At the conclusion of this class the students will be able to do the following.
1. Solve analytically the basic linear partial differential equations: Laplace equation, wave equation, heat equation.
2. Solve the same equations as well as their nonlinear extensions using numerical software (e.g., Matlab) and/or symbolic algebra software (e.g., Maple).
3. Use the PDE formalism in modeling and solving applied problems (e.g. applications to electromagnetic field, image processing, transportation problems etc.).
4. To be able to solve 2nd order linear differential equations by series methods.
5. To know the basic special functions (Bessel, Legendre, Chebyshev).
In addition, the students will have a clear intuitive understanding of the physical significance of partial differential equations as well as their connection to systems of algebraic equations.
Course Content (Syllabus)
Partial differential equations (PDE's): Laplace equation, heat equation, wave equation.
In the first part of this course solution methods will be taught.
1. Separation of variables.
2. Integral transforms (Fourier, Laplace).
3. Solution using symbolic algebra software (computer algebra systems, CAS) e.g. Maple, Mathematica.
4. Solution using numerical methods and introduction to the corrsponding software (e.g. Maple, Mathematica, Matlab PDE Toolbox, MathPDE).
In the second part of the course, the students will study and present papers from the current literature, related to the applications of PDE's (e.g., applications to electromagnetic field, image processing, transportation problems, stochastic processes etc.).
Partial differential equations (PDE's): Separation of variables, Laplace equation, heat equation, wave equation.