Document Sample

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. NM_Tutorial5.doc

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 2 |

posted: | 1/5/2013 |

language: | Unknown |

pages: | 1 |

OTHER DOCS BY pengxuebo

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.