Applied Mathamatics

·         Enable the students to handle field operators and their applications

·         Provide various solution methods for solving first order differential equations

·         Apply vector algebra to geometry

·         Solve the first order separable differential equations

·         Determine the solution of homogeneous linear differential equations with constant coefficients

·         Solve the first order exact differential equations

·         Make use of integrating factor method to solve first order linear differential equations

·         Compute partial derivatives

·         Define field operators (del, curl, and div)

·         Relate field operators with physical/ geometrical properties

·         Determine the characteristics (curvature, tangent, normal, binormal, etc.) of space curve

·         Compute line integrals

·         Familiarize computational techniques for solving ordinary differential equations

·         First order Ordinary Differential Equations (ODE): Variable separable, homogeneous equations, exact equations, linear equations and Bernoulli equations.

·         Vector Calculus: Ordinary derivatives of vectors, Space curves, Differentiation formula, Partial derivatives of vectors, Differential of vectors, Differential Geometry, The vector differential operators, del, Gradient, Divergence, Curl, Formula involving del, Invariance, Ordinary integral of vectors, Line integrals.

·         Computational Solution of First Order ODE: Getting started with MATLAB, Doing Mathematics using MATLAB. M-Files, MATLAB Interface, Workspace, Working directory, Command window Script m-files and function m-files. The built-in MATLAB commands for solving ordinary differential equation.

·         Practical Exam on MATLAB 05%

·         End-of-course Examination   70%

·         Introduction to Vector Analysis, Davis. H. F, and Snider. A. D, 3^rd Edition, Allyn and Bacon inc, Boston, 1979.

·         An Introduction to Ordinary Differential Equations, Earl. A. Coddington, Dover Publication, Inc., New York, 1989.

·         Elementary Differential Equations and Boundary Value Problems, William E. Boyce, Richard C. Diprima John Wiley & Sons, Inc, 1984.

·         Elementary Differential Equations with Applications, William R. Derrick, Stanley I. Grossman Addison-Wesley Publishing Company, 1981.

·         Familiarize with equilibrium problems of flexible strings and chains

·         Develop equation of motion for varying mass problems

·         Solve problems concerning varying mass anddamped motion on a space curve

·         Solve problems concerning oblique impact.

·         Recall the physical quantities associated with the system of particles

·         Derive equation of catenary for flexible string/chain

·         Solve the equilibrium problems of flexible chain

·         Dynamics of a Particle with one Degree of Freedom: Motion in a straight line under variable forces, Motion under gravity in a resisting medium, The rectilinear motion of bodies with variable mass, Particle motion on space curve, Damped harmonic Oscillations, Forced oscillations

·         Dynamics of a Particle with two Degrees of Freedom: Oblique impact of elastic bodies.

·         Moment of Inertia: Moments and product of inertia, Parallel axes theorem, Perpendicular axes theorem for moments and product and product of inertia, Principle axes and principle moments of inertia of a system of particles, Rotating coordinate systems, Infinitesimal rotation.

·         The Motion of a System of Particles-general Theorems: The motion of the centre of mass, Motion about the centre of mass, Motion generated by simultaneously applied impulses.

·         Flexible Chains and Strings: Equation of catenary, Various relations for common catenary, Equilibrium of light string on a smooth plane curve, light string on a rough plane curve, heavy string on a smooth plane curve and heavy string on a rough plane curve.

·         End of course Examination   70%

·         Classical Mechanics, Goldstein H., Norosa Publishing House, 2000.

·         Statics: A Text book for the use of the Higher Divisions in Schools and for First Year Students at the Universities, Ramsey A. S., Cambridge University Press, 2009.

·         Introduce integral theorems

·         Familiarize curvilinear coordinate systems

·         Acquainted with appropriate solution methods of linear differential equations

·         Characterize conservative and Solenoidal fields

·         Recall integral transformation theorems

·         Illustrate the integral theorems with examples

·         Illustrate the application of integral transformation theorems

·         Solve linear homogeneous ordinary differential equations

·         Solve first order linear partial differential equations

·         Integrals Theorems: The divergence theorem of Gauss, Stokes theorem, Green’s theorem in the plane, Related integral theorems, Integral operator form for del.

