Contents

• 1. Introduction

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  Introduction, Statement of an Optimization Problem, Various Optimization Techniques, Classification of Optimization Problems, Application of Operations Research in the Field of Engineering, Exercise 1 (A).
   
  2. Classical Optimization Techniques

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  Introduction, Single Variable Optimization, Concave and Convex Functions, Multivariable Optimization without Constraint, Semidefinte Case, Solution by Direct Substituion, Solution by the Method of Constrained Variation, Lagrange’s Multipliers Method, Multivariable Optimization with Inequality Constraint, Illustrative Eaxmples, Exercise 2 (A).
   
  3. Linear Programming

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  Introduction, Mathematical Preliminaries, Mathematical Description of a Linear Programming Problem (LPP), Mathematical Formulation of LPP, Definitions and Prerequisites, Reduction of FS to BFS, Graphical Method for Solving an LPP, Illustrative Examples, Exercise 3 (A), Simplex Method, Simplex Table, To Obtain an Initial BFS, Computational Procedure of Simplex Method, Artificial Variables Technique, Big M-Method (Charne’s M-Method), Two-Phase Method, Illustrative Examples, Exercise 3 (B).
   
  4. Duality

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  Introduction, Primal-Dual Relationship, Definition of Primal-Dual Problems, Algorithm for Converting Any Primal into its Dual, To Read the solution to the Dual From the Final Simplex Table of the Primal and Conversely, Illustrative Exmaples, Exercise 4 (A).
   
  5. Transportation Problem

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  Introduction, Important Definitions, Solution of A Transportation Problem, Some Useful Terminologies, Transportation Algorithm (Modified Distribution i.e. MODI Method or u – v method), Unbalanced Transportation Problem, Degeneracy and its Resolution, Resolution of Degeneracy Occuring at the Initial Stage, Resolution of Degeneracy at any Intermediate Stage, Illustrative Examples, Exercise 5 (A).
   
  6. Elements of Number Theory

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  Introduction, Well-Ordering Principle, Division Algorithm, Greatest Common Divisor, Properties of Greatest Common Divisors, Euclidean Algorithm, The Euclidean Algorithm, Prime and Composite Numbers, The Sieve of Eratosthenes, The Prime Number Theorem, Congruences, Linear Congruences, Systems of Linear Congruences, Modular Inverses, Finding an Inverse of a Modulo n, The Chinese Remainder Theorem, Fermat's Little Theorem, Euler's Phi Function, Euler Theorem, Quadratic Congruences, Quadratic Residue, Legendre Symbol, Properties of Legendre Symbol, Jacobi Symbol, Properties of Jacobi Symbol, Exercise 6 (A).
   
  7. Algebraic Structures: Group, Rings and Field

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  Introduction, Binary Operation, Properties of Binary Operations, Algebraic Structures, Semigroup, Monoid, Group, Abelian or Commutative Group, Finite and Infinite Groups, Elementary Theorems Based on Groups, Composition Table for Groups, Order of an Element, Subgroup, Cyclic Group, Illustrative Examples, Exercise 7(A), Rings, Ring with Unity, Commutative Rings, Elementary Properties of a Ring, Zero Divisors, Cancellation Laws in a Ring, Integral Domain, Field, Division Ring or Skew Field, Polynomial Over a Field, Finite Field or Galois Field, Characteristic of a Finite Field, Illustrative Examples, Exercise 7 (B).
   
  8. Laplace Transform

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  Introduction, Laplace Transform, Piecewise Continuous Function, Functions of Exponential Order, Sufficient Conditions for the Existence of Laplace Transform, Laplace Transform of Some Elementary Functions, Properties of Laplace Transform, Linearity Property, First Shifting (or First Translation) Theorem, Second Shifting (or Second Translation) Theorem, Change of Scale Property, Laplace Transform of Derivatives, Laplace Transform of Integrals, Laplace Transform of f(t) Multiplied by Some Positive Integral Power of t, Laplace Transform of f(t)/t, Unit Step Function, Properties of Unit Step Function Related to Laplace Transform, Laplace Transform of Periodic Functions, Laplace Transform of Bessel Function, Illustrative Examples, Exercise 8 (A), Inverse Laplace Transform, Table of Inverse Laplace Transform, Properties of Inverse Laplace Transform, Linearity Property, First Shifting (or First Translation) Theorem, Second Shifting (or Second Translation) Theorem, Change of Scale Property, Inverse Laplace Transform of Derivatives, Inverse Laplace Transform of Integrals, Inverse Laplace Transform of F(s) Multiplied by Powers of s, Inverse Laplace Transform of f s s ( ), Convolution Theorem For Laplace Transform, Partial Fraction Method to find Inverse Laplace Transform, Illustrative Examples, Exercise 8 (B), Application of Laplace Transform to Ordinary Differential Equations with Constant Coefficients, Application of Laplace Transform to the System of Simultaneous Differential Equations, Application of Laplace Transform to Electrical Circuits, Illustrative Examples, Exercise 8 (C), Application of Laplace Transform to Partial Differential Equations, The Complex Inversion Formula, Illustrative Examples, Exercise 8 (D).
   
  9. Finite Differences and Interpolation

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  Introduction, Finite Difference Calculus, Forward Differences, Backward Differences, Central Differences, Other Difference Operators, Relation Between Difference Operators, Exercise 9 (A), Errors in Polynomial Interpolation, Newton Gregory Forward Interpolation Formula, Newton-Gregory Backward Interpolation Formula, Gauss Forward Difference Formula, Gauss Backward Difference Formula, Central Difference Interpolation Formula, Lagrange’s Interpolation Formula for Unequal Spaced Points, Numerical Differentiation, Maximum and Minimum Value of a Tabulated Function, Numerical Integration, Illustrative Examples, Exercise 9 (B).
   
  10. Numerical Solution of Ordinary and Partial Differential Equations 

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  Introduction, Picard’s Method, Euler’s Method, Modified Euler’s Formula, Runge-Kutta Method of Fourth Order, Milne’s Predictor Corrector Method, Illustrative Examples, Exercise 10 (A), Difference Equations, Linear Difference Equation, Particular Integral, Illustrative Examples, Exercise 10 (B).
   
  P. Papers