{"id":2459,"date":"2018-09-12T19:47:19","date_gmt":"2018-09-13T00:47:19","guid":{"rendered":"http:\/\/blog.espol.edu.ec\/matg1013\/?p=2459"},"modified":"2026-04-05T21:01:53","modified_gmt":"2026-04-06T02:01:53","slug":"s3eva2018ti_t3-edp-parabolica-temperatura-en-varilla","status":"publish","type":"post","link":"https:\/\/blog.espol.edu.ec\/algoritmos101\/mn-s3eva20\/s3eva2018ti_t3-edp-parabolica-temperatura-en-varilla\/","title":{"rendered":"s3Eva2018TI_T3 EDP Parab\u00f3lica, temperatura en varilla"},"content":{"rendered":"\n<p><em><strong>Ejercicio<\/strong><\/em>: <a href=\"https:\/\/blog.espol.edu.ec\/algoritmos101\/mn-3eva20\/3eva2018ti_t3-edp-parabolica-temperatura-en-varilla\/\" data-type=\"post\" data-id=\"2451\">3Eva2018TI_T3 EDP Parab\u00f3lica, temperatura en varilla<\/a><\/p>\n\n\n\n<p>Se generan las ecuaciones usando diferencias finitas divididas centradas y hacia adelante<\/p>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> \\frac{d^2u}{dx^2} + \\frac{Kr}{\\rho C} = K\\frac{du}{dt} <\/span>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> \\frac{u[i-1,j]-2u[i,j]+u[i+1,j]}{\\Delta x^2}+<\/span>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\">+ \\frac{Kr}{\\rho C} = K \\frac{u[i,j+1]-u[i,j]}{\\Delta t}<\/span>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"589\" height=\"347\" src=\"http:\/\/blog.espol.edu.ec\/algoritmos101\/files\/2018\/09\/barrametalica3eit2018t3_01.png\" alt=\"barra met\u00e1lica 3eit2018t3_01\" class=\"wp-image-18599\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> \\frac{\\Delta t}{K\\Delta x^2} \\Big[u[i-1,j]-2u[i,j]+u[i+1,j] \\Big] + <\/span>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> + \\frac{Kr}{\\rho C} \\frac{\\Delta t}{K} = u[i,j+1]-u[i,j] <\/span>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<p>Se sustituye :<\/p>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> \\lambda = \\frac{\\Delta t}{K\\Delta x^2} <\/span>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> \\gamma = \\frac{Kr}{\\rho C} \\frac{\\Delta t}{K} = \\frac{r\\Delta t}{\\rho C} <\/span>\n\n\n\n<p>simplificando a:<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> \\lambda \\Big[u[i-1,j]-2u[i,j]+u[i+1,j] \\Big] +<\/span>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> +\\gamma = u[i,j+1]-u[i,j] <\/span>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<p>despejando para u[i,j+1]:<\/p>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> \\lambda u[i-1,j]-2\\lambda u[i,j]+\\lambda u[i+1,j] +<\/span>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> +\\gamma = u[i,j+1]-u[i,j] <\/span>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> u[i,j+1] =\\lambda u[i-1,j]+(1-2\\lambda) u[i,j]+ <\/span>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> +\\lambda u[i+1,j] +\\gamma <\/span>\n\n\n\n<p>con lo que se tiene una forma explicita de encontrar los valores de la ecuaci\u00f3n.<\/p>\n\n\n\n<p>La gr\u00e1fica se realiz\u00f3 para 20 valores de t con tama\u00f1o de paso dt<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"640\" height=\"480\" src=\"http:\/\/blog.espol.edu.ec\/algoritmos101\/files\/2018\/09\/s3eva2028TI_t3ParabolicaTemperatura.png\" alt=\"s3eva2028ti_t3 parab\u00f3lica temperatura\" class=\"wp-image-18601\" \/><\/figure>\n\n\n\n<pre class=\"wp-block-code\"><code>u&#091;:,t] para t = 0.07500000000000001\n&#091; 0.    0.81  1.23  1.35  1.23  0.81  0.  ]<\/code><\/pre>\n\n\n\n<p>algunos valores de u[i,j]<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Tabla de resultados\n&#091;&#091; 0.    0.    0.    0.    0.    0.    ...]\n &#091; 0.5   0.66  0.74  0.81  0.87  0.92  ...]\n &#091; 0.87  0.99  1.12  1.23  1.33  1.41  ...]\n &#091; 1.    1.11  1.23  1.35  1.47  1.57  ...]\n &#091; 0.87  0.99  1.12  1.23  1.33  1.41  ...]\n &#091; 0.5   0.66  0.74  0.81  0.87  0.92  ...]