Course Purpose
In this course, you will learn the basic properties of partial differential equations and the fundamental analytical techniques used to solve them.
Course Learning Outcomes
CLO 1: Concept of Partial Differential Equation.
CLO 2: Visualizing solutions Partial Differential Equation.
CLO 3: Derivation and Solution of the Transport Equation.
CLO 4: Method of Characteristics.Method of Characteristics.
Course Content
Module 1 - Introduction to Partial Differential Equations
- Definition of PDEs
- Difference between ODEs and PDEs
- Order and degree of PDEs
- Applications in science and computing
Module 2 - Classification of Partial Differential Equations
Key Topics:
- Linear vs nonlinear PDEs
- Homogeneous and non-homogeneous equations
- Elliptic, parabolic and hyperbolic PDEs
Module 3 - Solutions of Partial Differential Equations
Key Topics:
- General solution
- Particular solution
- Complete integral
- Singular solution
Module 4: Initial and Boundary Conditions
Key Topics:
- Boundary value problems
- Physical interpretation of conditions
- Uniqueness of solutions
Module 5: Visualizing Solutions of PDEs
Key Topics:
- Graphical interpretation
- Surface plots and contour plots
- Visualization tools (MATLAB/Python)
- Interpretation of solution behaviour
Module 6: Linear First-Order PDEs
Key Topics:
- Standard form
- Analytical solution methods
- Real-world interpretation
Module 7: Derivation of the Transport Equation
Key Topics:
- • Conservation laws
- • Mathematical derivation
- • Applications in fluid flow
Module 8: Method of Characteristics
Key Topics:
- Characteristic curves
- Geometric interpretation
- Solving first-order PDEs
Module 9: Advanced Applications of Characteristics
Key Topics:
- Nonlinear PDEs
- Shock formation concepts
- Piecewise solutions
Module 10: Applications of PDEs
Key Topics:
- Heat equation
- Wave equation
- Diffusion models
- Applications in computing and engineering