PMAT 42373: Advanced Mathematical Methods

Course Code    : PMAT 42373

Title                   : Advanced Mathematical Methods

Pre-requisites   : PMAT 22293, PMAT 22282

 Learning Outcomes:

At the end of this course, the student should be able to 

1. discuss the properties of special functions as solutions of differential equations

2. classify partial differential equations using various techniques learned

3. solve hyperbolic, parabolic and elliptic equations using fundamental principles

4. apply a range of techniques to find solutions of standard PDEs

5. demonstrate accurate and efficient use of Fourier analysis techniques and their applications in the theory of PDEs

6. solve real world problems by identifying them appropriately from the perspective of partial derivative equations

 Course Content:

Special Functions: 

• Legendre Polynomials, Bessel Functions.

Partial Differential Equations: 

• Classification of PDE, First order partial differential equations: Lagrange’s method and Charpit’s method, Second order partial differential equations: Linear Partial Differential Equations with Constant Coefficients, Partial Differential Equations of Order two with Variables Coefficients, Classification of Partial Differential Equations Reduction to Canonical or Normal Form: Parabolic, elliptic and hyperbolic partial differential equations.

Fourier Series: 

• Fourier Series, Even and Odd Functions, Half-Range Expansions, Fourier Integral, Fourier Cosine and Sine Transforms.

Solutions of PDEs: 

• Separations of variables, D’Alembert’s Solution and Characteristic solutions of the Wave Equation.

Integral Transforms: 

• Fourier Transforms, Laplace Transforms, Hankel Transforms.

 Method of Teaching and Learning : A combination of lectures, tutorial discussions and presentations.

 Assessment : Based on tutorials, tests, presentations and end of course examination.

 Recommended Reading:

1. Kreyszig, E. (10th Ed., 2011). Advanced Engineering Mathematics, Wiley.

2. Pinsky, M.A. (2011). Partial Differential Equations and Boundary Value Problems with Application, American Mathematical Soc.

3. Raisinghania, M.D. (1995). Advanced Differential Equations, S.Chands, India.