(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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It is useful to first define the function f:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f[t_, y_] = 1 - t + 4 y\)], "Input"], Cell[BoxData[ \(1 - t + 4\ y\)], "Output"] }, Open ]], Cell["The step size will be h.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(h = 0.1\)], "Input"], Cell[BoxData[ StyleBox["0.1`", StyleBoxAutoDelete->True, PrintPrecision->1]], "Output"] }, Open ]], Cell["Next we input the initial conditions.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(t = 0\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(y = 1\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell["\<\ Since the same thing is done at every step, we use a Do statement. \ The Do statement has the form Do[{list of statements},{i,max}]. The effect is \ that the list of statements is carried out max times, with the counter i \ starting from 1 and going to max in increments of 1. The statement Print[{t,y}] prints out t and y at each step.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Do[{y = y + h*f[t, y], t = t + h, Print[{t, y}]}, {i, 10}]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ StyleBox["0.1`", StyleBoxAutoDelete->True, PrintPrecision->1], ",", "1.5`"}], "}"}]], "Print"], Cell[BoxData[ \({0.2`, 2.18999999999999994`}\)], "Print"], Cell[BoxData[ \({0.300000000000000044`, 3.1459999999999999`}\)], "Print"], Cell[BoxData[ \({0.4`, 4.47440000000000015`}\)], "Print"], Cell[BoxData[ \({0.5`, 6.32416000000000089`}\)], "Print"], Cell[BoxData[ \({0.6`, 8.90382400000000195`}\)], "Print"], Cell[BoxData[ \({0.7`, 12.505353600000002`}\)], "Print"], Cell[BoxData[ \({0.799999999999999911`, 17.537495040000004`}\)], "Print"], Cell[BoxData[ \({0.9`, 24.5724930560000043`}\)], "Print"], Cell[BoxData[ \({0.999999999999999822`, 34.4114902784000076`}\)], "Print"] }, Open ]], Cell["\<\ We repeat the same calculation with some smaller step sizes. If \ there are many steps, you may not want to print the results at every step. \ The way to take care of this is an If statement. If you write \ If[condition,action], then the action is carried out only if the condition is \ true. This can be expanded into If[condition,action1,action2], which carries \ out action1 if the condition is true and action2 if it is false. Below, the \ condition is that i is divisible by 2 (i.e. i/2=Floor[i/2]), and the action \ is to print t and y.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(h = 0.05\)], "Input"], Cell[BoxData[ StyleBox["0.05`", StyleBoxAutoDelete->True, PrintPrecision->1]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(t = 0\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(y = 1\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Do[{y = y + h*f[t, y], t = t + h, If[i/2 == Floor[i/2], Print[{t, y}]]}, {i, 20}]\)], "Input"], Cell[BoxData[ \({0.1`, 1.54750000000000014`}\)], "Print"], Cell[BoxData[ \({0.2`, 2.32490000000000041`}\)], "Print"], Cell[BoxData[ \({0.3`, 3.43335600000000074`}\)], "Print"], Cell[BoxData[ \({0.399999999999999964`, 5.01853264000000098`}\)], "Print"], Cell[BoxData[ \({0.499999999999999911`, 7.29018700160000232`}\)], "Print"], Cell[BoxData[ \({0.6`, 10.550369282304004`}\)], "Print"], Cell[BoxData[ \({0.700000000000000088`, 15.2340317665177664`}\)], "Print"], Cell[BoxData[ \({0.800000000000000177`, 21.9675057437855825`}\)], "Print"], Cell[BoxData[ \({0.900000000000000177`, 31.6527082710512441`}\)], "Print"], Cell[BoxData[ \({1.00000000000000022`, 45.588399910313786`}\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(h = 0.025\)], "Input"], Cell[BoxData[ StyleBox["0.025`", StyleBoxAutoDelete->True, PrintPrecision->2]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(t = 0\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(y = 1\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Do[{y = y + h*f[t, y], t = t + h, If[i/4 == Floor[i/4], Print[{t, y}]]}, {i, 40}]\)], "Input"], Cell[BoxData[ \({0.1`, 1.57611875000000019`}\)], "Print"], Cell[BoxData[ \({0.199999999999999982`, 2.40801171187499996`}\)], "Print"], Cell[BoxData[ \({0.3`, 3.61438369735618714`}\)], "Print"], Cell[BoxData[ \({0.400000000000000088`, 5.36903042129919327`}\)], "Print"], Cell[BoxData[ \({0.500000000000000088`, 7.92640618982414935`}\)], "Print"], Cell[BoxData[ \({0.600000000000000177`, 11.6590575525215367`}\)], "Print"], Cell[BoxData[ \({0.700000000000000266`, 17.1124299126467827`}\)], "Print"], Cell[BoxData[ \({0.800000000000000355`, 25.0851098851061493`}\)], "Print"], Cell[BoxData[ \({0.900000000000000532`, 36.746308132783918`}\)], "Print"], Cell[BoxData[ \({1.00000000000000044`, 53.8078659872089293`}\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(h = 0.01\)], "Input"], Cell[BoxData[ StyleBox["0.01`", StyleBoxAutoDelete->True, PrintPrecision->1]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(y = 1\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(t = 0\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Do[{y = y + h*f[t, y], t = t + h, If[i/10 == Floor[i/10], Print[{t, y}]]}, {i, 100}]\)], "Input"], Cell[BoxData[ \({0.0999999999999999822`, 1.59529008834053342`}\)], "Print"], Cell[BoxData[ \({0.200000000000000044`, 2.46445873235218559`}\)], "Print"], Cell[BoxData[ \({0.300000000000000088`, 3.73903454315770122`}\)], "Print"], Cell[BoxData[ \({0.400000000000000177`, 5.61371199567477052`}\)], "Print"], Cell[BoxData[ \({0.500000000000000177`, 8.37668647370548669`}\)], "Print"], Cell[BoxData[ \({0.600000000000000355`, 12.4545575470627389`}\)], "Print"], Cell[BoxData[ \({0.700000000000000355`, 18.4787967909106818`}\)], "Print"], Cell[BoxData[ \({0.800000000000000532`, 27.3841363954744476`}\)], "Print"], Cell[BoxData[ \({0.900000000000000532`, 40.5542083432639977`}\)], "Print"], Cell[BoxData[ \({1.00000000000000066`, 60.037125968819911`}\)], "Print"] }, Open ]], Cell["For comparison, here is the exact solution.", "Text"], Cell[BoxData[ \(Clear[y, t]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(sol[t_] = y[t] /. DSolve[{\(y'\)[t] == 1 - t + 4 y[t], y[0] == 1}, y[t], t]\)], "Input"], Cell[BoxData[ \({1\/16\ \((\(-3\) + 19\ E\^\(4\ t\) + 4\ t)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Do[Print[{i*0.1, sol[i*0.1]}], {i, 10}]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ StyleBox["0.1`", StyleBoxAutoDelete->True, PrintPrecision->1], ",", \({1.60904182844900844`}\)}], "}"}]], "Print"], Cell[BoxData[ \({0.2`, {2.50532985258480556`}}\)], "Print"], Cell[BoxData[ \({0.300000000000000044`, {3.83013884574965111`}}\)], "Print"], Cell[BoxData[ \({0.4`, {5.7942260039691984`}}\)], "Print"], Cell[BoxData[ \({0.5`, {8.7120041174801468`}}\)], "Print"], Cell[BoxData[ \({0.600000000000000088`, {13.0525219520119062`}}\)], "Print"], Cell[BoxData[ \({0.700000000000000088`, {19.5155180406777546`}}\)], "Print"], Cell[BoxData[ \({0.8`, {29.1448796090673578`}}\)], "Print"], Cell[BoxData[ \({0.9`, {43.4979034018676102`}}\)], "Print"], Cell[BoxData[ \({1.