PDEs: New Applications
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en-us2015 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemFri, 09 Oct 2015 17:44:53 GMTFri, 09 Oct 2015 17:44:53 GMTNew applications in the PDEs categoryhttp://www.mapleprimes.com/images/mapleapps.gifPDEs: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=149
Jump-diffusion stochastic processes with Maple
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<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="/view.aspx?si=153516/Jump_image1.jpg" alt="Jump-diffusion stochastic processes with Maple" 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>153516Sat, 08 Mar 2014 05:00:00 ZIgor HlivkaIgor HlivkaGeneration and Interaction of Solitons
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<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="/view.aspx?si=141102/fig.jpg" alt="Generation and Interaction of Solitons" 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>141102Tue, 04 Dec 2012 05:00:00 ZS.I. ShyanS.I. ShyanClassroom Tips and Techniques: Fourier Series and an Orthogonal Expansions Package
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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.134198Mon, 14 May 2012 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Partial Derivatives by Subscripting
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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="/view.aspx?si=100266/thumb.jpg" alt="Classroom Tips and Techniques: Partial Derivatives by Subscripting" 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.100266Wed, 15 Dec 2010 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Diffusion with a Generalized Robin Condition
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<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="/view.aspx?si=96958/thumb.jpg" alt="Classroom Tips and Techniques: Diffusion with a Generalized Robin Condition" 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>96958Fri, 17 Sep 2010 04:00:00 ZRobert LopezRobert LopezAn Exact Solution For Diffusion Equation In Semiconductor Devices
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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="/view.aspx?si=7257/thumb.gif" alt="An Exact Solution For Diffusion Equation In Semiconductor Devices" 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.7257Wed, 18 Feb 2009 00:00:00 ZSeyed Mostafa AkramiSeyed Mostafa AkramiCylinder Heated by Induction
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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="/view.aspx?si=7240/A_Cylinder_heated_by_Induction_2009_2.gif" alt="Cylinder Heated by Induction" 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.7240Wed, 11 Feb 2009 00:00:00 ZMaplesoftMaplesoftTWO-DIMENSIONAL PARTIAL ELLIPTIC DIFFERENTIAL EQUATIONS IN MAPLE
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This work introduces functional programming method in MAPLE for boundary problem solving of two-dimensional partial elliptic differential equations in polar coordinates.<img src="/view.aspx?si=4972//applications/images/app_image_blank_lg.jpg" alt="TWO-DIMENSIONAL PARTIAL ELLIPTIC DIFFERENTIAL EQUATIONS IN MAPLE" align="left"/>This work introduces functional programming method in MAPLE for boundary problem solving of two-dimensional partial elliptic differential equations in polar coordinates.4972Wed, 30 May 2007 00:00:00 ZDr. Alexei TikhonenkoDr. Alexei TikhonenkoDemonstrating Soliton Interactions using 'pdsolve'
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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="/view.aspx?si=1733/solitonimage.jpg" alt="Demonstrating Soliton Interactions using 'pdsolve'" 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.1733Mon, 01 May 2006 00:00:00 ZDr. Frank WangDr. Frank WangFourier Series
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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="/view.aspx?si=4520//applications/images/app_image_blank_lg.jpg" alt="Fourier Series" 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. 4520Tue, 17 Aug 2004 15:45:06 ZAnton DzhamayAnton DzhamayHigher-dimensional PDE: Vibrating circular membranes and Bessel functions.
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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="/view.aspx?si=4519//applications/images/app_image_blank_lg.jpg" alt="Higher-dimensional PDE: Vibrating circular membranes and Bessel functions." 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 . 4519Tue, 17 Aug 2004 15:43:13 ZAnton DzhamayAnton DzhamayHigher-dimensional PDE: Vibrating rectangular membranes and nodes.
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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="/view.aspx?si=4518//applications/images/app_image_blank_lg.jpg" alt="Higher-dimensional PDE: Vibrating rectangular membranes and nodes." 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.4518Tue, 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="/view.aspx?si=4517//applications/images/app_image_blank_lg.jpg" alt="First-Order PDE: The method of characteristics" 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. 4517Tue, 17 Aug 2004 15:29:38 ZAnton DzhamayAnton DzhamayThe Heat Equation: Separation of variables and Fourier series
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In this worksheet we consider the one-dimensional heat equation diff(u(x,t),t) = k*diff(u(x,t),x,x) describint the evolution of temperature u(x,t) inside the homogeneous metal rod. We consider examples with homogeneous Dirichlet ( u(0,t) = 0 , u(L,t) = 0 ) and Newmann ( diff(u,x)(0,t) = 0, diff(u,x)(L,t) = 0 ) boundary conditions and various initial profiles f(x)<img src="/view.aspx?si=4515//applications/images/app_image_blank_lg.jpg" alt="The Heat Equation: Separation of variables and Fourier series" align="left"/>In this worksheet we consider the one-dimensional heat equation diff(u(x,t),t) = k*diff(u(x,t),x,x) describint the evolution of temperature u(x,t) inside the homogeneous metal rod. We consider examples with homogeneous Dirichlet ( u(0,t) = 0 , u(L,t) = 0 ) and Newmann ( diff(u,x)(0,t) = 0, diff(u,x)(L,t) = 0 ) boundary conditions and various initial profiles f(x)4515Fri, 09 Jul 2004 14:17:53 ZAnton DzhamayAnton DzhamayThe vibrating string: d'Alembert's formula
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The goal of this notebook is to illustrate the use of the method of characteristics and d'Alembert's formula for the one-dimentional wave equation (i.e., the vibrating string).<img src="/view.aspx?si=4514//applications/images/app_image_blank_lg.jpg" alt="The vibrating string: d'Alembert's formula" align="left"/>The goal of this notebook is to illustrate the use of the method of characteristics and d'Alembert's formula for the one-dimentional wave equation (i.e., the vibrating string).4514Thu, 08 Jul 2004 15:43:59 ZAnton DzhamayAnton DzhamayNon trivial Lie homomorphism
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This worksheet demonstrates the use of Maple for costructing a non-trivial vector field from a given matrix G and it's representation in canonical local coords.<img src="/view.aspx?si=4359//applications/images/app_image_blank_lg.jpg" alt="Non trivial Lie homomorphism" align="left"/>This worksheet demonstrates the use of Maple for costructing a non-trivial vector field from a given matrix G and it's representation in canonical local coords.4359Mon, 03 Feb 2003 14:32:03 ZYuri GribovYuri GribovFourier Approximate Solutions to PDE Boundary Value Problems
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In this session we find approximate solutions to boundary value problems for heat and wave equations using Fourier method. We solve problems from Numerical Solutions to PDE Boundary Value Problems in Maple 8.
