{"id":3863,"date":"2017-12-11T10:10:21","date_gmt":"2017-12-11T15:10:21","guid":{"rendered":"http:\/\/blog.espol.edu.ec\/matg1013\/?p=3863"},"modified":"2026-04-05T20:21:07","modified_gmt":"2026-04-06T01:21:07","slug":"s2eva2011tii_t3-edp-parabolica-explicito","status":"publish","type":"post","link":"https:\/\/blog.espol.edu.ec\/algoritmos101\/mn-s2eva20\/s2eva2011tii_t3-edp-parabolica-explicito\/","title":{"rendered":"s2Eva2011TII_T3 EDP Parab\u00f3lica, expl\u00edcito"},"content":{"rendered":"\n

Ejercicio<\/strong><\/em>: 2Eva2011TII_T3 EDP Parab\u00f3lica, expl\u00edcito<\/a><\/p>\n\n\n\n

La ecuaci\u00f3n a resolver es:<\/p>\n\n\n \\frac{\\delta u}{\\delta t} - \\frac{\\delta^2 u}{\\delta x^2} =2<\/span>\n\n\n\n

Como el m\u00e9todo requerido es expl\u00edcito se usan las diferencias divididas:<\/p>\n\n\n \\frac{d^2 u}{dx^2} = \\frac{u_{i+1,j} - 2 u_{i,j} + u_{i-1,j}}{(\\Delta x)^2} <\/span>\n\n\n \\frac{du}{dt} = \\frac{u_{i,j+1} - u_{i,j} }{\\Delta t} <\/span>\n\n\n\n

\"EDP<\/figure>\n\n\n\n

<\/p>\n\n\n\n

Reordenando la ecuaci\u00f3n al modelo realizado en clase:<\/p>\n\n\n\\frac{\\delta^2 u}{\\delta x^2} = \\frac{\\delta u}{\\delta t} - 2<\/span>\n\n\n\\frac{u_{i+1,j} - 2 u_{i,j} + u_{i-1,j}}{(\\Delta x)^2} = \\frac{u_{i,j+1} - u_{i,j} }{\\Delta t} - 2<\/span>\n\n\n\n

multiplicando cada lado por \u0394t<\/p>\n\n\n \\frac{\\Delta t}{(\\Delta x)^2} \\Big[u_{i+1,j} - 2 u_{i,j} + u_{i-1,j} \\Big]= u_{i,j+1} - u_{i,j} - 2\\Delta t <\/span>\n\n\n\n

Se establece el valor de<\/p>\n\n\n \\lambda = \\frac{\\Delta t}{(\\Delta x)^2}<\/span>\n\n\n\\lambda u_{i+1,j} - 2 \\lambda u_{i,j} + \\lambda u_{i-1,j} = u_{i,j+1} - u_{i,j} - 2\\Delta t <\/span>\n\n\n\n

obteniendo la ecuaci\u00f3n general, ordenada por \u00edndices de izquierda a derecha:<\/p>\n\n\n\\lambda u_{i-1,j} +(1- 2 \\lambda)u_{i,j} + \\lambda u_{i+1,j} + 2\\Delta t = u_{i,j+1} <\/span>\n\n\n\n

