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UNIVERSITY of ENGINEERING & MANAGEMENT,JAIPUR Lecture UNIVERSITY OF ENGINEERING & MANAGEMENT,JAIPUR Lecture-wise Plan Subject Name: Numerical Methods Subject Code-M(CS))401 Year: 2nd Year Semester: Fourth Module Topics Number of Lectures Number : Approximation in numerical computation: 4L Approximation of numbers 1 Types of errors 2 1 Calculation of errors 1 Interpolation: 6L Finite differences 1 2 Newton forward/backward interpolation 2 Lagrange’s method 1 Newton’s divided difference Interpolation 2 Numerical integration: 3L Trapezoidal rule 2 3. Simpson’s 1/3 rule 1 Numerical solution of a system of linear equations: 6L 4 Gauss elimination method 1 Matrix inversion 1 LU Factorization method 2 Gauss-Seidel iterative method 2 Numerical solution of Algebraic equation: 5L 5 Bisection method 2 Regula-Falsi method 1 Newton-Raphson method 2 Numerical solution of ordinary differential equation: 8L 6 Euler’s method 2 Runge-Kutta methods 2 Predictor- Corrector methods 2 FiniteDifference method 2 Assignment: Module-1: 1. Find the relative error if 2/3 is approximated to 0.667. 2. Find the percentage error if 625.483 is approximated to three significant figures. 3. Find the relative error in taking = 3.141593 as 22/7. 4. The height of an observation tower was estimated to be 47 m, whereas its actual height was 45 m. calculate the percentage relative error in the measurement. 5. Two numbers are 3.5 and 47.279 both of which are correct to the significant figures given. Find their product. Module-2: 1. Apply Newton’s backward Interpolation to the data below, to obtain a polynomial of degree4 in x x : 1 2 3 4 5 f(): x 1 -1 1 -1 1 2. Using Newton’s backward Interpolation, find the value of f (2) from the following table: x : 1 3 4 5 6 7 f(): x 2.68 3.04 3.38 3.68 3.96 4.21 3. Using Newton’s Forward Interpolation, the area A of a circle of diameter d . d : 80 85 90 95 100 A: 5026 5674 6362 7088 7854 Calculate the area of a circle of diameter 105. 4. Estimate the value of f (22) and f (42) from the following available data: x : 20 25 30 35 40 45 f(): x 354 332 291 260 231 204 Using Newton’s Forward Interpolation 5. Find f() x as a polynomial in x for the following data by Newton’s divided difference method: x : -4 -1 0 2 5 f(): x 1245 33 5 9 1335 6. Using Newton’s divided difference method to find f() x from the following available data: x : 0 1 2 4 5 6 f(): x 1 14 15 5 6 19. Module-3: 1 1. Apply trapezoidal rule to find the integral I sin x dx . 0 2. Find, from the following table the area bounded by the curve and the x-axis from x 7.47 to x 7.52 , f7.47 1.93, f 7.48 1.95, f 7.49 1.98, f 7.50 2.01, f7.51 2.03, f 7.52 2.06. 1 1 3. Evaluate I dx, correct to three decimal places and also find the approximate value 2 0 1 x of . 4. A solid of revolution is formed by rotating about the x-axis the area between the x-axis, the lines x 0 and x 1and a curve through the points with the following coordinates: 0,1 , 0.25,0.9896 0.5,0.9589 , 0.75,0.9089 , 1,0.8415 . Module-4: 1. Solve the following system of equations: UNIVERSITY OF ENGINEERING & MANAGEMENT,JAIPUR Lecture-wise Plan 4x y z 4 x4 y 2 z 4 , 3x 2 y 4 6 by matrix-inversion method. 2. Solve the above system by matrix-inversion method: x y z 2 2x 3 y 5 z 3 3x 2 y 3 z 6 3. The following system of equations are given: 4x y z 4 x4 y 2 z 4 , Solve the above system by LU decomposition method. 3x 2 y 4 6 4. Solve the given system of equations by LU decomposition method: x y z 2 2x 3 y 5 z 3 3x 2 y 3 z 6 Module-5: 1. Find the root of the following equations correct three decimal places by the Regulafalsi method: x3 x 1 0 . 2. Using Regulafalsi method, compute the real root of the following equation correct to four decimal places: xex 2 . 3. Find the root of the following equations correct three decimal places by the Regulafalsi method: x6 x 4 x 3 1 0 . 4. Find the root of the following equations correct three decimal places by the bisection method : x ex 0 5. Find the root of the following equations, using the bisection method correct three decimal places: xcos x 0 6. Using the bisection method to find a root of the equation to four decimal places: x3 9 x 1 0 Module-6: dy 1 1. Using Runge-kutta method of order 4, find y(0.2) given that 3x y , y (0) 1taking dx 2 h 0.1. 