ࡱ> 463E h3jbjb Tl8$4.>6t(!######,. NO@O>!.6d6! d[ *   Numerical Methods: Some Simulation Algorithms Quotes of the Day "It is the mark of an instructed mind to rest satisfied with the degree of precision which the nature of the subject permits and not to seek an exactness where only an approximation of the truth is possible. --- Aristotle I. Background (1) Why Numerical Methods? Numerical methods are needed for solving complex differential equations used to model physical, biological, and socioeconomic systems even though some of them (but few) may be analytically solvable. Numerical methods are imperative for differential equations that can not be solved analytically. (2) A Theoretical Basis for Numerical Methods According to the Taylor Theorem, a function Y(x) can be expanded, about a point x = xm, into a series, which is known as the Taylor series:  EMBED Equation.2  where  EMBED Equation.2 ,  EMBED Equation.2  are the first, second, third, ..., and nth derivative of Y(x) evaluated at x = xm, and  EMBED Equation.2 . The Taylor series expansion provides a theoretical basis for evaluating and comparing commonly used numerical methods. We can use the series to approximate the solution Y(x) at x = xm+"x simply by replacing x by xm+"x in the Taylor series. In the context of differential equations describing temporal dynamics, we use t for x and thus have:  EMBED Equation.2  We can select the first few terms in the Taylor series to approximate the solution of a function, but this will result in truncation error. Apparently, the more terms a procedure uses, the more accurate the approximation is. (1) First-order approximation:  EMBED Equation.2 ,  EMBED Equation.2  where  EMBED Equation.2  is the truncation error, and K is a constant. (Note: Compare the above equation with the definition of the first-order derivative!) (2) Second-order approximation:  EMBED Equation.2  where the truncation error is reduced to  EMBED Equation.2 . (3) kth-order approximation:  EMBED Equation.2  in which the truncation error is reduced by several orders of magnitude as compared to that in the first-order approximation.  EMBED Equation.2  II. Eulers and Runge-Kutta Methods There exist many different numerical methods for solving differential equations (see Dorn and McCracken 1972, Press et al. 1992). Two types of numerical methods can be distinguished: (1) one-step, self-starting methods (e.g., Eulers and modified Eulers methods, Runge-Kutta methods) and (2) Multistep, non-self-starting methods (e.g., predictor-corrector methods). Amongst the simplest and most commonly used are Euler s method and Runge-Kutta methods. These numerical techniques are all one time-step methods; that is, only information at one time interval is used to estimate change in state variables. They agree with the Taylor series through terms in ("t)k, where k is the order of the method. For example, k is 2 in the 2nd-order Runge-Kutta methods and 4 in the 4th-order Runge-Kutta method. Thus, Euler s method may be considered as a 1st-order Runge-Kutta method. 