BE Computer Engineering (IOE, TU) Numerical Methods (IOE, SH 553) Question Paper 2079

(a) Derive the iterative formula for the Newton-Raphson method for finding a root of a nonlinear equation $f(x)=0$, and state its order of convergence. Discuss two situations in which the method fails. [7]

(b) Using the Bisection method, find a root of the equation $x^3 - 4x - 9 = 0$ correct to three decimal places. Show the tabulation of iterations. [5]

(a) Derive Newton's divided difference interpolation formula and show that the divided differences are symmetric with respect to their arguments. [6]

(b) The following table gives the values of a function $f(x)$:

Using Lagrange's interpolation formula, estimate the value of $f(x)$ at $x = 10$. [6]

(a) Explain the Gauss-Seidel iterative method for solving a system of linear equations. State the condition of convergence (diagonal dominance) for the method. [5]

(b) Solve the following system of equations using the Gauss-Seidel iteration method, performing three iterations and starting with the initial approximation $(0,0,0)$:

[7]

(a) Derive the standard five-point finite-difference formula for the Laplace equation $\dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} = 0$ over a square mesh. [6]

(b) Classify the partial differential equation $A u_{xx} + B u_{xy} + C u_{yy} + f(x,y,u,u_x,u_y) = 0$ as elliptic, parabolic or hyperbolic, and give one physical example of each type. [6]

Find a real root of the equation $\cos x = 3x - 1$ correct to four significant figures using the Secant method. Take the initial approximations $x_0 = 0.5$ and $x_1 = 0.6$ and tabulate the successive iterations.

Evaluate the integral $\displaystyle\int_{0}^{6} \frac{dx}{1+x^2}$ by dividing the interval into six equal subintervals using (a) the Trapezoidal rule and (b) Simpson's 1/3 rule. Compare your results with the exact value.

From the following table of values, find the first and second derivatives of $f(x)$ at $x = 1.1$ using Newton's forward difference formula:

Fit a straight line $y = a + bx$ to the following data using the method of least squares and estimate the value of $y$ when $x = 2.5$:

Using the fourth-order Runge-Kutta method, find an approximate value of $y$ at $x = 0.2$ for the initial value problem $\dfrac{dy}{dx} = x + y^2$, $y(0) = 1$, taking a step size $h = 0.1$.

Using the LU (Doolittle) decomposition method, factorize the coefficient matrix and solve the system:

Using the explicit (Bender-Schmidt) finite-difference scheme, solve the one-dimensional heat conduction equation $\dfrac{\partial u}{\partial t} = \dfrac{\partial^2 u}{\partial x^2}$ subject to $u(0,t)=u(1,t)=0$, $u(x,0)=\sin\pi x$ for $0\le x\le 1$. Take $h=0.25$ and choose an appropriate time step satisfying the stability criterion; compute $u$ for the first two time levels.