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.
PreRequisites:
OrdinaryDifferential Equations
Intended Learning Outcomes
On successful completion of this course students will be able to:
 use knowledge of partial differential equations (PDEs), modelling, the general structure of solutions, and analytic and numerical methods for solutions.
 formulate physical problems as PDEs using conservation laws.
 understand analogies between mathematical descriptions of different (wave) phenomena in physics and engineering.
 classify PDEs, apply analytical methods, and physically interpret the solutions.
 solve practical PDE problems with finite difference methods, implemented in code, and analyse the consistency, stability and convergence properties of such numerical methods.
 interpret solutions in a physical context, such as identifying travelling waves, standing waves, and shock waves.
Contents
 First order PDEs: Introduction, formation of PDEs
 Solutions of PDEs of first order
 The Cauchy’s problem for quasilinear first order PDEs
 First order nonlinear equations, Special types of first order equations
 Second order PDEs: Basic concepts and definitions
 Mathematical problems, Linear operators, Superposition
 Mathematical models: The classical equations
 The vibrating string, the vibrating membrane, conduction of heat solids
 Canonical forms and variable, PDEs of second order in two independent variables with constant and variable coefficients
 Cauchy’s problem for second order PDEs in two independent variables
 Methods of separation of variables: Solutions of elliptic, parabolic and hyperbolic PDEs in Cartesian and cylindrical coordinates
 Laplace transform: Introduction and properties of Laplace transform
 Transforms of elementary functions, periodic functions, error function and Dirac delta function
 Inverse Laplace transform, convolution theorem
 Solution of PDEs by Laplace transform, Diffusion and wave equations
 Fourier transforms: Fourier integral representation
 Fourier sine and cosine representation, Fourier transform pair, transform of elementary functions and Dirac delta function
 Finite Fourier transforms
 Solutions of heat and wave equations by Fourier transforms.
 Solution of Laplace equation by Fourier transforms.
Recommended Books

Humi M, Miller W.B; Boundary Value Problems and Partial Differential Equations. PWSKENT Publishing Company, 1991.

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.):
 Homework, Exercises, Attendance, Assignments: 20%
 Midterm: 30%, Final Term: 50 %
Key Dates and Time of Class Meeting
Monday: 9:30am11:00am (Regular) & 12:30pm2:00pm (Self Support)
Tuesday: 9:30am11:00am (Regular)
Friday: 11:00am12:30pm(Self Support)
Commencement of Classes October 12, 2020
Mid Term Examination December 1418, 2020
Final Term Examination February 0812, 2021
Declaration of Result Fenruary 19, 2021