`, {64.8978031643587804`}}\)], "Print"] }, Open ]], Cell["\<\ Here is the same equation solved with the implicit Euler \ method.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(y = 1\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(h = 0.1\)], "Input"], Cell[BoxData[ StyleBox["0.1`", StyleBoxAutoDelete->True, PrintPrecision->1]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(t = 0\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell["\<\ Note that for the implicit Euler method t is incremented \ first:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Do[{t = t + h, yold = y, y = First[s /. Solve[s == yold + h*f[t, s], s]], Print[{t, y}]}, {i, 10}]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ StyleBox["0.1`", StyleBoxAutoDelete->True, PrintPrecision->1], ",", "1.81666666666666678`"}], "}"}]], "Print"], Cell[BoxData[ \({0.2`, 3.16111111111111187`}\)], "Print"], Cell[BoxData[ \({0.300000000000000044`, 5.38518518518518618`}\)], "Print"], Cell[BoxData[ \({0.4`, 9.07530864197530995`}\)], "Print"], Cell[BoxData[ \({0.5`, 15.2088477366255192`}\)], "Print"], Cell[BoxData[ \({0.6`, 25.4147462277091973`}\)], "Print"], Cell[BoxData[ \({0.7`, 42.4079103795153322`}\)], "Print"], Cell[BoxData[ \({0.799999999999999911`, 70.7131839658589012`}\)], "Print"], Cell[BoxData[ \({0.9`, 117.871973276431529`}\)], "Print"], Cell[BoxData[ \({0.999999999999999822`, 196.453288794052518`}\)], "Print"] }, Open ]], Cell["\<\ The First is necessary to get rid of extraneous parentheses which \ would mess up the Solve command at the next step. If f is nonlinear, FindRoot \ must be used instead of Solve. The starting value would simply be the solution from the previous step. Here is an \ example. \ \>", "Text"], Cell[BoxData[ \(Clear[t, y]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(f[t_, y_] = t^2 + Sin[y]\)], "Input"], Cell[BoxData[ \(t\^2 + Sin[y]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(t = 0\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(y = 1\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Do[{t = t + h, yold = y, y = s /. FindRoot[s == yold + h*f[t, s], {s, yold}], Print[{t, y}]}, { i, 10}]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ StyleBox["0.1`", StyleBoxAutoDelete->True, PrintPrecision->1], ",", "1.08964633530037712`"}], "}"}]], "Print"], Cell[BoxData[ \({0.2`, 1.18634684892102647`}\)], "Print"], Cell[BoxData[ \({0.300000000000000044`, 1.29147103067437507`}\)], "Print"], Cell[BoxData[ \({0.4`, 1.40611816865630867`}\)], "Print"], Cell[BoxData[ \({0.5`, 1.53103917846012134`}\)], "Print"], Cell[BoxData[ \({0.6`, 1.66658083773446419`}\)], "Print"], Cell[BoxData[ \({0.7`, 1.81266998054022732`}\)], "Print"], Cell[BoxData[ \({0.799999999999999911`, 1.96885169313594143`}\)], "Print"], Cell[BoxData[ \({0.9`, 2.13438603758345202`}\)], "Print"], Cell[BoxData[ \({0.999999999999999822`, 2.30839467143454069`}\)], "Print"] }, Open ]], Cell["\<\ We do not need the First, because FindRoot does not insert an \ extraneous bracket, while Solve does. The same ideas work for systems, all we have to do is make y and f into \ vectors. Here is an example for the system y1'=y1+y2, y2'=y1^2.\ \>", "Text"], Cell[BoxData[ \(Clear[y]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(f[t_, y_] = {y[\([1]\)] + y[\([2]\)], y[\([1]\)]^2}\)], "Input"], Cell[BoxData[ \(Part::"partd" \( : \ \) "Part specification \!