Chapter 2: Numerical Methods
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1 Chapter : Numerical Methods. Some Numerical Methods for st Order ODEs I this sectio, a summar of essetial features of umerical methods related to solutios of ordiar differetial equatios is give. I geeral, the umerical solutio of a differetial equatio cosists of a table (graph of values of the depedet variable for correspodig values of the idepedet variable. Page Numerical Methods for ODEs.. The Improved Euler Method for st ODEs The basic idea of Euler s method for solvig st ODEs is to covert the differetial (which is cotiuous to a differece equatio (which is discrete. The geeral form for a first-order differetial equatio will ow be writte as d (,. d = f (4 From the defiitio of a derivative, we ma write eq. (4 as d Δ ( +Δ ( lim = lim = f (,. (43 d Δ Δ Δ Δ Page 3 Numerical Methods for ODEs B rearragig this equatio, we get ( +Δ ( = f (,, Δ ( +Δ = ( + f (, Δ. Therefore, the basic idea of the fiite differece method ivolves writig the st order ODE (4 as ( + = ( + f (, Δ. (44 This is the Euler algorithm for solvig st order ODE, + = +Δ, =,,, To appl Euler s method, first select the iterval size Δ, the evaluate ( at, ad after that evaluate f (, at (,. The result for ( is ( = ( + f (, Δ. A secod iteratio with iput ( ields ( = ( + f (, Δ. Page 4 Numerical Methods for ODEs Page 5 Numerical Methods for ODEs The iteratio is cotiued to ield a umerical solutio of the st order ODE. To improve the simple Euler method, the solutio of st order ODE ca be epaded i the secod order approimatio of a Talor series. INSERT 5 ( + = ( + f(, Δ ( Δ f (, f (, + f (, + (45 The above equatio is referred to as the improved Euler method. Page Numerical Methods for ODEs INSERT 5 Talor Series For oe idepedet variable Talor series of g( at = : g ( = g ( +Δ = g( + g ( ( + g ( ( + (4!! st order approimatio d order approimatio Page 7 INSERT5: Talor Series
2 For two idepedet variables Talor series of g(, at (, : g g g (, = g + ( + ( g g ( + ( ( (47 +! + g + ( Prove eq. (45, b applig eq. (4 ad eglectig terms of 3 order ( ad smaller. Oe obtais g ( + Δ = g ( + g ( ( Δ + g ( ( Δ +!! ( + Δ ( + ( ( Δ + ( ( Δ Substitute d = = f (,, (from eq. (4 d ( = f (,, Page 8 INSERT5: Talor Series Page 9 INSERT5: Talor Series = f (,, f (, f d f d f (, = + d d partial derivative f f ( = f (, = + f (, We get ( +Δ f f ( + f (, ( Δ + + f (, ( Δ Chage variables + Δ + + Δ + ( + ( + f(, ( Δ ( Δ f (, f (, + f (, + Page INSERT5: Talor Series Page INSERT5: Talor Series EXAMPLE. Use the cocept of fiite differece to fid missig umbers of this sequece:,,?, 8, 4,, 3, 44,?, 74, 9,. SOLUTION Determie the differece betwee the two cosecutive umbers:,,?, 8, 4,, 3, 44,?, 74, 9,.,?,?,, 8,,,?,?, 8,.?,?,?,,,,?,?,?,. first differece secod differece Guess the secod differece:,,,,,,,,,. Guess the first differece:,, 4,, 8,,, 4,, 8,.,,,,,,,,,. Guess the origial sequece:,, 4, 8, 4,, 3, 44, 58, 74, 9,.,, 4,, 8,,, 4,, 8,.,,,,,,,,,. Page EXAMPLE. Page 3 EXAMPLE.