·         Curvilinear Coordinates: Transformation of coordinates, Orthogonal curvilinear coordinates, Unit vectors in curvilinear systems, Arc length and volume elements, Gradient, Divergent and Curl, Special orthogonal coordinates systems.

·         Higher Order Differential Equations: Second order equations with constant coefficients, computation of particular integrals using differential operators, undetermined coefficient method and variation of parameter method.

·         Partial Differential Equations: Introduction to partial differential equations, Formation and classification of partial differential equations, Solution methods for first order partial differential equations.

·         End-of-course Examination  70%

·         Introduction to Vector Analysis, Davis. H. F, and Snider. A. D, 3^rd Edition, Allyn and Bacon inc, Boston, 1979.

·         An Introduction to Ordinary Differential Equations, Earl. A. Coddington, Dover Publication, Inc., New York, 1989.

·         Elementary Differential Equations and Boundary Value Problems, William E. Boyce, Richard C. Diprima John Wiley & Sons, Inc, 1984.

·         Elementary Differential Equations with Applications, William R. Derrick, Stanley I. Grossman Addison-Wesley Publishing Company, 1981.

·         Familiarize the concept of shearing force and bending moment

·         Provide the techniques to compute the deflection of an elastic beam

·         Classify the equation of angular motion of a rigid body

·         Formulate general motion of a rigid lamina in its own plane

·         Determine bending moment and shearing force of a straight beam

·         Use Bernoulli’s Euler’s law to find the deflection of an elastic beam

·         Apply Clapeyron’s equation to find the bending moment of a beam

The motion of connected particles, two dimensional motion of a projectile in a resisting medium

·         An Introduction to the Dynamics of a Rigid Body: Rotation of a lamina about a fixed axis, Momentum and energy equations for angular motion of a lamina, Compound pendulum, Force exerted on the axis of rotation, Impulse and angular momentum, Relation between the equations of angular motion of a rigid body and the equations of motion of a particle moving in a straight line, Motion of a lamina in its own plane-instantaneous centre of rotation, General motion of a rigid lamina in its own plane.

·         Bending of Beams: Shearing force of a beam and shearing force diagram, Bending moment of a beam and bending moment diagram, Relation between shearing force and bending moment, Equilibrium of slightly elastic beams, Bending moment of slightly elastic beam, Bernoulli’s Euler’s law for bending moment of slightly elastic beam, Differential equations of equilibrium of thin beams with various types of loads, Clapeyron’s equation of three moments.

·         End of course Examination  70%

·         Engineering Mechanics: Statics, Meriam. L. J. and Kraige. G. L., Willey, 2004. Hibbeler. R.C., Engineering Mechanics: Statics, Pearson, 2013.

Hourly Breakdown

·         Introduce the concepts of P.D.E’s and solve linear P.D.E’s

·         Imparting the fundamentals and applications of Fourier series, Fourier transform and Laplace transforms

·         Recall the elementary special functions

·         Solve linear P.D.E’s with constant coefficients

·         Determine the Fourier series representation of periodic functions

·         Recall the properties of Fourier series

·         Define the Fourier transform and inverse transform

·         Discuss the fundamental properties of the Fourier transform

·         Solve Initial Boundary Value Problems using the Fourier series / Fourier transform

·         Define the Laplace transform.

·         Use the Laplace transform / techniques to solve Initial Boundary Value Problems

Special functions: an Introduction

Gamma functions, Bessel functions, Legendre polynomials and functions.

Partial Differential Equations: Introduction of P.D.E: Linear P.D.E, Superpossion of solutions, Auxiliary equation, Complementary function, Particular Integral, Examples and Applications.

Fourier series and integrals: Fourier series, Half range Fourier sine and cosine series,   Parseval’s identity, Fourier integrals, Fourier transforms, Properties of Fourier transforms, Applications to Boundary Value Problems.

Laplace Transforms: Laplace transforms and its properties, Inverse Laplace transforms, Applications to solving ordinary and partial differential equations.

·         End-of-course Examination 70%

·         M.D.Raisinghania, Advance Differential Equations, Chand & Company Ltd. Ramnagar, New Delhi, 2001.

·           A. Donald, Mathematical Methods for Scientists and Engineers, McQuarrie, Univ Science Books; 1st edition, 2003.

