Engineering Mathematics-III(For CS and IT students)

Engineering Mathematics-III(For CS and IT students)

Dr. C.B. GuptaProfessor

Department of Mathematics,Birla Institute of Technology and Science

Pilani, (Rajasthan)

Dr. A.K. MalikLecturer

Department of Mathematics,B. K. Birla Institute of Engineering and Technology

Pilani, (Rajasthan)

Dr. Vipin KumarLecturer

Department of Mathematics,B. K. Birla Institute of Engineering and Technology

Pilani, (Rajasthan)

Asian Books Private LimitedAsian Books Private Limited7/28, Mahavir Lane, Vardan House, Ansari Road,

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Engineering Mathematics-IIIDr. C.B. Gupta, Dr. A.K. Malik and Dr. Vipin Kumar

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SyllabusEngineering Mathematics-III

(Common to Comp. Engg. and Info. Tech.)

UNIT-I

Introduction: Engineering application of optimization, Statement and classification of optimizationproblem, single variable and multivariable optimization with and without constraints.

UNIT-II

Linear Programming: Formulation of Linear Programming problem, Graphical Approach, GeneralLinear Programming problem, Simple Method. Duality in Linear Programming and TransportationProblems.

UNIT-III

Project Scheduling: Project Scheduling by PERT and CPM Network Analysis, Sequencing Theory,General Sequencing problem n-jobs through 2 machines and 3 machines and 2-jobs through m machines.

UNIT-IV

Laplace Transform: Laplace transform with its simple properties. Inverse Laplace transform, convolutiontheorem (without proof), solution of ordinary differential equation with constant coefficient, solution ofpartial differential equation having constant coefficient with special reference to diffusion, Heat conductionand wave equation. Boundary value problems.

UNIT-V

Numerical Analysis: (1) Difference operators forward, backward, control, shift and average operatorsand relation between them. (2) Newton’s and Gauss forward and backward interpolation formula forequal interval, Sterling and formula for control difference. Lagrange’s Interpolation formula. InverseInterpolation. (3) Numerical differentiation by Newton’s, Gauss and Sterling’s formula. (4) NumericalIntegration by Simpson’s one third and there eight rule. (5) Numerical Integration of ordinary differentialequation of first order by Picard’s method, Euler’s and modified Euler’s method, Milure’s method andRunga Kutta fourth order method. (6) Solution of difference equation.

Preface

We feel happy and honoured while presenting this book “Engineering Mathematics-III” for engineeringstudents studying in B.Tech., III semester (CS and IT branch) of Rajasthan Technical University and allIndian Universities. In this book we have presented the subject matter in very simple and precise manner.The treatment of the subject is systematic and the exposition easily understandable. All the standardexamples have been included and their model solutions have also been given. This book has been dividedinto five units.

Unit I deals with introduction to optimization, statement, application of optimization techniques andclassical optimization techniques.

Unit II describes the introduction of linear programming problem, formulation of linear programmingproblem, solution of linear programming problem by graphical method, simplex method, duality in linearprogramming problem and transportation problem.

Unit III provides a detailed coverage of project scheduling, CPM, PERT and sequencing problemsof n-jobs with two machines, n-jobs with k-machines and two jobs with k-machines.

Unit IV contains an introduction to laplace transform with its simple properties, inverse laplacetransform, convolution theorem, heaviside expansion formula and application of laplace transform tosolution of ordinary and partially differential equation with constant coefficient.

Unit V focuses on numerical methods, finite differences, Newton’s forward and backward differenceformula, Stirling’s formula, Lagrange’s interpolation, inverse interpolation formula, numerical differentiationand integration, and numerical solution of differential equation of first order.

We place our thanks on record to all those who have directly or indirectly helped us in completionof the project.

We shall be extremely happy to acknowledge the suggestions and constructive criticisms for furtherimprovement of the book.