con valores de:<\/p>\n\n\n\n

\u03bb = \u0394t\/(\u0394x)2<\/sup> = (0.04)\/(0.25)2<\/sup> = 0.64\nP = \u03bb = 0.64\nQ = (1-2\u03bb) = -0.28\nR = \u03bb = 0.64<\/code><\/pre>\n\n\n0.64 u_{i-1,j} - 0.28 u_{i,j} + 0.64 u_{i+1,j} + 0.08 = u_{i,j+1} <\/span>\n\n\n\n
Se realizan 3 iteraciones:<\/p>\n\n\n\n
i= 1, j=0<\/p>\n\n\nu_{1,1} = 0.64 u_{0,0} -0.28u_{1,0}+0.64 u_{2,0}+0.08<\/span>\n\n\n u_{1,1} = 0.64 [\\sin(\\pi*0)+0*(1-0)]- <\/span>\n\n\n 0.28[\\sin(\\pi*0.25)+0.25*(1-0.25)]<\/span>\n\n\n+0.64[\\sin(\\pi*0.5)+ 0.5*(1-0.5)]+0.08 <\/span>\n\n\n\n
u[1,1] =0.89<\/p>\n\n\n\n
i = 2, j=0<\/p>\n\n\n0.64 u_{1,0} - 0.28 u_{2,0} + 0.64 u_{3,0} + 0.08 = u_{2,1} <\/span>\n\n\n\n
u[1,0] = 1.25<\/p>\n\n\n\n
i = 3, j=0<\/p>\n\n\n0.64 u_{2,0} - 0.28 u_{3,0} + 0.64 u_{4,0} + 0.08 = u_{3,1} <\/span>\n\n\n\n
u[3,1] = 0.89<\/p>\n\n\n\n
\"EDP<\/figure>\n\n\n\n
<\/p>\n\n\n\n
\"EDP<\/figure>\n\n\n\n
\n\n\n\n
Algoritmo en Python<\/h2>\n\n\n\n
con los valores y ecuaci\u00f3n del problema<\/p>\n\n\n
\n# EDP parab\u00f3licas d2u\/dx2  = K du\/dt\n# m\u00e9todo expl\u00edcito, usando diferencias finitas\n# Referencia: Chapra 30.2 p.888 pdf.912\n#       Rodriguez 10.2 p.406\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# INGRESO\n# Valores de frontera\nTa = 0\nTb = 0\nT0 = lambda x: np.sin(np.pi*x)+x*(1-x)\n# longitud en x\na = 0\nb = 1\n# Constante K\nK = 1\n# Tama\u00f1o de paso\ndx = 0.1\ndt = dx\/10\/2\n# iteraciones en tiempo\nn = 50\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[x,t]\nu = np.zeros(shape=(m,n), dtype=float)\n\n# valores iniciales de u[:,j]\nj=0\nultimot = n-1\nu[0,j]= Ta\nu[1:ultimox,j] = T0(xi[1:ultimox])\nu[ultimox,j] = Tb\n\n# factores P,Q,R\nlamb = dt\/(K*dx**2)\nP = lamb\nQ = 1 - 2*lamb\nR = lamb\n\n# Calcula U para cada tiempo + dt\nj = 0\nwhile not(j>=ultimot):\n    u[0,j+1] = Ta\n    for i in range(1,ultimox,1):\n        u[i,j+1] = P*u[i-1,j] + Q*u[i,j] + R*u[i+1,j]+2*dt\n    u[m-1,j+1] = Tb\n    j=j+1\n\n# SALIDA\nprint('Tabla de resultados')\nnp.set_printoptions(precision=2)\nprint(u)\n\n# Gr\u00e1fica\nsalto = int(n\/10)\nif (salto == 0):\n    salto = 1\nfor j in range(0,n,salto):\n    vector = u[:,j]\n    plt.plot(xi,vector)\n    plt.plot(xi,vector, '.r')\nplt.xlabel('x[i]')\nplt.ylabel('t[j]')\nplt.title('Soluci\u00f3n EDP parab\u00f3lica')\nplt.show()\n<\/pre><\/div>","protected":false},"excerpt":{"rendered":"
Ejercicio: 2Eva2011TII_T3 EDP Parab\u00f3lica, expl\u00edcito La ecuaci\u00f3n a resolver es: Como el m\u00e9todo requerido es expl\u00edcito se usan las diferencias divididas: Reordenando la ecuaci\u00f3n al modelo realizado en clase: multiplicando cada lado por \u0394t Se establece el valor de obteniendo la ecuaci\u00f3n general, ordenada por \u00edndices de izquierda a derecha: con valores de: Se realizan […]<\/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":[48],"tags":[58,54],"class_list":["post-3863","post","type-post","status-publish","format-standard","hentry","category-mn-s2eva20","tag-ejemplos-python","tag-mnumericos"],"_links":{"self":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/3863","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=3863"}],"version-history":[{"count":3,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/3863\/revisions"}],"predecessor-version":[{"id":23876,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/3863\/revisions\/23876"}],"wp:attachment":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/media?parent=3863"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/categories?post=3863"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/tags?post=3863"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}