PDEs: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=149
en-us2021 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemSun, 11 Apr 2021 22:05:54 GMTSun, 11 Apr 2021 22:05:54 GMTNew applications in the PDEs categoryhttps://www.maplesoft.com/images/Application_center_hp.jpgPDEs: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=149
Burgers' Vortex
https://www.maplesoft.com/applications/view.aspx?SID=154424&ref=Feed
The Burgers vortex is a well known solution of the Navier Stokes equations that combines vorticity and shear. It allows for the study of a realistic flow in analytical form, thereby offering intuition for more complex flows. However, the cylindrical coordinate system makes certain calculations cumbersome to carry out by hand. This worksheet allows the user to explore aspects of the flow without having to carry out the calculations.<img src="https://www.maplesoft.com/view.aspx?si=154424/burgersvortex.PNG" alt="Burgers' Vortex" style="max-width: 25%;" align="left"/>The Burgers vortex is a well known solution of the Navier Stokes equations that combines vorticity and shear. It allows for the study of a realistic flow in analytical form, thereby offering intuition for more complex flows. However, the cylindrical coordinate system makes certain calculations cumbersome to carry out by hand. This worksheet allows the user to explore aspects of the flow without having to carry out the calculations.https://www.maplesoft.com/applications/view.aspx?SID=154424&ref=FeedFri, 23 Mar 2018 04:00:00 ZEmilee CarsonEmilee CarsonPolarization of Dielectric Sphere .....
https://www.maplesoft.com/applications/view.aspx?SID=154296&ref=Feed
In this worksheet, we investigate the polarization of a dielectric sphere (dot) with a relative permittivitty "epsilon[Dot]" embedded in a dielectric matrix with a relative permittivitty "epsilon[Matrix]" and submitted to an uniform electrostatic field F oriented in z-axis direction. It's a fondamental and popular problem present in most of electromagnetism textbooks. First of all, we express Poisson equation in appropriate coordinates system:
"Delta V(r,theta,phi) = 0". We proceed to a full separation of variables and derive general expression of scalar electrostatic potential V(r,theta,phi). Then we particularize to a dielectric sphere surrounded by a dielectric matrix and give expressions of electrostatic potential V(r,theta) in the meridian plane (x0z) inside and outside the sphere by taking into account:
i) invariance property of the system under rotation around z-axis,
ii) choice of the plane z=0 as a reference of scalar electrostatic potential,
iii) regularity of V(r,theta) at the origine and very far from the sphere,
iv) continuity condition of scalar electrostatic potential V(r,theta) at the sphere surface,
v) continuity condition of normal components of electric displacement field D at the sphere surface.
The obtained expressions of V(r,theta) inside and outside the sphere allows as to derive expressions of electrostatic field F, electric displacement field D and polarization field P inside and outside dielectric dot in spherical coordinates as well as in cartesian rectangular coordinates. The paper is a proof of Maple algebraic and graphical capabilities in tackling the resolution of Poisson equation as a second order partial differential equation and also in displaying scalar electrostatic potential contourplot, electrostatic field lines as well as fieldplots of F, D and P inside and outside dielectric sphere.<img src="https://www.maplesoft.com/view.aspx?si=154296/fieldplot.PNG" alt="Polarization of Dielectric Sphere ....." style="max-width: 25%;" align="left"/>In this worksheet, we investigate the polarization of a dielectric sphere (dot) with a relative permittivitty "epsilon[Dot]" embedded in a dielectric matrix with a relative permittivitty "epsilon[Matrix]" and submitted to an uniform electrostatic field F oriented in z-axis direction. It's a fondamental and popular problem present in most of electromagnetism textbooks. First of all, we express Poisson equation in appropriate coordinates system:
"Delta V(r,theta,phi) = 0". We proceed to a full separation of variables and derive general expression of scalar electrostatic potential V(r,theta,phi). Then we particularize to a dielectric sphere surrounded by a dielectric matrix and give expressions of electrostatic potential V(r,theta) in the meridian plane (x0z) inside and outside the sphere by taking into account:
i) invariance property of the system under rotation around z-axis,
ii) choice of the plane z=0 as a reference of scalar electrostatic potential,
iii) regularity of V(r,theta) at the origine and very far from the sphere,
iv) continuity condition of scalar electrostatic potential V(r,theta) at the sphere surface,
v) continuity condition of normal components of electric displacement field D at the sphere surface.
