Applied Mathematics

Course Description

Physical backgrounds of the linear 2nd order
PDEs, derivation of the wave equation, derivation of the heat equation, Fourier series, finite domain problems in rectangular coordinates, series solutions, inhomogeneous boundary conditions and equations. separation of variable in polar, cylindrical and spherical coordinates. Helmholtz equations and problems.
Sturm-Liouville eigen value problems, self-adjoint operators, orthonormal functions.
Integral transforms, Fourier transform, infinite domain problems with Fourier transform, Laplace transform, applications of Laplace transform.
Green's function, Green's function solutions to inhomogeneous problems.

Course Objectives & Outcomes

Objectives:

• develop skills to solve linear second order partial differential equations (in particular, Laplace, wave and diffusion equations) using the methods of separation of variables in polar, cylindrical and spherical coordinates and also using the integral transforms.
• develop the solution techniques which are hugely dependent on the type of problem and boundary conditions imposed.

Outcomes:
On successful completion of this course student will be able to:

•  Identify different types of linear 2nd order PDEs and their general solutions, and recognize the physical origin of the mathematical equations, Distinguish the physical meaning of various boundary conditions and choose the boundary conditions in the respective coordinate frames.
• Distinguish between many of integral transforms
• Present Green's function kernels for a typical problems.

References

1. Text book
2. Haberman, R., (2004), Applied partial differential equations: With Fourier series and boundary value problems, Pearson Prentice Hall, ISBN 0130652431

Optional

1. Trim, D. W., (1990), Applied Partial Differential Equations, PWS, ISBN 0534982433
2.  Hillen, T. , Leonard, I. , Roessel, H.V., (2012), Partial Differential Equations: Theory and Completely Solved Problems, Wile, ISBN: 978-1-118-06330-9

Course ID: MATH 511