aplicación permite variar el valor de c, mientras que los gráficos ayudan al estudiante a visualizar y comprender el proceso de búsqueda de soluciones de la ecuación trigonométrica de interés.<img src="/view.aspx?si=154110/232a3a3435a381a76ee84170be3fcee2.gif" alt="Interpretación geométrica del proceso de solución de una ecuación trigonométrica" align="left"/>Esta aplicación tiene como objetivo ayudar al estudiante a comprender el significado geométrico de resolver la ecuación trigonométrica sen(theta) = c en un intervalo de longitud 2Pi.
La barra deslizante de la aplicación permite variar el valor de c, mientras que los gráficos ayudan al estudiante a visualizar y comprender el proceso de búsqueda de soluciones de la ecuación trigonométrica de interés.154110Tue, 24 May 2016 04:00:00 ZRanferi GutierrezRanferi GutierrezDifferential Equation Solver
http://www.maplesoft.com/applications/view.aspx?SID=154102&ref=Feed
The application allows you to solve Ordinary Differential Equations. Enter an ODE, provide initial conditions and then click solve.
<BR><BR>
An online version of this <A HREF="http://maplecloud.maplesoft.com/application.jsp?appId=5691363796451328">Differential Equation Solver</A> is also available in the MapleCloud.<img src="/view.aspx?si=154102/solver.PNG" alt="Differential Equation Solver" align="left"/>The application allows you to solve Ordinary Differential Equations. Enter an ODE, provide initial conditions and then click solve.
<BR><BR>
An online version of this <A HREF="http://maplecloud.maplesoft.com/application.jsp?appId=5691363796451328">Differential Equation Solver</A> is also available in the MapleCloud.154102Tue, 17 May 2016 04:00:00 ZMaplesoftMaplesoftCampo de direcciones: Un caso de estudio.
http://www.maplesoft.com/applications/view.aspx?SID=153954&ref=Feed
En esta hoja se aplica el campo de direcciones al estudio cualitativo de las soluciones de una ecuación diferencial que describe, utilizando un modelo sencillo, el fenómeno de la pesca.
En el modelo se asume que puede exitir sobrepoblación y/o captura, lo que da oportunidad al estudiante de lograr una mayor comprensión del fenómeno, así como de aprender cómo extraer información cualitativa de los campos de direcciones.<img src="/view.aspx?si=153954/Captura.PNG" alt="Campo de direcciones: Un caso de estudio." align="left"/>En esta hoja se aplica el campo de direcciones al estudio cualitativo de las soluciones de una ecuación diferencial que describe, utilizando un modelo sencillo, el fenómeno de la pesca.
En el modelo se asume que puede exitir sobrepoblación y/o captura, lo que da oportunidad al estudiante de lograr una mayor comprensión del fenómeno, así como de aprender cómo extraer información cualitativa de los campos de direcciones.153954Sat, 23 Jan 2016 05:00:00 ZDr. Ranferi GutierrezDr. Ranferi GutierrezDemo Worksheet for Numerical Delay Differential Equation Solution
http://www.maplesoft.com/applications/view.aspx?SID=153939&ref=Feed
<P>This application shows several examples of modeling using delay differential equations in Maple. These examples are from the webinar <A HREF="http://www.maplesoft.com/products/maple/demo/player/2015/solvingdelaydiffeq.aspx">Solving Delay Differential Equations</A>.</P>
<P>Note: Requires Maple 2015.2 or later.</P><img src="/view.aspx?si=153939/dde.PNG" alt="Demo Worksheet for Numerical Delay Differential Equation Solution" align="left"/><P>This application shows several examples of modeling using delay differential equations in Maple. These examples are from the webinar <A HREF="http://www.maplesoft.com/products/maple/demo/player/2015/solvingdelaydiffeq.aspx">Solving Delay Differential Equations</A>.</P>
<P>Note: Requires Maple 2015.2 or later.</P>153939Wed, 16 Dec 2015 05:00:00 ZAllan WittkopfAllan WittkopfThe Classic SIR Model
http://www.maplesoft.com/applications/view.aspx?SID=153877&ref=Feed
<P>This interactive application explores the classical SIR model for the spread of disease, which assumes that a population can be divided into three distinct compartments - S is the proportion of susceptibles, I is the proportion of infected persons and R is the proportion of persons that have recovered from infection and are now immune against the disease.</P>
<P>
