New CW, Solve all questionsAdvanced Mathematics for Civil Engineers

QEAD Rev:0 3 ver 01 Template B 1 F/QAP/021/001 Caledonian College of Engineering Course Work Name of the programme BEng (GCU) Name of Module with Code Advanced Mathematics for Civil Engineers (M3G124711 ) Level/Semester & AY Level 3 / Semester B AY:2017 -18 Name of the Module Leader/Tutor Dr. Javed Ali Dr A Senguttuvan Dr K R Karthikeyan K Amarender Reddy Coursework Type Assignment Assessment weightage 30% Type and d ate of submission Hard /Soft Copy (soft copy is preferred ) 8-5-2018 Aim Impart the student with the knowledge of applications of mathematics to engineering Learning Outcomes 1. Solve transcendental equations by numerical techniques . 2. Solve the system of equations using analytical and numerical techniques . 3. Finite difference s and th eir applications in interpolation. 4. Apply the knowledge for interpolation techniques . 5. Get the knowledge of numerical differentiation and applications . 6. Get the knowledge of numerical integration and applications . Task (s) Students are expected to s olv e assignment questions with the acquired knowledge from class room lectures, tutorial sessions and by referring CCE learn postings, indicative reading and e-brary resources and e -resources indicated at the end of the assignment questions individual ly. Marking scheme Component Description Weightage (%) 1 Knowledge and understanding of the topic 30% 2 Application and analysis of the topic (Module specific skill) 35% 3 Coherence and structure in terms of logic 25% 4 Extended reading 10% Total 100 QEAD Rev:0 3 ver 01 Template B 2 F/QAP/021/001 Instructions 1. Plagiarism is a serious offence. In case of any plagiarism detected, penalty will be imposed leading to zero mark. Policy and guidelines for dealing with plagiarism and malpractice in examination can be viewed by c licking: http://portal.cce.edu.om/member/contentdetails.aspx?id=490 2. The course work shall be subject to plagiarism software check . 3. Course work should be submitted on time . College guidelines o n late submission of coursework can be viewed by clicking: http://portal.cce.edu.om/member/contentdetails.aspx?id=565 4. Course work should be submitted with a n appropriate cover page, which can be obtained from the departmental assistant at the department. 5. Name, student identification and title of the course work to be written clearly and legibly on the cover page . 6. The completed course work is to be submitted to the departmental assistant on or before the deadline and record your name, date of submission and signature in the book with the departmental assistant. 7. For online submission of course work, pdf file with appropriate cover page mentioning name of student, student number and title of the course work should be uploaded using the submission link created and made available by the module leader. Referencing Harvard Referencing (CCE Style) First Edition 2013 should be followed for both in -text and listing references. This downloadable document can be found in our CCE portal at:

http://portal.cce.edu.om/member/contentdetails.aspx?cid=628 Name and Signature of Module leader Dr. Javed Ali Date: 7th March, 2018 QEAD Rev:0 3 ver 01 Template B 3 F/QAP/021/001 CALEDONIAN COLLEGE OF ENGINEERING, OMAN DEPARTMENT OF MATHEMATICS & STATISTICS M3G124711 : ADVENCED MATHEMATICS FOR CIVIL ENGINEERS Assignment –Sem. B -Feb 2017 -18 Note: In all questions, the letter “s” is the last nonzero digit of you r student ID. Q1(a) (b) (c) (d) (e) Graph using ezplot or fplot of MatLab. Find an interval [a, b] that contains the positive root of the equation by calculating values and Find that root (in part (b) ) correct to 4 digits by (i) Bisection method (ii) Newton Raphson method. Find three different ways to rewrite the equation in (b) as a fixed point problem . Discuss the possible convergence or divergence of the fixed p oint iteration for each . For what value of k, the equation given below can be solved by fixed point method. Use then fixed point iteration to approximate the root correct to 5 digits. [2] [5] [4+4] [3+3] [4] Q2(a) (b) Solve the following system using (i ) Gauss elimination (ii) Gauss -Jordon