Authors

Contents

UNIT-I: Introduction 1—811. Introduction to Optimization 3–17

1.1 Introduction 31.2 Origin of Operations Research 31.3 Definition of Operation Research 41.4 Optimization Techniques 51.5 Applications of Optimization in Engineering 71.6 Optimization Problem 71.7 Classification of Optimization Problems 9

1.7.1 Classification Based on the Existence of Constraints 91.7.2 Classification Based on the Nature of the Design Variables 91.7.3 Classification Based on the Physical Structure of the Problem 101.7.4 Classification Based on the Nature of the Equations Involved 101.7.5 Classification Based on the Permissible Values of the Design 101.7.6 Classification Based on the Deterministic Nature of the Variables 111.7.7 Classification Based on the Separability of the Functions 111.7.8 Classification Based on the Number of Objective Functions 11

Exercise 1.1 15

2. Classifical Optimization Techniques 18–81

2.1 Unconstrained Optimization Problems 182.1.1 Single Variable Optimization Problems 182.1.2 Condition for Local Maxima or Minima of Single Variable Function 192.1.3 Working Rule to Finding the Extreme Points 20

Exercise 2.1 35

(viii) Contents

2.2 Multivariable Optimization Problems 362.2.1 Working Rule to Find the Extreme Points of Functions of Two Variables 36

Exercises 2.2 492.3 Constrained Multivariable Optimization Problems with Equality Constraints 50

2.3.1 Direct Substitution Method 512.3.2 Lagrange Multipliers Method 51

2.4 Constrained Multivariable Optimization Problems with Inequality Constraints 522.4.1 Kuhn-Tucker Conditions 532.4.2 Convex and Concave Functions 55

Exercise 2.3 78

UNIT-II: Linear Programming 83—1911. Linear Programming Problem and Graphical Method 85–113

1.1 Introduction 851.2 Linear Programming Problem 851.3 General Mathematical form of Linear Programming Problem 851.4 Formulation of Linear Programming Model 86

Exercise 1.1 931.5 Some Important Terms 991.6 Solution of a Linear Programming Problems 1001.7 Graphical Method for the Solution of a LPP 100

2. The Simplex Method 114–139

2.1 Introduction 1142.2 Standard Form of a Linear Programming Problem 1142.3 Computational Procedure of Simplex Method 1152.4 Artificial Variable Method 1222.5 Big-M Method 1222.6 Two-Phase Method 1252.7 Solution of Simultaneous Equations by Simplex Method 1322.8 Inverse of a Matrix by Simplex Method 1322.9 LPP with Unrestricted Variable 133

Exercise 2.1 135

3. Duality in Linear Programming 140–163

3.1 Introduction 1403.2 symmetric Dual Problem 1403.3 Unsymmetric Primal Dual Problem 141

Contents (ix )

3.4 Fundamental Properties of Duals 1423.5 Algorithm for Finding the Dual Form its Primal Problem 1423.6 Prime and Dual Correspondence 1433.7 To Read the Solution to the Dual from the Final Simplex Table of the

Primal and Vice-Versa 144Exercise 3.1 161

4. Transportation Problem 164–1914.1 Introduction 1644.2 Tabular Representation of Transportation Problem 1644.3 Mathematical Formulation of Transportation Problem 1654.4 Definitions 1654.5 Solution of a Transportation Problem 1684.6 North-West Corner Rule 1684.7 Lowest Cost Entry Method (Matrix Minima Method) 1684.8 Vogel’s Approximation Method (Unit Cost-Penalty Method) 1694.9 Modified Distribution Method (Modi Method) 175

4.10 Degeneracy in Transportation Problem 1754.11 Unbalanced Transportation Problems 176

Exercise 4.1 187

UNIT-III: Project Scheduling 193—2911. Networking with CPM & PERT 195–268

1.1 Introduction 1951.2 Network Diagram 1971.3 Critical Path Method (CPM) 198

Exercise 1.1 2221.4 Programme Evaluation and Review Technique (PERT) 2331.5 Optimum Scheduling by CPM (Time-Cost Relationship) 2331.6 Time-cost Optimization Algorithm 235

Exercise 1.2 258

2. Sequencing 269–291

2.1 Assumptions, Notations and Terminology 2692.1.1 Assumptions 2692.1.2 Notations 2702.1.3 Terminology 270

Exercise 2.1 275Exercise 2.2 282Exercise 2.3 289

(x) Contents

UNIT-IV: Laplace Transform 293—4011. Laplace Transform 295–346

1.1 Laplace Transform 2951.2 Piecewise (or Sectionally) Continuous Function 2951.3 Existence of Laplace Transform 2961.4 Linear Property of Laplace Transformation 2971.5 Functions of Exponential Order 2971.6 A Function of Class A 2971.7 Some Elementary Functions of Laplace Transforms 2971.8 First Translation or Shifting Theorem of Laplace Transform 3021.9 Second Translation or Shifting Theorem of Laplace Transform 302

