## Third Semester | Second Year | Tribhuvan University

Old Question Collection | Question Bank

Numerical Method, Year: 2069

Computer Science and Information Technology (CSc 204)

Full Marks: 60 | Pass Marks: 24 | Time: 3 hours

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Candidates are required to give their answer in their own words as far as practicable.

**The figures in the margin indicate full marks.**

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- Derive a formula to solve nonlinear equation using secant method. Using your formula estimate a real root of following nonlinear equation using secant method correct up to decimal places x
^{2}+ ln x = 3. (3+5) - Estimate f(3) from the following data using Cubic Spline Interpolation. (8)

x 1 2.5 4 5.7 f(x) -2.0 4.2 14.4 31.2 **OR**

Find the best fitting quadratic polynomial from following data using least square approximation.x -2 -1.2 0 1 1.2 2.5 3 4.5 6.3 f(x) 10.39 2.96 -2.0 -2.63 -2.46 0.83 3.1 12.8 30.4 - (a) For the function ?(?) = estimate ?′(6.3) and ?”(6.3) [take ℎ = 0.01] (4)

(b) Evaluate using Gauss integration 3 point formula. (4) - Solve the following system of linear equations using Gauss-elimination or Gauss Jordan method.

3x_{1}+ 5x_{2}– 3x_{3}+ x_{4}= 16

2x_{1}+ x_{2}+ x_{3}+ 4x_{4}= 9

43 – 4x_{2}– x_{4}= 1

2x_{1}+ x_{2}– 3x_{3}+ 9x_{4}= 5 (8) - How can you solve higher order differential equation? Explain. Solve the following differential within 0 ≤ ? ≤ 1 using Heun’s method. (3+5)

+ 2xy = 1 with ?(0)=1 and ?′(0)=1 [?ake ℎ=0.5] - (a) How can you obtain numerical solution of a partial differential equation? Explain. (3)

(b) The steady state two dimensional heat-flow in a metal plate is defined by =0. Given the boundary conditions as sown in figure below, find the temperatures at interior points T1, T2, T3 and T4. (5) - Write an algorithm and C-program code to solve non-linear equation using Newton’s method. Your program should read an initial guess from keyboard and display the following if the solution is obtained.

• Estimated root of the equation

• Functional value at calculated root

• Required number of iterations (5+7)

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