{"id":9060,"date":"2023-08-31T08:35:28","date_gmt":"2023-08-31T13:35:28","guid":{"rendered":"http:\/\/blog.espol.edu.ec\/analisisnumerico\/?p=9060"},"modified":"2026-04-05T20:43:46","modified_gmt":"2026-04-06T01:43:46","slug":"s2eva2023paoi_t3-edp-eliptica-placa-rectangular-con-frontera-variable","status":"publish","type":"post","link":"https:\/\/blog.espol.edu.ec\/algoritmos101\/mn-s2eva30\/s2eva2023paoi_t3-edp-eliptica-placa-rectangular-con-frontera-variable\/","title":{"rendered":"s2Eva2023PAOI_T3 EDP el\u00edptica, placa rectangular con frontera variable"},"content":{"rendered":"\n

Ejercicio<\/strong>: 2Eva2023PAOI_T3 EDP el\u00edptica, placa rectangular con frontera variable<\/a><\/p>\n\n\n\n

Dada la EDP el\u00edptica,<\/p>\n\n\n \\frac{\\partial ^2 u}{\\partial x^2} + \\frac{\\partial ^2 u}{\\partial y^2} = \\Big( x^2 + y^2 \\Big) e^{xy} <\/span>\n\n\n 0 < x < 1<\/span>\n\n\n 0 < y < 0.5 <\/span>\n\n\n\n

Se convierte a la versi\u00f3n discreta usando diferencias divididas centradas, seg\u00fan se puede mostrar con la gr\u00e1fica de malla.<\/p>\n\n\n\n

literal b<\/h3>\n\n\n\n
\"EDP<\/figure>\n\n\n\n

literal a<\/h3>\n\n\n\n
\n\n\n \\frac{u[i-1,j]-2u[i,j]+u[i+1,j]}{\\Delta x^2} +<\/span>\n\n\n + \\frac{u[i,j-1]-2u[i,j]+u[i,j+1]}{\\Delta y^2} = \\Big( x^2 + y^2 \\Big) e^{xy}<\/span>\n\n\n\n
\n\n\n\n

literal c<\/h3>\n\n\n\n

Se agrupan los t\u00e9rminos \u0394x, \u0394y semejante a formar un \u03bb al multiplicar todo por \u0394y2<\/sup><\/p>\n\n\n \\frac{\\Delta y^2}{\\Delta x^2}\\Big(u[i-1,j]-2u[i,j]+u[i+1,j] \\Big) +<\/span>\n\n\n + \\frac{\\Delta y^2}{\\Delta y^2}\\Big(u[i,j-1]-2u[i,j]+u[i,j+1]\\Big) = \\Big( x^2 + y^2 \\Big) e^{xy}\\frac{\\Delta y^2}{\\Delta x^2}<\/span>\n\n\n\n

\n\n\n\n

los tama\u00f1os de paso en ambos ejes son de igual valor, se simplifica la ecuaci\u00f3n<\/p>\n\n\n \\lambda= \\frac{\\Delta y^2}{\\Delta x^2} = 1<\/span>\n\n\n\n

\n\n\n\n

se simplifica el coeficiente en \u03bb =1<\/p>\n\n\n u[i-1,j]-2u[i,j]+u[i+1,j] +<\/span>\n\n\n + u[i,j-1]-2u[i,j]+u[i,j+1] = \\Big( x^2 + y^2 \\Big) e^{xy}<\/span>\n\n\n\n

\n\n\n\n

Se agrupan los t\u00e9rminos iguales<\/p>\n\n\n u[i-1,j]-4u[i,j]+ u[i+1,j] + u[i,j-1] +u[i,j+1]<\/span>\n\n\n = \\Big( x^2 + y^2 \\Big) e^{xy}<\/span>\n\n\n\n