\n &#091; 0.    0.    0.    0.    0.    0.    ...]]<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<p>Algoritmo en Python<\/p>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: python; title: ; notranslate\" title=\"\">\n# 3ra Evaluaci\u00f3n I T\u00e9rmino 2018 \n# Tema 3. EDP Parab\u00f3lica, Temperatura en varilla\n# m\u00e9todo expl\u00edcito, usando diferencias finitas\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# INGRESO\n# Constantes\nL = 1.5\nK = 1.04\nro = 10.6\nC = 0.056\nr = 5.0\n# Tama\u00f1o de paso\ndx = 0.25\ndt = 0.025\n# longitud en x\na = 0\nb = L\n# iteraciones en tiempo\nn = 20\n# Valores de frontera\nTa = 0\nTb = 0\nux0 = lambda x: np.sin(np.pi*x\/L)\n\n# PROCEDIMIENTO\n# iteraciones en longitud\nxi = np.arange(a,b+dx,dx)\nm = len(xi)\nultimox = m-1\n\n# Resultados en tabla u&#x5B;x,t]\nu = np.zeros(shape=(m,n), dtype=float)\n\n# valores iniciales de u&#x5B;:,j]\nj=0\nultimot = n-1\nu&#x5B;0,j]= Ta\nu&#x5B;1:ultimox,j] = ux0(xi&#x5B;1:ultimox])\nu&#x5B;ultimox,j] = Tb\n\n# factores P,Q,R\nlamb = dt\/(K*(dx**2))\ngama = r*dt\/(ro*C)\nP = lamb\nQ = 1 - 2*lamb\nR = lamb\n\n# Calcula U para cada tiempo + dt\nj = 0\nwhile not(j&gt;=ultimot):\n    u&#x5B;0,j+1] = Ta\n    for i in range(1,ultimox,1):\n        u&#x5B;i,j+1] = P*u&#x5B;i-1,j] + Q*u&#x5B;i,j] + R*u&#x5B;i+1,j] + gama\n    u&#x5B;m-1,j+1] = Tb\n    j=j+1\n\n# SALIDA\nprint('Tabla de resultados')\nnp.set_printoptions(precision=2)\nprint(u)\nprint('u&#x5B;:,t] para t =', 3*dt)\nprint(u&#x5B;:,3])\n\n# Gr\u00e1fica\nsalto = 1 # int(n\/10)\nif (salto == 0):\n    salto = 1\nfor j in range(0,n,salto):\n    vector = u&#x5B;:,j]\n    plt.plot(xi,vector)\n    plt.plot(xi,vector, '.r')\nplt.xlabel('x&#x5B;i]')\nplt.ylabel('U&#x5B;x,tj]')\nplt.title('Soluci\u00f3n EDP parab\u00f3lica')\nplt.show()\n<\/pre><\/div>\n\n\n<p>algunos valores de u[i,j]<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Tabla de resultados\n&#091;&#091; 0.    0.    0.    0.    0.    0.    ...]\n &#091; 0.5   0.66  0.74  0.81  0.87  0.92  ...]\n &#091; 0.87  0.99  1.12  1.23  1.33  1.41  ...]\n &#091; 1.    1.11  1.23  1.35  1.47  1.57  ...]\n &#091; 0.87  0.99  1.12  1.23  1.33  1.41  ...]\n &#091; 0.5   0.66  0.74  0.81  0.87  0.92  ...]\n &#091; 0.    0.    0.    0.    0.    0.    ...]]<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Ejercicio: 3Eva2018TI_T3 EDP Parab\u00f3lica, temperatura en varilla Se generan las ecuaciones usando diferencias finitas divididas centradas y hacia adelante Se sustituye : simplificando a: despejando para u[i,j+1]: con lo que se tiene una forma explicita de encontrar los valores de la ecuaci\u00f3n. La gr\u00e1fica se realiz\u00f3 para 20 valores de t con tama\u00f1o de paso [&hellip;]<\/p>\n","protected":false},"author":8043,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"wp-custom-template-entrada-mn-ejemplo","format":"standard","meta":{"footnotes":""},"categories":[51],"tags":[58,54],"class_list":["post-2459","post","type-post","status-publish","format-standard","hentry","category-mn-s3eva20","tag-ejemplos-python","tag-mnumericos"],"_links":{"self":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/2459","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/users\/8043"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/comments?post=2459"}],"version-history":[{"count":3,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/2459\/revisions"}],"predecessor-version":[{"id":23929,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/2459\/revisions\/23929"}],"wp:attachment":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/media?parent=2459"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/categories?post=2459"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/tags?post=2459"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}