(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 11.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 158093, 4038] NotebookOptionsPosition[ 144548, 3822] NotebookOutlinePosition[ 145088, 3842] CellTagsIndexPosition[ 145045, 3839] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Equa\[CCedilla]\[ATilde]o de onda unidimensional e condi\[CCedilla]\ \[OTilde]es de contorno", "Title", CellChangeTimes->{{3.688514867217433*^9, 3.688514889878685*^9}, { 3.688576663583355*^9, 3.688576687443556*^9}, {3.6886004417348623`*^9, 3.688600442077608*^9}, {3.6886038565916348`*^9, 3.688603885046373*^9}, { 3.7785026150817204`*^9, 3.778502622720298*^9}}, FontSize->24,ExpressionUUID->"89e313b4-adf5-46fb-a48d-eeafd966d027"], Cell[CellGroupData[{ Cell["Davi C. 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Em alguns casos de interesse, a separa\[CCedilla]\ \[ATilde]o de vari\[AAcute]veis pode apenas ser aplicada de forma parcial. \ Veremos a seguir casos de EDP\[CloseCurlyQuote]s que s\[ATilde]o \ (completamente) separ\[AAcute]veis.\ \>", "Text", CellChangeTimes->{{3.6885149640933447`*^9, 3.688515058354587*^9}, { 3.688515203396367*^9, 3.6885152527316313`*^9}},ExpressionUUID->"ae6eee3b-04cb-4fb2-a121-\ 1e2699c4e863"], Cell["", "Text", CellChangeTimes->{{3.6885149640933447`*^9, 3.688515006299344*^9}},ExpressionUUID->"2f3b5ee6-9bf9-49ba-bf43-\ 2c3326dfe40b"], Cell[CellGroupData[{ Cell["Equa\[CCedilla]\[ATilde]o de onda em tr\[EHat]s dimens\[OTilde]es: \ apresenta\[CCedilla]\[ATilde]o. 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Ademais, o sinal \ relativo \[EAcute] muito importante, pois caso fosse o contr\[AAcute]rio, n\ \[ATilde]o seria poss\[IAcute]vel encontrar solu\[CCedilla]\[OTilde]es \ oscilat\[OAcute]rias simultanemanete para o tempo e para o espa\[CCedilla]o \ (requisito essencial para descrever, por exemplo, uma corda que oscila).\ \>", "Text", CellChangeTimes->{{3.68851586320566*^9, 3.688515935899336*^9}, { 3.688571593238785*^9, 3.6885715936452217`*^9}, {3.688571728271163*^9, 3.688571862523759*^9}, {3.688574631991776*^9, 3.688574632762002*^9}},ExpressionUUID->"cc9a8de4-dfa9-48df-9c5b-\ fa6af247e8da"] }, Open ]], Cell[CellGroupData[{ Cell["Equa\[CCedilla]\[ATilde]o de onda unidimensional", "Subsection", CellChangeTimes->{{3.68851543947579*^9, 3.6885154480006037`*^9}, { 3.688516169219349*^9, 3.688516170517495*^9}, 3.688600552247254*^9},ExpressionUUID->"34edd064-04a5-474e-a582-\ abbbeb3dc8d7"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["\[PartialD]", "2"], RowBox[{"\[PartialD]", " ", SuperscriptBox["x", "2"]}]], "\[Phi]"}], " ", RowBox[{"(", RowBox[{"x", ",", "t"}], ")"}]}], " ", "-", " ", RowBox[{ FractionBox["1", SuperscriptBox["v", "2"]], RowBox[{ OverscriptBox["\[Phi]", ".."], "(", RowBox[{"x", StyleBox[",", FontWeight->"Plain"], StyleBox["t", FontWeight->"Plain"]}], StyleBox[")", FontWeight->"Plain"]}]}]}], StyleBox[" ", FontWeight->"Plain"], "=", "0"}], TraditionalForm]], "EquationNumbered", CellChangeTimes->{{3.688515729176704*^9, 3.688515847247272*^9}, { 3.688516117365444*^9, 3.688516145622039*^9}},ExpressionUUID->"e1e837f5-7a74-4ad4-913b-\ 5af9331fa8fc"], Cell["\<\ Um importante indicativo sobre se o uso de separa\[CCedilla]\[ATilde]o de \ vari\[AAcute]veis \[EAcute] adequado ou n\[ATilde]o para um problema s\ \[ATilde]o as condi\[CCedilla]\[OTilde]es de contorno. 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Essa EDO \[EAcute] \ trivial, mas \[EAcute] interessante ver como o Mathematica a resolve." }], "Text", CellChangeTimes->{{3.688571226141501*^9, 3.6885712500863247`*^9}, { 3.688572677733673*^9, 3.688572698333713*^9}},ExpressionUUID->"2a3e12d1-7bb5-4cb6-9ed2-\ 8ba13cca1c8c"], Cell["", "Text", CellChangeTimes->{{3.722358933790097*^9, 3.7223589508611317`*^9}},ExpressionUUID->"e2daeccb-c445-4a82-ac0c-\ 240d3111ff6a"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"Clear", "[", "T", "]"}], "\[IndentingNewLine]", RowBox[{"DSolve", "[", RowBox[{ RowBox[{ FractionBox[ RowBox[{ RowBox[{"T", "''"}], "[", "t", "]"}], RowBox[{"T", "[", "t", "]"}]], " ", "\[Equal]", " ", RowBox[{"\[Alpha]", " ", SuperscriptBox["v", "2"]}]}], ",", " ", RowBox[{"T", "[", "t", "]"}], ",", " ", "t"}], "]"}]}], "Input", CellChangeTimes->{{3.688571316687504*^9, 3.688571356815072*^9}, { 3.7223592465921183`*^9, 