The objective of this course in differential equations is to provide the students with a formal approach to several problems related with physics and engineering. The topics discussed along the subject are usually applied in several disciplines related with the engineering, for example the method of characteristics together with the wave equation is used to model the shock waves experienced by aircrafts after trespassing the sonic barrier or to model the pressure wave generated within a pipe when a valve is suddenly closed in a flowing liquid. The heat equation is of interest to model and design all kind of aerospace devices such us the thermal engines mappings or the dimension of the aircraft heat needs in cabin. The Laplace equation is of help to model all kind of stationary problems, for instance to determine the velocity profile generated by an aerodynamic profile in an homogeneous liquid like air. Additionally the differential equations, both ordinary and partial, are used as well in elasticity, vibrations, solid mechanics, electrical circuits and electronics to model different aspects from the deformation of a solid to the transitory phenomena in a circuit.
SPECIFIC LEARNING OBJECTIVES (PO a):
- To understand the Linear operators and the principle of superposition for solving differential equations.
- To solve elementary differential equations by separation of variables and other methods.
- To understand the different application scopes of the differential equation to engineering and physics.
- To distinguish between elliptic, hyperbolic and parabolic partial differential equations and which initial or boundary conditions are appropriate for them.
- To understand how to apply separation of variables and the Fourier method to solve initial-boundary value problems for the equations of Mathematical Physics.
- To understand the separation of variables technique, the role of the resulting eigenvalue problems and the principle of superposition to solve initial-boundary value problems for the equations of Mathematical Physics.
- To understand when and how to use the method of characteristics to solve different cases of partial differential equations.
GENERAL ABILITIES (PO a, g, k):
- To understand the necessity of abstract thinking and formal mathematical proofs.
- To acquire communicative skills in mathematics.
- To acquire the ability to model real-world situations mathematically, with the aim of solving practical problems.
- To improve problem-solving skills.