\n\n\n\n

Se desarrollan las iteraciones para tres rombos y se genera el sistema de ecuaciones a resolver.
para i=1,j=1<\/p>\n\n\n u[0,1]-4u[1,1]+ u[2,1] + u[1,0] +u[1,2] <\/span>\n\n\n = \\Big( x[1]^2 + y[1]^2 \\Big) e^{x[1]y[1]} <\/span>\n\n\n 1-4u[1,1]+ u[2,1] + 1 + \\frac{1}{8} <\/span>\n\n\n = \\Big( 0.25^2 + 0.25^2 \\Big) e^{(0.25) (0.25)} <\/span>\n\n\n -4u[1,1]+ u[2,1] = \\Big( 0.25^2 + 0.25^2 \\Big) e^{(0.25) (0.25)} - \\frac{1}{8}<\/span>\n\n\n\n

para i=2, j=1<\/p>\n\n\n u[1,1]-4u[2,1]+ u[3,1] + u[2,0] +u[2,2]<\/span>\n\n\n = \\Big( x[2]^2 + y[1]^2 \\Big) e^{x[2]y[1]}<\/span>\n\n\n u[1,1]-4u[2,1]+ u[3,1] + 1 + 0.25<\/span>\n\n\n = \\Big( 0.5^2 + 0.25^2 \\Big) e^{(0.5)(0.25)}<\/span>\n\n\n u[1,1]-4u[2,1]+ u[3,1] = \\Big( 0.5^2 + 0.25^2 \\Big) e^{(0.5)(0.25)} -1.25<\/span>\n\n\n\n

para i=3, j=1<\/p>\n\n\n u[2,1]-4u[3,1]+ u[4,1] + u[3,0] +u[3,2]<\/span>\n\n\n = \\Big( x[3]^2 + y[1]^2 \\Big) e^{x[3]y[1]}<\/span>\n\n\n u[2,1]-4u[3,1]+ 0.25 + 0 + \\frac{3}{8}<\/span>\n\n\n = \\Big( 0.75^2 + 0.25^2 \\Big) e^{(0.75)(0.25)}<\/span>\n\n\n u[2,1]-4u[3,1] = \\Big( 0.75^2 + 0.25^2 \\Big) e^{(0.75)(0.25)} - 0.25 - \\frac{3}{8}<\/span>\n\n\n\n

con lo que se puede crear un sistema de ecuaciones y resolver el sistema para cada punto desconocido<\/p>\n\n\n \\begin{pmatrix} -4 & 1 & 0 & \\Big| & 0.008061807364732415 \\\\ 1 & -4 & 1 & \\Big| & -0.8958911084166168 \\\\0 & 1 & -4 &\\Big| & 0.1288939058881129 \\end{pmatrix} <\/span>\n\n\n\n

se obtiene los resultados para:<\/p>\n\n\n\n

u[1,1] = 0.05953113
u[2,1] = 0.24618634
u[3,1] = 0.02932311<\/p>\n\n\n\n

>>> import numpy as np\n>>> (0.25**2+0.25**2)*np.exp(0.25*0.25) - 1\/8\n0.008061807364732415\n>>> (0.5**2+0.25**2)*np.exp(0.5*0.25) - 1.25\n-0.8958911084166168\n>>> (0.75**2+0.25**2)*np.exp(0.75*0.25) - 0.25 -3\/8\n0.1288939058881129\n>>> A=[[-4,1,0],[1,-4,1],[0.0,1.0,-4.0]]\n>>> B = [0.008061807364732415, -0.8958911084166168, 0.1288939058881129]\n>>> np.linalg.solve(A,B)\narray([0.05953113, 0.24618634, 0.02932311])<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"
Ejercicio: 2Eva2023PAOI_T3 EDP el\u00edptica, placa rectangular con frontera variable Dada la EDP el\u00edptica, Se convierte a la versi\u00f3n discreta usando diferencias divididas centradas, seg\u00fan se puede mostrar con la gr\u00e1fica de malla. literal b literal a literal c Se agrupan los t\u00e9rminos \u0394x, \u0394y semejante a formar un \u03bb al multiplicar todo por \u0394y2 los tama\u00f1os […]<\/p>\n","protected":false},"author":8043,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"wp-custom-template-entrada-mn-ejemplo","format":"standard","meta":{"footnotes":""},"categories":[49],"tags":[58,54],"class_list":["post-9060","post","type-post","status-publish","format-standard","hentry","category-mn-s2eva30","tag-ejemplos-python","tag-mnumericos"],"_links":{"self":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/9060","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=9060"}],"version-history":[{"count":3,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/9060\/revisions"}],"predecessor-version":[{"id":23890,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/9060\/revisions\/23890"}],"wp:attachment":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/media?parent=9060"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/categories?post=9060"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/tags?post=9060"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}