Mathematical Methods Course File

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SCIENCE & HUMANITIES

Course File

.

Department: SCIENCE AND HUMANITIES

Name of the Subject: MATHEMATICAL METHODS

Subject Code: 51002

JOGINPALLY B.R. ENGINEERING COLLEGE

YENKAPALLY(V),MOINABAD(M),R.R.DIST,HYDERABAD

Year:2009-2010

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Department of

SCIENCE &

HUMANITIES

Course File

1. Department: S&H

2. Name of the Subject: Mathematical Methods

3. Subject Code: 51002

JOGINPALLY B.R. ENGINEERING COLLEGE

YENKAPALLY(V),MOINABAD(M),R.R.DIST,HYDERABAD

Department of

SCIENCE &

Course Status Paper(Target, Course Plan,

objectives, Guidelines

Year: 2010-2011

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HUMANITIES etc.)

Target:

1. Percentage Pass: __________98%______________

2. Percentage above 70% of marks: 90%____________

Course Plan:

(Please write how you intend to cover the contents: that is, coverage of units

by lectures, guest lectures, design exercises, solving numerical problems,

demonstration of models, model preparation, or by assignments etc.)

a. Design exercises,b. Solving numerical problems,c. Model preparation by assignments etc

3. On completion of the course the student shall be able to:

Understand the importance of Mathematics in EngineeringCourse.

To apply the mathematical knowledge and logical thinking inother subjects

4. Method of Evaluation:

3.1. Continuous Assessment Examination: Yes

3.2. Assignments: Yes

3.3. Questions in class room: Yes

3.4. Quiz as per University Norms: Yes

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3.5. Others: Assigning some technical topics to student to write usingthe all possible resources in the college.

5. List out any new topic(s) or any innovation you would like to

introduce in teaching the subject in this semester:

6. Guidelines to study the subject:

1. Mathematics has played a fundamental role in the formulation ofmodern

Science since the very beginning; a scientific theory is a theory that has an

adequate mathematical model.

2. The Mathematics that can be applied today covers all the fields of the

mathematical science and not only some special topics; it concerns

Mathematics of all levels of difficulty and not only simple results and

arguments.

3. The sciences continue to require today new results from ongoing research

and

present multiple new directions of inquiry to the researchers, but the rhythm

of the

contemporary society makes the time lapse substantially shorter and the

request more urgent.

4. The capabilities of scientific computation have made numericalsimulation, an indispensable tool in the design and control of industrialprocesses.

5. To develop an understanding of the basic principles governing theconditions of rest and motion of particles and rigid bodies subjected tothe action of forces; to develop the ability to analyze and solve problemsin a simple and logical manner

Expected

date of completion of the course and remarks, if any:

Unit Number: 1 30/10/2010

Unit Number: 2 11/11/2010

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Unit Number: 3

Unit Number: 4

Unit Number: 5

Unit Number: 6

Unit Number:7

Unit Number: 8

Remarks (if any):

Schedule of instruction

Unit No: 1

S.No

Date Numberof Hours

Subject Topics Reference

120/10/10

1 Introduction to Matrices, and system of Equations.

Dr Iyengar etal. S Chand.

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2 21/10/10 1 Definition of Rank of a Matrix and problemsto illustrate to find the rank of a given

matrix

-do-

3 22/10/10,

23/10/10

2 Finding the rank of a matrix using Echelon

Form and Normal Form

-do-

4 25/10/10 1 Numerical problems illustrating the

previous methods

-do-

5 26/10/10

27/10/10

2 Consistence of a system of equations:

Necessary and Sufficient Conditions.

-do-

6 28/10/10 1 Solving the System of Equations using

various matrix methods

-do-

7 29/10/10

30/10/10

1 LU-Decomposition theorem -do-

Unit No: 2

S. No Date Number

of Hours


1 1 Introduction to the concept of

Characteristic equation and its roots.

Dr Iyengar et al.

S Chand.

2 1 Problem of finding Eigen values and

Eigen vectors.

-do-

3 1 Statement and Proof of Cayley

Hamilition Theorem and Problems

-do-

5 1 Finding inverse and powers of a

matrix using Cayley-Hamiliton

Theorem.

-do-

6 1 Diagonalization of a matrix. -do-

7 1 Properties of Eigen Values and EigenVectors.

-do-

Unit No: 3

S. No Date Numberof

Hours


1 1 Introduction to Complex

matrices.

Dr Iyengar et al.

2 1 Basic Properties of Hermitian,

Skew-Hermitian and UnitaryMatrices.

-do-

3 1 On Eigen values andEigenvectors of Hermitian, Skew-

Hermitian and Unitary Matrices.

-do-

4 1 Canonical Forms. -do-

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5 2 Explaining the concept of converting to canonical form by

orthogonal Transformation

-do-

6 1 Singular Value Decomposition -do-

7 1 Problems. -do-

8 1 Revision to first internal exams. -do-

Unit No: 4

S. No Date Numberof

Hours


1 1 Introduction to Numerical

methods to find the roots ofAlgebraic and Transcendental

equations.

Dr Iyengar et al.

2 1 Bisection method --do--

3 1 Method of False position: Theoryand Problems

-do-

4 3 Iteration method (Fixed point

Iteration method): Theory andProblems

-do-

5 1 Newton-Raphson method:Theory and Problems

-do-

6 2 Introduction to Finite differenceoperators, and relation between

them.

-do-

7 1 Introduction to Interpolation:

Newton forward interpolationformula: Problems

-do-

8 1 Newton Backward interpolationformula: Problems

-do-

9 1 Gauss Central forward andbackward formulae: Problems

-do-

10 3 Lagranges General Interpolationformula and Newton Divided

Difference interpolation formula:Problems.

-do-

11 2 Spline Interpolation: Theory andProblems

-do-

Unit No: 5

S.No

Date Numberof Hours


1 2 Introduction to Curve fitting:Method of Least square

approximation.

Dr. Iyengar et al.S Chand Company.

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2 1 Fitting a straight line to agive data using Least square

method

3 1 Fitting a Parabola to a give

data using Least Squaremethod.

4 1 Fitting different type of exponential curves to give

data using Least squaremethod.

5 1 Introduction to Numericaldifferentiation and

Integration.

6 1 Trapezoidal Rule:

Introduction with problems

7 1 Simpsons 1/3 and 3/8th rules

with Problems

8 1 Gaussian Integration.

Unit No: 6

S.

No

Date Number

of Hours


1 1 Introduction to solve Differentialequations with initial conditions.

Dr. Iyengar et al.S Chand company

2 2 Taylor series method: theory and

problems

-do-

3 2 Picards method: Theory and Problems -do-

4 2 Eulers method: Theory and Problems -do-

5 2 Modified Eulers method: Theory andProblems

-do-

6 1 Runge-Kutta Method (Fourth order):

Theory and Problems

-do-

7 1 Predictor-Corrector methods: Milnes

Method: Problems

-do-

8 1 Adam-Bashforth-Moulton Method:

Problems

-do-

9 1 Conclusions and Revision -do-

Unit No: 7

S. No Date Number

of Hours


1 1 Introduction of Fourier Series Dr Iyengar et al.S Chand Company

2 1 Definitions and Problems -do-

3 1 Dirichlets Conditions and EulersFormulae

-do-

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4 1 Fourier Sine and Cosine Series -do-

5 1 Half Range Fourier Series -do-

6 1 Fourier Series of a Function over

an interval a to +a

-do-

7 1 Fourier Sine and Cosine Series -do-

8 1 Half-Range Fourier Series -do-

Unit No: 8

S. No Date Number

of Hours


1 1 Introduction to Partial Differential

Equations

Dr. Iyengar et al.

S Chand Company

2 1 Formation of Partial Differential

Equations

-do-

3 1 Lagranges method to findgeneral solution of PDE -do-

4 1 Complete integral of some

special Partial differentialequations.