The obtained expressions of V(r,theta) inside and outside the sphere allows as to derive expressions of electrostatic field F, electric displacement field D and polarization field P inside and outside dielectric dot in spherical coordinates as well as in cartesian rectangular coordinates. The paper is a proof of Maple algebraic and graphical capabilities in tackling the resolution of Poisson equation as a second order partial differential equation and also in displaying scalar electrostatic potential contourplot, electrostatic field lines as well as fieldplots of F, D and P inside and outside dielectric sphere.https://www.maplesoft.com/applications/view.aspx?SID=154296&ref=FeedMon, 18 Sep 2017 04:00:00 ZE. H. EL HAROUNY, A. IBRAL, S. NAKRA MOHAJER and J. EL KHAMKHAMIE. H. EL HAROUNY, A. IBRAL, S. NAKRA MOHAJER and J. EL KHAMKHAMIMathematics for Chemistry
https://www.maplesoft.com/applications/view.aspx?SID=154267&ref=Feed
This interactive electronic textbook in the form of Maple worksheets comprises two parts.
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Part I, mathematics for chemistry, is supposed to cover all mathematics that an instructor of chemistry might hope and expect that his students would learn, understand and be able to apply as a result of sufficient courses typically, but not exclusively, presented in departments of mathematics. Its nine chapters include (0) a summary and illustration of useful Maple commands, (1) arithmetic, algebra and elementary functions, (2) plotting, descriptive geometry, trigonometry, series, complex functions, (3) differential calculus of one variable, (4) integral calculus of one variable, (5) multivariate calculus, (6) linear algebra including matrix, vector, eigenvector, vector calculus, tensor, spreadsheet, (7) differential and integral equations, and (8) probability, distribution, treatment of laboratory data, linear and non-linear regression and optimization.
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Part II presents mathematical topics typically taught within chemistry courses, including (9) chemical equilibrium, (10) group theory, (11) graph theory, (12a) introduction to quantum mechanics and quantum chemistry, (14) applications of Fourier transforms in chemistry including electron diffraction, x-ray diffraction, microwave spectra, infrared and Raman spectra and nuclear-magnetic-resonance spectra, and (18) dielectric and magnetic properties of chemical matter.
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Other chapters are in preparation and will be released in due course.
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Last updated on March 19, 2019<img src="https://www.maplesoft.com/view.aspx?si=154267/molecule.PNG" alt="Mathematics for Chemistry" style="max-width: 25%;" align="left"/>This interactive electronic textbook in the form of Maple worksheets comprises two parts.
<BR><BR>
Part I, mathematics for chemistry, is supposed to cover all mathematics that an instructor of chemistry might hope and expect that his students would learn, understand and be able to apply as a result of sufficient courses typically, but not exclusively, presented in departments of mathematics. Its nine chapters include (0) a summary and illustration of useful Maple commands, (1) arithmetic, algebra and elementary functions, (2) plotting, descriptive geometry, trigonometry, series, complex functions, (3) differential calculus of one variable, (4) integral calculus of one variable, (5) multivariate calculus, (6) linear algebra including matrix, vector, eigenvector, vector calculus, tensor, spreadsheet, (7) differential and integral equations, and (8) probability, distribution, treatment of laboratory data, linear and non-linear regression and optimization.
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Part II presents mathematical topics typically taught within chemistry courses, including (9) chemical equilibrium, (10) group theory, (11) graph theory, (12a) introduction to quantum mechanics and quantum chemistry, (14) applications of Fourier transforms in chemistry including electron diffraction, x-ray diffraction, microwave spectra, infrared and Raman spectra and nuclear-magnetic-resonance spectra, and (18) dielectric and magnetic properties of chemical matter.
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Other chapters are in preparation and will be released in due course.