<B>Also:</B> <A HREF="http://maplecloud.maplesoft.com/application.jsp?appId=4837052487041024">View and interact with this app in the MapleCloud!</A></P><img src="/view.aspx?si=153877/sir_classic.png" alt="The Classic SIR Model" align="left"/><P>This interactive application explores the classical SIR model for the spread of disease, which assumes that a population can be divided into three distinct compartments - S is the proportion of susceptibles, I is the proportion of infected persons and R is the proportion of persons that have recovered from infection and are now immune against the disease.</P>
<P>
<B>Also:</B> <A HREF="http://maplecloud.maplesoft.com/application.jsp?appId=4837052487041024">View and interact with this app in the MapleCloud!</A></P>153877Wed, 16 Sep 2015 04:00:00 ZGünter EdenharterGünter EdenharterThe SIR model with births and deaths
http://www.maplesoft.com/applications/view.aspx?SID=153878&ref=Feed
<P>This interactive application explores a variation of the classic SIR model for the spread of disease. The classical SIR model assumes that a population can be divided into three distinct compartments: S is the proportion of susceptibles, I is the proportion of infected persons and R is the proportion of persons that have recovered from infection and are now immune against the disease. One extension to the classic SIR model is to add births and deaths to the model. Thus there is an inflow of new susceptibles and an outflow from all three compartments.</P>
<P>
<B>Also:</B> <A HREF="http://maplecloud.maplesoft.com/application.jsp?appId=6584880737550336">View and interact with this app in the MapleCloud!</A></P><img src="/view.aspx?si=153878/sir_births_deaths.png" alt="The SIR model with births and deaths" align="left"/><P>This interactive application explores a variation of the classic SIR model for the spread of disease. The classical SIR model assumes that a population can be divided into three distinct compartments: S is the proportion of susceptibles, I is the proportion of infected persons and R is the proportion of persons that have recovered from infection and are now immune against the disease. One extension to the classic SIR model is to add births and deaths to the model. Thus there is an inflow of new susceptibles and an outflow from all three compartments.</P>
<P>
<B>Also:</B> <A HREF="http://maplecloud.maplesoft.com/application.jsp?appId=6584880737550336">View and interact with this app in the MapleCloud!</A></P>153878Wed, 16 Sep 2015 04:00:00 ZGünter EdenharterGünter EdenharterThe SEIR model with births and deaths
http://www.maplesoft.com/applications/view.aspx?SID=153879&ref=Feed
<P>This interactive application explores the SEIR model for the spread of disease. The SEIR model is an extension of the classical SIR (Susceptibles, Infected, Recovered) model, where a fourth compartment is added that contains exposed persons which are infected but are not yet infectious. The SEIR (Susceptibles, Exposed, Infectious, Recovered) model as presented here covers also births and deaths.</P>
<P>
<B>Also:</B> <A HREF="http://maplecloud.maplesoft.com/application.jsp?appId=6407056173039616">View and interact with this app in the MapleCloud!</A></P><img src="/view.aspx?si=153879/seir.png" alt="The SEIR model with births and deaths" align="left"/><P>This interactive application explores the SEIR model for the spread of disease. The SEIR model is an extension of the classical SIR (Susceptibles, Infected, Recovered) model, where a fourth compartment is added that contains exposed persons which are infected but are not yet infectious. The SEIR (Susceptibles, Exposed, Infectious, Recovered) model as presented here covers also births and deaths.</P>
<P>
<B>Also:</B> <A HREF="http://maplecloud.maplesoft.com/application.jsp?appId=6407056173039616">View and interact with this app in the MapleCloud!</A></P>153879Wed, 16 Sep 2015 04:00:00 ZGünter EdenharterGünter EdenharterThe Comet 67P/Churyumov-Gerasimenko, Rosetta & Philae
http://www.maplesoft.com/applications/view.aspx?SID=153706&ref=Feed