1. Euler s Method Euler s method is one of the oldest and best known numerical methods for integrating differential equations. It simply uses the estimate of the rate of change (Y ) at time t to calculate the change in the state variable over the time interval ("t), assuming the rate of change is constant over "t. The formula for Euler s method is simply  EMBED Equation.2   A geometrical representation of Euler s method: a series of Euler approximations (straight lines) to a true solution (curved line) over (t solution intervals (Haefner 1996) Although it is extremely simple and computationally efficient, Euler s method may introduce significant truncation and integration errors especially when "t is relatively large. Besides, it is often unstable, meaning that small roundoff and truncation errors may propagate as simulation time progresses. This error propagation may lead to artifactual behaviors that do not really come from the differential equations. A general procedure of numerical approximation of solutions to differential equations  2. Runge-Kutta Methods To reduce truncation and integration errors, Runge-Kutta methods require additional evaluations within each iteration. Two most commonly used Runge-Kutta methods are the second- and fourth-order Runge-Kutta Methods. As compared to Eulers method, the 2nd-order Runge-Kutta method requires the evaluation of the rate of change at two points within a time interval, whereas the 4th-order requires the evaluation at four points within a time interval. The accuracy of numerical solutions is substantially improved at the expense of computational time. Let  EMBED Equation.2 , a commonly used formula for the 2nd-order Runge-Kutta is  EMBED Equation.2 , which simply means that "Y is calculated based on an average of the two estimates of the rate of change at two points, t and t+"t (namely, the beginning and end of a time interval). The fourth-order Runge-Kutta method uses a weighted average of 4 estimates of the rate of change to calculate the value of the state variable over each "t. A widely used 4th-order Runge-Kutta formulation is as follows:  EMBED Equation.2  where  EMBED Equation.2  3. Accuracy Comparison of Euler s and Runge-Kutta Methods Runge-Kutta methods are usually much more accurate than Euler s method, but higher-order Runge-Kutta techniques may become rather time-consuming when a small "t is used. However, with larger time steps and thus with about the equivalent amount of computational time, integration errors produced by Runge-Kutta methods are usually significantly smaller than those by Eulers method. These can be demonstrated through the following example on simulating the coffee cooling process (model demo). III. Notes on Simulation Algorithms in STELLA STELLA includes three numerical techniques: Eulers method, and the second- and fourth-order Runge-Kutta methods. Here are some very important points (cf. Technical Documentation of STELLA), which one needs to know when using STELLA for any serious, scientific purposes (not just for fun!). 1. For modeling discrete dynamics or discrete objects, use Eulers method only. This includes situations in which you use any of STELLAs discrete objects (queues, conveyors, and ovens), or built-in functions to generate integer values (e.g., IF-THEN-ELSE logic to set 0-1 flags, INT, ROUND, SWITCH, etc). STELLA can be used for simulating discrete systems, but in this case "t is usually set to 1 and only Euler s method has the appropriate mathematical formulation. Runge-Kutta methods are designed for continuous systems, and inappropriate for simulating discrete dynamics. 