<img src="/view.aspx?si=4267//applications/images/app_image_blank_lg.jpg" alt="Fourier Approximate Solutions to PDE Boundary Value Problems " align="left"/>In this session we find approximate solutions to boundary value problems for heat and wave equations using Fourier method. We solve problems from Numerical Solutions to PDE Boundary Value Problems in Maple 8.
4267Tue, 23 Apr 2002 14:26:04 ZAleksas DomarkasAleksas DomarkasThe VectorCalculus Package
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Maple 8 provides a new package called VectorCalculus for computing with vectors, vector fields, multivariate functions and parametric curves. Computations include vector arithmetic using basis vectors, "del"-operations, multiple integrals over regions and solids, line and surface integrals, differential-geometric properties of curves, and many others. You can easily perform these computations in any coordinate system and convert results between coordinate systems. The package is fully compatible with the Maple LinearAlgebra package. You can also extend the VectorCalculus package by defining your own coordinate systems.<img src="/view.aspx?si=1382/veccalc.gif" alt="The VectorCalculus Package" align="left"/>Maple 8 provides a new package called VectorCalculus for computing with vectors, vector fields, multivariate functions and parametric curves. Computations include vector arithmetic using basis vectors, "del"-operations, multiple integrals over regions and solids, line and surface integrals, differential-geometric properties of curves, and many others. You can easily perform these computations in any coordinate system and convert results between coordinate systems. The package is fully compatible with the Maple LinearAlgebra package. You can also extend the VectorCalculus package by defining your own coordinate systems.1382Mon, 15 Apr 2002 16:13:38 ZMaplesoftMaplesoftPDE Boundary Value Problems Solved Numerically with pdsolve
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Maple 8 can now compute numerical solutions to linear PDE systems over rectangular regions. One application of this feature is the solution of classical boundary-value problems from physics, such as the heat conduction equation and the wave equation.<img src="/view.aspx?si=4259/pdes.gif" alt="PDE Boundary Value Problems Solved Numerically with pdsolve" align="left"/>Maple 8 can now compute numerical solutions to linear PDE systems over rectangular regions. One application of this feature is the solution of classical boundary-value problems from physics, such as the heat conduction equation and the wave equation.4259Mon, 15 Apr 2002 16:04:37 ZMaplesoftMaplesoftPeriodic solutions for a first order PDE with periodic forcing function
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This Maple worksheet provides a method for finding solutions for non-homogeneous partial differential equation of the form
<br><img src="http://www.mapleapps.com/categories/mathematics/pdes/html/images/PeriodSum/PeriodSum1.gif" width=41 height=58 alt="diff(u,t)" align=middle>
<font color=#000000> = </font>
<img src="http://www.mapleapps.com/categories/mathematics/pdes/html/images/PeriodSum/PeriodSum2.gif" width=53 height=76 alt="diff(u,`$`(x,2))" align=middle>
<font color=#000000> + </font>
<img src="http://www.mapleapps.com/categories/mathematics/pdes/html/images/PeriodSum/PeriodSum3.gif" width=60 height=32 alt="F(t,x)" align=middle>
<font color=#000000>, u(t, 0) = 0 = u(t, 1).</font>
<br>More important, if the function F(t,x) is periodic as a function of t, it provides a method for finding a periodic solution for the partial differential equation. Even more. Techniques are used to emphasize a general structure suitable for finding a periodic solution for ordinary differential equations having the same general form of Y' = AY + F
where F is periodic.
<img src="/view.aspx?si=4158//applications/images/app_image_blank_lg.jpg" alt="Periodic solutions for a first order PDE with periodic forcing function" align="left"/>This Maple worksheet provides a method for finding solutions for non-homogeneous partial differential equation of the form
<br><img src="http://www.mapleapps.com/categories/mathematics/pdes/html/images/PeriodSum/PeriodSum1.gif" width=41 height=58 alt="diff(u,t)" align=middle>
<font color=#000000> = </font>
<img src="http://www.mapleapps.com/categories/mathematics/pdes/html/images/PeriodSum/PeriodSum2.gif" width=53 height=76 alt="diff(u,`$`(x,2))" align=middle>
<font color=#000000> + </font>
<img src="http://www.mapleapps.com/categories/mathematics/pdes/html/images/PeriodSum/PeriodSum3.gif" width=60 height=32 alt="F(t,x)" align=middle>
<font color=#000000>, u(t, 0) = 0 = u(t, 1).</font>
<br>More important, if the function F(t,x) is periodic as a function of t, it provides a method for finding a periodic solution for the partial differential equation. Even more. Techniques are used to emphasize a general structure suitable for finding a periodic solution for ordinary differential equations having the same general form of Y' = AY + F
where F is periodic.
4158Tue, 30 Oct 2001 10:39:16 ZJim HerodJim Herod