\(y \\[LeftDoubleBracket] 1 \ \\[RightDoubleBracket]\) is longer than depth of object."\)], "Message"], Cell[BoxData[ \(Part::"partd" \( : \ \) "Part specification \!\(y \\[LeftDoubleBracket] 2 \ \\[RightDoubleBracket]\) is longer than depth of object."\)], "Message"], Cell[BoxData[ \(Part::"partd" \( : \ \) "Part specification \!\(y \\[LeftDoubleBracket] 1 \ \\[RightDoubleBracket]\) is longer than depth of object."\)], "Message"], Cell[BoxData[ \(General::"stop" \( : \ \) "Further output of \!\(Part :: \"partd\"\) will be suppressed during \ this calculation."\)], "Message"], Cell[BoxData[ \({y\[LeftDoubleBracket]1\[RightDoubleBracket] + y\[LeftDoubleBracket]2\[RightDoubleBracket], y\[LeftDoubleBracket]1\[RightDoubleBracket]\^2}\)], "Output"] }, Open ]], Cell["Ignore the error message, things will work just fine.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(t = 0\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(y = {1, 0}\)], "Input"], Cell[BoxData[ \({1, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(h = 0.1\)], "Input"], Cell[BoxData[ StyleBox["0.1`", StyleBoxAutoDelete->True, PrintPrecision->1]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Do[{y = y + h*f[t, y], t = t + h, Print[{t, y}]}, {i, 10}]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ StyleBox["0.1`", StyleBoxAutoDelete->True, PrintPrecision->1], ",", RowBox[{"{", RowBox[{"1.10000000000000008`", ",", StyleBox["0.1`", StyleBoxAutoDelete->True, PrintPrecision->1]}], "}"}]}], "}"}]], "Print"], Cell[BoxData[ \({0.2`, {1.22000000000000019`, 0.22100000000000004`}}\)], "Print"], Cell[BoxData[ \({0.300000000000000044`, {1.36410000000000031`, 0.369840000000000035`}} \)], "Print"], Cell[BoxData[ \({0.4`, {1.5374940000000004`, 0.555916881000000095`}}\)], "Print"], Cell[BoxData[ \({0.5`, {1.74683508810000054`, 0.792305661003600203`}}\)], "Print"], Cell[BoxData[ \({0.6`, {2.00074916301036065`, 1.09744894350533384`}}\)], "Print"], Cell[BoxData[ \({0.7`, {2.31056897366193014`, 1.49774866483399975`}}\)], "Print"], Cell[BoxData[ \({0.799999999999999911`, {2.6914007375115232`, 2.03162156303891405`}} \)], "Print"], Cell[BoxData[ \({0.9`, {3.1637029675665671`, 2.75598535602667116`}}\)], "Print"], Cell[BoxData[ \({0.999999999999999822`, {3.75567179992589084`, 3.75688700272562137`}} \)], "Print"] }, Open ]] }, Open ]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 640}, {0, 460}}, WindowSize->{520, 365}, WindowMargins->{{4, Automatic}, {Automatic, 1}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, MacintoshSystemPageSetup->"\<\ 00<0004/0B`000002n88o?mooh<" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1731, 51, 77, 0, 92, "Title"], Cell[1811, 53, 172, 4, 38, "Text"], Cell[CellGroupData[{ Cell[2008, 61, 57, 1, 22, "Input"], Cell[2068, 64, 46, 1, 22, "Output"] }, Open ]], Cell[2129, 68, 40, 0, 25, "Text"], Cell[CellGroupData[{ Cell[2194, 72, 40, 1, 22, "Input"], Cell[2237, 75, 103, 3, 22, "Output"] }, Open ]], Cell[2355, 81, 53, 0, 25, "Text"], Cell[CellGroupData[{ Cell[2433, 85, 38, 1, 22, "Input"], Cell[2474, 88, 35, 1, 22, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2546, 94, 38, 1, 22, "Input"], Cell[2587, 97, 35, 1, 22, "Output"] }, Open ]], Cell[2637, 101, 360, 6, 64, "Text"], Cell[CellGroupData[{ Cell[3022, 111, 91, 1, 22, "Input"], Cell[3116, 114, 169, 5, 20, "Print"], Cell[3288, 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