2. For modeling continuous systems, the Runge-Kutta methods are generally preferable. For those that exhibit oscillations the Runge-Kutta methods must be used. Runge-Kutta methods are not only more accurate, but also are more stable. On the other hand, error propagation in Eulers method may turn a sustained oscillation into an ever expanding oscillation! 3. The aspects of accuracy and simulation time should be simultaneously considered in choosing a proper "t. References Dorn, W. S. and D. D. McCracken. 1972. Numerical Methods with Fortran IV Case Studies. Wiley, New York. Press, W. H., W.A. Teukolsky, W. T. Vetterling, and B. P. Flannery. 1992. Numerical Recipes in C: The Art of Scientific Computing. 2nd Ed. Cambridge University Press, Cambridge. 994 pp. Lecture Notes for Ecological Modeling --- Jianguo (Jingle) Wu  Page  PAGE 2 lsc 405 - Simulation Modeling Lecture Notes J. 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"%'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQTUXZ[\]^_`abcdefghilorstuvy{|}~LJ>iP# Y(x)=Y m +Y m (x-x m )+Y m (x-x m ) 2 2!+Y m (x-x mE dxpr  "  currentpoint ",Times .+Y)()x)), Symbol)=) Y +m (2+) Y +m( B' +()x)-) x +m (j))+) Y +m( ')' ( ()x)-) x +m ( ) (2 (2)!",(+) Y +m( ')')' ( ()x) -) x +m ( ) (3 (3)!",(+)......)+) Y +m( -()n)) ( 9()x) -) x +m ( \) (an (Kn)!"9,(i+) Y +x( z()n)+)1)) ( ()x)-) x +m ( ) (n)+)1 (()n) +)1))!"4/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 14464 div 1024 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 4278 466 moveto 1439 0 rlineto stroke 6611 466 moveto 1426 0 rlineto stroke 9968 466 moveto 1439 0 rlineto stroke 12665 466 moveto 1685 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (Y) -16 565 sh (x) 383 565 sh (Y) 1093 565 sh (Y) 1839 565 sh (x) 2375 565 sh (x) 2951 565 sh (Y) 3826 565 sh (x) 4449 324 sh (x) 5025 324 sh (Y) 6096 565 sh (x) 6782 324 sh (x) 7358 324 sh (Y) 9335 565 sh (x) 10139 324 sh (x) 10715 324 sh (n) 10556 858 sh (Y) 11786 565 sh (x) 12836 324 sh (x) 13412 324 sh (n) 13109 858 sh 224 ns (m) 1272 661 sh (m) 2018 661 sh (m) 3131 661 sh (m) 4005 661 sh (m) 5205 420 sh (m) 6275 661 sh (m) 7538 420 sh (m) 9514 661 sh (n) 9680 393 sh (m) 10895 420 sh (n) 11238 152 sh (n) 12131 393 sh (m) 13592 420 sh (n) 13935 152 sh 384 /Times-Roman f1 (\() 226 565 sh (\)) 574 565 sh (\() 2218 565 sh (\)) 3338 565 sh (\() 4292 324 sh (\)) 5412 324 sh (!) 5034 858 sh (\() 6625 324 sh (\)) 7745 324 sh (!) 7350 858 sh (......) 8406 565 sh (\() 9982 324 sh (\)) 11102 324 sh (!) 10725 858 sh (\() 12679 324 sh (\)) 13799 324 sh (\() 12971 858 sh (\)!) 13812 858 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 224 /MT_Times-Roman f1 (\251) 2078 393 sh (\251) 4065 393 sh (\251) 4128 393 sh (\251) 6335 393 sh (\251) 6398 393 sh (\251) 6461 393 sh (\() 9587 393 sh (\)) 9809 393 sh (\() 12038 393 sh (\)) 12506 393 sh 384 /Symbol f1 (=) 791 565 sh (+) 1553 565 sh (-) 2639 565 sh (+) 3540 565 sh (-) 4713 324 sh (+) 5810 565 sh (-) 7046 324 sh (+) 8130 565 sh (+) 9049 565 sh (-) 10403 324 sh (+) 11500 565 sh (-) 13100 324 sh (+) 13382 858 sh 224 ns (+) 12266 393 sh (+) 14070 152 sh 224 /MT_Times-Roman f1 (2) 5547 152 sh (3) 7877 152 sh (1) 12394 393 sh (1) 14198 152 sh 384 ns (2) 4863 858 sh (3) 7196 858 sh (1) 13642 858 sh r/f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 224 /Symbol f2 (x) 11953 680 sh end MTsave restore dMATHPT Y(x)=Y m +Y m (x-x m )+Y m (x-x m ) 2 2!