1.10 Change of Scale Property of Laplace Transform 303Exercise 1.1 316

1.11 Laplace Transform of Derivatives 3181.12 Laplace Transform of Integrals 3191.13 Laplace Transform of Function Multiplication by Powers of t 321

1.14 Laplace Transform of the Function F(t) Division by t i.e. ( )F tLt

322

Exercise 1.2 3351.15 Laplace Transform of Periodic Function 3361.16 Laplace Transform of Some Special Functions 337

Exercise 1.3 345

2. The Inverse Laplace Transform 347–379

2.1 Inverse Laplace Transform 3472.2 Linear properties of Laplace Transform 3472.3 First Translation or Shifting Theorem of Inverse Laplace Transform 3482.4 Second Translation or Shifting Theorem of Inverse Laplace Transform 3482.5 Change of Scale Property of Inverse Laplace Transform 3492.6 Inverse Laplace Transform of Some Standard Function 350

Exercise 2.1 3572.7 Inverse Laplace Transform of Derrivatives 3592.8 Inverse Laplace Transform of Integrals 3592.9 Inverse Laplace Transform of Function Multiplication by Powers of s 359

2.10 Inverse Laplace Transform of Function Division by s 359Exercise 2.2 365

2.11 The Convolution Theorem 3662.12 Heaviside Expansion Formula 368

Exercise 2.3 378

Contents (xi )

3. Applications of Laplace Transform to Solve Differential Equation 380–401

3.1 Solution of an Ordinary Differential Equation with Constant and Varaible Coefficients 3803.2 Solution of Simultaneous Ordinary Differential Equations 380

Exercise 3.1 3923.3 Solution of Partial Differential Equations with Constant Coefficients 393

Exercise 3.2 401

UNIT-V: Numerical Methods 403—4981. Calculus of Finite Differences 405–421

1.1 Introduction 4051.2 Finite Differences 4051.3 Shift operator E 4051.4 The Operator E–1 4061.5 Forward Differences 4061.6 Backward Differences 4061.7 Central Differences 4071.8 The Difference Table 4071.9 Relation Between the Operators 408

1.10 Fundamental Theorem of the Difference Calculus 4101.11 Factorial Function 4111.12 Computation of Missing Term 412

1.12.1 Another Method 4121.12.2 To Find Two Missing Term 413

Exercise 1.1 418

2. Interpolation 422–443

2.1 Introduction 4222.2 Newton-Gregory’s Formula For Forward Interpolation with Equal Intervals 4222.3 Newton-Gregory’s Formula for Backward Interpolation with Equal Intervals 4232.4 Gauss’s Forward Interpolation Formula 4252.5 Gauss’s Backward Difference Formula 4262.6 Stirling’s Difference Formula 4272.7 Lagrange’s Interpolation Formula for Unequal Interval 4282.8 Inverse Interpolation 429

Exercise 2.1 440

(xii) Contents

3. Numerical Differentiation 444–455

3.1 Introduction 4443.2 Numerical Differentiation by Using Newton’s Forward Difference Formula 4443.3 Numerical Differentiation by Using Newton’s Backward Difference Formula 4453.4 Numerical Differentiation by Using Stirling Difference Formula 446

Exercise 3.1 454

4. Numerical Integration 456–466

4.1 Introduction 4564.2 Newton-cotes Quadrature Formula 4564.3 The Trapezoidal Rule 457

4.3.1 Simpson’s “1/3” Rule 4584.3.2 Simpson’s “3/8” Rule 458

Exercise 4.1 465

5. Ordinary Differential Equations of First Order 467–484

5.1 Introduction 4675.2 Euler’s Method 4675.3 Euler’s Modified Method 4685.4 Picard’s Method 4685.5 Runge-Kutta Method of Fourth Order 4695.6 Milne’s Method 469

Exercise 5.1 483

6. Difference Equation 485–498

6.1 Introduction 4856.2 Order of Difference Equation 4856.3 Degree of Difference Equation 4856.4 Linear Difference Equation 4866.5 Solution of Homogenious Linear Difference Equation with Constant Coefficient 4866.6 Solution of Non-homogenious Linear Difference Equation with Constant Coefficients 4876.7 Solution of Difference Equation 4896.8 Formation of Difference Equation 489

Exercise 6.1 496

Paper 499–502

UNIT-I

INTRODUCTION

In this unit, we shall discuss the introduction to optimization techniques, statement,application of optimization techniques and classical optimization techniques.