·         E. Zauderer, Partial Differential Equations of Applied Mathematics, third edition, John Wiley, 2006.

Hourly Breakdown

·         Prove the equation of continuity and Euler’s equation of motion

·         Apply Bernoulli’s equation to flow problems

·         Discuss possible motion and bounding surfaces

·         Solve expanding bubble problems

·         Discuss 2D ideal irrotational flow

·         Determine complex potential in the presence of  circular cylinder and long wall

·         Applying Blasius’s theorem to compute the net force

·         Discuss the water wave problems

Uniform flow past a circular cylinder, flow past a bubble, circulation, drag andlift, flow past a flat wing.

Water Waves: 1D free boundary problems, progressive waves, infinite and finite depth, wave speed, dispersion and group velocity.

Incompressible Newtonian fluids: introduction of concepts, viscosity, Reynolds number and turbulence.

·         End of course Examination    70%

·         J.Williams,Fluid Mechanics, problem solvers, 2014.

·         G.K.Batchelor, An introduction to Fluid Dynamics, Cambridge university press, 2000.

·         J. Alexandre, Mathematical Introduction to Fluid Mechanics, springer publication, 2000.

Hourly Breakdown

·         Enable the students to solve transportation, assignment and network problems and perform sensitive analysis

·         Solve simple linear programming problems by graphical method

·         Demonstrate skills in applying simplex methods

·         Discuss primal and dual relationships with applications

·         Solve real world problems by applying acquired techniques

·         Perform sensitivity analysis

·         Utilize the special algorithms to solve transportation and assignment problems

·         Find the shortest route and maximum flow in Network problems

Simplex method: un boundedness, degeneracy, big-M method, duality, primal-dual relationship, dual simplex methods, the revised simplex methods, sensitive analysis.

Transportation and Assignment problems: formulating transportation problems, finding basic solutions, simplex method for transportation problems, assignment problems, transshipment problems and special algorithms.

Network: shortest route problems, maximum flow problems, project scheduling, spanning trees.

·         End of course Examination 70%

·         W.L.Winston and M.Venkataramanan, Introduction to Mathematical Programming, Thomson Brooks, 2003.

·         V. Steven, Mathematical Programming, Dover Publications Ins.,2009.

·         W.L.Walter, Introduction to Mathematical Programming, Pearson Education Company, 1999.

Hourly Breakdown

·         Develop knowledge and skills in computer Arithmetic, iterative methods to solve non-linear equations, interpolating techniques to find zeros of polynomials and Numerical Integration methods

·         Determine the quotient and remainder in a polynomial division using Hornor algorithm

·         Recall rounding and chopping

·         Establish the relationship between the number of real zeros of a polynomial and the Sturm sequence

·         Apply bisection method to find roots

·         Discuss the convergence of the fixed point iteration

·         Establish the convergence of Newton’s iteration

·         Apply Newton’s iteration to compute the zero of a given function

·         Discuss interpolation methods

·         Establish the error bound in interpolation

·         Recall Numerical integrating methods

·         Evaluate definite integrals using numerical integration methods

·         Evaluate error bounds integration methods

Solution of Equations in one variable: Sturm sequences: Construction, Locating zeros; Bisection Method: Convergence, Error bounds, Fixed Point iteration: Contraction mapping theorem, Existence and uniqueness of fixed point, Convergence; Newton’s method: Convergence, Error bounds; Rate of Convergence: Comparison of rate of convergence, Accelerating the convergence.

Interpolation and Polynomial Approximation: Interpolation: Existence and Uniqueness, Lagrange Polynomial, Lagrange interpolation, Divided differences, Newton’s interpolation, Error analysis in interpolation.

Numerical Integration: Trapezoidal rule, Simpson’s rule, Composite Trapezoidal rule, Composite Simpson’s rule, Error Analysis in numerical integration.

·         End-of-course Examination: 70%

·         E. Isaacson and H. B. Keller, Analysis of Numerical Methods, Dover, 1994.

·         F. Scheid, Numerical Analysis: Schaum’s Outline Series, McGraw-Hill, 1989.

·         H. F. Hildebrand, Introduction to Numerical Analysis, Dover, 1987.

Hourly Breakdown

·         Enable to investigate and apply standard linear and nonlinear programming problems.