+Y m (x-x m ) 3 3!+......+Y m(n) (x-x m ) n n!+Y x(n+1) (x-x m ) n+1 (n+1)!BsObjInfo Equation Native _922825730 FvMvMOle   ) 3 3!+......+Y m(n) (x-x m ) n n!+Y x(n+1) (x-x m ) n+1 (n+1)!LL,,7dxpr  7"7PIC   LPICT ,ObjInfo  Equation Native !Q currentpoint ",Times .+ Y +m, Symbol ( =) Y)()x +m ( 2)/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1760 div 480 3 -1 roll exch div scale currentpoint translate 64 -1098 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (Y) -16 1418 sh (Y) 761 1418 sh (x) 1160 1418 sh 224 ns (m) 163 1514 sh (m) 1340 1514 sh 384 /Symbol f1 (=) 459 1418 sh 384 /Times-Roman f1 (\() 1003 1418 sh (\)) 1547 1418 sh end MTsave restore dAMATH5 Y m =Y(x m ) 15 Y m =Y(x m )LM XhT Rdxpr  R"R currentpoint ",Times .+ Y +_922825989FvMvMOle #PIC $LPICT & m(' +,)Y +m(')' +,)Y +m((')')' +,)...,) Y +m(G()n))/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2624 div 544 3 -1 roll exch div scale currentpoint translate 64 -1688 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (Y) -16 2072 sh (Y) 482 2072 sh (Y) 1001 2072 sh (Y) 1986 2072 sh 224 ns (m) 163 2168 sh (m) 661 2168 sh (m) 1180 2168 sh (m) 2165 2168 sh (n) 2331 1900 sh f/mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 224 /MT_Times-Roman f1 (\251) 223 1900 sh (\251) 721 1900 sh (\251) 784 1900 sh (\251) 1240 1900 sh (\251) 1303 1900 sh (\251) 1366 1900 sh (\() 2238 1900 sh (\)) 2460 1900 sh 384 ns (,) 357 2072 sh (,) 876 2072 sh (,) 1458 2072 sh (...,) 1573 2072 sh end MTsave restore dMATH| Y m ,Y m ,Y m ,...,Y m(n)mB| Y m ,Y m ,Y m ,...,Y m(n)L*4@ObjInfoREquation Native S_922826083 RFvMvMOle VPIC WLPICT Y#ObjInfojEquation Native kA#4dxpr  4"4 currentpoint ",Times .+ x +m, Symbol ( ) x) ) x//MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1664 div 512 3 -1 roll exch div scale currentpoint translate 64 -2344 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (x) 11 2696 sh (x) 1389 2696 sh 224 ns (m) 191 2792 sh 384 /Symbol f1 (\243) 480 2696 sh (\243) 1065 2696 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (x) 769 2696 sh end MTsave restore d1MATH% x m xxts% x m xxL|%"@X_922794401FvMvMOle mPIC  nLPICT V##dxpr  " currentpoint ",Times .+Y +t, Symbol)+)D)t (=) Y +t (-+) Y +t( >' +()D)t)))+) Y +t( h')' ( o()D)t)) (2 (w2)!"n(+) Y +t( ')')' ( ()D)t)) (3 (3)!"(+) ......)+) Y +t( n ( ()D)t)) ( n (n)!":/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8704 div 960 3 -1 roll exch div scale currentpoint translate 64 -1109 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 3478 1618 moveto 827 0 rlineto stroke 5199 1618 moveto 814 0 rlineto stroke 7769 1618 moveto 827 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (Y) -16 1717 sh (Y) 1045 1717 sh (Y) 1693 1717 sh (t) 2409 1717 sh (Y) 3026 1717 sh (t) 3871 1476 sh (Y) 4684 1717 sh (t) 5592 1476 sh (Y) 7311 1717 sh (t) 8162 1476 sh (n) 8051 2010 sh 224 ns (t) 157 1815 sh (t) 539 1815 sh (t) 1218 1813 sh (t) 1866 1813 sh (t) 3199 1813 sh (t) 4857 1813 sh (t) 7484 1813 sh (n) 7568 1545 sh (n) 8427 1304 sh 224 /Symbol f1 (+) 251 1815 sh (D) 402 1815 sh 384 ns (=) 743 1717 sh (+) 1407 1717 sh (D) 2174 1717 sh (+) 2740 1717 sh (D) 3636 1476 sh (+) 4398 1717 sh (D) 5357 1476 sh (+) 6106 1717 sh (+) 7025 1717 sh (D) 7927 1476 sh ;/mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 224 /MT_Times-Roman f1 (\251) 1932 1545 sh (\251) 3265 1545 sh (\251) 3328 1545 sh (\251) 4923 1545 sh (\251) 4986 1545 sh (\251) 5049 1545 sh 384 ns (\() 2030 1717 sh (\)) 2538 1717 sh (\() 3492 1476 sh (\)) 4000 1476 sh (!) 