Numerical Method
This course contains the concepts of numerical method techniques for solving linear and nonlinear equations, interpolation and regression, differentiation and integration, and partial differential equations. The main objective of the course is to provide the knowledge of numerical method techniques for mathematical modeling.
Tribhuvan University
Institute of Science and Technology
2081
Bachelor Level / third-semester / Science
Computer Science and Information Technology( CSC212 )
Numerical Method
Full Marks: 60 + 20 + 20
Pass Marks: 24 + 8 + 8
Time: 3 Hours
Candidates are required to give their answers in their own words as far as practicable.
The figures in the margin indicate full marks.
Group A
Attempt any TWO questions:
What are inherent errors? Derive the Newton Raphson method for solving non-linear equation and using this method solve
\[x^2 – 5x + 6 = 0\]. Calculate upto 3 decimal places.
What are the limitations of direct methods for solving a system of linear equations? How Gauss Seidel method differs from Jacobi iteration? Solve the following system of linear equation using Jacobi iteration method.
2x-7y-10z=-17
5x+y+3z=14
x+10y+9z=7
Write an algorithm and program to implement Lagrange interpolation method.
Group B
Attempt any EIGHT questions
Consider the following data points estimate the f(0.6) using Newton’s interpolation formula.
| x | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 |
| f(x) | 2.68 | 3.04 | 3.38 | 3.69 | 3.97 |
What is regression analysis? Fit a second order polynomial for the following data values.
| x | 2 | 4 | 6 | 8 | 10 |
| y | 1.4 | 2.0 | 2.4 | 2.6 | 2.8 |
What is numerical differentiation? The table below gives the values of distance travelled by a vehicle at various time interval, estimate the velocity and acceleration at x = 4.
| Time(x) | 1 | 2 | 4 | 8 | 10 |
| Distance(y) | 0 | 1 | 5 | 21 | 27 |
What is an application of numerical integration? Find the value of the integral
\[
\int_1^2 \frac{e^x}{x} \, dx
\]
using Simpson’s \( \frac{3}{8} \) rule with \( n = 6 \).
Solve the following system of linear equations using Gauss-Jordan elimination method.
x+2y-3z=4
2x+4y-6z=8
x-2y+5z=4
Given the data points below
| X | 1.0 | 3.0 | 4.0 |
| f(x) | 1.5 | 4.5 | 9.0 |
Find cubic spline which belongs to 1<=x<=3 and estimate f(2) using cubic splines.
What is differential equation? Differentiate between ODE and PDE with example.
Solve \(\frac{dy}{dx} = \frac{x}{y}, \quad y(0) = 1\), at \(x = 0.4\) using Runge–Kutta’s \(4^{\text{th}}\) order method.
Solve the Poisson equation
\[
\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = -64xy, \quad 0 \leq x \leq 1,\; 0 \leq y \leq 1
\]
with boundary conditions:
\[
u(0, y) = 0,\quad u(x, 0) = 0,\quad u(1, y) = 150,\quad u(x, 1) = 150 \quad \text{and} \quad h = \frac{1}{3}.
\]