3.722359252475807*^9}}, CellLabel-> "In[146]:=",ExpressionUUID->"b240c911-e5b8-4f1e-b1a2-3c8e0765e899"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"T", "[", "t", "]"}], "\[Rule]", RowBox[{ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"t", " ", "v", " ", SqrtBox["\[Alpha]"]}]], " ", TemplateBox[{"1"}, "C"]}], "+", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", "t"}], " ", "v", " ", SqrtBox["\[Alpha]"]}]], " ", TemplateBox[{"2"}, "C"]}]}]}], "}"}], "}"}]], "Output", CellChangeTimes->{{3.6885713534791193`*^9, 3.6885713571381474`*^9}, 3.688574888740362*^9, 3.688575655527897*^9, 3.6889112211780577`*^9, 3.721500601519691*^9, {3.722359258730309*^9, 3.7223592611824217`*^9}, 3.778502012646834*^9}, CellLabel-> "Out[147]=",ExpressionUUID->"1842cd57-24d8-4384-bb52-22412e803a74"] }, Open ]], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{"C", "[", "1", "]"}], TraditionalForm]],ExpressionUUID-> "f2016dea-7e03-4fed-87b1-8cd8ffac2555"], StyleBox[" ", "Input"], "e ", Cell[BoxData[ FormBox[ RowBox[{"C", "[", "2", "]"}], TraditionalForm]],ExpressionUUID-> "621ead4d-6801-4f19-87ce-ef5ad650b6a0"], " s\[ATilde]o constantes de integra\[CCedilla]\[ATilde]o. Claramente se ", Cell[BoxData[ FormBox[ RowBox[{"k", ">", "0"}], TraditionalForm]],ExpressionUUID-> "08582207-e0db-4748-ada2-797aea824b06"], " teremos crescimento exponencial, enquanto ", Cell[BoxData[ FormBox[ RowBox[{"k", "<", "0"}], TraditionalForm]],ExpressionUUID-> "4550ae84-80bb-4c88-9c0a-d46363f58356"], StyleBox[" ", "Input"], Cell[BoxData[ FormBox[ Cell["leva a ",ExpressionUUID->"5ceea84c-fa32-4405-baf5-ce16844be4e2"], TraditionalForm]],ExpressionUUID->"c4eea15f-a4ed-4651-bc3d-89ccc4a4b2d6"], "comportamento oscilat\[OAcute]rio. S\[ATilde]o as \ condi\[CCedilla]\[OTilde]es de contorno que determinam qual deve ser o sinal \ de ", Cell[BoxData[ FormBox["k", TraditionalForm]],ExpressionUUID-> "a0112a28-591a-430b-b7e1-30e9a6aa6d9e"], ". Neste exemplo, consideraremos a solu\[CCedilla]\[ATilde]o \ oscilat\[OAcute]ria, a fim de descrever, por exemplo, uma corda que vibra." }], "Text", CellChangeTimes->{{3.688571392379488*^9, 3.688571416301827*^9}, { 3.688571447252727*^9, 3.68857144957157*^9}, {3.6885714892651033`*^9, 3.6885717140634327`*^9}, {3.688572049138011*^9, 3.688572049138373*^9}},ExpressionUUID->"29ce0708-bf32-4e1f-a4d9-\ b27b237256ab"], Cell["\<\ Neste contexto, \[EAcute] conveniente fixar as condi\[CCedilla]\[OTilde]es de \ contorno tais que \ \>", "Text", CellChangeTimes->{{3.688571392379488*^9, 3.688571416301827*^9}, { 3.688571447252727*^9, 3.68857144957157*^9}, {3.6885714892651033`*^9, 3.6885717140634327`*^9}, {3.688572049249834*^9, 3.688572060553492*^9}, { 3.7785030406065893`*^9, 3.7785030406953077`*^9}},ExpressionUUID->"fcbdf00d-2270-48a3-961b-\ cc77388daf02"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"T", "[", "t_", "]"}], " ", "=", " ", RowBox[{"A", " ", RowBox[{"Cos", "[", RowBox[{ RowBox[{ 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"Input", CellChangeTimes->{{3.722359858869195*^9, 3.722359972054467*^9}, { 3.722360036890251*^9, 3.722360039307808*^9}, 3.72236014318125*^9, { 3.77850266897073*^9, 3.778502673627508*^9}}, CellLabel-> "In[215]:=",ExpressionUUID->"32e48f71-b759-4b13-bcfc-bffc8dae08d2"], Cell[BoxData[ RowBox[{"B", " ", RowBox[{"Sin", "[", RowBox[{"x", " ", OverscriptBox["k", "~"]}], "]"}]}]], "Output", CellChangeTimes->{{3.7223598804904947`*^9, 3.722359973403496*^9}, 3.722360072504788*^9, {3.7223601176739273`*^9, 3.722360146114932*^9}, 3.77850201278645*^9, 3.778502674586419*^9}, CellLabel-> "Out[217]=",ExpressionUUID->"172372ab-dde7-442f-8213-fa5b2babb353"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ OverscriptBox["k", "~"], "\[Rule]", RowBox[{"ConditionalExpression", "[", RowBox[{ FractionBox[ RowBox[{"2", " ", "\[Pi]", " ", TemplateBox[{"1"}, "C"]}], "l"], ",", RowBox[{ TemplateBox[{"1"}, "C"], "\[Element]", TemplateBox[{}, "Integers"]}]}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{ OverscriptBox["k", "~"], "\[Rule]", RowBox[{"ConditionalExpression", "[", RowBox[{ FractionBox[ RowBox[{"\[Pi]", "+", RowBox[{"2", " ", "\[Pi]", " ", TemplateBox[{"1"}, "C"]}]}], "l"], ",", RowBox[{ TemplateBox[{"1"}, "C"], "\[Element]", TemplateBox[{}, "Integers"]}]}], "]"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{{3.7223598804904947`*^9, 3.722359973403496*^9}, 3.722360072504788*^9, {3.7223601176739273`*^9, 3.722360146114932*^9}, 3.77850201278645*^9, 