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# Class Note for MATH 3331 with Professor He at UH

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This 11 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 15 views.

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Date Created: 02/06/15

Lecture 22 s mm Methnd J W H 2 Wm M WM WW M mm m KVLuNnvmmzn Basic Idea Examles Errors Section 61 iir Euler s Method Basic Idea Basic Idea 0 ODE y fty 0 Assume yt is known o For small h approximate z h y th quoterror fa ya ap tangent line gt w h e gape h ya i h 39 where gt t t th yapt h yt hft 3105 o Truncation Error yt h yapthl H Jiwen He University of Houston Math 3331 Section 19470 Lecture 22 March 23 2009 2 11 Section 61 Basic Idea Examles Errors Euler s Method Iteration Scheme Iteration Scheme IVPI y fty 31050 yo Approximate ytk yk at tk yl y0hft07y07 751 toh y2 y1hft17y1a 752751lh ykl l yk h Him yk tk1 75k h Jiwen He University of Houston Math 3331 Section 19470 Lecture 22 March 23 2009 3 11 Section 61 iw i Example 1 I Ex Approximate the solution to y2 125 025125 15625 I 12 025 025 05 y 97 M0 1 y3 15625 025 15625 if 0 S t S 1 Start to 0 yo 1 1953125 1 153 05 025 075 1953125 025 1953125 2 0 1 1 1 1 2 94 31 3012507131 1 244140525 1 0 t4 075 025 1 h 05 3 I I I 311 1 05 1 15 Euler approximation t1 O y1yI I y2 15 05 15 225 2539 1 12 05 05 1 exact 0 25 gt 2 h 25 II3quotquot 0 y1 1 025 1 125 t1 O 025 025 15 4 34quot Jiwen He University of Houston Math 3331 Section 19470 Lecture 22 March 23 2009 4 11 Section 61 Example 2 Ex Approximate the solution to 3 t y y0 05 in 032531 using hO25 Start yo 05 to Z O 05 025 0 05 0375 311 t1 0 025 025 y2 0375 025 025 0375 03438 152 025 025 05 y3 03438 025 05 03438 2 03828 153 05 025 075 y4 03828 025 075 03828 2 04746 2 t4 075 025 1 I Jiwen He University of Houston Math 3331 Section 19470 Lecture 22 March 23 2009 5 11 Section 61 39 a 1 Basic Idea Examles Errors Euler s Method Errors exact solution Errors y solution for yt0hy1 Three error sources l TE truncation error o Truncation error at each y PTEi LOnp fg irmr Euler step 2 quotquotquotquotquotquotquotquotquotquot o Propagated accumulated truncation error Y1 I o Roundoff error i 5 not controlable yo 39 h A to t0h t02h Jiwen He University of Houston Math 3331 Section 19470 Lecture 22 March 23 2009 6 11 Section 61 7 Basic Idea Examles Errors Errors in Euler s Method First Order EX y t y y0 5 Approximate y1 for stepsizes h 1m m 12481632 Exact Value y1 05518 Error Eh y1 ym Theorem There 3 C gt 0 st h ym Em Eh 3 Ch 1 0 05518 12 0 375 O 1768 Euler method is first order 14 04746 00772 method 18 05154 00364 116 05341 00177 132 05431 00087 Eh2 Eh2 gt Eh Ch Jiwen He University of Houston Math 3331 Section 19470 Lecture 22 March 23 2009 7 11 InClass Exercises Exercise 611 1 EX 1 y ty y0 1 Compute five Euler iterates for h 01 Arrange computation and results in a table k 15k M 1001273910 tkyk h 13051672011 0 0 1 0 01 0 1 01 1 01000 01 00100 2 02 10100 02020 01 00202 3 03 10302 03091 01 00309 4 04 10611 04244 01 00424 5 05 11036 05518 01 00552 Jiwen He University of Houston Math 3331 Section 19470 Lecture 22 March 23 2009 8 11 ll n In Class Exercises Exercise 617 EX 7 y 2133 x 310 2 8 i Compute Euler approximations in 0 g a g 1 for h 02 h 01 h 005 ii Find exact solution iii Plot exact solution as curve and Euler approximations as points IhO2 m1hx0y8 xvxsyvy D In Nthb Eumw appmmuna on for k1zm for h 02 is computed and stored f2XyX H1anaysgtdL2y02vm hf39 v v 39 yyh 3 y JET Analogoust for h 01 and h 005 XX XV XV X arrays x01 y01 and X005 y005 end X02Xvy02yv Iiii Jiwen He University of Houston Math 3331 Section 19470 Lecture 22 March 23 2009 9 ll ln Class Exercises Exercise 617ltii EX 7 y 22163 L y0 8 i Compute Euler approximations in 0 g 13 g 1 for h 02 h 01 h 005 ii Find exact solution iii Plot exact solution as curve and Euler approximations as points ii Variation of Parameter Man 2 yh 8f yh d 312 2y gt 862 902 6 502 ge 2d 0 311200 exp fa lumen 6 862 562 6 901er 12 O 152e x2 12 Jiwen He University of Houston Math 3331 Section 19470 Lecture 22 March 23 2009 10 11 39 i In Class Exercises Exercise 617 iii EX 7 y 2mg 2 9310 8 i Compute Euler approximations in 0 g 1 g 1 for h 02 h 01 h 005 H F nd exactsoh jon iii Plot exact solution as curve and Euler approximations as points iii Matlab plot commands xlinspace01100 y12152expX 2 plotXO2yO2 ko XO1yO1 k XO05yO05 k Xy k Xlabel X ylabel y aXisO 1 35 8 Jiwen He University of Houston Math 3331 Section 19470 Lecture 22 March 23 2009

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