3 EXAMPLE. Fid the umerical solutio of the equatio d 3, (, d = = for =. to.5 i steps of., b usig the d ad 5 th order approimatio. Compare it with the eact solutio obtaied aalticall. SOLUTION For the d order approimatio, we ca appl the improved Euler method eq. (45 ( + = ( + f(, Δ ( Δ f (, f (, + f (, + Page 4 EXAMPLE. Substitute Page 5 EXAMPLE. 3 f (, =, f (, f (, =, ad = 3. Therefore, 3 ( Δ 3 ( + = ( + Δ + ( (3 or 3 + = + Δ + 3 ( Δ d ( order For the 5th order approimatio, we eed to epad the Talor series to the 5th order, g ( = g ( +Δ g( + g ( ( + g ( (!! g ( ( + g ( ( 3! 4! 5 + g ( ( 5! Page EXAMPLE. Page 7 EXAMPLE. We get ( Δ + ( Δ, Δ + ( Δ + ( Δ 3 =, 3 = (3 ( ( =, 3 5 = ( ( = 4, = ( ( =, 3 7 = = (3 ( th (5 order Page 8 EXAMPLE. Page 9 EXAMPLE. 3
4 Compare the umerical results with the eact solutio obtaied aalticall d 3, (, d = = 3 d = d, ( = + C, = + C, = + C, Appl ( =, Therefore, For eample, = +C, C =. = +, ( =. ( (.5 = = 4. (.5 Page EXAMPLE. Page EXAMPLE.... d ( order th (5 order (eact The for st ODEs Basic Idea To simulate the (accurate Talor series for. ( i + h b calculatig, ot ol at poit i, but at a umber of poits (aroud. i % error % error Advatage No fuctios other tha f (, are used, o differetiatio is eeded. Disadvatage Uses complicated argumets for each step. Page EXAMPLE. Page 3 Calculatio Scheme Let s cosider the case of secod-order accurac Talor epasio, ( +Δ ( + ( ( Δ + ( ( Δ (48 From the geeral form of first-order differetial equatio d = = f (,, d ad settig Δ = h, eq. (48 becomes h f f + + hf + + f. (49 The we assume that eq. (49 ca be simulated b the form + = + α hf + α hf ( + βh, + β hf. (5 Remember that the Ruge-Kutta method wat to use ol fuctio. f (,, o differetiatio is eeded. α α So, our task is to fid the values of,, β ad β, such that eq. (5 coicides with eq. (49. Cosider the last term of eq. (5. Epad it b usig the Talor series for two idepedet variables, eq (47. Page 4 Page 5 4
5 From eq. (47, we obtai f ( + βh, + β hf f f = f (, + ( βh + ( βhf f f ( β h + ( βh( βhf + + f + ( β hf Substitute it back ito eq. (5, + = + α hf f f f + ( βh + ( βhf f f + αh ( β h + ( βh( βhf + + f + ( β hf (5 Page Page 7 B comparig eq. (5 with eq. (49, we obtai the values of α, α, β ad β as α + α =, α = α, β = β =, β = β =. α ( α Oe possible (ad simple choice is α =.5, α =.5, β = β =. Therefore, eq. (5 ca be reduced to + = + hf + hf ( + h, + hf. Page 8 (5 Or it ca be rewritte as + = + ( a + a, a = hf (,, a = hf ( + h, + a. Page 9 (53 This is the secod-order Ruge-Kutta approimatio, which is accordig to the secod-order accurac Talor epasio i eq. (49. The similar schemes ca be applied to get a higher-order accurac. The third-order Ruge-Kutta approimatio The fourth-order Ruge-Kutta approimatio + = + ( b + 4 b + b3, b = hf (,, b = hf ( + h, + b, b = hf ( + h, + b b. 3 (54 + = + ( c + c + c3 + c4, c = hf (,, c = hf ( + h, + c, c3 = hf ( + h, + c, c = hf ( + h, + c. 4 3 (55 Page 3 Page 3 5
6 HOMEWORK 3 Prove the fourth-order Ruge-Kutta approimatio, eq. (55. Hit: assume + = + αc + αc + α3c3 + α4c4, c = hf (,, c = hf ( + ah, + bc, c = hf ( + a h, + b c + bc, 3 3 c = hf ( + a h, + b c + bc + bc EXAMPLE.3 Fid the umerical solutio of the equatio d 3, (, d = = for =. to.5 i steps of., b usig the fourth-order Ruge-Kutta approimatio. SOLUTION Appl the fourth-order Ruge-Kutta approimatio eq. (55 + = + ( c + c + c3 + c4, For the first calculatio, =, 3 3 c = hf (, = h( = (.(( =., Page 3 Page 33 3 c = hf ( + h, + c = h(( + c, 3 = (.(( + (.5(. =.37, 3 c3 = hf ( + h, + c = h(( + c, 3 = (.(( + (.5(.37 =.35, c = hf ( + h, + c = h(( + c, = (.(( +.35 =.747, = + (. + (.37 + ( , = HOMEWORK 4 Complete this table as a homework. Hit: You ma use a computer program to do calculatio. Page 34 Page 35 Note: We ca compare the accuracies of the Euler method ad the Ruge- Kutta method, b cosiderig the secodorder approimatio of the both cases.
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