Teaching Partial Differential Equations with CAS

Partial Differential Equations (PDE) are one of the topics where Engineering students find more difficulties when facing Math subjects. A basic course in Partial Differential Equations (PDE) in Engineering, usually deals at least, with the following PDE problems: 1. Pfaff Differential Equations 2. Quasi-linear Partial Differential Equations 3. Using Lagrange-Charpit Method for finding a complete integral for a given general first order partial differential equation 4. Heat equation 5. Wave equation 6. Laplace’s equation In this talk we will describe how we introduce CAS in the teaching of PDE. The tasks developed combine the power of a CAS with the flexibility of programming with it. Specifically, we use the CAS DERIVE. The use of programming allows us to use DERIVE as a Pedagogical CAS (PECAS) in the sense that we do not only provide the final result of an exercise but also display all the intermediate steps which lead to find the solution of a problem. This way, the library developed in DERIVE serves as a tutorial showing, step by step, the way to face PDE exercises. In the process of solving PDE exercises, first-order Ordinary Differential Equations (ODE) are needed. The programs developed can be grouped within the following blocks: - First-order ODE: separable equations and equations reducible to them, homogeneous equations and equations reducible to them, exact differential equations and equations reducible to them (integrating factor technique), linear equations, the Bernoulli equation, the Riccati equation, First-order differential equations and nth degree in y’, Generic programs to solve first order differential equations. - First-order PDE: Pfaff Differential Equations, Quasi-linear PDE, Lagrange-Charpit Method for First-order PDE. - Second-order PDE: Heat Equation, Wave Equation, Laplace’s Equation. We will remark the conclusions obtained after using these techniques with our Engineering students.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

A partial differential equation for the strictly quasiconvex envelope

In a series of papers Barron, Goebel, and Jensen studied Partial Differential Equations (PDE)s for quasiconvex (QC) functions \cite{barron2012functions, barron2012quasiconvex,barron2013quasiconvex,barron2013uniqueness}. To overcome the lack of uniqueness for the QC PDE, they introduced a regularization: a PDE for \e-robust QC functions, which is well-posed. Building on this work, we introduce a stronger regularization which is amenable to numerical approximation. We build convergent finite difference approximations, comparing the QC envelope and the two regularization. Solutions of this PDE are strictly convex, and smoother than the robust-QC functions.Comment: 20 pages, 6 figures, 1 tabl

On integration of multidimensional version of nn-wave type equation

We represent a version of multidimensional quasilinear partial differential equation (PDE) together with large manifold of particular solutions given in an integral form. The dimensionality of constructed PDE can be arbitrary. We call it the $n$-wave type PDE, although the structure of its nonlinearity differs from that of the classical completely integrable (2+1)-dimensional $n$-wave equation. The richness of solution space to such a PDE is characterized by a set of arbitrary functions of several variables. However, this richness is not enough to provide the complete integrability, which is shown explicitly. We describe a class of multi-solitary wave solutions in details. Among examples of explicit particular solutions, we represent a lump-lattice solution depending on five independent variables. In Appendix, as an important supplemental material, we show that our nonlinear PDE is reducible from the more general multidimensional PDE which can be derived using the dressing method based on the linear integral equation with the kernel of a special type (a modification of the $\bar\partial$-problem). The dressing algorithm gives us a key for construction of higher order PDEs, although they are not discussed in this paper.Comment: 36 pages, 2 figure

PDE Face: A Novel 3D Face Model

YesWe introduce a novel approach to face models, which exploits the use of Partial Differential Equations (PDE) to generate the 3D face. This addresses some common problems of existing face models. The PDE face benefits from seamless merging of surface patches by using only a relatively small number of parameters based on boundary curves. The PDE face also provides users with a great degree of freedom to individualise the 3D face by adjusting a set of facial boundary curves. Furthermore, we introduce a uv-mesh texture mapping method. By associating the texels of the texture map with the vertices of the uv mesh in the PDE face, the new texture mapping method eliminates the 3D-to-2D association routine in texture mapping. Any specific PDE face can be textured without the need for the facial expression in the texture map to match exactly that of the 3D face model

All the lowest order PDE for spectral gaps of Gaussian matrices

Tracy-Widom (TW) equations for one-matrix unitary ensembles (UE) (equivalent to a particular case of Schlesinger equations for isomonodromic deformations) are rewritten in a general form which allows one to derive all the lowest order equations (PDE) for spectral gap probabilities of UE without intermediate higher-order PDE. This is demonstrated on the example of Gaussian ensemble (GUE) for which all the third order PDE for gap probabilities are obtained explicitly. Moreover, there is a {\it second order} PDE for GUE probabilities in the case of more than one spectral endpoint. This approach allows to derive all PDE at once where possible, while in the method based on Hirota bilinear identities and Virasoro constraints starting with different bilinear identities leads to different subsets of the full set of equations.Comment: 22 pages, references corrected, remark adde

A personal, distributed exposimeter: procedure for design, calibration, validation, and application

This paper describes, for the first time, the procedure for the full design, calibration, uncertainty analysis, and practical application of a personal, distributed exposimeter (PDE) for the detection of personal exposure in the Global System for Mobile Communications (GSM) downlink (DL) band around 900 MHz (GSM 900 DL). The PDE is a sensor that consists of several body-worn antennas. The on-body location of these antennas is investigated using numerical simulations and calibration measurements in an anechoic chamber. The calibration measurements and the simulations result in a design (or on-body setup) of the PDE. This is used for validation measurements and indoor radio frequency (RF) exposure measurements in Ghent, Belgium. The main achievements of this paper are: first, the demonstration, using both measurements and simulations, that a PDE consisting of multiple on-body textile antennas will have a lower measurement uncertainty for personal RF exposure than existing on-body sensors; second, a validation of the PDE, which proves that the device correctly estimates the incident power densities; and third, a demonstration of the usability of the PDE for real exposure assessment measurements. To this aim, the validated PDE is used for indoor measurements in a residential building in Ghent, Belgium, which yield an average incident power density of 0.018 mW/m(2)