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It is very common that Engineering students nd difficulties when studying advanced Mathematics subjects. To help in the teaching and learning process of such subjects, the teacher can use an adequate mathematical software. But not always the used given to these specific pieces of software is the right one. The use of a Computer Algebra System (CAS) to achieve this goal is a good idea mainly because programming with a CAS, the solution to a problem can be obtained step by step. This way, the student can check all the intermediate steps to get the solution and can nd the step or steps where the student made a mistake. In this talk, we introduce SPDES, a Stepwise Partial Differential Equation Solver (an extension of SFOPDES). SPDES deals with some second order PDE in addition to the first order PDE considered in SFOPDES. SPDES can be used as a self tutorial for PDE since it solves, step by step, the typical exercises within the topic. The type of PDE that SPDES can solve are: Pfaff Differential Equations, Quasi-linear PDE, Lagrange-Charpit Method for first order PDE, Heat equation, Wave equation and Laplace's equation. SPDES has been developed using the programming capabilities of the CAS Derive, providing not only the final result but also, optionally, the display of all the steps needed to solve typical exercises on PDE. Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech.

In this paper we introduce SODES (Step-wise Ordinary Differential Equations Solver) which is a new solver for Ordinary Dfferential Equations (ODE). SODES can be used not only as a solver but also as a tutorial for the teaching and learning process of ODE, since it provides the solution displaying all the the steps needed to obtain it. In [1] we introduced SFOPDES (Step-wise First Order Partial Differential Equations Solver) developed in Derive, a Computer Algebra System (CAS). Two of the future works related considered in [1] were the development of new step-wise solvers for different math topics and the use of other CAS and programming with Python. This way, it is possible to develop web applications, freely available, providing different step-wise solvers. In this sense, SODES is the first work related with both ideas. Therefore, SODES is a web solver for ODE, developed in Python (using its CAS module SymPy). SODES increases the types of ODE that SFOPDES can deal with. Specifically, it provides, step by step, the solution of the following types of ODE: 1. 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. 2. Higher-order ODE: higher order linear equations with constant coefficients. As a final remark, actually, SODES can be used in English, French or Spanish, but it can be easily extendable to other languages. Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech.

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.

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.

The residue theorem is one of the most interesting result in Complex Analysis which allows not only computations in C, the Field of Complex Numbers, but also provides many applications in the Field of Real Numbers R. In this talk we present the library ResidueApplications, that was initially developed in DERIVE since Engineering students in the University of Málaga are still using this software in computer lectures. However, we are migrating this library to PYTHON using the symbolic mathematics library SYMPY. This way it will be also possible to use this package in other CAS as SAGEMATH. The main goals of the ResidueApplications library are not only to provide some important applications of the Residue theorem but also to use it as a pedagogical tool for Engineering students. ResidueApplications can be used as a tutorial in the teaching and learning process of this topic since it provides the results step by step allowing the students to check their computations when they solve an exercise. When developing this package, we were not interesting only in the computations of residues and their applications (which can be easily done using standards functions in different CAS) but mainly on its pedagogical use. In addition of the step by step facility, using this library, the students also can develop their own programs to deal with different applications. This way, the student are the protagonist of their selflearning process. For example, If the students develop a program to compute the residues of a function, they will be better prepared to understand this topic. The programs developed in this tutorial can be grouped in the following blocks: 1. Compute of residues. 2. Compute of complex integrals using the residue theorem. 3. Applications of the residue theorem to compute integrals in R: (a) Trigonometric integrals. (b) Improper integrals. Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech.

Our group introduced random samples generation using a CAS, specifically Derive, in the Derive session of ACA 2009. This talk was extended in a paper published in The Derive Newsletter. In the year 2017 we presented a new version focused on research of the random samples generator in STATA that was communicated in [2]. Now a new version of the generator is being implemented in Python with the symbolic pack Simpy. In this case, our we are mainly focused in education. We are now researching in extensions of the Conway’s Game of Life, specifically using probabilistic cellular automata. Our long-term goal is the simulation of the growth of cancerous tissues. We have supervised a master thesis with the preliminary works in this topic. For the generation of random numbers involved in the probabilistic automata of this work, we have used the results developed in the previous mentioned work. A summary of this work is about to be published in Advances in Computational Mathematics [3]. This work is both, an education and a research experience for the student who carried out the master thesis. Moreover, the experience of teaching the subject Statistics to engineering students using the material developed for random samples generation has turned out to be useful in the teaching and learning process. The marks obtained in this subject by the students are statistically significantly better than the marks obtained by students of the same subject in others group of engineering where this material has not been used. A brief statistical study about the situation is carried out. Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech.