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Last updated on March 19, 2019https://www.maplesoft.com/applications/view.aspx?SID=154267&ref=FeedTue, 30 May 2017 04:00:00 ZJohn OgilvieJohn OgilvieODEs, PDEs and Special Functions
https://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="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="ODEs, PDEs and Special Functions" style="max-width: 25%;" 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>.https://www.maplesoft.com/applications/view.aspx?SID=154164&ref=FeedFri, 30 Sep 2016 04:00:00 ZDr. Edgardo Cheb-TerrabDr. Edgardo Cheb-TerrabNew developments on exact solutions for PDEs with Boundary Conditions
https://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="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="New developments on exact solutions for PDEs with Boundary Conditions" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=154169&ref=FeedFri, 30 Sep 2016 04:00:00 ZDr. Edgardo Cheb-TerrabDr. Edgardo Cheb-TerrabODEs, PDE solutions: when are they "general"?
https://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="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="ODEs, PDE solutions: when are they "general"?" style="max-width: 25%;" 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>.https://www.maplesoft.com/applications/view.aspx?SID=154165&ref=FeedFri, 30 Sep 2016 04:00:00 ZDr. Edgardo Cheb-TerrabDr. Edgardo Cheb-TerrabFactorizing with non-commutative variables
https://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="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Factorizing with non-commutative variables" style="max-width: 25%;" 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>.https://www.maplesoft.com/applications/view.aspx?SID=154166&ref=FeedFri, 30 Sep 2016 04:00:00 ZDr. Edgardo Cheb-TerrabDr. Edgardo Cheb-TerrabJump-diffusion stochastic processes with Maple
https://www.maplesoft.com/applications/view.aspx?SID=153516&ref=Feed
<p>The application presents and definition, creation and handling of jump-diffusion processes. In general, jump-diffusion is an extension to the theory of stochastic processes where the underlying parameters exhibit shocks and "jump" to their new values. Stochasticity with jumps is well recognised in several scientific branches including physics, chemistry, biology, but also economic and finance. The application looks at the example of the last-mentioned fields where the theory of jump-diffusions has been particularly actively researched and applied.</p><img src="https://www.maplesoft.com/view.aspx?si=153516/JD_Process.jpg" alt="Jump-diffusion stochastic processes with Maple" style="max-width: 25%;" align="left"/><p>The application presents and definition, creation and handling of jump-diffusion processes. In general, jump-diffusion is an extension to the theory of stochastic processes where the underlying parameters exhibit shocks and "jump" to their new values. Stochasticity with jumps is well recognised in several scientific branches including physics, chemistry, biology, but also economic and finance. The application looks at the example of the last-mentioned fields where the theory of jump-diffusions has been particularly actively researched and applied.</p>https://www.maplesoft.com/applications/view.aspx?SID=153516&ref=FeedSat, 08 Mar 2014 05:00:00 ZIgor HlivkaIgor HlivkaGeneration and Interaction of Solitons
https://www.maplesoft.com/applications/view.aspx?SID=141102&ref=Feed
<p>Classic computer experiments demonstrating the generation of solitons first time, has been published by N. J. Zabusky and M. D. Kruskal in 1965. Considered that was an earlier idea of Enrico Fermi. In 2006, Frank Wang has created a demonstration on the same subject with Maple tools . We would like to show both the origin and the interaction of Korteweg de Vries solitons as a development of approach of above cited publications.</p><img src="https://www.maplesoft.com/view.aspx?si=141102/fig.jpg" alt="Generation and Interaction of Solitons" style="max-width: 25%;" align="left"/><p>Classic computer experiments demonstrating the generation of solitons first time, has been published by N. J. Zabusky and M. D. Kruskal in 1965. Considered that was an earlier idea of Enrico Fermi. In 2006, Frank Wang has created a demonstration on the same subject with Maple tools . We would like to show both the origin and the interaction of Korteweg de Vries solitons as a development of approach of above cited publications.</p>https://www.maplesoft.com/applications/view.aspx?SID=141102&ref=FeedTue, 04 Dec 2012 05:00:00 ZS.I. ShyanS.I. ShyanClassroom Tips and Techniques: Fourier Series and an Orthogonal Expansions Package
https://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="https://www.maplesoft.com/view.aspx?si=134198/thumb.jpg" alt="Classroom Tips and Techniques: Fourier Series and an Orthogonal Expansions Package" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=134198&ref=FeedMon, 14 May 2012 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Partial Derivatives by Subscripting