·         Solve deterministic programming problems using dynamic programming algorithm

·         Construct the goal programming model  problems

·         Solve goal programming problems using modified simplex method

·         Discuss the solution method of unconstrained nonlinear extremum problems

·         Apply Lagrangian Multipliers method and Karush-Kuhn-Tucker (KKT) conditions to find local minimizers

·         Solve fractional programming problems by modifying them into linear programming problems

·         Utilize Wolfe’s algorithm for solving quadratic programming problems

·         Apply separable programming algorithm to solve nonlinear programming problems

Dynamic programming: Stage, State, Recursive equation, Developing optimal decision policy, Bellman’s principal of optimality, characteristics of deterministic dynamic programming, solving linear programming problem using dynamic programming approach.

Goal programming: Difference between linear programming and Goal programming approach, Model formulation, objectives of pre-emptive and non-pre-emptive programming, Modified Simplex method.

Nonlinear programming: Fundamentals of optimization, types of problems, Optimality conditions for unconstrained and constrained problems, Lagrangian Multipliers method and Karush-Kuhn-Tucker conditions, Fractional programming, convert the Fractional programming problem into linear programming problem, Quadratic programming, Wolfe’s algorithm, Wolfe’s modified simplex method, Separable programming, separable function, piece wise linear approximation of separable nonlinear programming problem, Separable programming algorithm.

·         End-of-course Examination 70%

·         F. S. Hillier and G. J. Lieberman, Introduction to Operations Research, 7^th edition, McGrawHill, New York, 2001.

·         H. A. Taha, Operations Research an Introduction, 8^th edition, Pearson Prentice Hall, New Jersey, 2007.

Hourly Breakdown

·         Enable to derive the governing equations for standard systems, including the Lagrangian and Hamiltonian formulations

·         Introduce the application of classical mechanics to standard rigid body problems and rotating systems

• Analyze  kinematics of the three dimensional particle motion in different coordinate systems
• Discuss translational/ rotational motion in three dimensions
• Formulate expressions for velocity and acceleration of a particle in a rotating frame
• Discuss motion of a particle moving near earth’s surface
• Recall the angular momentum, moments of inertia, principal moment of inertia, moment of inertia tensor
• Discuss the motion of a rigid body in three dimensions with angular momentum and moments of inertia
• Apply Euler’s equations of motion to rigid body motion
• Describe the motion of a mechanical system using Lagrange / Hamilton formalism
• Utilize  the canonical transformations
• Explain the  motion of a spinning top by deriving and solving the equations of motion

Motion of a particle: Equation of motion, components of velocity and   acceleration in cylindrical coordinates and in spherical polar coordinates, Finite displacements, Infinitesimal displacements, rate of change of displacement referred to rotating axes and motion relative to earth.

Motion of a rigid body: Definitions of moments of inertia and products of inertia, Radius of gyration, Moment of inertia with respect to a variable line, The momental ellipsoid , Principal axes and principal moments of inertia, Equations of motion of a rigid body, Euler’s equation of motion and applications.

Lagrange’s equations: Degrees of freedom, Generalized coordinates, Generalized velocity, Holonomic and non-Holonomic systems, virtual displacement, virtual work, D’Alembert’s principle, Generalized force,  Expression for Lagrangian, solving Lagrange’s equations of motion, Application.

Lagrange’s equations for impulsive motion: Formulation and elementary applications.

Hamilton’s Equations: Equations of motion, Applications, Canonical transformations.

Motion of spinning top: Eulerian angles, type of motions – Precession, Nutation and   Spin, Equations of motion.

• Lectures, Tutorials, Handouts, Problem solving, e-resources.

·         End-of-course Examination 70%

·         R. Douglas Gregory, Classical Mechanics, Cambridge University Press, 2006.

·         H. Goldstein, C. Poole and J. Safko, Classical Mechanics, 3^rd edition, Addison Wesley, 2000.

·         D. A. Wells, Lagrangian Dynamics, Schaum’s outline series, McGraw-Hill, Inc., 1967.