3928 2010 sh (\() 5213 1476 sh (\)) 5721 1476 sh (!) 5632 2010 sh (......) 6382 1717 sh (\() 7783 1476 sh (\)) 8291 1476 sh (!) 8220 2010 sh 224 /MT_Times-Roman f1 (2) 4135 1304 sh (3) 5853 1304 sh 384 ns (2) 3757 2010 sh (3) 5478 2010 sh end MTsave restore d?MATH34 Y t+Dt =Y t +Y t (Dt)+Y t (Dt) 2 2!+Y t (Dt) 3 3!+......+Y tn (Dt) n n!reObjInfo!pEquation Native qO_922794897($FvMvMOle w3 Y t+Dt =Y t +Y t (Dt)+Y t (Dt) 2 2!+Y t (Dt) 3 3!+......+Y tn (Dt) n n!PIC #&xLPICT zj ObjInfo%'Equation Native LXpT jldxpr  l"l currentpoint ",Times .+ Y +t, Symbol)+)D)t ( =) Y +t ( -+) Y +t(>' +()D)t)))+) e +T(f1g/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3456 div 544 3 -1 roll exch div scale currentpoint translate 64 -3358 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (Y) -16 3742 sh (Y) 1045 3742 sh (Y) 1693 3742 sh (t) 2409 3742 sh 224 ns (t) 157 3840 sh (t) 539 3840 sh (t) 1218 3838 sh (t) 1866 3838 sh (T) 3216 3838 sh 224 /Symbol f1 (+) 251 3840 sh (D) 402 3840 sh 384 ns (=) 743 3742 sh (+) 1407 3742 sh (D) 2174 3742 sh (+) 2740 3742 sh G/mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 224 /MT_Times-Roman f1 (\251) 1932 3570 sh 384 ns (\() 2030 3742 sh (\)) 2538 3742 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) 3026 3742 sh 224 /MT_Times-Roman f1 (1) 3219 3570 sh end MTsave restore dzMATHnS Y t+Dt =Y t +Y t (Dt)+e T1enn Y t+Dt =Y t +Y t (Dt)+e T1LE4@<dxpr  <"<_922794924*FvMvMOle PIC ),LPICT  currentpoint ", Symbol .+ e,Times +T(1 + @) K)()D)t)) (62/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1920 div 512 3 -1 roll exch div scale currentpoint translate 64 -2706 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) -16 3090 sh 224 /Times-Italic f1 (T) 174 3186 sh 384 ns (K) 771 3090 sh (t) 1431 3090 sh 224 /Times-Roman f1 (1) 177 2918 sh (2) 1695 2918 sh 384 /Symbol f1 (@) 448 3090 sh (D) 1196 3090 sh 384 /Times-Roman f1 (\() 1052 3090 sh (\)) 1560 3090 sh end MTsave restore dGMATH; e T1 @K(Dt) 2nt; e T1 @K(Dt) 2L4@ObjInfo+-Equation Native W_922795069"F0FvMvMOle PIC /2LPICT ObjInfo13Equation Native 8 dxpr   " currentpoint ", Symbol .+ e,Times +T(1/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 512 3 -1 roll exch div scale currentpoint translate 64 -2706 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) -16 3090 sh 224 /Times-Italic f1 (T) 174 3186 sh 224 /Times-Roman f1 (1) 177 2918 sh end MTsave restore d(MATH e T1 e T1L " X Odxpr  " currentpoint ",Times .+Y +_9227951136FvMvMOle PIC 58LPICT O       !"#$%&'()*+,-./2589:;<?ABCDEFGHIJKLMNOPQRSTWZ\]^_`abcdefghijklmnopqrstuvwxyz{|}~t, Symbol)+)D)t (=) Y +t (-+) Y +t( >' +()D)t)))+) Y +t( h')' ( o()D)t)) (2 (w2)!"n(+) e +T( 2/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5120 div 960 3 -1 roll exch div scale currentpoint translate 64 -533 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 3478 1042 moveto 827 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (Y) -16 1141 sh (Y) 1045 1141 sh (Y) 1693 1141 sh (t) 2409 1141 sh (Y) 3026 1141 sh (t) 3871 900 sh 224 ns (t) 157 1239 sh (t) 539 1239 sh (t) 1218 1237 sh (t) 1866 1237 sh (t) 3199 1237 sh (T) 4874 1237 sh 224 /Symbol f1 (+) 251 1239 sh (D) 402 1239 sh 384 ns (=) 743 1141 sh (+) 1407 1141 sh (D) 2174 1141 sh (+) 2740 1141 sh (D) 3636 900 sh (+) 4398 1141 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 224 /MT_Times-Roman f1 (\251) 1932 969 sh (\251) 3265 969 sh (\251) 3328 969 sh 384 ns (\() 2030 1141 sh (\)) 2538 1141 sh (\() 3492 900 sh (\)) 4000 900 sh (!) 