https://www.maplesoft.com/applications/view.aspx?SID=100266&ref=Feed
As output, Maple can display the partial derivative ∂/∂<em>x f</em>(<em>x,y</em>) as <em>f</em><sub>x</sub>; that is, subscript notation can be used to display partial derivatives, and it can be done with two completely different mechanisms. This article describes these two techniques, and then investigates the extent to which partial derivatives can be calculated by subscript notation.<img src="https://www.maplesoft.com/view.aspx?si=100266/thumb.jpg" alt="Classroom Tips and Techniques: Partial Derivatives by Subscripting" style="max-width: 25%;" align="left"/>As output, Maple can display the partial derivative ∂/∂<em>x f</em>(<em>x,y</em>) as <em>f</em><sub>x</sub>; that is, subscript notation can be used to display partial derivatives, and it can be done with two completely different mechanisms. This article describes these two techniques, and then investigates the extent to which partial derivatives can be calculated by subscript notation.https://www.maplesoft.com/applications/view.aspx?SID=100266&ref=FeedWed, 15 Dec 2010 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Diffusion with a Generalized Robin Condition
https://www.maplesoft.com/applications/view.aspx?SID=96958&ref=Feed
<p>The one-dimensonal heat equation with a generalized Robin condition is solved on [0, 1] by a finite-difference scheme and by the Laplace transform, with the inversion implemented numerically. The left end is insulated and the initial temperature is zero. The Robin condition at the right end is driven by a function governed by an ODE, that is in turn, driven by the endpoint temperature.</p><img src="https://www.maplesoft.com/view.aspx?si=96958/thumb.jpg" alt="Classroom Tips and Techniques: Diffusion with a Generalized Robin Condition" style="max-width: 25%;" align="left"/><p>The one-dimensonal heat equation with a generalized Robin condition is solved on [0, 1] by a finite-difference scheme and by the Laplace transform, with the inversion implemented numerically. The left end is insulated and the initial temperature is zero. The Robin condition at the right end is driven by a function governed by an ODE, that is in turn, driven by the endpoint temperature.</p>https://www.maplesoft.com/applications/view.aspx?SID=96958&ref=FeedFri, 17 Sep 2010 04:00:00 ZRobert LopezRobert LopezAn Exact Solution For Diffusion Equation In Semiconductor Devices
https://www.maplesoft.com/applications/view.aspx?SID=7257&ref=Feed
An analytical solution for diffusion equation in semiconductor devices has been presented. The complete solution has been found in terms of following processes: Drift, Diffusion, Generation, Recombination and Carrier Trapping, using Maple.
Legal Notice: The copyright for this application is owned by the author. Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities.<img src="https://www.maplesoft.com/view.aspx?si=7257/thumb.gif" alt="An Exact Solution For Diffusion Equation In Semiconductor Devices" style="max-width: 25%;" align="left"/>An analytical solution for diffusion equation in semiconductor devices has been presented. The complete solution has been found in terms of following processes: Drift, Diffusion, Generation, Recombination and Carrier Trapping, using Maple.
Legal Notice: The copyright for this application is owned by the author. Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities.https://www.maplesoft.com/applications/view.aspx?SID=7257&ref=FeedWed, 18 Feb 2009 00:00:00 ZSeyed Mostafa AkramiSeyed Mostafa AkramiCylinder Heated by Induction
https://www.maplesoft.com/applications/view.aspx?SID=7240&ref=Feed
In this worksheet we consider a long metal cylinder that has a magnetic field applied parallel to the axis and a constraint on the current density at a particular depth. We demonstrate how the temperature depends on time and depth from the surface in a very long cylinder. This is calculated using the current density, the power density and the partial differential equation of heat conduction. In induction, the heating is caused by eddy currents, which themselves give rise to alternating magnetic fields. Because of the skin effect and depending on the frequency of the magnetic field, the highest current density exists directly under the surface of the heated work piece. It decreases rapidly with increasing depth. We calculate the effective magnetic field intensity on the surface required to reach a given temperature in a given time.