elimination (iii) Jacobi Iteration for six iterations (iv) Gauss Seidel Iteration for six iterations. For the last two methods choose appropriate starting point (initial approximate solution) other than and use six digits rounding. Fill in the following table using the results obtained in (a): Note that is the exact solution and is the distance betw een , given by . n is the number of iterations. n for Jacobi -Iteration for Gauss -Seidel Iteration 0 1 2 3 4 5 6 [3(4 ] [4] x f ( x ) Cos x s xe  0 x Cos x s xe  ()fa ( ).fb () x g x ()gx 0 x k Cos x xe s  0(0, 0, 0) P 1 2 3 4 2 1 0 0 3 1 4 1 0 0 1 4 1 1 0 0 1 2 2 x s x s x s x s                                    1 2 3 4( , , , )P x x x x n PP and n PP 2 2 2 2 1 1 2 2 3 3 4 4( ) ( ) ( ) ( ) n n n n n PP x x x x x x x x         n PP n PP QEAD Rev:0 3 ver 01 Template B 4 F/QAP/021/001 (c) (d) Interchange first two rows in the matrix above and then apply Gauss -Seidel method for six iterations with the starting point Use six digits rounding. Fill in the following table using the results obtained in (c): Note: is the distance between two consecutive values. n for Gauss -Seidel 1 2 3 4 5 6 [4] [5] Q3(a) (b) (c) (d) Q4(a) (b) (c) Develop a formula (Iteration scheme) to find the root of “ “ using Newton -Raphson Method. Set then and , to perform five iterations with six digits rounding. Approximate the integral using Simpson’s rule for n=1, 2, 3. Find the true error and error bound in each case: Find area under a parabola that passes through the points: Hint: You can use Lagrange Interpolation formula to find interpolating polynomial. Integration of this polynomial gives an approximation for the exact area. Compare your result with Simpson’s rule for n=1. Extend the result obtained in (c), to the sum of areas under 4 parabolas, where each parabola passes through three consecutive points. Points u nder consideration are: Compare your result with Simpson’s rule for n=4. Draw the graph of for using MatLab. Construct 1 , 2 , 3 , 4 and 5 trapeziums over separately on one sheet and approximate area under the curve of by finding area of the trapeziums. Trapezoidal rule finds area of a trapezium. Find the exact area by evaluating the below integral using MatLab, and compare this exact area with the areas of trapeziums found in part (b). [3+3] [3+3 +3] [5] [5] [2] [4] [2]   0 0, 3, 0.5, 1.25 . P 1 nniixx   1 11nnxx   1 22nnxx   1 33nnxx   1 44nnxx   n PP rth A rs 1 As  3 2 1 x s dx   0 0 1 1 2 2 0 ( , ), ( , ), ( , ), where . i x y x y x y x x ih  0 0 1 1 2 2 8 8 0 ( , ), ( , ), ( , ) , ... , ( , ), where . i x y x y x y x y x x ih  () Sin x x [ , 1] x s s [ , 1]ss  ( ) / Sin x x 1 () s s Sin x dx x   QEAD Rev:0 3 ver 01 Template B 5 F/QAP/021/001 (d) (e) (g) (h) What should be the value of n so that the error is less than for the approximation of by trapezoidal rule. There are other ways to approximate the above integral (in part d). We approximate using 1. Taylor’s series for of order for . 2. Lagrange interpolation formula to find interpolating polynomial at : 3. Newton forward interpolation formula to fin d interpolating polynomial at the same points as above in (2). In each case the interpolating polynomial is then integrated to approximate the above integral. Compare your results in each case with exact value of the integral and the result obtained using trapezoidal rule for n=5. Compare the graphs of interpolating polynomials ( part (e) 1,2,3) with the graph of . Which method of approximation is better, state your comments. Is there any o ther meth od to get interpolating pol ynomial for ? Explain the method. [5] [2] [2] [2] [2] [2] [2] For detailed study of pedagogical things concerning this part of coursework, refer to the following Ebrary resources. 1. An Introduction to Numerical Methods and Analysis Epperson, James F. John Wiley & Sons, Incorporated 2013 . 2. Numerical Methods Iyengar, S.R.K. ; Jain, R.K. New A ge In ternational Pvt, Ltd., Publishers 2009 3. Numerical Methods : Using MATLAB Lindfield, George ; Penny, John Elsevier Science 2012 ********** 6 s .s 10   1 () s s Sin x dx x   ( ) / Sin x x ( ) / Sin x x 6 ()Ox [ , 1] x s s , 0.2, 0.4, 0.6, 0.8, 1. x s s s s s s      ( ) / Sin x x ( ) / Sin x x