Tutorial 5 - Nepal Engineering College

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					                                                 Nepal Engineering College
                                                 Changunarayan, Bhaktapur
                                                  email: info@nec.edu.np
Subject: Numerical Methods                                                                     Teacher: Hari K. Shrestha
Subject Code: MTH 317.3                                                                                  Tutorial No.: 5
Title: Solution of Ordinary Differential Equations                                                Date: August 10, 2007
One Step methods (Taylor, Picard, Euler, Heun, Runge Kutta) to solve first order initial value problems.
1.     Use the Taylor method to solve the equation y’ = x² + y² for x = 0.25 and x = 0.5 given y(0) = 1.
2.     Find y(0.1) and y(0.2) using Taylor method to solve the equation y’ = x²y - 1, y(0) = 1.
3.     Use Picard’s method to solve the following equations, and find y(0.1), y(0.2) and y(1).
          a) y’(x) = x² + y², y(0) = 0
          b) y’(x) = xey, y(0) = 0
4.     Use Euler’s method to solve y’ = 3x² + 1, y(1) = 2. Take step size = 0.5 and estimate y(2).
5.     Solve the equation y’ = x + y, y(0) = 1 by Euler’s method. Taking step size = 0.5 and 0.25, find y(1). Compute error
       in both cases. Compare the results with exact solution (y = 2ex – x – 1).
6.     The current I in a circuit having a resistance R and inductance L is given by the differential equation as,
       dI/dt = (E sin t - RC) / L where E = 100 V, L = 1.5 H,  = 500, C = 1 and R = 90 Ohm. Initially, I = 0 at t = 0. By
       the use of a suitable numerical method, write a flow chart and a computer program in C language to determine
       current at various times and print.
7.     Use Heun’s method to solve y’ = 2y/x, y(1) = 2. Take step size = 0.25 and estimate y(2).
8.     A simply supported beam of effective span 6 m is loaded by a system of load q(x) = h x² / l², where h = 10, l = 6 m.
       Using a suitable numerical method, flowchart and suitable program to determine shear force V at x = 0(1)6. Use
       differential equation EI (dv / dx) = q(x).
9.     Solve the differential equation y’ = x² y + 2x, given y(0) = 0 using Runge Kutta method.
10. Solve the initial value problem y’ = 0.5 (1 + x) y², y(0) = 1, for x = 0(0.1)0.6 using midpoint method formula.
11. Find y(0.1) using Euler’s method when dy/dx = x² + y² and y(0) = 1. Also obtain y(0.1) using Runge-Kutta method
       using above equation with given initial condition. State the reasons for the difference in the result.
12. Solve the initial value problem y’ = x y + y² given that y(0) = 1 for x = 0(0.1)0.3 using RK4 method.
Higher order initial value problems:
13. Solve the following 2nd order differential equation by fourth order Runge-Kutta method.
        d 2q    dq q
    L      2
              R  5       where L=1, C= 0.25, R= 0.5 for t = 0 to 0.2 with a step of 0.1, given q(0) = 0 and q ' (0) = 0
        dt      dt c
14. Using Runge-Kutta (RK-4) method solve y” = x y’ + y², y(0) = 1, y’(0) = 2, use step size of 0.1 to find y(0.2).
15. Solve the following system of simultaneous equation with associated initial conditions using fourth order Runge-Kutta
   (RK-4) nethod.
   y’ = y – z            y(0) = 0
   z’ = -y + z           z (0) = 0
   in the range 0  x  1 with step size of 0.5.
16. Solve y” – x² y’ – 2xy = 0; y(0) = 1, y’(0) = 0 for y(0.1) using Runge Kutta (RK4) method.
Shooting method to solve Boundary Value Problems (BVP):
17. Using shooting method, solve the equation
   a) d²y/dx² = 6x, y(1) = 2, y(2) = 9 in the interval (1,2).
   b) d²y/dx² = 3x + 4y, y(0) = 1, y(1) = 1. Take step size = 0.25
   c) d²y/dx² + 2 (dy/dx) – y/2 = 2.5, y(0) = 10, y(10) = 6.
   d) x² (d²y/dx²) – 2x (dy/dx) + x² sin x = 2y, y(1) = 1, y(2) = 2. Estimate y(1.25), y(1.5) and y(1.75).
18. Using the shooting method, find the solution of the BVP x² y” + x y’ = 1, given y(1) = 0, y(1.4) = 0.0566
                        2
19. Solve d²y/dx² = e x , y(0) = 0, y(1) = 0 for x = 0.25, 0.5, and 0.75, using the shooting method.
                                                             d4y
20. The deflection of a beam is governed by the equation             81 y  f(x)
                                                             dx 4
    where f(x) is given by the table
                                                  x     1/3    2/3        1
                                                f(x)    81     162      243
   and boundary condition y(0) = y’(0) = y”(1) = y’”(1) = 0. Evaluate the deflection at the pivotal points of the beam using
   three sub-intervals.
21. Write an algorithm and computer program in any one of the high level programming language to find the solution of
    second order differential equation y” = f(x,y) by RK-4 method as boundary value problem. Adopt boundary values
    y(xo) = yo and y’(xo) = y’o.
Submission Deadline: August 15, 2007. Attempt only even numbered problems for submission.



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