3928 1434 sh 224 /MT_Times-Roman f1 (2) 4135 728 sh (2) 4895 969 sh 384 ns (2) 3757 1434 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) 4684 1141 sh end MTsave restore dMATH Y t+Dt =Y t +Y t (Dt)+Y t (Dt) 2 2!+e T2le Y t+Dt =Y t +Y t (Dt)+Y t (Dt) 2 2!+e ObjInfo79Equation Native _9227952224@<FvMvMOle T2L!4@;dxpr  ;"; currentpoint ", Symbol .+ e,Times +T(2 +@) K)()D)t)) (63/MTsave save def 40 dict begin currentpPIC ;>LPICT ObjInfo=?0Equation Native 1Woint 3 -1 roll sub neg 3 1 roll sub 1888 div 512 3 -1 roll exch div scale currentpoint translate 64 -2706 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) -16 3090 sh 224 /Times-Italic f1 (T) 174 3186 sh 384 ns (K) 771 3090 sh (t) 1431 3090 sh 224 /Times-Roman f1 (2) 195 2918 sh (3) 1692 2918 sh 384 /Symbol f1 (@) 448 3090 sh (D) 1196 3090 sh 384 /Times-Roman f1 (\() 1052 3090 sh (\)) 1560 3090 sh end MTsave restore dGMATH; e T2 @K(Dt) 30 ; e T2 @K(Dt) 3L("XI Y t+Dt =_922795374BFvMvMOle 3PIC AD4LPICT &dxpr  &"& currentpoint ",Times .+Y +t, Symbol)+)D)t (=) Y +t (-+) Y +t( >' +()D)t)))+) Y +t( h')' ( o()D)t)) (2 (w2)!"n(+) Y +t( ')')' ( ()D)t)) (3 (3)!"(+) ......)+) Y +t( k ( ()D)t)) ( k (k)!"(+) e +T( !km/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 9408 div 960 3 -1 roll exch div scale currentpoint translate 64 -1629 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 3478 2138 moveto 827 0 rlineto stroke 5199 2138 moveto 814 0 rlineto stroke 7769 2138 moveto 827 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (Y) -16 2237 sh (Y) 1045 2237 sh (Y) 1693 2237 sh (t) 2409 2237 sh (Y) 3026 2237 sh (t) 3871 1996 sh (Y) 4684 2237 sh (t) 5592 1996 sh (Y) 7311 2237 sh (t) 8162 1996 sh (k) 8054 2530 sh 224 ns (t) 157 2335 sh (t) 539 2335 sh (t) 1218 2333 sh (t) 1866 2333 sh (t) 3199 2333 sh (t) 4857 2333 sh (t) 7484 2333 sh (k) 7570 2065 sh (k) 8429 1824 sh (T) 9165 2333 sh (k) 9189 2065 sh 224 /Symbol f1 (+) 251 2335 sh (D) 402 2335 sh 384 ns (=) 743 2237 sh (+) 1407 2237 sh (D) 2174 2237 sh (+) 2740 2237 sh (D) 3636 1996 sh (+) 4398 2237 sh (D) 5357 1996 sh (+) 6106 2237 sh (+) 7025 2237 sh (D) 7927 1996 sh (+) 8689 2237 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 224 /MT_Times-Roman f1 (\251) 1932 2065 sh (\251) 3265 2065 sh (\251) 3328 2065 sh (\251) 4923 2065 sh (\251) 4986 2065 sh (\251) 5049 2065 sh 384 ns (\() 2030 2237 sh (\)) 2538 2237 sh (\() 3492 1996 sh (\)) 4000 1996 sh (!) 3928 2530 sh (\() 5213 1996 sh (\)) 5721 1996 sh (!) 5632 2530 sh (......) 6382 2237 sh (\() 7783 1996 sh (\)) 8291 1996 sh (!) 8219 2530 sh 224 /MT_Times-Roman f1 (2) 4135 1824 sh (3) 5853 1824 sh 384 ns (2) 3757 2530 sh (3) 5478 2530 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) 8975 2237 sh end MTsave restore dUMATHI8& Y t+Dt =Y t +Y t (Dt)+Y t (Dt) 2 2!+Y t (Dt) 3 3!+......+Y tk (Dt) k k!