<img src="https://www.maplesoft.com/view.aspx?si=7240/A_Cylinder_heated_by_Induction_2009_2.gif" alt="Cylinder Heated by Induction" style="max-width: 25%;" align="left"/>In this worksheet we consider a long metal cylinder that has a magnetic field applied parallel to the axis and a constraint on the current density at a particular depth. We demonstrate how the temperature depends on time and depth from the surface in a very long cylinder. This is calculated using the current density, the power density and the partial differential equation of heat conduction. In induction, the heating is caused by eddy currents, which themselves give rise to alternating magnetic fields. Because of the skin effect and depending on the frequency of the magnetic field, the highest current density exists directly under the surface of the heated work piece. It decreases rapidly with increasing depth. We calculate the effective magnetic field intensity on the surface required to reach a given temperature in a given time.https://www.maplesoft.com/applications/view.aspx?SID=7240&ref=FeedWed, 11 Feb 2009 00:00:00 ZMaplesoftMaplesoftTWO-DIMENSIONAL PARTIAL ELLIPTIC DIFFERENTIAL EQUATIONS IN MAPLE
https://www.maplesoft.com/applications/view.aspx?SID=4972&ref=Feed
This work introduces functional programming method in MAPLE for boundary problem solving of two-dimensional partial elliptic differential equations in polar coordinates.<img src="https://www.maplesoft.com/view.aspx?si=4972//applications/images/app_image_blank_lg.jpg" alt="TWO-DIMENSIONAL PARTIAL ELLIPTIC DIFFERENTIAL EQUATIONS IN MAPLE" style="max-width: 25%;" align="left"/>This work introduces functional programming method in MAPLE for boundary problem solving of two-dimensional partial elliptic differential equations in polar coordinates.https://www.maplesoft.com/applications/view.aspx?SID=4972&ref=FeedWed, 30 May 2007 00:00:00 ZDr. Alexei TikhonenkoDr. Alexei TikhonenkoDemonstrating Soliton Interactions using 'pdsolve'
https://www.maplesoft.com/applications/view.aspx?SID=1733&ref=Feed
The term "soliton" was introduced in a paper by Zabusky and Kruskal published in Physical Review Letters.[1] By solving the Korteweg-de Vries equation (KdV equation) numerically, solitary-wave pulses propagating in nonlinear dispersive media are observed. This worksheet demonstrates soliton interactions using pdsolve.<img src="https://www.maplesoft.com/view.aspx?si=1733/solitonimage.jpg" alt="Demonstrating Soliton Interactions using 'pdsolve'" style="max-width: 25%;" align="left"/>The term "soliton" was introduced in a paper by Zabusky and Kruskal published in Physical Review Letters.[1] By solving the Korteweg-de Vries equation (KdV equation) numerically, solitary-wave pulses propagating in nonlinear dispersive media are observed. This worksheet demonstrates soliton interactions using pdsolve.https://www.maplesoft.com/applications/view.aspx?SID=1733&ref=FeedMon, 01 May 2006 00:00:00 ZDr. Frank WangDr. Frank WangFourier Series
https://www.maplesoft.com/applications/view.aspx?SID=4520&ref=Feed
In this worksheet we define a number of Maple commands that make it easier to compute the Fourier coefficients and Fourier series for a given function and plot different Fourier polynomials (i.e., finite approximations to Fourier Series). We illustrate how to use these commands (and also the Fourier series themselves) by a number of examples. <img src="https://www.maplesoft.com/view.aspx?si=4520//applications/images/app_image_blank_lg.jpg" alt="Fourier Series" style="max-width: 25%;" align="left"/>In this worksheet we define a number of Maple commands that make it easier to compute the Fourier coefficients and Fourier series for a given function and plot different Fourier polynomials (i.e., finite approximations to Fourier Series). We illustrate how to use these commands (and also the Fourier series themselves) by a number of examples. https://www.maplesoft.com/applications/view.aspx?SID=4520&ref=FeedTue, 17 Aug 2004 15:45:06 ZAnton DzhamayAnton DzhamayHigher-dimensional PDE: Vibrating circular membranes and Bessel functions.
https://www.maplesoft.com/applications/view.aspx?SID=4519&ref=Feed
In this worksheet we consider some examples of vibrating circular membranes. Such membranes are described by the two-dimensional wave equation. Circular geometry requires the use of polar coordinates, which in turn leads to the Bessel ODE , and so the basic solutions obtained by the method of separations of variables (product solutions or standing waves) are described with the help of Bessel functions . <img src="https://www.maplesoft.com/view.aspx?si=4519//applications/images/app_image_blank_lg.jpg" alt="Higher-dimensional PDE: Vibrating circular membranes and Bessel functions." style="max-width: 25%;" align="left"/>In this worksheet we consider some examples of vibrating circular membranes. Such membranes are described by the two-dimensional wave equation. Circular geometry requires the use of polar coordinates, which in turn leads to the Bessel ODE , and so the basic solutions obtained by the method of separations of variables (product solutions or standing waves) are described with the help of Bessel functions . https://www.maplesoft.com/applications/view.aspx?SID=4519&ref=FeedTue, 17 Aug 2004 15:43:13 ZAnton DzhamayAnton DzhamayHigher-dimensional PDE: Vibrating rectangular membranes and nodes.