3.778502674588498*^9}, CellLabel-> "Out[218]=",ExpressionUUID->"ef2bc4c2-cfad-436a-8617-18eb8179b51f"] }, Open ]], Cell[TextData[{ "Acima temos a express\[ATilde]o mais geral, sem usar ainda as condi\ \[CCedilla]\[OTilde]es de contorno, exceto por exigir a presen\[CCedilla]a de \ solu\[CCedilla]\[OTilde]es oscilat\[OAcute]rias (ou seja, exigir ", Cell[BoxData[ FormBox[ RowBox[{"k", "<", "0"}], TraditionalForm]],ExpressionUUID-> "519151e5-0cdd-4a3f-b8af-9da00582f475"], ")." }], "Text", CellChangeTimes->{{3.688572511177456*^9, 3.688572612634179*^9}, { 3.6885726466437807`*^9, 3.688572646644125*^9}},ExpressionUUID->"a63742c0-7096-4b54-82a0-\ 5d07a9bcc900"], Cell["\<\ Par continuar o exemplo, usemos agora a condi\[CCedilla]\[ATilde]o de \ contorno dado pela eq. (3). Esta condi\[CCedilla]\[ATilde]o de contorno n\ \[ATilde]o imp\[OTilde]e diretamente nenhuma restri\[CCedilla]\[ATilde]o para \ a fun\[CCedilla]\[ATilde]o T, mas para X(x) temos\ \>", "Text", CellChangeTimes->{{3.688572511177456*^9, 3.688572612634179*^9}, { 3.688572648211998*^9, 3.688572656165536*^9}, {3.6885728219539022`*^9, 3.6885728969307957`*^9}},ExpressionUUID->"3963987c-e04d-4d06-991e-\ 9deb54abfac9"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"\[Kappa]", " ", "=", " ", RowBox[{ RowBox[{"-", "\[Pi]"}], "/", "2"}]}], ";"}], " ", RowBox[{"(*", RowBox[{ RowBox[{ RowBox[{"\[Pi]", "/", "2"}], " ", "tamb\[EAcute]m", " ", "\[EAcute]", " ", "poss\[IAcute]vel"}], ",", " ", RowBox[{ "mas", " ", "por", " ", "fim", " ", "leva", " ", "ao", " ", "mesmo", " ", "resultado"}], ",", " ", RowBox[{ "a", " ", "menos", " ", "de", " ", "uma", " ", "mudan\[CCedilla]a", " ", "no", " ", "sinal", " ", "da", " ", "constante", " ", "arbitr\[AAcute]ria", " ", "B"}]}], "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ OverscriptBox["k", "~"], " ", "=", " ", RowBox[{"n", " ", FractionBox["\[Pi]", "l"]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"X", "[", "x", "]"}]}], "Input", CellChangeTimes->{{3.6885729142269697`*^9, 3.688573172093877*^9}, { 3.688573203197204*^9, 3.6885732069190617`*^9}}, CellLabel-> "In[158]:=",ExpressionUUID->"bbba761b-1540-4cd6-88cf-97c4f8fad758"], Cell[BoxData[ RowBox[{"B", " ", RowBox[{"Sin", "[", FractionBox[ RowBox[{"n", " ", "\[Pi]", " ", "x"}], "l"], "]"}]}]], "Output", CellChangeTimes->{{3.6885730604217787`*^9, 3.6885730778296957`*^9}, { 3.6885731308730803`*^9, 3.688573136619329*^9}, 3.688573174806058*^9, 3.688573207703862*^9, 3.688574902443782*^9, 3.6885756610452623`*^9, 3.688576057002644*^9, 3.688595981515664*^9, 3.688911226562183*^9, 3.721500608860005*^9, 3.722359539021708*^9, 3.722359634903002*^9, 3.722360441831859*^9, 3.778502012798724*^9}, CellLabel-> "Out[160]=",ExpressionUUID->"980f0d13-2296-4f9a-965e-e1d36bdf1b37"] }, Open ]], Cell["\<\ Na \[UAcute]ltima linha o Mathematica retornou o atual valor da \ fun\[CCedilla]\[ATilde]o X(x). Verifiquemos se a condi\[CCedilla]\[ATilde]o \ de contorno \[EAcute] satisfeita.\ \>", "Text", CellChangeTimes->{{3.688573139631805*^9, 3.688573141971651*^9}, { 3.688573179807103*^9, 3.688573219484022*^9}},ExpressionUUID->"af0ac91a-e3b9-4693-a310-\ 961c5c460db6"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"X", "[", "0", "]"}]], "Input", CellChangeTimes->{{3.6885732218623133`*^9, 3.6885732338225803`*^9}}, CellLabel-> "In[161]:=",ExpressionUUID->"6fbbe31f-a749-4c63-8db1-a57a2cdcdbaf"], Cell[BoxData["0"], "Output", CellChangeTimes->{3.688573234293276*^9, 3.688574904161784*^9, 3.688595988941681*^9, 3.722360447788295*^9, 3.778502012820459*^9}, CellLabel-> "Out[161]=",ExpressionUUID->"6f63891f-66a7-4965-8398-57b10a0c0984"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"X", "[", "l", "]"}]], "Input", CellChangeTimes->{{3.688573235473765*^9, 3.688573241360392*^9}}, CellLabel-> "In[162]:=",ExpressionUUID->"d4fc01ce-010f-41d6-b9f1-460df2c1f056"], Cell[BoxData[ RowBox[{"B", " ", RowBox[{"Sin", "[", RowBox[{"n", " ", "\[Pi]"}], "]"}]}]], "Output", CellChangeTimes->{ 3.6885732418099203`*^9, 3.688574904999455*^9, 3.688575662901506*^9, { 3.688595986385066*^9, 3.688595990353367*^9}, 3.722360449441304*^9, 3.7785020128314238`*^9}, CellLabel-> "Out[162]=",ExpressionUUID->"63ee3d12-5ca5-4136-ba90-704a64d46532"] }, Open ]], Cell["\<\ Note que o Mathematica poderia ter simplificado mais, mas n\[ATilde]o o fez. \ Podemos exigir que simplifique mais atrav\[EAcute]s de \ \>", "Text", CellChangeTimes->{{3.688573262243616*^9, 3.6885732840856752`*^9}},ExpressionUUID->"1e12cbac-1f84-4246-9302-\ 72489a51bab7"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{"X", "[", "l", "]"}], "]"}]], "Input", CellChangeTimes->{{3.6885732856853333`*^9, 3.688573302665938*^9}}, CellLabel-> "In[163]:=",ExpressionUUID->"9882ef38-1183-4dbc-b36d-b4503636708b"], Cell[BoxData[ RowBox[{"B", " ", RowBox[{"Sin", "[", RowBox[{"n", " ", "\[Pi]"}], "]"}]}]], "Output", CellChangeTimes->{{3.6885733031250677`*^9, 3.688573322725761*^9}, 3.6885749061446342`*^9, 3.688575664028481*^9, 3.688595984489007*^9, 3.7223604632130413`*^9, 3.778502012853196*^9}, CellLabel-> "Out[163]=",ExpressionUUID->"221a40ee-9c17-4255-8f59-07cb897dcc09"] }, Open ]], Cell["\<\ Note que mesmo pedindo a simplifica\[CCedilla]\[ATilde]o, ainda ele \ n\[ATilde]o conseguiu avan\[CCedilla]ar. O motivo \[EAcute] simples, em \ nenhum momento foi enviado para o Mathematica a informa\[CCedilla]\[ATilde]o \ de que n \[EAcute] n\[UAcute]mero inteiro. Podemos desde j\[AAcute] atribuir \ esta informa\[CCedilla]\[ATilde]o para ele, e ainda apresentar outras que \ estamos usando mas n\[ATilde]o enviamos a ele. Isto \[EAcute] feito da \ seguinte forma:\ \>", "Text", CellChangeTimes->{{3.688573325985609*^9, 3.688573436324452*^9}},ExpressionUUID->"74ae339f-6e28-4ce5-abb1-\ 062e8b8d47b0"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"$Assumptions", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{"n", "\[Element]", "Integers"}], ",", RowBox[{"B", "\[Element]", "Reals"}], ",", RowBox[{"A", "\[Element]", "Reals"}], ",", RowBox[{"l", "\[Element]", "Reals"}], ",", " ", RowBox[{"\[Delta]", " ", "\[Element]", " ", "Reals"}], ",", " ", RowBox[{"x", "\[Element]", " ", "Reals"}], ",", " ", RowBox[{"t", " ", "\[Element]", " ", "Reals"}]}], "}"}]}]], "Input", CellChangeTimes->{{3.68857343729288*^9, 3.688573452065043*^9}, { 3.6885735719453793`*^9, 3.688573596990152*^9}}, CellLabel-> "In[164]:=",ExpressionUUID->"dc71f2e7-0f20-43a6-9146-6353bc81a6b8"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"n", "\[Element]", TemplateBox[{}, "Integers"]}], ",", RowBox[{"B", "\[Element]", TemplateBox[{}, "Reals"]}], ",", RowBox[{"A", "\[Element]", TemplateBox[{}, "Reals"]}], ",", RowBox[{"l", "\[Element]", TemplateBox[{}, "Reals"]}], ",", RowBox[{"\[Delta]", "\[Element]", TemplateBox[{}, "Reals"]}], ",", RowBox[{"x", "\[Element]", TemplateBox[{}, "Reals"]}], ",", RowBox[{"t", "\[Element]", TemplateBox[{}, "Reals"]}]}], "}"}]], "Output", CellChangeTimes->{ 3.688573453614826*^9, {3.688573577120798*^9, 3.688573597347135*^9}, 3.688574907761221*^9, 3.688575665401187*^9, 3.6885760596767883`*^9, 3.68859599311936*^9, 3.688911230819549*^9, 3.721500613383017*^9, 3.722360507231436*^9, 3.778502012864605*^9}, CellLabel-> "Out[164]=",ExpressionUUID->"ac2b7a75-502e-4bce-81e9-941c42aefcf8"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{"X", "[", "l", "]"}], "]"}]], "Input", CellChangeTimes->{{3.6885734568019114`*^9, 3.688573459176794*^9}}, CellLabel-> "In[165]:=",ExpressionUUID->"46a6668c-1ab3-404e-b91a-3fbd7ea1fbdb"], Cell[BoxData["0"], "Output", CellChangeTimes->{3.688573461482396*^9, 3.688574909386641*^9, 3.6885756670729313`*^9, 3.6885959950000668`*^9, 3.688911232612461*^9, 3.7215006153830023`*^9, 3.722360515857946*^9, 3.778502012886613*^9}, CellLabel-> "Out[165]=",ExpressionUUID->"8ebeee26-46f5-417c-a7cc-f4d4c4366181"] }, Open ]], Cell["\<\ Podemos juntar tudo e descobrir como a fun\[CCedilla]\[ATilde]o \[Phi](x,t) \ ficou. Primeiro definimos ela,\ \>", "Text", CellChangeTimes->{{3.688573475243011*^9, 3.6885734983086643`*^9}},ExpressionUUID->"269a5ab9-8c45-476b-ade8-\ 21d873349f9c"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"\[Phi]", "[", RowBox[{"x_", ",", "t_"}], "]"}], " ", "=", " ", RowBox[{ RowBox[{"X", "[", "x", "]"}], " ", RowBox[{"T", "[", "t", "]"}]}]}]], "Input", CellChangeTimes->{{3.6885735134272614`*^9, 3.6885735250857697`*^9}}, CellLabel-> "In[166]:=",ExpressionUUID->"87b15862-b1e0-4bd8-9d48-3627a0259339"], Cell[BoxData[ RowBox[{"A", " ", "B", " ", RowBox[{"Cos", "[", RowBox[{ FractionBox[ RowBox[{"n", " ", "\[Pi]", " ", "t", " ", "v"}], "l"], "+", "\[Delta]"}], "]"}], " ", RowBox[{"Sin", "[", FractionBox[ RowBox[{"n", " ", "\[Pi]", " ", "x"}], "l"], "]"}]}]], "Output", CellChangeTimes->{3.6885735284851522`*^9, 3.6885735993860598`*^9, 