Traffic simulations usually require the search of a path to join two different points. Dijkstra’s algorithm [1] is one of the most commonly used for this task due to its easiness and quickness. In [2, 3] we developed an accelerated time simulation of car traffic in a smart city using Dijkstra’s algorithm to compute the paths. Dijkstra’s algorithm provides a shortest path between two different points but this is not a realistic situation for simulations. For example, in a car traffic situa- tion, the driver may not know the shortest path to follow. This ignorance can be produced, among others, because one of the following two facts: the driver may not know the exact length of the lanes, or, even knowing the exact length, the driver may not know how to find the shortest path. Even more, in many cases, a mixture of both facts occurs. A more realistic simulation should therefore consider these kind of facts. The algorithm used to compute the path from one point to another in a traffic simulation might consider the possibility of not using the shortest path. In this talk, we use a new probabilistic extension of Dijkstra’s algorithm which covers the above two situations. For this matter, two different modifications in Di- jkstra’s algorithm have been introduced: using non-exact length in lanes, and the choice of a non-shortest path between two different points. Both modifications are used in a non-deterministic way by means of using probability distributions (classi- cal distributions such as Normal or Poisson distributions or even "ad hoc" ones). A precise, fast, natural and elegant way of working with such probability distributions is the use of a CAS in order to deal with exact and explicit computations. As an example of use of this extension of Dijkstra’s algorithm, we will show the ATISMART+ model. This model provides more realistic accelerated time sim- ulations of car traffics in a smart city and was first introduced in [4] and extended in [5]. This model was developed combining J AVA for the GUI and M AXIMA for the mathematical core of the algorithm. The studies developed in the above mentioned works, dealt with Poisson, Ex- ponential, Uniform and Normal distributions. In this talk we will introduce, as a novelty, the possibility of using other continuous probability distributions such as: Lognormal, Weibul, Gamma, Beta, Chi-Square, Student’s t, Z, Pareto, Lo- gistic, Cauchy or Irwin-Hall, and other discrete distributions such as: Bernouille, Rademacher, Binomial, Geometric, Negative Binomial or Hypergeometric. Even 1 more, this new version allows to deal with any “ad-hoc” continuous, discrete or mixed user’s distributions. This fact improves the flexibility of ATISMART+ model. Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech.

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.

Partial Differential Equations (PDE) is a very important topic in advance Mathematics for Engineering. The three main first-order PDE problems that a basic course must deal with are: \begin{enumerate} \item {\bf Pfaff Differential Equations}, which consists on finding the general solution for: $$ P(x,y,z)\:{\rm d}x + Q(x,y,z)\:{\rm d}y + Q(x,y,z)\:{\rm d}z = 0 $$ \item {\bf Quasi-linear Partial Differential Equations}, which consists on finding the general solution for: $$ P(x,y,z)\:p + Q(x,y,z)\:q = R(x,y,z) $$ where $\displaystyle \quad p=\frac{\partial\: z}{\partial\: x}\quad$ and $\displaystyle \quad q=\frac{\partial\: z}{\partial\: y}$. \item {\bf Lagrange-Charpit Method} for finding a {\em complete integral} for a given general first order partial differential equation: $\quad F(x,y,z,p,q)=0$. \end{enumerate} \medskip In order to help the teaching and learning process of this topic, we have developed, using {\sc Derive} 6 {\sc Cas}, the file \textbf{\tt FOPDE.mth}. The use of this file allows the user to solve these three problems (their general cases and the particular cases) stepwise. Since these three problems requires several steps for their resolution, the programs developed in {\tt FOPDE.mth} show step by step all the resolution procedure providing in this way a powerful tool as a tutorial for teaching how to solve these types of equations. This way the use of {\sc Derive} 6 is done as a {\sc PeCas} (Pedagogical {\sc Cas}) providing not only the final result but also all partial results. \medskip On the other hand, in the resolution process of such equations, first-order Ordinary Differential Equations (ODEs) are needed. Therefore, {\tt FOPDE.mth} loads the package {\tt FOODE.mth}, which is part of the {\sc Derive} package that was introduced in TIME 2010. \medskip The programs contained in the file \textbf{\tt FOPDE.mth} can be grouped within the following blocks: \begin{itemize} \item \textbf{First-order ODEs}: 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. \item \textbf{First-order differential equations and nth degree in y'}. \item \textbf{Generic programs to solve first order differential equations}. \item \textbf{Pfaff Differential Equations}. \item \textbf{Quasi-linear PDE}. \item \textbf{Lagrange-Charpit Method for First-Order PDE}. \end{itemize} \medskip Finally, we will state the conclusions obtained after using this file with our students and also some future work on this and other related subjects. Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech.