Finite Difference Methods-AM301
Paper-I
Course Outcomes
1.Identify whether a PDE is in the form of a conservation law, describe the characteristic of a conservation law and how the solution behaves along the characteristic.
2.Explain the CFL condition and determine the timestep constraints resulting from the CFL condition.
3.The learner should be able to classify the given p.d.e into parabolic, hyperbolic or elliptic type.
4.The learner should be able to understand and analyse various difference methods used to solve one-dimensional and two-dimensional parabolic, hyperbolic and elliptic equations.
Unit I
Partial Differential Equations.
Introduction.
Classification of Second order PDE's.
Difference Methods.
Routh Hurwitz criterion.
Domain of Dependence of Hyperbolic Equations.
(1.1to1.4)
Unit II
Difference Methods for Parabolic Partial Differential Equations.
Introduction.
One Space Dimension.
Two Space Dimensions.
Spherical and Cylindrical Coordinate System.
(2.1to2.3,2.5,2.6).
Unit III
Difference Methods for Hyperbolic Partial Differential Equations
Introduction.
One Space Dimensions.
Two Space Dimensions.
System of First order equations.
(3.1to3.5).
Unit-IV
Numerical Methods for Elliptic Partial Differential Equations.
Introduction.
Difference Methods for linear boundary value problems.
General second order linear equation.
Equation in polar coordinates.
(4.1to 4.5).
TextBook:
M. K.Jain,S.R.K.Iyengar,R.K.Jain,Computational Methods for Partial Differential Equations,Wiley
Eastern Limited,New Age International(P)Limited,NewDelhi.
Viscous Flows-AM302
Paper-II
Course Outcomes
1.Know the definitions of fundamental concepts of fluid mechanics including: continuum, velocity field; viscosity, surface tension and pressure (absolute and gage); flow visualization using timelines, pathlines, streaklines, and streamlines; flow regimes: laminar, turbulent and Department! transitional flows; compressibility and incompressibility; viscous and inviscid.
2.Apply the basic equation of fluid statics to determine forces on planar and curved surfaces that are submerged in a static fluid; to manometers; to the determination of buoyancy and stability; and to fluids in rigid-body motion.
3.Use of conservation laws in integral form and apply them to determine forces and moments on surfaces of various shapes and simple machines.
4.Use of conservation laws in differential forms and apply them to determine velocities, pressures and acceleration in a moving fluid. Understand the kinematics of fluid particles, including the concepts of substantive derivatives, local and convective accelerations, vorticity and circulation.
Unit I
Vortex Motion.
Vorticity – Vortex Line.
Vortex Tube and Vortex Filament.
Properties of Vortex Filament.
Kelvin’s Proof .
Helmholtz’s Vorticity Theorems.
Strength of Vortex.
Rectilinear Vortices.
Complex Potential of a Two-Dimensional Vortex Motion.
Centre of Vortices.
Two Vortex Filaments.
Vortex Pair .
Vortex Doublet or Dipole.
Image of a Vortex Filament in a Plane.
Vortex inside and outside Circular Cylinder.
Rectilinear Vortices.
Rectilinear Vortex with Circular Section.
Unit II
General Theory of Stress and Rate of Strain.
Viscosity – Measurement of Viscosity .
Stress Tensor.
Stress components in a Real Fluid.
Relations between Cartesian Components of Stress .
Translation motion of Fluid Element.
Stress Analysis in Fluid Motion.
The coefficient of Viscosity and Laminar Flow .
The Navier-Stokes Equations of motion of a viscous fluid.
Unit III
Laminar Flow of Viscous Incompressible Fluid:
Steady motion between two parallel plates:
Plane Couette flow.
Generalized plane Couette flow .
Plane Poiseuille flow.
Flow through a circular pipe .
The Hagen Poiseuille flow.
Steady visous flow in Tubes of uniform cross section.
A Uniqueness Theorem.
Steady motion in Tube having uniform Elliptic cross section and Tube having Equilateral Triangle cross section.
Unsteady flow over a flat plate.
Unit-IV
Dynamical Similarity:
Dynamical similarity.
Flow similarity.
Dimensional Analysis.
Buckingham πTheorem.
Non-dimensional parameters in fluid mechanics .
Reynolds number.
Significance of Reynolds number.
Boundary Layer Theory:
Prandtl’s Boundary Layer Theory .
Boundary Layer thickness.
Displacement thickness.
Momentum thickness.
Energy thickness.
Boundary Layer equations in two dimensions.
The Boundary Layer along a flat plate (Blasius solution) .
Approximate solutions of Boundary Layer Equations .
Von Karman’s Integral Relation .
Von Karman’s Integral Relation by Momentum Law.
Text Books: [1] Frank M.White, Viscous Fluid Flow,McGraw-Hill, Inc. [2] FRANK CHORLTON, Textbook of Fluid Dynamics, CBS-Publishers, New Delhi, India. [3] M.D.RAISINGHANIA, Fluid Dynamics S.Chand & Company, New Delhi. [4] M.M.RAHAMAN, Hydodynamics, Atlantic Pub. New Delhi.
Linear Algebra-AM303
Paper-III
Course Outcomes
1.Solving linear equations, working with matrices, in particular eigenvalues and eigenvectors, and applying the techniques to real life problems like graph theory, computer science, electronics and applied mathematics. Spectral theorems, prevalent in many branches of mathematics.
2.Use computational techniques and algebraic skills essential for the study of systems of linear equations, matrix algebra, vector spaces, eigenvalues and eigenvectors, orthogonality and diagonalization. (Computational and Algebraic Skills
3.Use visualization, spatial reasoning, as well as geometric properties and strategies to model, solve problems, and view solutions, especially in R2 and R3 , as well as conceptually extend these results to higher dimensions. (Geometric Skills).
4.Use technology, where appropriate, to enhance and facilitate mathematical understanding, as well as an aid in solving problems and presenting solutions (Technological Skills)
Unit I
Elementary Canonical forms Introduction
Characteristic Values,Annihilating Polynomials.
Invariant Sub-spaces.
Simultaneous Triangulation and Simultaneous Diagonalization
Unit II
Direct sum Decomposition,Invariant Direc tsums.
The Primary DecompositionTheorem.
The Rational and Jordan Forms:Cyclic Subspaces and Annihilators.
Unit III
Cyclic Decompositions and the RationalForm.
TheJordanForm.
Computation of Invariant Factors.
Semi Simple Operators.
Unit-IV
Bilinear Forms:BilinearForms, Symmetric Bilinear Forms.
Skew-Symmetric Bilinear Forms.
GroupsPre-serving BilinearForms.
TextBook:
Linear AlgebrabyKennethHo manandRayKunze(2e)PHI
References:
[1] AdvancedLinearAlgebrabyStevenRoman(3e)
[2] LinearAlgebrabyDavidCLay
[3] LinearAlgebrabyKuldeepSingh
Operations Research-AM304
Paper-IV
Course Outcomes
1.Define and formulate linear programming problems and appreciate. their limitations.
2.Solve linear programming problems using appropriate techniques and optimization solvers, interpret the results obtained and translate solutions into directives for action.
3.Conduct and interpret post-optimal and sensitivity analysis and explain the primal-dual relationship.
4.Develop mathematical skills to analyse and solve integer programming and network models arising from a wide range of applications
Unit I
Formulation of Linea rProgramming problems.
Graphical solution of Linear Programming problem.
General formulation of Linear Programming problems, Standard and Matrix forms of Linear Programming problems.
Simplex Method,Two-phase method,Big-M method.
Method to resolve degeneracy in Linear Programming problem.
Alternative optimal solutions.
Solution of simultaneous equations by simplex Method.
Inverse of a Matrix by simplex Method.
Concept of Duality in Linear Programming.
Comparison of solutions of the Dual and its primal.
Unit II
Mathematical formulation o fAssignmentproblem.
Reduction theorem.
Hungarian Assignment Method.
Travelling sales man problem, Formulation of Travelling Sales man problem.
Assignment problem,Solution procedure.
Mathematical formulation of Transportation problem,Tabular representation.
Methods to find initial basic feasible solution, North West corner rule, Lowest cost entry method, Vogel's approximation methods,
Opti-malitytest, Method of finding optimal solution.
Degeneracy in transportation problem, Method to resolve degeneracy.
Unbalanced transportation problem.
Unit III
Concept of Dynamic programming.
Bellman's principle of optimality.
characteristics of Dynamic programming problem.
Backward and Forward recursive approach.
Minimum path problem.
Single Additive constraint and Multiplicatively separable return.
Single Additive constraint and Additively separable return.
Single Multiplicatively constraint and Additively separable return.
Unit-IV
Historical development of CPM/PERT Techniques-Basicsteps.
Network diagram representation.
Rules for drawing networks.
Forwardpass andBack ward pass computations.
Determination of floats.
Determination of critical path.
Project evaluation and review techniques.
TextBooks:
[1] S.D.Sharma,OperationsResearch.
[2] KantiSwarup,P.K.GuptaandManmohan,OperationsResearch.
[3] H.A.Taha,OperationsResearch{AnIntroduction.
[4] G.I.Gauss,LinearProgramming.
Numerical Analysis-AM305
Paper-V
Course Outcomes
1. Explain the least square method
2.Find the lagrange polynomial passing through the given points
3.Find the hermite polynomial passing through the given points
4.Find the cubic spline passing through the given points
Unit I
Transcendental and Polynomial Equations: Introduction.
Bisection Method-Iteration.
Methods Based on First Degree Equation:SecantMethod, Regula FalsiMethod, Newton-Raphson Method.
Iteration Methods Based on Second Degree Equation: Muller'sMethod, ChebyshevMethod, Multipoint IterationMethods.
Rate of convergence-Iteration Methods.
Unit II
System of Linear Algebraic Equations: Introduction.
DirectMethods: GaussEliminationMethod,Gauss Jordan Elimination Method, Triangularization Method, Cholesky Method,Partition Method.
Iteration Methods: Jacobi Iteration Method, Gauss Seidel Iteration Method, SOR Method.
Unit III
Interpolation and Approximation: Interpolation: Introduction.
Lagrange and Newton Interpolations.
Finite Di erence Operators-Interpolating Polynomials using FiniteDi erences.
Hermite Interpolations.
Piecewise and Spline Interpolation.
Approximation: Least Squares Approximation.
Unit-IV
Numerical Integration: Methods Based on Interpolation: Newton Cotes Methods.
Methods Based on Undetermined Coe cients: Guass Legendre Integration Methods-Composite Integration Methods.
Numerical Solution of ODE's: Introduction-Numerical Methods: EulerMethods-Midpoint Method- Single Step Methods: Taylor series method, Runge-Kutta Method.
Multi step Methods: Adam Bash forth Method-Adams Moulton Method.
Milne-Simpson Method.
Predictor Corrector Methods.
TextBooks:
[1] NumericalMethodsforScienti candEngineeringComputationbyM.K.Jain,S.R.K.Iyengar,R.K.
Jain, NewAgeInt.Ltd.,NewDelhi.