Being a mediocre student in school and understanding the education how it suppose to be via training of the mind, Prof. Banerjee thinksthat the biggest intellectual challenge is not the implementation of a new education system butinjection of 'critical thinking' process in the students and creating an environment that fosters 'critical thinking', while being in any education system. Following graduate and undergraduate courses are taught by Prof. Banerjee and he continuously experiments with new educational methods that canenhance 'critical thinking' process in our students.

### EMCH 501: Engineering Analysis I(Advanced Mathematical Methods for Engineers)

#### Every Fall Semesters Since 2012

##### Topics Covered

- Review of ordinary differential equations, their applications and methods of solution.
- Review of ODEs and PDEs; Different type of ODEs and PDEs commonly used in engineering problems (Mechanical, Civil, Electrical, Chemical Engineering); introduction to the necessary tools required to solve ODEs and PDEs.
- Necessary Linear Algebra: Orthogonality, Projection, Linear mapping, Fredholm alternative, eigen values, eigen vectors, Jordon's form, application to linear ODE.
- Necessary Vector Calculus and Calculus of Variation
- Introduction to Advanced PDEs
- The heat equation: derivation in 1-D, extension to higher dimensions, boundary conditions, steady state heat conduction (Laplace's equation), solution and qualitative properties.
- The wave equation: vibrating strings and membranes, boundary conditions, wave propagation in 1D and 2D.
- Analytic solution techniques: Separation of variables, Series solutions, Fourier series and Fast Fourier transform methods, Laplace transform method, eingen function expansion method, Green's function based solution method.
- Numerical Solution techniques/Computational tools: implementation of analytic solutions using libraries and packages

### EMCH 561C: Advanced Numerical Methods for Engineers

#### Alternate Spring Semesters

##### Topics Covered

- Eigen Analysis: Eigen Values and Eigen Vectors
- Error Analysis: Taylor series, stability conditions.
- Interpolation and Extrapolation; first and second order splines, B-splines.
- Higher Order Numerical Integration; Gauss quadrature rules.
- Numerical Solutions of Ordinary Differential Equations: Euler, Heun's&Runge-Kutta method, explicit and implicit methods
- Solution of Systems of Nonlinear Equations: Newton-Raphson (NR) method, Modified NR method.
- Numerical Solutions of Boundary Value Problem; Finite Difference Methods
- Miscellaneous Advanced Topics; Calculus of Variation, Introduction to Finite Element Methods, Rayleigh-Ritz method, Weighted residual method, Bubnov-Galerkin, Petrov-Galarkin methods, Kantorwich method, Boundary Initial Value problems.
- Each technique will be taught with follow up programming in MATLAB.

### EMCH 201/PHYS 311: Introduction to the Application of Numerical Methods for Engineers

#### Every Semester Fall and Spring (Two sections with >35 students in each)

##### Topics Covered

**Topic 1**: Fundamentals

- Numbers, Algebraic Operations ; Applications
- Linear Interpolations and Extrapolation ; Applications
- Solution of Nonlinear
- Equations of one variable: Bisection method & Newton-Raphson method

**Topic 2**: Necessary Calculus

- Introduction to differentiation. Why it is necessary?
- Applications of derivatives in Engineering.
- Discussion on derivatives (up to second order) mathematically and numerically.
- Error analysis and Introduction to Taylor series

**Topic 3**: Numerical Package (for the Lab classes only)

- Introduction to MATLAB
- Introduction to Number, Algebraic, Trigonometric, Vector and Matrix operations using MATLAB
- Introduction to graph plot (visual representation) in MATLAB
- Introduction to Programming in MATLAB
- For fun: make your own software with a GUI (Graphics User Interface)

**Topic 4**: Polynomial and Curve Fitting

- Introduction to Polynomials; Application of Polynomials in Engineering
- Solution of Polynomials using MATLAB
- Curve fitting using Polynomials in MATLAB
- Application of curve fitting in engineering experiments

**Topic 5**: Linear Algebra

- Application of Linear Algebra in various engineering problems
- Formulation of various Linear Algebra problems, System of linear equations
- Solution of simultaneous linear equations using Gaussian Elimination method
- Solution of simultaneous linear equations using Gauss-Jordan method
- Solution of simultaneous linear equations using MATLAB
- Introduction to the system of nonlinear equations and their solution method using the approach listed in section 3 under
**Topic 1**.

**Topic 6**: Differential Equations

- Application of first order and second order differential equations in Engineering
- Solution of first order differential equations using Eulerâ€™s method
- Solution of first order and second order differential equations using 4th order Runge-Kutta method
- Solution of differential equations using MATLAB

**Topic 7**: Integration

- Application of integration in Engineering
- Numerical integration using trapezoidal rule
- Numerical integration using Simpson 1/3 rule
- Numerical integration using MATLAB