https://www.maplesoft.com/applications/view.aspx?SID=4518&ref=Feed
In this worksheet we consider some examples of the vibrating patterns of rectangular membranes. Such membranes are described by the 2-dimensional wave equation diff(u(x,y,t),t,t) = c^2*Delta(u(x,y,t)) . we are mainly interested in the product solution obtained by the method of separation of variables, such product solutions of the wave equations are also called standing waves . In particular, we consider intricate patterns of nodal curves appearing when there is more than one eigenfunction corresponding to the same eigenvalue (this happens, for example, for a square membrane.<img src="https://www.maplesoft.com/view.aspx?si=4518//applications/images/app_image_blank_lg.jpg" alt="Higher-dimensional PDE: Vibrating rectangular membranes and nodes." style="max-width: 25%;" align="left"/>In this worksheet we consider some examples of the vibrating patterns of rectangular membranes. Such membranes are described by the 2-dimensional wave equation diff(u(x,y,t),t,t) = c^2*Delta(u(x,y,t)) . we are mainly interested in the product solution obtained by the method of separation of variables, such product solutions of the wave equations are also called standing waves . In particular, we consider intricate patterns of nodal curves appearing when there is more than one eigenfunction corresponding to the same eigenvalue (this happens, for example, for a square membrane.https://www.maplesoft.com/applications/view.aspx?SID=4518&ref=FeedTue, 17 Aug 2004 15:33:56 ZAnton DzhamayAnton DzhamayFirst-Order PDE: The method of characteristics
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In this worksheet we give some examples on how to use the method of characteristics for first-order linear PDEs of the form
a(x,t)*diff(u(x,t),t)+b(x,t)*diff(u(x,t),x)+c(x,t)*u(x,t) = h(x,t) . The main idea of the method of characteristics is to reduce a PDE on the ( x, t )-plane to an ODE along a parametric curve (called the characteristic curve) parametrized by some other parameter tau . The characteristic curve is then determined by the condition that diff(u(x(tau),y(tau)),tau) = diff(t(tau),tau)*diff(u(x,t),t)+diff(x(tau),tau)*diff(u(x,t),x) = a(x,t)*diff(u(x,t),t)+b(x,t)*diff(u(x,t),x) and so we need to solve another ODE to find the characteristic. In the examples below we always take a(x,t) = 1 , and so we can use t instead of tau . In this case the characteristics are given by the equation diff(x,t) = b(x,t) . On the characteristic we then get an equation diff(u(t),t)+c(t)*u(t) = h(t) , which is again an ODE. Solving both ODEs, choosing the constants of integration to match the initial data, and going from the characteristic to the whole plane then gives us the solution u(x,t) of the PDE. <img src="https://www.maplesoft.com/view.aspx?si=4517//applications/images/app_image_blank_lg.jpg" alt="First-Order PDE: The method of characteristics" style="max-width: 25%;" align="left"/>In this worksheet we give some examples on how to use the method of characteristics for first-order linear PDEs of the form
a(x,t)*diff(u(x,t),t)+b(x,t)*diff(u(x,t),x)+c(x,t)*u(x,t) = h(x,t) . The main idea of the method of characteristics is to reduce a PDE on the ( x, t )-plane to an ODE along a parametric curve (called the characteristic curve) parametrized by some other parameter tau . The characteristic curve is then determined by the condition that diff(u(x(tau),y(tau)),tau) = diff(t(tau),tau)*diff(u(x,t),t)+diff(x(tau),tau)*diff(u(x,t),x) = a(x,t)*diff(u(x,t),t)+b(x,t)*diff(u(x,t),x) and so we need to solve another ODE to find the characteristic. In the examples below we always take a(x,t) = 1 , and so we can use t instead of tau . In this case the characteristics are given by the equation diff(x,t) = b(x,t) . On the characteristic we then get an equation diff(u(t),t)+c(t)*u(t) = h(t) , which is again an ODE. Solving both ODEs, choosing the constants of integration to match the initial data, and going from the characteristic to the whole plane then gives us the solution u(x,t) of the PDE. https://www.maplesoft.com/applications/view.aspx?SID=4517&ref=FeedTue, 17 Aug 2004 15:29:38 ZAnton DzhamayAnton Dzhamay