ESO218, 03-04 II Sem, Assignment 7

(Not to be submitted)

1.Steady state temperature distribution in a rectangular plate is governed by the Laplace equation , subject to the boundary conditions T(0,y) = T(1,y) = T(x,0) =0 and T(x,1) = sin x. The exact solution of the problem is given by . Develop a computer code for the numerical solution of the problem using central difference approximations and graphically compare the numerical solution with the exact solution at x = 0.5 for x = y = 0.1.

2.Consider the heat equation:

0 x 1; t  0

with initial condition T(x,0) = sin x and boundary conditions T(0,t) = T(1,t) = 0.

a)Write a computer program to solve the equation using Euler explicit-Central difference approximations. For  = 1, x = 0.05 and t = 0.001, plot T(x) vs. x at t = 0.0, 0.5, 1.0, 1.5 and 2.0 in one graph.

b)Take t = 0.0015 (and same and x as in (a)) and plot T(x) vs. x at t = 0.0, 0.15, 0.153, 0.1545 and 0.156 in another graph.

c)Explain the results obtained in (a) and (b).

ESO218, 03-04 II Sem, List of Formulae to be memorized for End-sem

List of Formulae which you are expected to memorize for the End-sem (If a question is asked involving some other formula, that formula will be given in the question paper):

  1. Numerical Errors: Expressions for true and approximate errors, Taylor series
  2. Roots of equations: Bisection, False-position, Fixed-point iteration, Newton-Raphson, Secant method, Modified secant method,Muller’s method
  3. Linear algebraic equations: Gauss elimination, Gauss Jordan, LU decomposition, Thomas, Cholesky, Gauss-Seidel
  4. Analysis of experimental data: Curve Fitting: Linear and polynomial regression, Newton divided difference and Lagrange interpolation, Quadratic spline, Legendre, Gram’s, and Tchebycheff polynomials up to order 3, Fourier series.
  5. Numerical differentiation and integration: Newton-Cotes closed formula and their errors for 1 and 2 segments (Trapezoidal and Simpson’s 1/3), Newton-Cotes open formula and their errors for 2 and 3 segments (1 and 2 points), Romberg integration, Gauss quadrature weights and abscissa for 2 and 3 points. First and second derivatives using forward, backward, and central difference of the lowest order (h for forward and backward and h2 for central), Richardson extrapolation.
  6. Ordinary differential equations:
  7. Initial Value Problems (IVP): Euler’s, Improved Euler, and Modified Euler, Second, Third, and Fourth order R-K method, Non-self-starting Heun’s method, Adams-Bashforth for order 1 and 2, Adams-Moulton for order 2 and 3.
  8. Boundary Value Problems (BVP)
  9. Eigenvalue Problems: Fadeev-LeVerrier method, Power and Inverse power method.
  10. Partial Differential Equations: Finite difference method: Explicit, Implicit, Crank-Nicolson, and ADI methods.