utf-8''Chap2 Lec3 Euler Methhod (Student)

utf-8''Chap2 Lec3 Euler Methhod (Student) - H U T D EPART M...

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HUT – DEPARTMENT OF MATH. APPLIED -------------------------------------------------------------------------------------------------------- -- I DE – ADVANCED PROGRAM CHAPTER 2: Mathematical Models and Numerical Methods (1 weeks) LEC 3: EULER’S METHOD (FOR STUDENT) PhD. NGUYEÃN QUOÁC LAÂN (9/2007)
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CONTENTS ------------------------------------------------------------------------------------------------------------------------------- 3.3 Mathematical model: Radioactive decay 3.1 Euler’s Method 3.2 Example
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I DEA ---------------------------------------------------------------------------------------------------------------------------------------- Answe r: a/ The problem is non – linear Generally, we can’t solve it! The initial value (Cauchy’s) problem: a/ The problem is linear or non – linear? Can you solve it? - = + - = 2 ) 0 ( 0 , 2 ' 2 y x e y y y x b/ Approximate its solution at x = 0.5, x = 1 with step h = 0.5 b/ To approximate its solution: We must approximate y’(x 0 ): ( 29 ( 29 ( 29 h x y h x y x y 0 0 0 ' - + ( 29 ( 29 ( 29 ( 29 0 0 2 0 0 0 2 e x y x y h x y h x y + - = - + 0 0 = x 5 . 0 1 0 = = + x h x 1 2 1 = = + x h x
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THE NUMERI CAL SOLUTI ON ---------------------------------------------------------------------------------------------------------------------------------------- b/ At the point x = 0, (*) The complete procedure: ( 29 ( 29 ( 29 0 2 0 0 2 0 ' e y y y + - = Approximate the derivative: ( 29 ( 29 ( 29 ( 29 ? 5 . 0 0 0 0 ' = - + y h y h y y At the point x = 0.5, (*) ( 29 ( 29 ( 29 5 . 0 2 5 . 0 5 . 0 2 5 . 0 ' e y y y + - = Approximate the derivative: ( 29 ( 29 ? 1 5 . 0 ' = y y x k Value y k 0 –2 0.
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