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

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

*Prerequisites: Nill*

** Contents**

- First-order PDEs: Introduction, Formation of PDEs, Solutions of PDEs of the first order
- The Cauchy’s problem for quasi-linear first-order PDEs, First order nonlinear equations
- Special types of first-order equations Second-order PDEs
- Basic concepts and definitions, Mathematical problems, Linear operator
- 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
- The 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, Wave and Laplace equations by Fourier transforms

*Recommended Books *

- Humi M, Miller W.B;
*Boundary Value Problems and Partial Differential Equations*. (PWS- KENT Publishing Company, 1991) - Myint UT, Partial
*Differential Equations for Scientists and Engineers*, (3rdedition, North Holland, Amsterdam, 1987)

*Suggested Books *

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

*RESEARCH PROJECT /PRACTICALS/LABS/ASSIGNMENTS*

- Classification of Partial differential equation.
- Solve PDEs using Laplace and Fourier Transform.
- Students are assigning to solve heat and wave equations using Laplace and Fourier Transform.
- 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

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