The unit is divided into two chapters:

Chapter one deals with introduction to optimization, applications of optimizationtechniques, statement of optimization techniques and classification ofoptimization techniques.Chapter two deals with classical optimization techniques, unconstrained andconstrained multivariable optimization problems with equality and inequalityconstraints. Unconstrained single optimization problems are also discussed.

Introduction toOptimization

In this chapter, we shall discuss the introduction to optimization techniques, applications of optimizationtechniques in engineering, statement of optimization techniques and classification of optimizationtechniques.

1

1.1 INTRODUCTIONOptimization techniques are used in electric, electronics, civil, mechanical, automotive, aerospace, medicalscience, banking, education, management, computer, manufacturing industries, biotechnology andinformation technology industries etc. Optimization means getting the best output which may be maximumor minimum value of the critierion.

The optimization techniques were used in solving various mathematical, physical, chemical andbiological problems etc. from the time of Newton, Cauchy and Lagrange. Newton and Leibnitz developedsome important optimization methods in differential calculus. Lagrange developed a method of optimizationfor constrained problems, which involves the addition of unknown multipliers. Known by the name ofLagrange’s method of undetermined multipliers. Cauchy was the first mathematician who gives the firstapplication of unconstrained minimization problems solved by steepest descent method. Optimizationtechniques are generally studied as a part of operations research. Operations research is a branch ofmathematics concerned with the application of scientific methods and techniques to decision makingproblems and with the best or optimal solution.

1.2 ORIGIN OF OPERATIONS RESEARCHThe formal initiation of operational research was given in the Second World War and arose from a studyof the effectiveness of military operations, primarily in the royal air force. Operations Research hasbecome the most important, instrument in the organization and management in various institutions. Inrecent years service organizations such as hospitals, airlines, banks, railways, and libraries have startedrecognizing the usefulness of Operations Research for improving efficiency.

4 Engineering Mathematics-III

The application of mathematical models for solving such business problems first attracted theattention of the Russian mathematician, L. V. Kantorovich, who published a monograph “MathematicalMethods in the Organization and Planning of Production”, in 1939. In this monologue he pointed out thatthere are many classes of production problems that can be defined mathematically and that can therefore,be solved numerically.

The term of operations Research was first used by Mc Clospy and Trefthen in the year 1940. Theycoined the term ‘Operations Research’ as a result of research on military operations during World WarII. During the years 1914-1915 Thomas Edison made an effect to use a tactical game board to find away to minimize shipping losses from enemy submarines instead of risking ships in actual war conditions.He used a particular model and techniques of Operations Research for this purpose.

An OR club was established in England in 1948. Later it came to be known as the OperationalResearch society of U.K. It started publishing a quarterly journal and its first issue appeared in 1950. Abig step in the field of Operational Research was the establishment of Operations Research Society ofAmerica (ORSA) in 1952 and the publication of its first journal Operations Research in 1953.

Operations Research in India came into existence in the year 1949 with the establishment of an ORunit at Regional Research laboratory, Hyderabad for the purpose of planning and organizing research.OR equal importance was the setting up of an OR team at Defence Science Laboratory by Prof. R. S.Verma for the specific purpose of solving the problems of store, purchase and planning. OR techniquesprovide effective base for management decisions. Simplex algorithm developed by G.B. Dantzig is apractical tool for decision making.

OR received a further boost with the setting up of OR team in the Indian Statistical Institute, Kolkataby Prof. P.C. Mahalanobis in 1953 for the purpose of solving the problems related to national planningand survey. Later the OR society of India (ORSI) was established in 1957. This society started publishingits journal OPSEARCH from 1964.

1.3 DEFINITION OF OPERATION RESEARCHA proper precise definition of OR is difficult because of its big scope of applications. However, a fewdefinitions of OR which are commonly used are as follows:

According to Churchman, Ackoff and ArnoffOR is the application of scientific methods, Techniques and tools to problems involving the operation ofa system so as to provide those in control of the system with optimum solution to the problem.

According to H.M. WagnerOR is a scientific approach to problems solving for executive management.

According to T.L. SaatyOR is the art of giving bad answers to problems which otherwise have worse answers.

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