3.688574911335622*^9, 3.688575667769107*^9, 3.688576061244376*^9, 3.6885959967289*^9, 3.6889112341735992`*^9, 3.7215006170691*^9, 3.7223605395260687`*^9, 3.778502012897862*^9}, CellLabel-> "Out[166]=",ExpressionUUID->"5b982713-d7e8-455d-8462-46167757d6f0"] }, Open ]], Cell[TextData[{ "V\[EHat]-se que \[Delta] continua arbitr\[AAcute]rio, contudo isso \ \[EAcute] natural. Para x n\[ATilde]o temos liberdade semelhante pois as \ condi\[CCedilla]\[OTilde]es de contorno fixaram valores particulares de x nos \ quais \[Phi] se torna nula, logo transla\[CCedilla]\[OTilde]es em x \ n\[ATilde]o preservam a solu\[CCedilla]\[ATilde]o, pois as condi\[CCedilla]\ \[OTilde]es de contorno n\[ATilde]o s\[ATilde]o invariantes por transla\ \[CCedilla]\[ATilde]o em x. Para t isso n\[ATilde]o ocorre, as condi\ \[CCedilla]\[OTilde]es de contorno n\[ATilde]o indicam nenhum valor de t \ especial, ademais n\[ATilde]o h\[AAcute] na equa\[CCedilla]\[ATilde]o de onda \ nenhum t que apare\[CCedilla]a explicitamente e poderia fornecer a t um valor \ especial. Como n\[ATilde]o h\[AAcute] valor especial para t, \[EAcute] de se \ esperar que a solu\[CCedilla]\[ATilde]o seja invariante por \ transla\[CCedilla]\[ATilde]o temporal. Ou seja, podemos trocar t por t + ", Cell[BoxData[ FormBox[ SubscriptBox["t", "0"], TraditionalForm]],ExpressionUUID-> "34c2232e-a274-43a5-9898-caaf06b83522"], " a vontade que continuaremos tendo uma solu\[CCedilla]\[ATilde]o (note que \ depois de fazer essa transla\[CCedilla]\[ATilde]o pode-se sempre selecionar \ um \[Delta] tal que ", Cell[BoxData[ FormBox[ SubscriptBox["t", "0"], TraditionalForm]],ExpressionUUID-> "c35c409b-1e41-4d53-b918-f3814b7e9158"], " desapare\[CCedilla]a da solu\[CCedilla]\[ATilde]o).\[LineSeparator]\ \[LineSeparator]Podemos por exemplo fixar \[Delta] = \[Pi]/2. Neste caso, \ para t =0 temos que T = 0, ou seja, toda a corda est\[AAcute] reta, sem \ oscila\[CCedilla]\[OTilde]es. Esse instante em que n\[ATilde]o h\[AAcute] \ oscila\[CCedilla]\[OTilde]es em nenhuma parte da corda se repete \ periodicamente." }], "Text", CellChangeTimes->{{3.688573750573185*^9, 3.688574061886468*^9}, { 3.688574667266098*^9, 3.688574693819655*^9}, {3.6885747240362597`*^9, 3.688574798520747*^9}, {3.688574842104426*^9, 3.688574860743203*^9}},ExpressionUUID->"4bb8d0a2-8cfa-406b-be98-\ 47721de0c83a"], Cell[BoxData[ RowBox[{ RowBox[{"\[Delta]", " ", "=", RowBox[{"\[Pi]", "/", "2"}]}], ";"}]], "Input", CellChangeTimes->{{3.688574862244385*^9, 3.688574871683413*^9}, { 3.688575356663224*^9, 3.6885753587685328`*^9}}, CellLabel-> "In[167]:=",ExpressionUUID->"030e9a77-503f-4082-be51-e2557aa82b4c"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"\[Phi]", "[", RowBox[{"x", ",", " ", "t"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.688574917601335*^9, 3.688574930068149*^9}}, CellLabel-> "In[168]:=",ExpressionUUID->"95ab2c78-53a8-4e1c-b4b4-6324019501a6"], Cell[BoxData[ RowBox[{ RowBox[{"-", "A"}], " ", "B", " ", RowBox[{"Sin", "[", FractionBox[ RowBox[{"n", " ", "\[Pi]", " ", "t", " ", "v"}], "l"], "]"}], " ", RowBox[{"Sin", "[", FractionBox[ RowBox[{"n", " ", "\[Pi]", " ", "x"}], "l"], "]"}]}]], "Output", CellChangeTimes->{3.688574930610907*^9, 3.688575361395817*^9, 3.688575671562231*^9, 3.688576064687825*^9, 3.688596001835732*^9, 3.6889112368233747`*^9, 3.7215006212338142`*^9, 3.722360588213484*^9, 3.778502012928504*^9}, CellLabel-> "Out[168]=",ExpressionUUID->"8b6af8a1-5669-4fd7-a551-0f30e59c1731"] }, Open ]], Cell["Equivalentemente, podemos escrever,", "Text", CellChangeTimes->{{3.688575039002927*^9, 3.688575042633423*^9}},ExpressionUUID->"9142edef-f779-4aad-bb89-\ 75d7b5e5583e"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Expand", "[", RowBox[{"TrigReduce", "[", RowBox[{"\[Phi]", "[", RowBox[{"x", ",", " ", "t"}], "]"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.688573673496265*^9, 3.688573703082075*^9}, { 3.6885761039815598`*^9, 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Por se tratar de EDP linear, a solu\[CCedilla]\ \[ATilde]o geral ser\[AAcute] uma combina\[CCedilla]\[ATilde]o linear de \ todas as poss\[IAcute]veis solu\[CCedilla]\[OTilde]es. Assim vamos chamar a \ solu\[CCedilla]\[ATilde]o que encontramos anteriormente de ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["c", "n"], RowBox[{ SubscriptBox["\[Phi]", "n"], "(", RowBox[{"x", ",", "t"}], ")"}]}], TraditionalForm]],ExpressionUUID-> "2ad7a780-89ce-4605-b5f1-c306c38a9738"], ", assim a solu\[CCedilla]\[ATilde]o geral \[EAcute] ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["c", "n"], RowBox[{ SubscriptBox["\[Phi]", "n"], "(", RowBox[{"x", ",", "t"}], ")"}]}], TraditionalForm]],ExpressionUUID-> "0a0e977c-cf36-4452-9930-d12dcb1a31b3"] }], "Text", CellChangeTimes->{{3.688576817462768*^9, 3.688576879916171*^9}, { 3.6885769755216713`*^9, 3.688577032065482*^9}},ExpressionUUID->"9943497c-0eda-4257-a644-\ 9449a91f2385"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Phi]", "(", RowBox[{"x", ",", "t"}], ")"}], " ", "=", " ", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "0"}], "\[Infinity]"], RowBox[{ SubscriptBox["c", "n"], RowBox[{ SubscriptBox["\[Phi]", "n"], "(", RowBox[{"x", ",", "t"}], ")"}]}]}]}], TraditionalForm]], "EquationNumbered", CellChangeTimes->{{3.688577042973543*^9, 3.6885770817332497`*^9}},ExpressionUUID->"72ea8923-73e8-4b5e-89fe-\ 405eee124902"], Cell["Ou seja,", "Text", CellChangeTimes->{{3.6885771278240232`*^9, 3.68857712868849*^9}},ExpressionUUID->"9ee9254b-9618-47ca-b169-\ 05b85ebd2a95"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"Clear", "[", RowBox[{"\[Phi]", ",", " ", "n", ",", " ", "l", ",", "v"}], "]"}], "\[LineSeparator]", RowBox[{ RowBox[{"\[Phi]", "[", RowBox[{"n_", ",", "x_", ",", "t_"}], "]"}], " ", "=", " ", RowBox[{ RowBox[{"c", "[", "n", "]"}], " ", RowBox[{"Sin", "[", FractionBox[ RowBox[{"n", " ", "\[Pi]", " ", "t", " ", "v"}], "l"], "]"}], " ", RowBox[{"Sin", "[", FractionBox[ RowBox[{"n", " ", "\[Pi]", " ", "x"}], "l"], "]"}]}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[Phi]", "[", RowBox[{"x_", ",", "t_"}], "]"}], " ", "=", " ", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "1"}], "\[Infinity]"], RowBox[{"\[Phi]", "[", RowBox[{"n", ",", "x", ",", "t"}], "]"}], " "}]}]}], "Input", CellChangeTimes->{{3.6885771303496428`*^9, 3.688577205075328*^9}, { 3.688577262483824*^9, 3.688577262855698*^9}, {3.688577393468686*^9, 3.6885774252736073`*^9}, {3.688577474940385*^9, 3.6885775061271877`*^9}}, CellLabel-> "In[181]:=",ExpressionUUID->"77bc2190-b2c1-49c1-b822-46e9c5725d69"], Cell[BoxData[ RowBox[{ RowBox[{"c", "[", "n", "]"}], " ", RowBox[{"Sin", "[", FractionBox[ RowBox[{"n", " ", "\[Pi]", " ", "t", " ", "v"}], "l"], "]"}], " ", RowBox[{"Sin", "[", FractionBox[ RowBox[{"n", " ", "\[Pi]", " ", "x"}], "l"], "]"}]}]], "Output", CellChangeTimes->{ 3.688577211195924*^9, 3.6885772632076178`*^9, 3.6885774294695787`*^9, { 3.688577498871043*^9, 3.688577507477068*^9}, 3.721500676150289*^9, 3.722361407612554*^9, 3.77850201386862*^9}, CellLabel-> "Out[182]=",ExpressionUUID->"aeac41d8-8150-4d67-b790-93d99cc885bc"], Cell[BoxData[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "1"}], "\[Infinity]"], RowBox[{ RowBox[{"c", "[", "n", "]"}], " ", RowBox[{"Sin", "[", FractionBox[ RowBox[{"n", " ", "\[Pi]", " ", "t", " ", "v"}], "l"], "]"}], " ", RowBox[{"Sin", "[", FractionBox[ RowBox[{"n", " ", "\[Pi]", " ", "x"}], "l"], "]"}]}]}]], "Output", CellChangeTimes->{ 3.688577211195924*^9, 3.6885772632076178`*^9, 3.6885774294695787`*^9, { 3.688577498871043*^9, 3.688577507477068*^9}, 3.721500676150289*^9, 3.722361407612554*^9, 3.778502015030588*^9}, CellLabel-> "Out[183]=",ExpressionUUID->"f325450f-398d-4079-a13e-3ea5c0b95bc9"] }, Open ]], Cell[TextData[{ "Esta \[UAcute]ltima acima \[EAcute] a solu\[CCedilla]\[ATilde]o geral, \ sendo v\[AAcute]lida a separa\[CCedilla]\[ATilde]o de vari\[AAcute]veis. A \ condi\[CCedilla]\[ATilde]o de contorno dada n\[ATilde]o \[EAcute] suficiente \ para fixar completamente a solu\[CCedilla]\[ATilde]o, h\[AAcute] ainda ", Cell[BoxData[ FormBox[ SubscriptBox["c", "n"], TraditionalForm]],ExpressionUUID-> "0c2fb323-fd43-41fe-b483-7dde768257c7"], " coeficientes reais arbitr\[AAcute]rios. Isto \[EAcute] natural pois h\ \[AAcute] infintos modos de vibra\[CCedilla]\[ATilde]o da corda: o modo \ fundamental, com n=1, \[EAcute] o \[UAcute]nico em que os extremos x=0 e x=l \ constituem os \[UAcute]nicos pontos nos quais a corda est\[AAcute] \ im\[OAcute]vel. Para cada unidade maior, aumenta-se o n\[UAcute]mero de \ pontos nos quais a corda est\[AAcute] fixa em qualquer instante t \ (propriedade de onda estacion\[AAcute]ria). " }], "Text", CellChangeTimes->{{3.688577438311034*^9, 3.688577464135109*^9}, { 3.688595937491008*^9, 