Hourly Breakdown

·         Familiarize the underlying mathematical concepts of computer aided numerical algorithms

·         Enable to solve differential equations by using numerical methods

• Outline the fundamental concepts in numerical linear algebra
• Apply the matrix factorization algorithms to solve system of linear equations
• Determine bounds for relative error in the solution of a system of linear equations
• Examine the convergence of iterative methods for solving system of linear equations
• Use iterative methods to solve a system of linear equations
• Apply the numerical methods for certain types of differential equations
• Discuss the convergence of a numerical method applied to a differential equation
• Formulate basic algorithms in mathematical software

Error Analysis: Vector and Matrix norms, perturbation, perturbation bounds, residual, residual bounds, iterative refinement

Iterative Methods: Construction of iterative algorithms, Jacobi Method, Gauss Seidel Method, convergence, conditioning

Numerical Methods for Differential Equations: Differential Equations Review, Initial Value problems, Euler Methods, Linear Multistep methods, order, consistency, convergence, stability.

·         End-of-course Examination    70%

• N. Trefethen and D. Bau, Numerical Linear Algebra, SIAM, 1997.
• G. Golub and V. L. Charles, Matrix Computations, John Hopkins University Press, 1996.
• R. Dormand, Numerical Methods for Differential Equations: A computational Approach, Taylor and Francis, 1996.
• D. Griffiths and D. Higham, Numerical Methods for Ordinary Differential Equations, Springer, 2010. 

Hourly Breakdown

·                     Formulate the constitutive equations of Hydrodynamics

·                     Provide the solution methods for fluid mechanics problems

·                     Determine center of pressure and pressure on immersed bodies

·                     Recall the elements of Hydrodynamics

·                     Outline the laws of thermodynamics

·                     Formulate the constitutive equations of Fluid motion

·                     Explain the driven flows and rotating flows

·                     Analyze the fundamentals of water waves

·                     Discuss the nature of incompressible Newtonian turbulent flows

Definition of center of pressure, Formula for finding center of pressure, Center of pressure on certain type of objects and illustrative examples, effect of further immersion, Liquids more than one, The center of pressure of a plane area lies vertically beneath the center of the superincumbent fluid.

Hydrodynamics:

Fundamentals of fluid mechanics: Microscopic and macroscopic properties of liquids and gases, the continuum hypothesis, review of thermodynamics, general equations of motion, surface gravity waves, buoyancy – driven flows, rotating flows.

Water Waves: 1D free boundary problems, progressive waves, infinite and finite depth, wave speed, dispersion and group velocity.

Incompressible Newtonian fluids: introduction of concepts, viscosity, Reynolds number and turbulence.

·         End-of-course Examination    70%

·         J. Williams, Fluid Mechanics, problem solvers, George Allen & Unwin Ltd, 1974.

·         J. Alexandre, A mathematical Introduction to Fluid Mechanics, Springer, 1993.

·         M. Ray and H. S. Sharma, A text book of Hydrostatics Sultan Chand & Company, 2000.

·         P. Kundu and I. Cohen, Fluid mechanics, Academic Press, 2011.

Hourly Breakdown

·         Describe the mathematical model for given word problems

·         Sketch the solution of the formulated model problems involving Differential equations

·         Analyze the solution and behavior of Differential equation models

·         Modify the simple models for the change of environment

·         Solve single species population models

·         Discuss interacting two species population models

·         Analyze selected Models and Case studies

Differential Equations: Direction field, solution sketch, phase portrait, equilibrium solution, stability.

Single species population models: Basic concepts, Exponential growth model, formulation, solution, interpretation and limitations, Compensation and dispensation, Logistic growth model, formulation, solution, interpretation and limitations.

Two species population models: Types of interaction between two species. Lotka-Volterra prey-predator model, formulation, solution, interpretation and limitations. Lotka-Volterra model of two competing species, formulation, solution, interpretation and limitations.

Selected Models and Case studies: Models from Biology/Medicine/Social Science/Engineering/Physics – Especially epidemic model, SIR model, diffusion model, combat model, discrete models, election model.

·         End of course Examination     70%

·         W. Meyer, Concepts of Mathematical Modeling, McGraw Hill, New York, 1994.

·         Moghadas, S. M. and Douraki, M. J., Mathematical Modeling: A Graduate Textbook, Wiley, 2018.

·         Tung, K. K., Topics in Mathematical Modeling, illustrated edition, Princeton University Press, 2007.

·         Meerschaert, M. M., Mathematical Modeling, 2^nd edition, Academic Press, 1999.