Course

One-Dimensional Modeling: Origin of Ordinary Differential Equations (1^st and 2^nd Order); First Order OD: Direct Integration, Integrating Factor – Linear and Nonlinear Equations; Systems of First Order ODEs. Stability. (12 hrs)

Second Order ODE: Homogeneous and Non-homogeneous – Linear equations with constant coefficients; Laplace Transforms: Definition, Properties and Inverse Laplace Transforms; Solution of Linear First and Second Order ODEs using Laplace Transforms. Fixed points, stability of fixed points.

Numerical methods for solving ODE: Euler’s method, Improved Euler’s method and Runge-Kutta method. (15 hrs)

Two-Dimensional Modeling: Partial Differential Equations, classifications of PDE, Separation of Variables: Fourier Series, arbitrary period, even and odd expressions, half range expressions. Fourier serious solutions of one dimensional Heat and wave Equations. Numerical methods for solving PDE: Finite difference method, solution of Laplace equation by FDM, Crank-Nicolson method. (15 hrs)

Course Objectives:

• To model spatiotemporal variations in engineering systems and processes using differential equations
• To analyze and solve ordinary differential equations (ODE)
• To analyze stability of systems of first order ordinary differential equations
• To define Laplace transforms and utilize them to solve linear first and second order ODEs
• To understand partial differential equations and its applications in engineering.
• To Apply the numerical techniques for solving ODE and PDE.

Course Outcomes:

CO1: Define first-order ordinary differential equations and demonstrate ability to use techniques to solve them and apply these solutions in engineering contexts.

CO2: Reduce higher-order ordinary differential equations to a system of first-order differential equations, solve them using the method of eigenvector expansions and apply the solutions to engineering problems.

CO3: Define second-order ordinary differential equations and demonstrate ability to use techniques to solve them and apply these solutions in engineering contexts.

CO4: Define Laplace transforms and their inverses, apply their properties to solve linear ordinary differential equations.

CO5: Understand the types of partial differential equations arising from two-dimensional modeling. Use separation of variables to solve linear partial differential equations.

Textbook:

1. Erwin Kreyszig, Advanced Engineering Mathematics, 10th Edition, Wiley-India Pvt. Ltd., 2011

References:

1. Michael Greenberg, Advanced Engineering Mathematics, 2nd Edition, Pearson, 2011
2. Bruce A. Finlayson, Introduction to Chemical Engineering Computing, John Wiley & Sons, 2006.
3. Engineering Mathematics, Srimanta Pal and Subodh C Bhunia, Oxford university press, 2015.
4. Advanced Engineering Mathematics, Wylie and Barrett, 6^th Edition, McGraw Hall India, 2015.