3.688595954925528*^9}, {3.6885995795211077`*^9, 3.688599673502411*^9}, {3.6885997253129187`*^9, 3.6885998311931057`*^9}, { 3.688599911158202*^9, 3.6886000289983597`*^9}},ExpressionUUID->"8fd36929-2e6a-48e8-8a6e-\ eb1185283748"], Cell[TextData[{ "No contexto de mec\[AHat]nica qu\[AHat]ntica, a equa\[CCedilla]\[ATilde]o \ diferencial relevante \[EAcute] a eq. de Schroedinger, contudo encontra-se \ solu\[CCedilla]\[OTilde]es semelhantes no contexto de uma part\[IAcute]cula \ presa numa caixa. Nesse contexto, fixar os ", Cell[BoxData[ FormBox[ SubscriptBox["c", "n"], TraditionalForm]],ExpressionUUID-> "fd4cf18a-4525-4188-903b-6803c658d15a"], " corresponde a saber a energia da part\[IAcute]cula. Se a \[UAcute]nica \ informa\[CCedilla]\[ATilde]o dada foi a de que a part\[IAcute]cula est\ \[AAcute] na caixa, os coeficientes ", Cell[BoxData[ FormBox[ SubscriptBox["c", "n"], TraditionalForm]],ExpressionUUID-> "10d6a121-dc48-4c59-910f-923017e344da"], " ser\[ATilde]o necessariamente arbitr\[AAcute]rios." }], "Text", CellChangeTimes->{{3.688577438311034*^9, 3.688577464135109*^9}, { 3.688595937491008*^9, 3.688595954925528*^9}, {3.6885995795211077`*^9, 3.688599673502411*^9}, {3.6885997253129187`*^9, 3.688599883352042*^9}},ExpressionUUID->"1133fb6c-1655-47a0-ad62-\ 6eb05284af27"], Cell[BoxData[""], "Input", CellLabel-> "In[184]:=",ExpressionUUID->"8b20e63a-6cb4-4471-95dd-6f396d392349"] }, Open ]], Cell[CellGroupData[{ Cell["Soma de modos de vibra\[CCedilla]\[ATilde]o.", "Subsubsection", CellChangeTimes->{{3.688600376995863*^9, 3.688600400111887*^9}},ExpressionUUID->"52679116-a51b-4852-b71b-\ 49dca62b495d"], Cell[TextData[{ "Vejamos como fica a soma de dois modos de vibra\[CCedilla]\[ATilde]o, n =1 \ com n=3 (ou seja, o caso em que todos os ", Cell[BoxData[ FormBox[ SubscriptBox["c", "n"], TraditionalForm]],ExpressionUUID-> "f2203b05-48f0-4425-b583-af1b22fbbec4"], " s\[ATilde]o nulos, exceto por ", Cell[BoxData[ FormBox[ SubscriptBox["c", "1"], TraditionalForm]],ExpressionUUID-> "699e9b9e-e35f-4dba-996f-355c07dece4a"], " e ", Cell[BoxData[ FormBox[ SubscriptBox["c", "3"], TraditionalForm]],ExpressionUUID-> "c202b48b-ab45-4a7e-8b32-d7391fb09492"], ", os quais tomamos como iguais a 1)" }], "Text", CellChangeTimes->{{3.688600049426015*^9, 3.688600070442296*^9}, { 3.688600466328355*^9, 3.688600510990635*^9}},ExpressionUUID->"8e2d4c1c-ae77-4a51-8df7-\ cdf4ba245092"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"l", " ", "=", " ", "5"}], ",", " ", RowBox[{"v", " ", "=", " ", "1"}], ",", " ", RowBox[{"n", "=", "1"}]}], "}"}], ",", RowBox[{"Animate", "[", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", " ", RowBox[{ RowBox[{ RowBox[{"-", " ", RowBox[{"Sin", "[", FractionBox[ RowBox[{"n", " ", "\[Pi]", " ", "t", " ", "v"}], "l"], "]"}]}], " ", RowBox[{"Sin", "[", FractionBox[ RowBox[{"n", " ", "\[Pi]", " ", "x"}], "l"], "]"}]}], " ", "-", " ", RowBox[{ RowBox[{"Sin", "[", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"n", "+", "2"}], ")"}], " ", "\[Pi]", " ", "t", " ", "v"}], "l"], "]"}], " ", RowBox[{"Sin", "[", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"n", "+", "2"}], ")"}], " ", "\[Pi]", " ", "x"}], "l"], "]"}]}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", "0", ",", "l"}], "}"}], ",", " ", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{"Automatic", ",", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "2"}], "}"}]}], "}"}]}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{"t", ",", "0", ",", " ", RowBox[{"2", FractionBox["l", "v"]}]}], "}"}], ",", RowBox[{"AnimationRunning", "\[Rule]", "False"}]}], "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.688600072875515*^9, 3.688600135017372*^9}, { 3.688911299803739*^9, 3.688911303368245*^9}}, CellLabel-> "In[185]:=",ExpressionUUID->"9b2c2872-382f-4754-a24e-98b04529b06a"], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`t$$ = 0.5221014022827148, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`t$$], 0, 10}}, Typeset`size$$ = {720., {225., 234.