(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 13.3' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 265861, 5715] NotebookOptionsPosition[ 262505, 5653] NotebookOutlinePosition[ 262943, 5670] CellTagsIndexPosition[ 262900, 5667] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[ RowBox[{"ClearAll", "[", "\"\\"", "]"}]], "Input", CellChangeTimes->{{3.92615667517485*^9, 3.926156686424818*^9}}, CellLabel->"In[1]:=",ExpressionUUID->"1e6115ea-cb4c-429c-be4f-790d4d1d31c1"], Cell[CellGroupData[{ Cell["Einf\[UDoubleDot]hrung in die Computer Algebra - 2024 - \ \[CapitalUDoubleDot]bungsblatt 4", "Chapter", CellChangeTimes->{{3.9270239720010033`*^9, 3.927024011562078*^9}, { 3.927277035400279*^9, 3.927277036245101*^9}, {3.9289036792676163`*^9, 3.928903680025075*^9}, {3.9294897222784357`*^9, 3.9294897236673803`*^9}}, TextAlignment->Center,ExpressionUUID->"d64facf9-cde0-44d8-9a21-ec4ca1aa32c4"], Cell[CellGroupData[{ Cell["\<\ 1. 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Aufgabe: Euler Methode", "Subsection", CellChangeTimes->{{3.927702464160159*^9, 3.92770246812077*^9}, { 3.9289186256528997`*^9, 3.9289186284477577`*^9}, {3.929179983074396*^9, 3.9291799839427*^9}, {3.9294312418644733`*^9, 3.929431242765588*^9}},ExpressionUUID->"43cba288-ba28-4e34-a72b-\ 8a11b306f21a"], Cell[TextData[{ " Numerische Methoden zur L\[ODoubleDot]sung von DGLs basieren darauf, \ Ableitungen als diskrete Differenzen zu approximieren. Die einfachste dieser \ Methoden ist das Euler-Verfahren. Betrachten Sie eine DGL erster Ordnung: ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ FractionBox[ RowBox[{ StyleBox["d", "TI"], StyleBox["x", "TI"], RowBox[{"(", StyleBox["t", "TI"], ")"}]}], RowBox[{ StyleBox["d", "TI"], StyleBox["t", "TI"]}]], "\[LongEqual]", StyleBox["f", "TI"], RowBox[{"(", RowBox[{ StyleBox["x", "TI"], RowBox[{"(", StyleBox["t", "TI"], ")"}], ",", StyleBox["t", "TI"]}], ")"}]}], TraditionalForm], "errors" -> {}, "input" -> "\\frac{d x(t)}{dt}= f(x(t),t)", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "bea52b4d-938e-4104-8453-6b9d6d179cc1"], " . Die Ableitung kann als Differenzquotient geschrieben werden ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ FractionBox[ RowBox[{"\[CapitalDelta]", StyleBox["x", "TI"], RowBox[{"(", StyleBox["t", "TI"], ")"}]}], RowBox[{"\[CapitalDelta]", StyleBox["t", "TI"]}]], "\[LongEqual]", StyleBox["f", "TI"], RowBox[{"(", RowBox[{ StyleBox["x", "TI"], RowBox[{"(", StyleBox["t", "TI"], ")"}], ",", StyleBox["t", "TI"]}], ")"}], " ", "\[DoubleLongRightArrow] ", StyleBox["x", "TI"], RowBox[{"(", RowBox[{ SubscriptBox[ StyleBox["t", "TI"], "1"], "+", "\[CapitalDelta]", StyleBox["t", "TI"]}], ")"}], "-", StyleBox["x", "TI"], RowBox[{"(", SubscriptBox[ StyleBox["t", "TI"], "1"], ")"}], "\[LongEqual]", "\[CapitalDelta]", StyleBox["t", "TI"], "\[CenterDot]", StyleBox["f", "TI"], RowBox[{"(", RowBox[{ StyleBox["x", "TI"], RowBox[{"(", SubscriptBox[ StyleBox["t", "TI"], "1"], ")"}], ",", SubscriptBox[ StyleBox["t", "TI"], "1"]}], ")"}]}], TraditionalForm], "errors" -> {}, "input" -> "\\frac{\\Delta x(t)}{\\Delta t}= f(x(t),t) \\quad \\implies \\quad \ x(t_1+\\Delta t)-x(t_1)=\\Delta t \\cdot f(x(t_1),t_1)", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "cdd941eb-22b2-49af-a4ea-0a2d1c54e96a"], "\nF\[UDoubleDot]r kleine Werte von ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"\[CapitalDelta]", StyleBox["t", "TI"]}], TraditionalForm], "errors" -> {}, "input" -> "\\Delta t", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "73f4a12a-3a5d-4e53-838a-8acd0be936ba"], " kann der Funktionswert ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["x", "TI"], "(", SubscriptBox[ StyleBox["t", "TI"], RowBox[{ StyleBox["n", "TI"], "+", "1"}]], ")"}], TraditionalForm], "errors" -> {}, "input" -> "x(t_{n+1})", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "d768db21-5d39-482a-94cc-d529a047eb2c"], " durch ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["x", "TI"], RowBox[{"(", SubscriptBox[ StyleBox["t", "TI"], RowBox[{ StyleBox["n", "TI"], "+", "1"}]], ")"}], "\[LongEqual]", StyleBox["x", "TI"], RowBox[{"(", StyleBox["n", "TI"], ")"}], "+", "\[CapitalDelta]", StyleBox["t", "TI"], "\[CenterDot]", StyleBox["f", "TI"], RowBox[{"(", RowBox[{ StyleBox["x", "TI"], RowBox[{"(", SubscriptBox[ StyleBox["t", "TI"], StyleBox["n", "TI"]], ")"}], ",", SubscriptBox[ StyleBox["t", "TI"], StyleBox["n", "TI"]]}], ")"}]}], TraditionalForm], "errors" -> {}, "input" -> "x(t_{n+1})= x({n})+ \\Delta t \\cdot f(x(t_{n}),t_{n})", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "b7384af2-d41f-4e57-8e56-c643dc43ed72"], " approximiert werden. Durch iteratives Anwenden dieser Relation kann die L\ \[ODoubleDot]sung f\[UDoubleDot]r ein Zeitintervall ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["t", "TI"], "\[Epsilon]", StyleBox["t", "TI"], RowBox[{"[", RowBox[{ SubscriptBox[ StyleBox["t", "TI"], StyleBox["i", "TI"]], ",", SubscriptBox[ StyleBox["t", "TI"], StyleBox["f", "TI"]]}], "]"}]}], TraditionalForm], "errors" -> {}, "input" -> "t \\epsilon t [t_i,t_f]", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "15742bfc-5bd1-4396-bbcb-854830cdde16"], " numerisch bestimmt werden, wenn die Anfangsbedingung ", Cell[BoxData[ FormBox[ RowBox[{ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["x", "TI"], RowBox[{"(", SubscriptBox[ StyleBox["t", "TI"], StyleBox["i", "TI"]], ")"}], "\[LongEqual]", SubscriptBox[ StyleBox["x", "TI"], "0"]}], TraditionalForm], "errors" -> {}, "input" -> "x(t_i)=x_0", "state" -> "Boxes"|>, "TeXAssistantTemplate"], " "}], TraditionalForm]],ExpressionUUID-> "b413aa8b-fd76-4c40-93cf-727467f78738"], " bekannt ist. \n\nBetrachten Sie nun die Ihnen bekannte DGL eines \ harmonischen Oszillators. ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["m", "TI"], OverscriptBox[ StyleBox["x", "TI"], "\:0308"], RowBox[{"(", StyleBox["t", "TI"], ")"}], "\[LongEqual]", "-", StyleBox["k", "TI"], StyleBox["x", "TI"], RowBox[{"(", StyleBox["t", "TI"], ")"}]}], TraditionalForm], "errors" -> {}, "input" -> "m \\ddot x(t)=- k x(t)", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "2a867f0d-549a-4bac-8118-115bf249d5d8"], " ist eine DGL zweiter Ordnung. Um das Euler-Verfahren anzuwenden, ist es \ praktisch, diese Gleichung als zwei gekoppelte DGLs erster Ordnung f\ \[UDoubleDot]r die Ortskoordinate und die Geschwindigkeit umzuschreiben:\n\t\ 1. Gl: ", Cell[BoxData[ FormBox[ RowBox[{ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["m", "TI"], FractionBox[ RowBox[{ StyleBox["d", "TI"], StyleBox["v", "TI"], RowBox[{"(", StyleBox["t", "TI"], ")"}]}], RowBox[{ StyleBox["d", "TI"], StyleBox["t", "TI"]}]], "\[LongEqual]", "-", StyleBox["k", "TI"], StyleBox["x", "TI"], RowBox[{"(", StyleBox["t", "TI"], ")"}]}], TraditionalForm], "errors" -> {}, "input" -> "m \\frac{ d v(t)}{dt}=- k x(t)", "state" -> "Boxes"|>, "TeXAssistantTemplate"], " "}], TraditionalForm]],ExpressionUUID-> "90c22620-3402-4450-a356-43325b8112e2"], " \n\t2. Gl: ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["v", "TI"], RowBox[{"(", StyleBox["t", "TI"], ")"}], "\[LongEqual]", FractionBox[ RowBox[{ StyleBox["d", "TI"], StyleBox["x", "TI"], RowBox[{"(", StyleBox["t", "TI"], ")"}]}], RowBox[{ StyleBox["d", "TI"], StyleBox["t", "TI"]}]]}], TraditionalForm], "errors" -> {}, "input" -> "v(t)= \\frac{ d x(t)}{dt}", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "855b6987-c40d-4ee6-883f-198d4febf4d7"], " \nWie