Download GTU BE/B.Tech 2019 Winter 4th Sem New 2140105 Numerical Methods Question Paper

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GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER- IV (New) EXAMINATION - WINTER 2019
Subject Code: 2140105 Date: 07/12/2019
Subject Name: Numerical Methods
Time: 10:30 AM TO 01:00 PM Total Marks: 70

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Instructions:

1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.

Q.1

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1. State the numerical methods for solving initial value differential equations. [03]
2. Implement bisection method to solve x² —4x—9 =0 [04]
3. Describe the fitting of a straight line y=ae^bx and fit it for the data: [07]

   X	2.30	3.10	4.00	4.92	5.91	7.20
   y	33.0	39.1	50.3	67.2	85.6	125.0

Q.2

1. State the formulae for Lagrange's interpolation methods. [03]

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2. Using the Lagrange's formula find the polynomial and evaluate f(9). [04]

   X	5	7	11	13	17
   y	150	392	1452	2366	5202

3. Obtain cubic spline for every subinterval from the following data: [07]

   X	0	1	2	3
   y	2	-6	-8	2

OR

1. Use Stirling’s formulae for finding y(12.2) from the data: [07]

   X	10	11	12	13	14
   y	23967	28060	31788	35209	38368

Q.3

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1. Use Gauss elimination solve x+4y-z=-5, x+y-6z=-12, 3x-y-z=4. [03]
2. Use Trapezoidal rule to evaluate ?₀^1.4 1/x dx taking h=0.2, step length. [04]
3. Describe the Newton Raphson method in brief and evaluate vN for N=10. [07]

OR

1. Use Gauss Jordan method to solve 3x+y+2z=3, 2x-3y-z=-3, x+2y+z=4. [03]

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2. Use Simpson’s 3/8 rule to evaluate, taking h=0.2 and n=6 for ?_0.2^1.4 (sinx —log x + e^x)dx [04]
3. Describe method of False position and solve cos x — xe^x within the interval (0,1). [07]

Q.4

1. State the finite difference method for Laplace equation [03]
2. Solve heat equation ?u/?t + ?²u/?x² =0 over a rectangular slab that is 20 cm wide and 10 cm high. All edges are kept at 0°except the right edge which is maintained at 100°. There is no heat gain or lost from the surface of the slab. Place nodes with step length of 5 cm to generate grids and solve using finite difference method. [04]

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3. State the Taylor's method and solve equation dy/dx = (x+y) with x₀=0, y₀=1. Let h=0.1 and find four iterations. [07]

OR

1. State the finite difference quotients for first and second order derivatives. [03]
2. Solve y”+4y+1=0 with y(0)=0, y(1)=0, Using h=0.5 implement finite difference approach. [04]
3. State the Picard’s formula and solve the equation for x=0.1 dy/dx = (y-x)/(y+x) with y(0)=1. [07]

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Q.5

1. Discuss in brief finite difference and finite element approach [03]
2. Describe the Galerkin method in brief. [04]
3. Solve using Runge Kutta 4^th order method dy/dx = (x²+y²) , y(0)=1 for x=0.2,x=0.4. [07]

OR

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1. Discuss the shooting approach for boundary value problems. [03]
2. Solve u”=u, u’(1)=1.1752, u’(3)=10.01787 using appropriate method. [04]
3. Implement shooting method to solve u"—(1-x²)u =x with u(1)=2, u(3)=-1. [07]

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