ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA DEL DISEÑO
AERONAUTIC ENGINEERING, 2ND YEAR, 3RD SEMESTER
ADVANCED MATHEMATICS (AMPLIACIÓN DE MATEMÁTICAS)
Objectives
Advanced Mathematics for Aeronautic Engineering is a one-semester subject of 9 credits that
demonstrates the modelling process in the context of differential equations and complex variables.
They are split into:
4.5 credits Theoretical/Practical Teaching Classroom
1.5 credits Problem Solving Teaching Classroom
3 credits Labwork
We will use case studies from a number of areas such as motion and geometric problems, population
dynamics, electric circuits, mechanical systems and physic laws. It is designed as a course in basic
theory from both quantitative and qualitative methods of ordinary differentiale equations and complex
variables as illustrated by standard examples from engineering and physics. On completion of the
course the student should be able to:
- Create and analyze mathematical models based on ordinary differential equations and complex
variables;
- Determine the type of a given differential equation, determine the existence of a solution and if a
solution can be obtained, select the appropriate analytical technique for finding the solution;
- Use technology tools to find graphical, analytic or numeric solutions;
- Solve linear sytems of equations;
- Find an approximate solution to a given differential equation by Euler and Runge-Kutta’s numerical
methods;
- Use the fundamentals of complex variables and its application to evaluate improper real integrals
- Solve differential equations using Laplace and Fourier transforms;
- Study dinamical systems with the aid of the transfer function;
- Solve differencial equations by using Zeta transforms.
Syllabus
Ordinary Differential Equations. Problem-solving techniques involving First-order equations: direction
field, separation of variables, homogeneous equations, linear equations, higher-order linear equations,
method of undetermined coefficients, variation of parameters, systems of linear equations and
introduction to numerical methods. Applications: geometric and motion problems, electric circuits,
growth and decay problems, harmonic oscillators, Newton's law of gravity. Systems of linear differential
equations.
Real Laplace Transform. Properties and applications to solve initial value problems and systems of
differential equations. Application to dynamic systems.
Complex variables. Holomorhic functions, Complex Integral, Laurent Series. Fourier and Laplace
Transforms. Definition and properties. Periodic and antiperiodic functions. Applications: linear
differential equations and systems, transfer function, stability analysis. Zeta transform.
Laboratory work of 3 credits with mathematical software covering all the above topics is included within
the course.
Textbooks
- W.E. Boyce and R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems - 5th
Edition, Wiley and Sons, New York, 1992.
- R. Derrick and S.I. Grossman, Introduction to Differential Equations with Boundary Value Problems.
West Publishing.
- G.B. Folland. "Fourier Analysis and its applications". Wadsworth & Brooks/Cole 1992.
- B.R. Hunt, R.L. Lipsman, J.E. Osborn, J.M. Rosenberg. “Differential Equations with Matlab”. John
Wiley and Sons 2005.
- E. Kreyszig, Advanced Engineering Mathematics. Wiley.
- C.B. Moler. “Numerical Computing with Matlab”. SIAM 2000
- R.K. Nagle and E.B. Saff, Fundamentals of Differential Equations and Boundary Value Problems.
Addison and Wesley.
- L.M. Sánchez Ruiz, M. Legua. Ecuaciones diferenciales y Transformada de Laplace con Aplicaciones.
SPUPV 798
- L.M. Sánchez Ruiz, M. Legua. Fundamentos de Variable Compleja y Aplicaciones. SPUPV 625
- S. Wolfram. “The MATHEMATICA Book”. Wolfram Media/Cambridge University Press.
- D.G. Zill, Differential Equations with Boundary-Value Problems, PWS-KENT.
Assessment
One partial exam (Differential equations and Real Lplace Transform) will be held during the semester.
Approximate timing for this is Week 9. Assessment of the final exam will include the whole course
except for those students that succeeded in the partial exam and take only the Complex Variable part.
Final grade will take into account the Lab work and assignments developed during the course.
No credit will be kept after the final exam.
Instructor
Prof. Luis M. Sánchez Ruiz, Department of Applied Mathematics
phone 76651, email: LMSR@mat.upv.es
Consultation with the course convenor is possible during 6 hours per week at his office on the thirf floor
at ETSID. If you cannot come at that time you may contact the course convenor by email or phone and
make an appointment