### Partial Differential Equations (MATH-405)

Partial Differential Equations (PDEs) are in 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.

Learning Outcomes

After the completion of the course, students will be able to

1. Classify partial differential equations and transform them into canonical form.
2. Solve linear partial differential equations of both first and second order.
3. Apply partial derivative equation techniques to predict the behavior of certain phenomena.
4. Solve partial differential equations by Laplace and Fourier Transform.

Prerequisites: Nill

Contents

1. First-order PDEs: Introduction, Formation of PDEs, Solutions of PDEs of the first order
2.  The Cauchy’s problem for quasi-linear first-order PDEs, First order nonlinear equations
3.  Special types of first-order equations Second-order PDEs
4. Basic concepts and definitions, Mathematical problems, Linear operator
5. Superposition, Mathematical models
6.  The classical equations. The vibrating string
7. The vibrating membrane
8. Conduction of heat solids
9.  Canonical forms and variable
10.  PDEs of second order in two independent variables with constant and variable coefficients
11.  Cauchy’s problem for second-order PDEs in two independent variables
12. Methods of separation of variables, Solutions of elliptic
13.  Parabolic and hyperbolic  PDEs in Cartesian and cylindrical coordinates
14. Laplace transform: Introduction and properties of Laplace transform
15. Transforms of elementary functions
16. Periodic functions, error function and Dirac delta function
17.  Inverse Laplace transform, Convolution Theorem
18. The solution of PDEs by Laplace transform, Diffusion, and wave equations
19. Fourier transforms, Fourier integral representation
20.  Fourier sine and cosine representation
21.  Fourier transform pair
22.  Transform of elementary functions and Dirac delta function, Finite Fourier transforms
23. Solutions of heat, Wave and Laplace equations 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)

Suggested Books

1. Dennis G. Zill, Michael R. Cullen, Differential equations with boundary value   problems, (Brooks Cole, 2008)
2. John Polking, Al Boggess, Differential Equations with Boundary Value Problems, (2nd  Edition, Pearson, July 28, 2005)

RESEARCH PROJECT /PRACTICALS/LABS/ASSIGNMENTS

1. Classification of Partial differential equation.
2. Solve PDEs using Laplace and Fourier Transform.
3. Students are assigning to solve heat and wave equations using Laplace and Fourier Transform.
4. Students are assigning to solve PDEs of First and Second-order.

ASSESSMENT CRITERIA

Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)

Mid-Term Exam:   30

Final-Term Exam: 50

### Key Dates and Time of Class Meeting

Wednesday-Thursday                                  15:30 pm-17:00 pm (Ex-PPP sub-Campsus)

Commencement of Classes                                                   October 12, 2020

Mid Term Examination                                                            December 14 to 18, 2020

Final Term Examination                                                          February 08 to 12, 2020

Declaration of Result                                                              February 19, 2020