Unit 1 Differential Equations

9 lecture hours

• Differential Equations
• Ordinary and Partial differentiation
• Order of the equation
• Existence and Uniqueness of Solution
• Initial Value problem
• Boundary Value Problem
• Linear and Non-Linear Equation. 1st Order ODE: Geometric Meaning of y'= ƒ(x
• Direction Fields
• Euler’s Method
• Separable ODEs
• Exact ODEs (Integrating Factors Method
• Linear ODEs(Homogeneous and Non-Homogeneous
• Reduction to Linear problems)
• Orthogonal Trajectories. 2nd Order ODE: Linear Dependence and Linear Independence
• Homogeneous Linear ODEs of Second Order (Principal of Superposition
• Homogeneous Linear ODEs with Constant Coefficients (Euler's formula and review of the circular and hyperbolic function
• Exponential Solutions
• Repeated Roots and Stability)
• Differential Operator
• Modelling of Free Oscillations of Spring-Mass System
• Homogeneous Linear ODEs with Non-constant Coefficient (Cauchy-Euler Equation
• Existence and Uniqueness of Solutions)

Unit 2 Laplace Transforms

5 lecture hours

• Laplace Transform
• First Shifting Theorem (s-Shifting)
• Transforms of Derivatives and Integrals, ODE
• Unit Step Function (Heaviside Function)
• Second Shifting Theorem (t- Shifting)
• Short Impulses
• Dirac’s Delta Function
• Partial Fractions
• Integral Equations
• Differentiation and Integration of Transforms
• ODEs with Variable Coefficients
• Systems of ODEs

Unit 3 Linear Algebra

8 lecture hours

• Matrices and Vectors: Vectors in 2-Space and 3- Space
• Addition and Scalar Multiplication
• Matrix Multiplication
• Linear Systems of Equations and Gauss Elimination
• Linear Independence
• Rank of a Matrix
• Solutions of Linear Systems: Existence and Uniqueness
• Determinants and Cramer’s Rule
• Inverse of a Matrix
• Gauss–Jordan Elimination
• Solution by Iteration
• Vector Spaces
• Inner Product Spaces
• Linear Transformations
• Matrix Eigenvalues
• Determining Eigenvalues-Eigenvectors and their Program Name: Master of Engineering Level: PG Branch: Mechanical Engineering (Thermal Engineering) Course / Subject Code : ME01000421 Course / Subject Name : Computational Methods for Mechanical Engineers applications
• Power Method for Eigenvalues
• Skew- Symmetric, and Orthogonal Matrices

Unit 4 Vector Calculus

6 lecture hours

• Vector Product
• Vector and Scalar Functions and Their Fields
• Vector Calculus: Derivatives
• Arc Length
• Gradient of a Scalar Field
• Directional Derivative
• Divergence of a Vector Field
• Curl of a Vector Field
• Line Integrals
• Path Independence of Line Integrals
• Green’s Theorem in the Plane
• Surfaces for Surface Integrals
• Surface Integrals
• Triple Integrals
• Divergence Theorem of Gauss
• Further Applications of the Divergence Theorem
• Stokes’ Theorem

Unit 5 Fourier Analysis and PDE

8 lecture hours

• Fourier Series
• Arbitrary Period
• Even and Odd Functions
• Half-Range Expansions
• Forced Oscillations
• Approximation by Trigonometric Polynomials
• Sturm–Liouville Problems
• Orthogonal Functions
• Orthogonal Series
• Generalized Fourier Series
• Fourier Integral
• Fourier Cosine and Sine Transforms
• Fourier Transform
• Discrete and Fast Fourier Transforms
• Basic Concepts of PDEs
• Modeling: Vibrating String
• Wave Equation
• Solution by Separating Variables
• Use of Fourier Series
• D’Alembert’s Solution of the Wave Equation
• Modelling: Heat Flow from a Body in Space
• Heat Equation: Solution by Fourier Series
• Steady Two-Dimensional Heat Problems

Unit 6 Numeric Analysis

• Solution of Equations by Iteration
• Newton’s Divided-Difference Interpolating Polynomials
• Lagrange Interpolating Polynomials
• Coefficients of an Interpolating Polynomial
• Inverse Interpolation
• Spline Interpolation
• Numeric Integration and Differentiation
• Numeric Methods for: First-Order ODEs
• Multistep Methods
• Systems and Higher (up to second) Order ODEs
• Elliptic PDEs * 10%

Unit 7 Probability

3 lecture hours

• Data Representation
• Permutations and Combinations
• Random Variables
• Probability Distributions
• Mean and Variance of a Distribution
• Poisson, and Normal Distribution

Unit 8 Statistics

6 lecture hours

• Random Sampling
• Point Estimation of Parameter
• Confidence Intervals
• Testing Hypotheses
• Goodness of Fit, χ2 - Test
• Regression: Linear Regression
• Polynomial Regression
• Nonlinear Regression