+e TktiObjInfoCE6Equation Native 7e_922795432:HFvMvMOle =Y t +Y t (Dt)+Y t (Dt) 2 2!+Y t (Dt) 3 3!+......+Y tk (Dt) k k!+e TkL< 4<@PIC GJ>LPICT @ObjInfoIKUEquation Native V]Cdxpr  C"C currentpoint ", Symbol .+ e,Times +T(k +@) K)()D)t)) (7k)+)1/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2144 div 512 3 -1 roll exch div scale currentpoint translate 64 -2708 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) -16 3092 sh 224 /Times-Italic f1 (T) 174 3188 sh (k) 198 2920 sh (k) 1698 2920 sh 384 ns (K) 771 3092 sh (t) 1431 3092 sh 384 /Symbol f1 (@) 448 3092 sh (D) 1196 3092 sh 224 ns (+) 1831 2920 sh 384 /Times-Roman f1 (\() 1052 3092 sh (\)) 1560 3092 sh 224 /Times-Roman f1 (1) 1959 2920 sh end MTsave restore dMMATHA  e Tk @K(Dt) k+1\)A e Tk @K(Dt) k+1_922800818NFvMvMOle XPIC MPYLPICT [ L XT Vdxpr  V"V currentpoint ",Times .+ Y +t, Symbol)+)D)t ( =) Y +t ( -+) Y +t(>' +()D)t))E/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2752 div 544 3 -1 roll exch div scale currentpoint translate 64 -3360 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (Y) -16 3744 sh (Y) 1045 3744 sh (Y) 1693 3744 sh (t) 2409 3744 sh 224 ns (t) 157 3842 sh (t) 539 3842 sh (t) 1218 3840 sh (t) 1866 3840 sh 224 /Symbol f1 (+) 251 3842 sh (D) 402 3842 sh 384 ns (=) 743 3744 sh (+) 1407 3744 sh (D) 2174 3744 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 224 /MT_Times-Roman f1 (\251) 1932 3572 sh 384 ns (\() 2030 3744 sh (\)) 2538 3744 sh end MTsave restore ddMATHX Y t+Dt =Y t +Y t (Dt)X Y t+Dt =ObjInfoOQEquation Native t_922828480X^TFvMvMOle Y t +Y t (Dt)L 0dxpr  " currentpoint ",Times .+Y, Symbol) =) F)()t))) and ( DdY+dt" C(V=) f)()t),)Y)PIC SVLPICT ObjInfoUWEquation Native p)/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4064 div 896 3 -1 roll exch div scale currentpoint translate 64 -986 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 2101 1431 moveto 481 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (Y) -16 1530 sh (F) 646 1530 sh (t) 1023 1530 sh (dY) 2128 1289 sh (f) 3064 1530 sh (t) 3367 1530 sh (Y) 3608 1530 sh 384 /Symbol f1 (=) 322 1530 sh (=) 2690 1530 sh 384 /Times-Roman f1 (\() 891 1530 sh (\)) 1152 1530 sh (\() 3235 1530 sh (,) 3483 1530 sh (\)) 3857 1530 sh 384 /Times-Roman f1 ( and ) 1291 1530 sh (dt) 2187 1823 sh end MTsave restore d`MATHT Y=F(t) and dYdt=f(t,Y)pT Y=F(t) and dYdt=f(t,Y)L8L0US_922828188ZFvMvMOle PIC Y\LPICT dxpr  " currentpoint ",Times .+Y +t, Symbol)+)D)t (=) Y +t (-+( 81*2" 7(Ef)()t),)Y +t (_))+) f)()t)+) D)t),)Y +t)+)D)t () (?[)s]/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5856 div 896 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1709 403 moveto 235 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (Y) -16 502 sh (Y) 1045 502 sh (f) 2175 502 sh (t) 2478 502 sh (Y) 2719 502 sh (f) 3567 502 sh (t) 3870 502 sh (t) 4608 502 sh (Y) 4849 502 sh 224 ns (t) 157 600 sh (t) 539 600 sh (t) 1218 598 sh (t) 2892 598 sh (t) 5022 600 sh (t) 5404 600 sh 224 /Symbol f1 (+) 251 600 sh (D) 402 600 sh (+) 5116 600 sh (D) 5267 600 sh 384 ns (=) 743 502 sh (+) 1407 502 sh (+) 3209 502 sh (+) 4073 502 sh (D) 4373 502 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1455 /Symbol f3 ([) 1975 548 sh (]) 5646 548 sh 384 /Times-Roman f1 (1) 1730 261 sh (2) 1730 795 sh 384 /Times-Roman f1 (\() 2346 502 sh (,) 2594 502 sh (\)) 3007 502 sh (\() 3738 502 sh (,) 4724 502 sh (\)) 5519 502 sh end MTsave restore dMATH+ Y t+Dt =Y t +12f(t,Y t )+f(t+Dt,Y t+Dt )[]` Y t+Dt =Y t +12f(t,Y t )+f(t+Dt,Y t+Dt )[]ObjInfo[]Equation Native _922829069d`FvMvMOle PIC _bLPICT ObjInfoacEquation Native L 0YSdxpr  " currentpoint ",Times .