### Partial differential equations (MATH-405)(BS-VII))(Regular & Self Support)

Partial Differential Equations (PDEs) are at the heart of applied mathematics and many other scientific disciplines. The course aims at developing understanding about fundamental concepts of PDEs theory, identification and classification of their different types, how they arise in applications, and analytical methods for solving them. Special emphasis would be on wave, heat and Laplace equations.

Pre-Requisites:

Ordinary-Differential Equations

Intended Learning Outcomes

On successful completion of this course students will be able to:

1. use knowledge of partial differential equations (PDEs), modelling, the general structure of solutions, and analytic and numerical methods for solutions.
2. formulate physical problems as PDEs using conservation laws.
3. understand analogies between mathematical descriptions of different (wave) phenomena in physics and engineering.
4. classify PDEs, apply analytical methods, and physically interpret the solutions.
5. solve practical PDE problems with finite difference methods, implemented in code, and analyse the consistency, stability and convergence properties of such numerical methods.
6. interpret solutions in a physical context, such as identifying travelling waves, standing waves, and shock waves.

Contents

1. First order PDEs: Introduction, formation of PDEs
2. Solutions of PDEs of first order
3. The Cauchy’s problem for quasilinear first order PDEs
4. First order nonlinear equations, Special types of first order equations
5. Second order PDEs: Basic concepts and definitions
6. Mathematical problems, Linear operators, Superposition
7. Mathematical models: The classical equations
8. The vibrating string, the vibrating membrane, conduction of heat solids
9. Canonical forms and variable, PDEs of second order in two independent variables with constant and variable coefficients
10. Cauchy’s problem for second order PDEs in two independent variables
11. Methods of separation of variables: Solutions of elliptic, parabolic and hyperbolic PDEs in Cartesian and cylindrical coordinates
12. Laplace transform: Introduction and properties of Laplace transform
13. Transforms of elementary functions, periodic functions, error function and Dirac delta function
14. Inverse Laplace transform, convolution theorem
15. Solution of PDEs by Laplace transform, Diffusion and wave equations
16. Fourier transforms: Fourier integral representation
17. Fourier sine and cosine representation, Fourier transform pair, transform of elementary functions and Dirac delta function
18. Finite Fourier transforms
19. Solutions of heat and wave equations by Fourier transforms.
20. Solution of Laplace equation by Fourier transforms.

Recommended Books

1.  Humi M, Miller W.B; Boundary Value Problems and Partial Differential Equations. PWS-KENT Publishing Company, 1991.

2.  Myint UT, Partial Differential Equations for Scientists and Engineers, 3rdedition, North Holland, Amsterdam, 1987.

Dennis G. Zill, Michael R. Cullen, Differential equations with boundary value problems, Brooks Cole, 2008.4.

Suggusted Books:

1. John Polking, Al Boggess, Differential Equations with Boundary Value Problems, 2nd  Edition, Pearson,July 28, 2005.

2. J. Wloka, Partial Differential Equations, Cambridge University press, 1987.

Description of system of evaluation (homework, midterms, final,  assignments etc.):

1. Homework, Exercises, Attendance, Assignments: 20%
2. Midterm: 30%, Final Term: 50 %

Key Dates and Time of Class Meeting

Tuesday: 11:00am-12:30 (Self Support)

Wednesday:  9:30am-11:00(Regular) & 12:30pm-2:00pm(Self Support)

Friday: 9:30am-11:00am(Regular)

Commencement of Classes                                                   October 12, 2020

Mid Term Examination                                                            December 14-18, 2020

Final Term Examination                                                          February 08-12, 2021

Declaration of Result                                                              Fenruary 19, 2021