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`t$46746$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`t$$ = 0}, "ControllerVariables" :> { Hold[$CellContext`t$$, $CellContext`t$46746$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Plot[{(-Sin[(($CellContext`n$46745 Pi) $CellContext`t$$) \ ($CellContext`v$46745/$CellContext`l$46745)]) Sin[($CellContext`n$46745 Pi) ($CellContext`x/$CellContext`l$46745)] - 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Ou seja, neste caso h\[AAcute] uma solu\[CCedilla]\[ATilde]o geral para \ \[Phi] sem impor a separa\[CCedilla]\[ATilde]o de vari\[AAcute]veis, mas \ qualquer fun\[CCedilla]\[ATilde]o (diferenci\[AAcute]vel) desses argumentos \ ser\[AAcute] solu\[CCedilla]\[ATilde]o da eq. diferencial original." }], "Text", CellChangeTimes->{{3.688600853974123*^9, 3.688600997157477*^9}},ExpressionUUID->"e8941504-8879-4923-8604-\ 533cd06d864a"], Cell["\<\ Um exemplo trivial de solu\[CCedilla]\[ATilde]o da eq. de onda que \ n\[ATilde]o pode ser encontrada por separa\[CCedilla]\[ATilde]o de vari\ \[AAcute]veis \[EAcute] \ \>", "Text", CellChangeTimes->{{3.68860327427553*^9, 3.688603292534216*^9}},ExpressionUUID->"523236ac-f033-460e-9f75-\ f918d77ffa9e"], Cell[BoxData[{ RowBox[{"Clear", "[", "\[Phi]", "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[Phi]", "[", RowBox[{"x_", ",", "t_"}], "]"}], " ", ":=", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"t", " ", "v"}], " ", "-", " ", "x"}], ")"}], "2"], SuperscriptBox["\[Sigma]", "2"]]}]]}]}], "Input", CellChangeTimes->{{3.688602859030423*^9, 3.688602914514151*^9}, { 3.688603025178301*^9, 3.6886030361470947`*^9}}, CellLabel-> "In[190]:=",ExpressionUUID->"77b30e15-4e94-42a3-a11c-2d0c1bf7eb77"], Cell["Abaixo verifica-se se essa \[EAcute] realmente uma solu\[CCedilla]\ \[ATilde]o,", "Text", CellChangeTimes->{{3.688603651664156*^9, 3.68860367867066*^9}},ExpressionUUID->"4281bc97-827a-4571-9b16-\ e257a2f332c3"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", RowBox[{"x", ",", "x"}]], RowBox[{"\[Phi]", "[", RowBox[{"x", ",", "t"}], "]"}]}], " ", "-", " ", RowBox[{ FractionBox["1", SuperscriptBox["v", "2"]], RowBox[{ SubscriptBox["\[PartialD]", RowBox[{"t", ",", "t"}]], RowBox[{"\[Phi]", "[", RowBox[{"x", ",", "t"}], "]"}]}]}]}], " ", "\[Equal]", "0"}], "]"}]], "Input", CellChangeTimes->{{3.688602921261353*^9, 3.6886029301824837`*^9}, 3.688603646484894*^9}, CellLabel-> "In[192]:=",ExpressionUUID->"ace5a804-9924-42f3-9767-d4d025ca63bf"], Cell[BoxData["True"], "Output", CellChangeTimes->{{3.688602925808515*^9, 3.688602930476275*^9}, 3.6886036809757*^9, 3.721500704369676*^9, 3.722361508596324*^9, 3.778502015729948*^9}, CellLabel-> "Out[192]=",ExpressionUUID->"d1bfa490-01ef-4c12-8fcd-416c00b8b8a0"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"v", " ", "=", " ", "1"}], ",", " ", RowBox[{"\[Sigma]", "=", RowBox[{"1", "/", "10"}]}]}], "}"}], ",", RowBox[{"Animate", "[", RowBox[{ RowBox[{"Plot", "[", " ", RowBox[{ RowBox[{"N", "[", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", RowBox[{"Rationalize", "[", RowBox[{ FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"t", " ", "v"}], " ", "-", " ", "x"}], ")"}], "2"], SuperscriptBox["\[Sigma]", "2"]], ",", "0"}], "]"}]}]], ",", "100"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", "0", ",", "10"}], "}"}], ",", " ", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{"Automatic", ",", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "2"}], "}"}]}], "}"}]}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{"t", ",", "0", ",", " ", "10", ",", RowBox[{"1", "/", "10"}]}], "}"}], ",", RowBox[{"AnimationRate", "\[Rule]", "30"}], ",", " ", RowBox[{"AnimationRunning", "\[Rule]", "False"}]}], "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.6886029452746887`*^9, 3.688603014913727*^9}, { 3.688603124279422*^9, 3.688603140008651*^9}, {3.6889113294702377`*^9, 3.6889113296512547`*^9}, {3.7785021381009293`*^9, 3.778502231037299*^9}, { 3.7785022688951073`*^9, 3.778502323447258*^9}, {3.7785024101735067`*^9, 3.778502492433505*^9}}, CellLabel-> "In[214]:=",ExpressionUUID->"59efba2a-17db-4915-96c5-ff64a28239f9"], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`t$$ = Rational[17, 10], Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`t$$], 0, 10, Rational[1, 10]}}, Typeset`size$$ = {720., {225., 234.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`t$92902$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`t$$ = 0}, "ControllerVariables" :> { Hold[$CellContext`t$$, $CellContext`t$92902$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Plot[ N[ E^(-Rationalize[($CellContext`t$$ $CellContext`v$92901 - 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