Initial Value Problems
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The document covers initial value problems and methods for solving them, particularly focusing on the Euler method and the Runge-Kutta methods. It provides examples and mathematical formulations for applying these methods to specific differential equations, along with convergence analysis. Additionally, it includes assignments for further practice on the discussed methods.
Initial Value Problems
- 1. Initial Value Problems Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Initial Value Problems Mohammad Tawfik
- 2. Initial Value Problems Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Objectives • Understand the applications of initial-value problems • Be able to apply the Euler method for solving initial value problems • Be able to apply the Runge-Kutta method for solving initial value problem
- 3. Initial Value Problems Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example Problem dt dv mmaF cvmgFFF UD cvmgvm dt dv m m cvmg v mct e c mg tv / 1
- 4. Initial Value Problems Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Exact Solution
- 5. Initial Value Problems Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Approximate Solution 12 12 tt vv t v dt dv m cvmg tt vv 12 12 m cvmg tt vv 1 12 12
- 6. Initial Value Problems Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Approximate Solution m cvmg ttvv 1 1212
- 7. Initial Value Problems Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Euler Method • Given the differential equation: • We may write: • Giving: tyf dt dy , tyf t yy t y dt dy ttt , tytfyy ttt ,
- 8. Initial Value Problems Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example • Given the differential equation: • The exact solution is: • At t=0, y=2 • Find y(4) using Euler method with step t=1 ye dt dy t 5.04 8.0 tt eety 5.08.0 08.108.3
- 9. Initial Value Problems Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solution ye dt dy t 5.04 8.0 yetyy t tttt 5.04 8.0 5142 5.04 0 0 01 yeyy 4.115.245 5.04 8.0 1 8.0 12 e yeyy 5.254.11*5.044.11 5.04 6.1 2 2*8.0 23 e yeyy 8.565.25*5.045.25 5.04 4.2 3 3*8.0 34 e yeyy
- 10. Initial Value Problems Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Convergence! 0 10 20 30 40 50 60 70 80 0 0.5 1 1.5 2 2.5 3 3.5 4 Time y(t) Exact Dt=1.0 Dt=0.5 Dt=0.1
- 11. Initial Value Problems Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Runge-Kutta Methods • The Runge-Kutta methods achieves the Taylor series accuracy • Many forms of the method are available; we will use 2nd order and 3rd order methods only
- 12. Initial Value Problems Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 2nd Order Runge-Kutta method • For the DE: • The 2nd order R.K. solution is: • Where: tyf dt dy , 21 2 kk t yy ttt tyfk t ,1 tttkyfk t ,12
- 13. Initial Value Problems Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example • Given the differential equation: • The exact solution is: • At t=0, y=2 • Find y(4) using 2nd order R.K. method with step t=1 ye dt dy t 5.04 8.0 tt eety 5.08.0 08.108.3
- 14. Initial Value Problems Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solution • At t=0 32*5.04 0 1 ek 4.61*325.04 108.0 2 ek 7.6 2 2101 kk t yy • Repeat for all t
- 15. Initial Value Problems Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solution yk2k1t 20 6.7010826.40216431 16.3197813.685785.5516232 37.1992530.106711.652243 83.3377766.7839625.493084 yk2k1t 20 3.8043254.21729930.5 6.3165385.9837174.0651361 9.9240688.6862255.7438951.5 15.196312.770498.3184342 22.9759418.9045812.213982.5 34.5149228.0876718.068263 51.6768141.8123226.835253.5 77.2385262.3066739.940184
- 16. Initial Value Problems Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Convergence 0 10 20 30 40 50 60 70 80 90 0 0.5 1 1.5 2 2.5 3 3.5 4 Time y(t) Exact Dt=1.0 Dt=0.5 Dt=0.1
- 17. Initial Value Problems Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 3rd Order Runge-Kutta method • For the DE: • The 3rd order R.K. solution is: • Where: tyf dt dy , 321 4 6 kkk t yy ttt tyfk t ,1 2 , 2 1 2 t t tk yfk t tttktkyfk t ,2 213
- 18. Initial Value Problems Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Assignment • Solve: • Given y(0)=1 1. Analytically 2. Using Euler method until t=2, with t=0.5 3. Repeat part 2 using 2nd order RK method 4. Repeat part 2 using 3rd order RK method 5. Repeat parts 2 through 4 using t=0.25 6. Compare results of all parts above yyt dt dy 2.12