+Y +t, Symbol)+)D)t (=) Y +t (-+( 81*6" 7(@()F +1 (P+) F +2 (f+) F +3 (|+) F +4 ()/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4800 div 896 3 -1 roll exch div scale currentpoint translate 64 -2593 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1709 3038 moveto 230 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (Y) -16 3137 sh (Y) 1045 3137 sh (F) 2137 3137 sh (F) 2818 3137 sh (F) 3535 3137 sh (F) 4239 3137 sh 224 ns (t) 157 3235 sh (t) 539 3235 sh (t) 1218 3233 sh 224 /Symbol f1 (+) 251 3235 sh (D) 402 3235 sh 384 ns (=) 743 3137 sh (+) 1407 3137 sh (+) 2510 3137 sh (+) 3227 3137 sh (+) 3931 3137 sh 384 /Times-Roman f1 (1) 1728 2896 sh (6) 1728 3430 sh 224 ns (1) 2297 3233 sh (2) 2996 3233 sh (3) 3710 3233 sh (4) 4421 3233 sh 384 /Times-Roman f1 (\() 1985 3137 sh (\)) 4577 3137 sh end MTsave restore dMATH Y t+Dt =Y t +16(F 1 +F 2 +F 3 +F 4 ) Y t+Dt =Y t +16(F 1 +F 2 +F 3 +F 4 )L) NL Endxpr4  n"n currentpoint ",Times .+ F +1, Symbol ( =) f)()t),)Y _922829116fFvMvMOle PIC ehLPICT E       !"#$%&'()*-./012456789;<=>?A+t ( 4))D)t(#F +2 (#=) f)()t)+(5D)t(,72"4 (#B,)Y +t (#Q+([1*2"[(#dF +1 (#m))D)t(CF +3 (C=) f)()t)+(<4D)t(L72"?3 (CA,)Y +t (CQ+(<[1*2"?Z(CdF +2 (Cn))D)t(cF +4 (c=) f)()t)+) D)t),)Y +t (cN+(\X1*2"`X(caF +3 (ck))D)t/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4064 div 3520 3 -1 roll exch div scale currentpoint translate 64 -3178 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1612 4211 moveto 407 0 rlineto stroke 2848 4211 moveto 235 0 rlineto stroke 1599 5241 moveto 407 0 rlineto stroke 2835 5241 moveto 235 0 rlineto stroke 2753 6268 moveto 235 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (F) 6 3475 sh (f) 768 3475 sh (t) 1071 3475 sh (Y) 1312 3475 sh (t) 1972 3475 sh (F) 6 4310 sh (f) 804 4310 sh (t) 1107 4310 sh (t) 1877 4069 sh (Y) 2184 4310 sh (F) 3153 4310 sh (t) 3824 4310 sh (F) 6 5340 sh (f) 791 5340 sh (t) 1094 5340 sh (t) 1864 5099 sh (Y) 2171 5340 sh (F) 3140 5340 sh (t) 3847 5340 sh (F) 6 6367 sh (f) 807 6367 sh (t) 1110 6367 sh (t) 1848 6367 sh (Y) 2089 6367 sh (F) 3058 6367 sh (t) 3752 6367 sh 224 ns (t) 1485 3571 sh (t) 2357 4406 sh (t) 2344 5436 sh (t) 2262 6463 sh 224 /Times-Roman f1 (1) 166 3571 sh (2) 184 4406 sh (1) 3313 4406 sh (3) 181 5436 sh (2) 3318 5436 sh (4) 188 6463 sh (3) 3233 6463 sh 384 ns (2) 1719 4603 sh (1) 2869 4069 sh (2) 2869 4603 sh (2) 1706 5633 sh (1) 2856 5099 sh (2) 2856 5633 sh (1) 2774 6126 sh (2) 2774 6660 sh 384 /Symbol f1 (=) 394 3475 sh (D) 1737 3475 sh (=) 430 4310 sh (+) 1310 4310 sh (D) 1642 4069 sh (+) 2546 4310 sh (D) 3589 4310 sh (=) 417 5340 sh (+) 1297 5340 sh (D) 1629 5099 sh (+) 2533 5340 sh (D) 3612 5340 sh (=) 433 6367 sh (+) 1313 6367 sh (D) 1613 6367 sh (+) 2451 6367 sh (D) 3517 6367 sh 384 /Times-Roman f1 (\() 939 3475 sh (,) 1187 3475 sh (\)) 1600 3475 sh (\() 975 4310 sh (,) 2059 4310 sh (\)) 3452 4310 sh (\() 962 5340 sh (,) 2046 5340 sh (\)) 3475 5340 sh (\() 978 6367 sh (,) 1964 6367 sh (\)) 3380 6367 sh end MTsave restore dMATH@  F 1 =f(t,Y t 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