- Structure of Vector Spaces: Subspaces, generating systems, span, linear independence, bases, dimension, codimension.
- Linear Operators: Matrix representation, similarity, change of bases. Kernel and range spaces, rank, nullity and invertibility. LU deocomposition.
- Spectral Theory: Characteristic quantities, algebraic and geometric multiplicity of eigenvalues, direct sum decomposition of vector spaces and of linear operators. Generalized eigenvectors and eigenspaces. Diagonal representations and Jordan canonical forms.
- Introduction to Perturbation Theory: Regular and singular perturbation theory, inner and outer solutions. Examples from chemical engineering problems.
- Partial Differential Equations: The methods of characteristics. Examples from chemical engineering problems.
- Green’s Functions: The fundamental solutions of partial differential equations.
- Potential Theory: Modeling of time invariant physical problems in isotropic and anisotropic spaces. Solution of inviscid flow and steady state problems.
- Stokes Flow: The mathematical peculiarities of elliptic equations of the fourth order.
- Diffusion Theory: Mathematical characteristics of non–invertible physical phenomena governed by parabolic equations. Fundamental solution and infinite speed of propagation. Solutions of transient diffusion problems.
- Wave Theory: Modeling the physical behavior of initial and boundary value problems governed by hyperbolic equations. Main characteristics of vibration, radiation, wave propagation and scattering problems.
Prerequisite for this course is knowledge of Advanced Calculus, Matrix Theory, Linear Algebraic Systems and Ordinary Differential Equations. The course will be adapted to the engineering point of view. That is, it will emphasize the mathematical interpretation and the corresponding formulation of the relative physical phenomena, avoiding any rigorous mathematical analysis. Numerous examples and applications will be the core of the course.
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