-do-

5 1 Char pits method to solvenonlinear partial differential

equations

-do-

6 1 Method of Separation of Variable. -do-

7 1 Solution of Heat equation -do-

8 2 Solution of Wave equation -do-

9 1 Solution of Laplace equation -do-10 2 problems -do-

This Assignment/Tutorial is concerned to Unit Number: 1

(Please write the questions/problems/exercises. Which you would like to give to thestudents)

Q1. Define Rank of a matrix. Find the Rank of the following matrix using the Normal form

=

134813748

3124

5312

A

Q2. Test for the Consistence of the following equations and solve them if possible23;932;42 ==+=++ zyxzyxzyx

Q3.Determine the value of for which the following set of equations may posses non-

trivial solution and solve them in each case

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042;0324;03 =++==+ zyxzyxzyx



Q1. Diagonalize the matrix

=

344

120

111

A and hence find A4.

Q2. Show that the matrix

=

210

321

221

A satisfies its characteristic equation. Hence find

A-1.

Q3.Verify the sum of eigen values is equal to its trace of A for the matrix

=

1312

6204

2210

A and find the corresponding eigen vectors.



Q1. Find a matrix P which diagonalize the matrix associated with the quadratic form

xyzxyzzyx 222353 222 +++ and reduce its to canonical form

Q2. Show that the eigen values of Hermitian matrix are real

Q3. Find the nature of the quadratic form, index and signature

yzxzxyzyx 61045210222

+++

This Assignment/Tutorial is concerned to Unit Number: 4(Please write the questions/problems/exercises. Which you would like to give to the

students)

Q1. Find the real root of 2=xxe using Regula False method and approximate with Newton

Raphson method.

Q2. Find f(22) form the following table using Gauss forward formula

x 20 25 30 35 40 45

f(x) 354 332 291 260 231 204

Q3.Construct difference table for the following data:

X 0.1 0.3 0.5 0.7 0.9 1.1 1.3

F(x) 0.003 0.067 0.148 0.248 0.370 0.518 0.697


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Q1. Derive normal equations to fit the straight line y=a+bx

Q2. Fit a parabola to the following data using least square method

X 10 15 20 25 30 35y 35.3 32.4 29.2 26.1 23.2 20.5

Q3. Evaluate +

1

01 x

dxtaking h=0.25 using Cubic spline.



Q1. Using Runge-Kutta Method to evaluate y(0.1) and y(0.2), given that

1)0(; =+= yyx

dx

dy.

Q2.Given 1)0(;sin =+= yyxdx

dycompute y(0.2) and y(0.4) with h=0.2 using Eulers

modified method

Q3. Find y(0.2) using Picards method given that 1)0(, == yxydx

dyusing Picards method of

successive approximation with stepsize h=0.1

This Assignment/Tutorial is concerned to Unit Number: 7(Please write the questions/problems/exercises. Which you would like to give to the

students)

Q1. Find the half-range Cosine and Sine series of f(x)=(x-1)2

in the interval 0


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S.No. Quiz Test MaximumMarks

Best Marks Worst Marks Remarks

1 Quiz Test 1 10

2 Quiz Test 2 10

3 Quiz Test 3 10

4 Quiz Test 4 10

5 Quiz Test 5 10

* indicate if any remedial tests were conducted, if any.

Please note:

1. The question papers in respect of quiz test 1, 2, 3, 4 and 5 of this subject should be

included in the course file.

2. Model question paper which you have distributed to the students in the beginning of

the semester for this subject should be included in the course file.

3. The list of seminar topics you have assigned, if any may also be included here.

4. The J. N. T. University end examination question paper for this subject must beincluded in the course file.

5. A record of the best and worst marks achieved by the students in every quiz tests

must be maintained properly.

6. A detailed / brief course material / lecture notes if prepared may be submitted in the

HODs office.

7. Xerox copies of at least 5 answer sheets, after duly signed by the student on

verification of the evaluated answer script.

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Mathematical Methods Course File