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

SFOPDES: A stepwise tutorial for teaching Partial Differential Equations using a CAS

Partial Differential Equations (PDE) are one of the most difficult topics that Engineering and Sciences students have to study in the different Math subjects in their degree. In this talk we introduce SFOPDES (Stepwise First Order Partial Differential Equations Solver) aimed to be used as a tutorial for helping both the teacher and the students in the teaching and learning process of PDE. The type of problems that SFOPDES solves can be grouped in the following three blocks: 1. Pfaff Differential Equations, which consists on finding the general solution for: P(x; y; z) dx + Q(x; y; z) dy + R(x; y; z) dz = 0 (a) General method. (b) Particular cases: i. Separable equations. ii. Exact Pfaff equations. iii. One-separated variable equations. 2. Quasi-linear Partial Differential Equations, which consists on finding the general solution for: P(x; y; x) p + Q(x; y; z) q = R(x; y; z) (a) General method. (b) Particular solution which contents a given curve. 3. Using Lagrange-Charpit Method for finding a complete integral for a given general first order partial differential equation: F(x; y; z; p; q) = 0. (a) General method. (b) Particular cases: i. F(p; q) = 0 ii. g1(x; p) = g2(y; q) iii. z = px + qy + g(p; q)Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Improving CAS Capabilities: New Rules for Computing Improper Integrals

There are diferent applications in Engineering that require to compute improper integrals of the first kind (integrals defined on an unbounded domain) such as: the work required to move an object from the surface of the earth to in nity (Kynetic Energy), the electric potential created by a charged sphere, the probability density function or the cumulative distribution function in Probability Theory, the values of the Gamma Functions(wich useful to compute the Beta Function used to compute trigonometrical integrals), Laplace and Fourier Transforms (very useful, for example in Differential Equations).Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

SPDES: A Stepwise Solver for Teaching Partial Differential Equations

Partial Differential Equations (PDE) are an important topic within the Engineering Degrees syllabus. In addition, many students find some dificulties in the learning process of this topic. Therefore, the use of didactical tools to improve the teaching and learning process of PDE is very helpful. In this talk, we introduce the solver SPDES (Stepwise Partial Differential Equation Solver), an extension of SFOPDES introduced in [1] where only first order PDE were considered. This new solver SPDES deals also with some second order PDE. It can be used as a self tutorial for PDE since it solves, step by step, the typical exercises within the topic. Specifically, the type of PDE that SPDES can solve are: 1. Pfaff Differential Equations. 2. Quasi-linear PDE. 3. Lagrange-Charpit Method for first order PDE. 4. Heat equation. 5. Wave equation. 6. Laplace's equation. where the fi rst three types where considered in SFOPDES and the last three types are new in SPDES. SPDES has been developed using the programming capabilities of a Computer Algebra System (CAS), displaying step by step the solution of the problem to be solved. This way, we potentiate the use of the CAS as a Pedagogical CAS (PeCAS). This fact makes SPDES to be an important tool for students which can use it as a tutorial for their learning process.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Advanced techniques to compute improper integrals using a CAS

Let us consider the following types of improper integrals: $$\int_0^\infty f(t)\:{\rm d}t \qquad ; \qquad \int_{-\infty}^0 f(t)\:{\rm d}t \qquad {\rm and} \qquad \int_{-\infty}^\infty f(t)\:{\rm d}t$$ \medskip Let $F$ be an antiderivative of $f$. The basic approach to compute such integrals involves the following computations: \medskip \begin{eqnarray*} \int_0^\infty f(t)\:{\rm d}t & = & \lim_{m\to\infty} \int_0^m f(t)\:{\rm d}t = \lim_{m\to\infty} \big(F(m)-F(0)\big) \\ \\ \int_{-\infty}^0 f(t)\:{\rm d}t & = & \lim_{m\to-\infty} \int_m^0 f(t)\:{\rm d}t = \lim_{m\to-\infty} \big(F(0)-F(m)\big) \\ \\ \int_{-\infty}^\infty f(t)\:{\rm d}t & = & \int_{-\infty}^0 f(t)\:{\rm d}t + \int_0^\infty f(t)\:{\rm d}t \qquad \mbox{or, in case of convergence,} \\ \\ \int_{-\infty}^\infty f(t)\:{\rm d}t & = & \lim_{m\to\infty} \int_{-m}^m f(t)\:{\rm d}t = \lim_{m\to\infty} \big(F(m)-F(-m)\big) \qquad \mbox{(Cauchy principal value)} \end{eqnarray*} \medskip \noindent But, what happens if an antiderivative $F$ for $f$ or the above limits do not exist? \medskip \noindent For example, for \quad $\displaystyle\int_0^\infty\frac{{\rm sin}(at)}{t}\:{\rm d}t$ \quad ; \quad $\displaystyle\int_0^\infty\frac{{\rm cos}(at)-{\rm cos}(bt)}{t}\:{\rm d}t$ \quad {\rm or} \quad $\displaystyle\int_{-\infty}^\infty\frac{{\rm cos}(at)}{t^2+1}\:{\rm d}t$ \qquad the antiderivatives can not be computed. Hence, the above procedures cannot be used for these examples. \medskip In this work we will deal with advance techniques to compute this kind of improper integrals using a {\sc Cas}. Laplace and Fourier transforms or Residue Theorem in Complex Analysis are some advance techniques which can be used for this matter. \medskip We will introduce the file \textbf{\tt ImproperIntegrals.mth}, developed in {\sc Derive} 6, which deals with such computations. \medskip Some {\sc Cas} use different rules for computing integrations. For example {\sc Rubi} system, a {\bf ru}le-{\bf b}ased {\bf i}ntegrator developed by Albert Rich (see {\tt http://www.apmaths.uwo.ca/\~{ }arich/}), is a very powerful system for computing integrals using rules. We will be able to develop new rules schemes for some improper integrals using {\tt ImproperIntegrals.mth}. These new rules can extend the types of improper integrals that a {\sc Cas} can compute.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Study of the fronts of Johnsons and Hurd Glaciers (Livingston Island, Antarctica) from 1957 to 2013, with links to shapefiles

The study of glacier fronts combines different geomatics measurement techniques as the classic survey using total station or theodolite, technical GNSS (Global Navigation Satellite System), using laser-scanner or using photogrammetry (air or ground). The measure by direct methods (classical surveying and GNSS) is useful and fast when accessibility to the glaciers fronts is easy, while it is practically impossible to realize, in the case of glacier fronts that end up in the sea (tide water glaciers). In this paper, a methodology that combines photogrammetric methods and other techniques for lifting the front of the glacier Johnsons, inaccessible is studied. The images obtained from the front, come from a non-metric digital camera; its georeferencing to a global coordinate system is performed by measuring points GNSS support in accessible areas of the glacier front side and applying methods of direct intersection in inaccessible points of the front, taking measurements with theodolite. The result of observations obtained were applied to study the temporal evolution (1957-2014) of the position of the Johnsons glacier front and the position of the Argentina, Las Palmas and Sally Rocks lobes front (Hurd glacier)

Digital elevation models for Johnsons and Hurd glaciers (1957-2013), Livingston Island, Antarctica, links to files in multiple formats

Digital elevation models (terrain elevation models and surface elevation models) are very important to evaluate numeric models for glaciers. In this collection of data, there are classical surveying models, photogrammetry models, restitution models, GNSS models and GPR models for different dates between 1957 and 2013