## Third Semester | Second Year | Tribhuvan University

Old Question Collection | Question Bank

Numerical Method, Year: 2067

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.**

**Attempt all Questions:**

- Discuss methods of Half Interval and Newton’s for solving the nonlinear equation f(x) = 0. Illustrate the methods by figures and compare them stating their advantages and disadvantages. (8)
- Derive the equation for Lagrange’s interpolating polynomial and find the value of f(x) at x = 1 for the following: (4+4)

X -1 -2 2 4 F -1 -9 11 69 - Write Newton-cotes integration formulas in basic form for x = 1, 2, 3 and give their composite rules. Evaluate dx using the Gaussian integration three point formula. (4+4)
- Solve the following algebraic system of linear equations by Gauss-Jordan algorithm. (8)

- Write an algorithm and program to solve system of linear equations using Gauss-Siedel iterative method. (4+8)
- Explain the Picard’s proves of successive approximation. Obtain a solution upto the fifth approximation of the equation

= y + x such that y=1 when x=0

using Picard’s process of successive approximations. (2+6) - Define a difference equation to represent a Laplace’s equation. Solve the following Laplace equation.

=0 within 0≤?≤3, 0≤?≤3.

For the rectangular plate given as: (3+5)

**OR**

Derive a difference equation to represent a Poison’s equation. Solve the Poison’s equation ∇^{2}?=2?^{2}?^{2}over the domain 0≤?≤3, 0≤?≤3 with ?=0 on the boundary and ℎ=1. (3+5)

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