<p> Abstract<br /><br />The Rosetta space probe launched 10 years ago by the European Space Agency (ESA) arrived recently (November 12, 2014) at the site of the comet known as 67P/Churyumov-Gerasimenco after a trip of 4 billions miles from Earth. After circling the comet, Rosetta released its precious load : the lander Philae packed with 21 different scientific instruments for the study of the comet with the main purpose : the origin of our solar system and possibly the origin of life on our planet.<br /><br />Our plan is rather a modest one since all we want is to get , by calculations, specific data concerning the comet and its lander.<br />We shall take a simplified model and consider the comet as a perfect solid sphere to which we can apply Newton's laws.<br /><br />We want to find:<br /><br />I- the acceleration on the comet surface ,<br />II- its radius,<br />III- its density,<br />IV- the velocity of Philae just after the 1st bounce off the comet (it has bounced twice),<br />V- the time for Philae to reach altitude of 1000 m above the comet .<br /><br />We shall compare our findings with the already known data to see how close our simplified mathematical model findings are to the duck-shaped comet already known results.<br />It turned out that our calculations for a sphere shaped comet are very close to the already known data.<br /><br />Conclusion<br /><br />Even with a shape that defies the application of any mechanical laws we can always get very close to reality by adopting a simplified mathematical model in any preliminary study of a complicated problem.<br /><br /></p><img src="/applications/images/app_image_blank_lg.jpg" alt="The Comet 67P/Churyumov-Gerasimenko, Rosetta & Philae" align="left"/><p> Abstract<br /><br />The Rosetta space probe launched 10 years ago by the European Space Agency (ESA) arrived recently (November 12, 2014) at the site of the comet known as 67P/Churyumov-Gerasimenco after a trip of 4 billions miles from Earth. After circling the comet, Rosetta released its precious load : the lander Philae packed with 21 different scientific instruments for the study of the comet with the main purpose : the origin of our solar system and possibly the origin of life on our planet.<br /><br />Our plan is rather a modest one since all we want is to get , by calculations, specific data concerning the comet and its lander.<br />We shall take a simplified model and consider the comet as a perfect solid sphere to which we can apply Newton's laws.<br /><br />We want to find:<br /><br />I- the acceleration on the comet surface ,<br />II- its radius,<br />III- its density,<br />IV- the velocity of Philae just after the 1st bounce off the comet (it has bounced twice),<br />V- the time for Philae to reach altitude of 1000 m above the comet .<br /><br />We shall compare our findings with the already known data to see how close our simplified mathematical model findings are to the duck-shaped comet already known results.<br />It turned out that our calculations for a sphere shaped comet are very close to the already known data.<br /><br />Conclusion<br /><br />Even with a shape that defies the application of any mechanical laws we can always get very close to reality by adopting a simplified mathematical model in any preliminary study of a complicated problem.<br /><br /></p>153706Mon, 17 Nov 2014 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyThe 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.153511Thu, 20 Feb 2014 05:00:00 ZProf. Michael MonaganProf. Michael MonaganThe 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 heat-flow 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 heat-flow in a house, and shows how he uses this model in his class.153491Wed, 22 Jan 2014 05:00:00 ZProf. Michael MonaganProf. Michael MonaganClassroom Tips and Techniques: Slider-Control of Parameters in an ODE
http://www.maplesoft.com/applications/view.aspx?SID=152112&ref=Feed
Several ways to provide slider-control 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 2-dimensional slider is constructed.<img src="/view.aspx?si=152112/thumb.jpg" alt="Classroom Tips and Techniques: Slider-Control of Parameters in an ODE" align="left"/>Several ways to provide slider-control 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 2-dimensional slider is constructed.152112Mon, 23 Sep 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom 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.149514Tue, 16 Jul 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Gems 31-35 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 initial-value 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 31-35 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 initial-value 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.147092Fri, 10 May 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezDynamics 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 explain--and to predict--how 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 explain--and to predict--how 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>146801Mon, 06 May 2013 04:00:00 ZDouglas LewitDouglas Lewit