Numerical Methods Micro Syllabus | Engineering Course

Numerical Methods Micro Syllabus (ENSH 202)

Marks Distribution

Chapter	Hours	Marks
1: Solution of Non-Linear Equations	7	10
2: Solution of System of Linear Algebraic Equations	8	10
3: Interpolation	9	10
4: Numerical Differentiation and Integration	6	10
5: Solution of Ordinary Differential Equations	8	10
6: Solution of Partial Differential Equations	7	10
Total	45	60

Depth Codes

A – Algorithm
C – Comparison with other methods
D – Derivation
F – Formula Derivation
N – Numerical
W – Working Mechanism
L – Advantages/Limitations

Numerical Methods Micro Syllabus Content

Unit	Topic / Subtopic	Depth Code	Description of Depth
1
	Unit 1: Solution of Non-Linear Equations (7 hours)
	1.1 Errors and accuracy in numerical computations	N, C	Accuracy Vs. Precision Sources or types of error Truncation, rounding-off, significant figures Error Calculation: absolute, relative, % errors
	1.2 Bisection Method	D, W, A, N, C, L
	1.3 Regula Falsi Method and Secant Method
	Regula Falsi Method	D, W, A, N, C, L
	Secant Method	D, W, A, N, C, L
	1.4 Newton Raphson Method	D, W, A, N, C, L	Only for real roots
	1.6 Comparison of the methods	C, L	Bracketing Vs. Open-ended methods
		D, C, L	Rates of Convergence
	1.7 Solution of system of non-linear equations
	1.7.1 Direct Approach	W, N,L	Using simple substitution
	1.7.2 Newton-Raphson Method	D, W, A, N, C, L
2
	Unit 2: Solution of System of Linear Algebraic Equations (8 hours)
	2.1 Direct Methods	C, L
	2.1.1 Gauss Jordan Method	W, A, N, C, L
	2.1.2 Gauss Elimination Method	D, W, A, N, C, L
	2.1.3 Matrix inverse using Gauss Jordan and elimination methods	D, W, A, N, C	Without pivoting
	2.1.4 Factorization Methods	D, W, A, N, C	Do-Little’s Method & Crout’s Method
	2.2 Iterative Methods	C, L
	2.2.1 Jacobi’s Method	W, A, N, C, L
	2.2.2 Gauss Seidal Method	W, A, N, C, L
	2.3 Determination of largest and smallest Eigen values and corresponding vectors using the power method	W, A, N, C, L	Review of Eigen values and Eigen vectors and their physical interpretation Importance of dominant Eigen values in realistic problems Smallest Eigen value using inverse Power Method
3
	Unit 3: Interpolation (9 hours)
	3.1 Polynomial Interpolation	L
	3.1.1 Finite Differences	W	Constructing forward, backward, central and divided difference tables and their notation Important properties of finite differences
	3.1.2 Interpolation with Equally Spaced Intervals
	Newton’s Forward and Backward difference interpolation	W, A, N, C, L
	Stirling’s and Bessel’s Central Difference interpolation	W, N, C, L
	3.1.3 Interpolation with unequally spaced intervals
	Newton’s Divided difference interpolation	D, W, A, N, C, L
	Lagrange interpolation	D, W, A, N, C, L
	3.2 Least square method of curve fitting	C, L
	3.2.1 Linear form and forms reducible to linear form	D, W, A, N	Forms reducible to linear form: power, exponential and logarithmic form
	3.2.2 Quadratic form and forms reducible to quadratic form	D, W, A, N
	3.2.3 Higher degree polynomials	D, W, A	Generalization from quadratic
	3.3 Cubic Spline Interpolation	W, A, N, C, L	Only Natural Cubic Spline
4
	Unit 4: Numerical Differentiation and Integration (6 hours)
	4.1 Numerical Differentiation
	4.1.1 Differentiation using polynomial interpolation formulae for equally spaced intervals	D, W, N, L	Up to second order derivatives using Newton’s Forward and Backward interpolation formulae
	4.1.2 Local maxima and minima from equally spaced data	D, W, N, L	By approximating the function using cubic polynomial such that the derivative reduces to quadratic Using Newton’s forward and backward difference interpolation formulae
	4.2 Numerical Integration
	4.2.1 Newton Cote’s general quadrature formula	D
	4.2.2 Trapezoidal Rule, Simpson’s 1/3 and 3/8 rules, Boole’s rule, Weddle’s rule	D, W, A, N	Geometric interpretation
	4.2.3 Romberg Integration	W, N	Using Trapezoidal rule
	4.2.4 Gausss-Legendre Integration	W, A, N	Up to 3-point formula only
5
	Unit 5: Solution of Ordinary Differential Equations (8 Hours)
	5.1 Initial Value Problems		Definition and significance of ODEs in modeling physical phenomena Comparison of numerical solution with analytical solution
	5.1.1 Solution of first order equations
	Taylor’s Series Method	W, N
	Euler’s Method	D, W, A, N, L
	Runge-Kutta second order (RK-2) method	W, A, N, L
	Runge-Kutta fourth order (RK-4) method	W, A, N, L
	5.1.2 Solution of system of first order ODEs via Runge-Kutta methods	W, A, N	Using RK-1 (Euler), RK-2 and RK-4
	5.1.3 Solution of second order ODEs via Runge-Kutta methods	W, A, N	By reduction to a system of first order ODEs Using RK-1 (Euler), RK-2 and RK-4
	5.2 Two-point boundary value problems
	5.2.1 Shooting Method	W, A, N, C, L	Up to second order equations
	5.2.2 Finite Difference Method	D, W, A, N, C, L	Up to second order equations
6
	Unit 6: Solution of Partial Differential Equations (7 hours)
	6.1 Introduction and Classification		Definition and role of PDEs in modeling physical phenomena ODE Vs. PDE comparison Analytical Vs. Numerical Solution Standard forms of second order PDEs Classification: Elliptic, Parabolic, Hyperbolic
	6.2 Finite Difference approximations of partial derivatives		Approximation of first and second order partial derivatives
	6.3 Solution of elliptic equations
	6.3.1 Laplace equation	D, W, A, N
	6.3.2 Poisson equation	D, W, A, N
	6.4 Solution of parabolic and hyperbolic equations
	6.4.1 One-dimensional heat equation
	Bendre-Schmidt method	D, W, A, N
	Crank-Nicolson method	D, W, A, N
	6.4.2 Wave Equation	D, W, N	Convergent case only

Practical Implementation

45 Hours

Python Programming

Key Requirements:

Results to be visualized wherever possible
Practical report must include:
Working Principle, Algorithm/Pseudocode, Source code, Test Cases

Unit 1: Basics of Programming in Python

Basic Data Types

int, float, string, Boolean

Type conversion between int/float and string

Basic math and string operations

Data Structures

Lists: Basic operations, arrays/matrices, sequences, list comprehension, map/zip, aggregate functions

Tuples, Dictionaries, Sets: Introduction and examples

Basic Input/Output

Formatted output with f-strings or .format()

Converting input strings to int/float and lists

Control Flow

Conditional statements: if…elif…else

Loops: while, for, break, continue

Exception handling with try…except

Functions and Modules

Defining functions with def and lambda

Evaluating mathematical functions from string input

Working with packages and modules

Numerical and Scientific Computation

Numerical arrays with numpy

Solving non-linear equations:

scipy.optimize.fsolve
scipy.optimize.root
numpy.roots

Marks Distribution

Depth Codes

Numerical Methods Micro Syllabus Content

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