im vorherigen Beispiel k\[ODoubleDot]nnen die Ableitungen als \ Differenzquotienten umgeschrieben werden, um die iterativen Relationen f\ \[UDoubleDot]r ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["x", "TI"], "(", SubscriptBox[ StyleBox["t", "TI"], RowBox[{ StyleBox["n", "TI"], "+", "1"}]], ")"}], TraditionalForm], "errors" -> {}, "input" -> "x(t_{n+1})", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "721bf034-5c73-4cdc-838c-fdb0413a75b8"], " und ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["v", "TI"], "(", SubscriptBox[ StyleBox["t", "TI"], RowBox[{ StyleBox["n", "TI"], "+", "1"}]], ")"}], TraditionalForm], "errors" -> {}, "input" -> "v(t_{n+1})", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "2c11a44c-eac1-4214-9728-d463e295b099"], " zu bestimmen.\nSchreiben Sie eine Funktion, die die DGL f\[UDoubleDot]r \ einen harmonischen Oszillator mithilfe des Euler-Verfahrens f\[UDoubleDot]r \ beliebige Anfangsbedingungen und Parameterwerte numerisch l\[ODoubleDot]st. \ Vergleichen Sie ihre Ergebnisse f\[UDoubleDot]r ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["x", "TI"], RowBox[{"(", "0", ")"}], "\[LongEqual]", "1"}], TraditionalForm], "errors" -> {}, "input" -> "x(0)=1", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "14118e44-7814-4f9f-8830-4f3aee2dd0e1"], " , ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["v", "TI"], RowBox[{"(", "0", ")"}], "\[LongEqual]", "0"}], TraditionalForm], "errors" -> {}, "input" -> "v(0)=0", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "c8d3feee-4d27-4ef3-9d48-ab526ea1e9f2"], ", ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["k", "TI"], "\[LongEqual]", "1.2"}], TraditionalForm], "errors" -> {}, "input" -> "k=1.2", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "232b36c2-7d56-4622-83bc-1ff46d6acb75"], " und ", Cell[BoxData[ FormBox[ RowBox[{ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["m", "TI"], "\[LongEqual]", "0.25"}], TraditionalForm], "errors" -> {}, "input" -> "m=0.25", "state" -> "Boxes"|>, "TeXAssistantTemplate"], " "}], TraditionalForm]],ExpressionUUID-> "92bbd003-78f6-4a81-aaea-7e6a2233b3c7"], "im Zeitintervall ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["t", "TI"], "\[Epsilon]", RowBox[{"[", RowBox[{"0", ",", "10"}], "]"}]}], TraditionalForm], "errors" -> {}, "input" -> "t \\epsilon [0,10]", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "0249a1ee-81e6-4a62-a957-ead2cfff3f7a"], " mit den analytischen Ausdr\[UDoubleDot]cken f\[UDoubleDot]r ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["x", "TI"], "(", StyleBox["t", "TI"], ")"}], TraditionalForm], "errors" -> {}, "input" -> "x(t)", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "9b6c9ca7-7589-453b-9768-664c5d5008f1"], " und ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["v", "TI"], "(", StyleBox["t", "TI"], ")"}], TraditionalForm], "errors" -> {}, "input" -> "v(t)", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "103a545e-268a-4d48-aef0-f0ed54805f74"], " . Setzen Sie dazu einmal ", Cell[BoxData[ FormBox[ RowBox[{ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"\[CapitalDelta]", StyleBox["t", "TI"], "\[LongEqual]", "0.4"}], TraditionalForm], "errors" -> {}, "input" -> "\\Delta t=0.4", "state" -> "Boxes"|>, "TeXAssistantTemplate"], " "}], TraditionalForm]],ExpressionUUID-> "7ab498ce-10ea-432c-a2ad-83d6253985fb"], "und einmal ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"\[CapitalDelta]", StyleBox["t", "TI"], "\[LongEqual]", "0.01"}], TraditionalForm], "errors" -> {}, "input" -> "\\Delta t = 0.01", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "58bebf0c-ed02-4932-a7dc-b4017cb3523d"], ". 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