2. Using Runge-kutta method of order 4, compute y(0.2) and y(0.4) from dy 10x2 y 2 , y (0) 1taking h 0.1. dx dy 3. Using Milne’s predictor-corrector method to obtain the solution of the equation x y2 dx at x 0.8 given that y(0) 0.0000 , y(2) 0.0200 , y(4) 0.0795 y(6) 0.1762 . dy 4. Given 2 (1 x2 ) y 2 and y(0) 1 , y(0.1) 1.06 , y(0.2) 1.12 , y(0.3) 1.21, dx evaluate y(0.4) by Milne’s predictor-corrector method. UNIVERSITY OF ENGINEERING & MANAGEMENT,JAIPUR Lecture-wise Plan Subject Name: Values & Ethics in Profession Subject Code-HU401 Year: 2nd Year Semester: Fourth Module Number Topics Number of Lectures Introduction: 19L Rapid Technological growth and depletion of 3 resources, Reports of the Club of Rome. 1 Limits of growth: Sustainable development Energy Crisis: Renewable Energy Resources 5 Environmental degradation and pollution. Eco- friendly Technologies. Environmental Regulations, Environmental Ethics Appropriate Technology Movement of Schumacher; 6 later developments Technology and developing notions. Problems of Technology transfer, Technology assessment impact analysis. Human Operator in Engineering projects and 5 industries. Problems of man, machine, interaction, Impact of assembly line and automation. Human centered Technology. Ethics of Profession: 9L 2 Engineering profession: Ethical issues in Engineering 3 practice, Conflicts between business demands and professional ideals. Social and ethical responsibilities of Technologists. Codes of professional ethics. Whistle blowing and 6 beyond. Profession and Human Values 8L Values Crisis in contemporary society 3. Nature of values: Value Spectrum of a good life 3 Psychological values: Integrated personality; mental health Societal values: The modern search for a 5 good society, justice, democracy, secularism, rule of law, values in Indian Constitution. Aesthetic values: Perception and enjoyment of beauty, simplicity, clarity Moral and ethical values: Nature of moral judgements; canons of ethics; ethics of virtue; ethics of duty; ethics of responsibility. UNIVERSITY OF ENGINEERING & MANAGEMENT, JAIPUR Lecture-wise Plan Subject Fluid mechanics & Hydraulic Machines Subject Code-ME401 Year: 2nd Year Semester: Fourth Module Topics Number of Number Lectures 1 1. Introduction to Fluid Mechanics 12L Review of fluid properties and fluid statics 2 Viscosity-dynamic and kinematic 1 Surface Tension. Fluid Statics: Basic equation of fluid 2 statics Manometers 2 Hydraulic forces on submerged surfaces; forces on 1 vertical,horizontal surfaces Hydraulic forces on inclined and curved surfaces 2 Buoyant force, Stability of floating and Submerged 2 bodies. 2 2. Kinematics of fluid flow: 3L fluid flow and classifications. Continuity equation in 1-D 1 Continuity equation in 3-D 1 Potential flow & Stream function; types of flow lines 1 3 3. Dynamics of fluid: 4L Equations of motion; Euler’s equation; 1 Bernoulli’s equation; Applications of Bernoulli’s 2 Problems on Bernoulli’s equation 1 4 4. Momentum Analysis of flow systems; 4L The linear momentum equation for steady flow 1 The linear momentum equation for steady flow 2 (differential approach) Problems 1 5. Flow through pipes; 4L 5 Darcy – Weisbach equation of friction loss; 1 Hydraulic grade line and total energy line 2 Problems 1 6 6. Basic principle for flow through orifices; 4L Derivation of discharge through V-notches 1 Derivation of discharge through weirs (rectangular) 1 Flow through open channels; use of Chezy’s formula 1 Problems 1 7. Dimensional Analysis & Model investigation 4L 7 applied to flow systems Introduction to dimensional analysis and its applications 1 Buckingham Pi theorem. 1 Dimensionless numbers in fluid flow 1 Problems 1 8 8. Application of fluid; 4L Hydraulic press, Hydraulic accumulator 1 Hydraulic Ram, Hydraulic lift, 1 Hydraulic coupling, Hydraulic torque convertor 1 Gear pump 1 09 09. Hydraulic Turbines; 6L Hydraulic Turbines; Principles and Classifications 1 working principle of a Pelton Wheel, Francis Turbine, 1 Kaplan Turbine working principle of Kaplan Turbine 1 Function of Draft Tube. Cavitation in Turbines. 2 Problems 1 Total Number Of Hours = 45 UNIVERSITY OF ENGINEERING & MANAGEMENT, JAIPUR Lecture-wise Plan Assignments 1 1. Define fluid with examples. 2. Define Ideal Fluid & Real Fluid. 3. Distinguish between Newtonian and non-Newtonian fluids. 4. Define the term centre of pressure of the plane area immersed in a fluid .Do the Centre of pressure and centre of gravity ever coincide and if so under what conditions? 5. Define the following terms with their units. i. Specific weight ii. Specific gravity iii. Surface Tension iv. Dynamic viscosity v. Specific weight or Weight density vi.Specific volume vii.Specific gravity or Relative density viii.
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