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# Differential Equations MATH 3331

UH

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This 11 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 3331 at University of Houston taught by Jiwen He in Fall. Since its upload, it has received 21 views. For similar materials see /class/208372/math-3331-university-of-houston in Mathmatics at University of Houston.

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Date Created: 09/19/15

Lectuv s mm Methnd J W H 2 Wm M WM WW M Nam KVLuNnvmmzn Basic Idea Examles Errors Section 61 M Euler s Method Basic Idea Basic Idea 0 ODE y fty o Assume yt is known o For small h approximate z h y th quot41 quoterror ap x xx tangent g line gt W h s We h ya i h where i i r t t th yapt h MD hft 3105 o Truncation Error iyt h yapthi 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 yak m yk at tk yl y0hft07y07 t1 toh y2 ylhft1y1a 752751lh ykl l 2916 h farm 9 75k1 75k h Jiwen He University of Houston Math 3331 7 Section 19470 Lecture 22 March 23 2009 3 ll Section 61 r i Example 1 I Ex Approximate the solution to y y y01 ihOStSl Start to0yo1 1 y1 yohf0l1ll2 t1 toh011 h05 y1 1O5115 t1 OIO5O5 y2 15O53915225 t2 O5O51 2025 y1 1O251125 t1 OO25O25 Jiwen He University of Houston Basic Idea Examles 25 15 Math 3331 Section 19470 Lecture 3922 Errors 125 025 125 15625 025 025 05 15625 025 15625 1953125 05 025 075 1953125 025 1953125 244140625 075 025 1 Euler39approXimation for y y y01 I exaC I i I v i i v v I 39 v v v 39 v vv I h5 h 1 v v March 3923 2009 411 Section 61 i w Basic Idea Examles Errors Example 2 Ex Approximate the solution to y t 31 y0 05 in 031531 using hO25 Start yo 05 to Z O 91 05 025 0 05 0375 151 0 025 025 y2 0375 025 025 0375 03438 752 025 025 05 y3 03438 025 05 03438 2 03828 753 05 025 075 34 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 i J v s Basic Idea Examles Errors Euler s Method Errors exact solution Errors y solution for yt0hy1 Three error sources i TE truncation error o Truncation error at each y PTEi LOanthfj irmr Euler step 2 quotquotquotquotquotquotquotquotquotquot o Propagated accumulated truncation error Y1 o Roundoff error not controlable yo I 94 to t h t02h Jiwen He University of Houston Math 3331 Section 19470 Lecture 22 March 23 2009 6 11 Section 61 39 Basic Idea Examles Errors Errors in Euler s Method First Order EX 3 t y y0 5 Approximate y1 for stepsizes h 1m m 12481632 Exact Value y1 05518 Error E00 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 gt i v InClass Exercises Exercise 611 EX 1 y ty y0 1 Compute five Euler iterates for h 01 Arrange computation and results in a table k tk yk 1005 yk tkyk h 18051672011 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 7 1 Section 19470 Lecture 22 March 23 2009 8 ll In Class Exercises Exercise 617 EX 7 y 2mg 39 y0 8 i Compute Euler approximations in 0 g 1 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 XVX3YVY D In Nthb Eumw appmmhna on for k1zm for h 02 is computed and stored f2XyX H1anaysgtdL2y02vm hf39 v v 39 yyh 3 y r Analogoust for h 01 and h 005 X X 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 617ii EX 7 y 233 13 y0 8 i Compute Euler approximations in 0 g 1 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 may 2 yhc 8f yh d y2 2Uy gt 6 86 562 6 502 ge 2dg O 8e m2 e2ex2 12 152e2 12 yh exp Am2dm 562 Jiwen He University of Houston Math 3331 Section 19470 Lecture 22 March 23 2009 10 11

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