{
"metadata": {
"name": "",
"signature": "sha256:2221c3190a93e10b0d60c9c13471ed510a28f24c08c158331f416508f81a0bfb"
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Chapter09:Numerical Solution of Partial Differential Equations"
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex9.1:pg-350"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#standard five point formula\n",
"#example 9.1\n",
"#page 350\n",
"\n",
"u2=5.0;u3=1.0;\n",
"for i in range(0,3):\n",
" u1=(u2+u3+6.0)/4.0;\n",
" u2=(u1/2.0)+(5.0/2.0);\n",
" u3=(u1/2.0)+(1.0/2.0);\n",
" print\" the values are u1=%d\\t u2=%d\\t u3=%d\\t\\n\\n\" %(u1,u2,u3)\n",
" \n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" the values are u1=3\t u2=4\t u3=2\t\n",
"\n",
"\n",
" the values are u1=3\t u2=4\t u3=2\t\n",
"\n",
"\n",
" the values are u1=3\t u2=4\t u3=2\t\n",
"\n",
"\n"
]
}
],
"prompt_number": 120
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex9.2:pg-351"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#solution of laplace equation by jacobi method,gauss-seidel method and SOR method\n",
"#example 9.2\n",
"#page 351\n",
"u1=0.25;u2=0.25;u3=0.5;u4=0.5;#initial values\n",
"print \"jacobis iteration process\\n\\n\"\n",
"print\"u1\\t u2\\t u3\\t u4\\t \\n\\n\"\n",
"print \"%f\\t %f\\t %f\\t %f\\t \\n\" %(u1,u2,u3,u4)\n",
"for i in range(0,7):\n",
" u11=(0+u2+0+u4)/4\n",
" u22=(u1+0+0+u3)/4\n",
" u33=(1+u2+0+u4)/4\n",
" u44=(1+0+u3+u1)/4\n",
" u1=u11;u2=u22;u3=u33;u4=u44;\n",
" print \"%f\\t %f\\t %f\\t %f\\t \\n\" %(u11,u22,u33,u44) \n",
"print \" gauss seidel process\\n\\n\"\n",
"u1=0.25;u2=0.3125;u3=0.5625;u4=0.46875;#initial values\n",
"print \"u1\\t u2\\t u3\\t u4\\t \\n\\n\"\n",
"print \"%f\\t %f\\t %f\\t %f\\t \\n\" %(u1,u2,u3,u4)\n",
"for i in range(0,4):\n",
"\n",
" u1=(0.0+u2+0.0+u4)/4.0\n",
" u2=(u1+0.0+0.0+u3)/4.0\n",
" u3=(1.0+u2+0.0+u4)/4.0\n",
" u4=(1.0+0.0+u3+u1)/4.0\n",
" print \"%f\\t %f\\t %f\\t %f\\t \\n\" %(u1,u2,u3,u4) \n",
"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"jacobis iteration process\n",
"\n",
"\n",
"u1\t u2\t u3\t u4\t \n",
"\n",
"\n",
"0.250000\t 0.250000\t 0.500000\t 0.500000\t \n",
"\n",
"0.187500\t 0.187500\t 0.437500\t 0.437500\t \n",
"\n",
"0.156250\t 0.156250\t 0.406250\t 0.406250\t \n",
"\n",
"0.140625\t 0.140625\t 0.390625\t 0.390625\t \n",
"\n",
"0.132812\t 0.132812\t 0.382812\t 0.382812\t \n",
"\n",
"0.128906\t 0.128906\t 0.378906\t 0.378906\t \n",
"\n",
"0.126953\t 0.126953\t 0.376953\t 0.376953\t \n",
"\n",
"0.125977\t 0.125977\t 0.375977\t 0.375977\t \n",
"\n",
" gauss seidel process\n",
"\n",
"\n",
"u1\t u2\t u3\t u4\t \n",
"\n",
"\n",
"0.250000\t 0.312500\t 0.562500\t 0.468750\t \n",
"\n",
"0.195312\t 0.189453\t 0.414551\t 0.402466\t \n",
"\n",
"0.147980\t 0.140633\t 0.385775\t 0.383439\t \n",
"\n",
"0.131018\t 0.129198\t 0.378159\t 0.377294\t \n",
"\n",
"0.126623\t 0.126196\t 0.375872\t 0.375624\t \n",
"\n"
]
}
],
"prompt_number": 51
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex9.4:pg-354"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#poisson equation\n",
"#exaample 9.4\n",
"#page 354\n",
"u2=0.0;u4=0.0;\n",
"print \" u1\\t u2\\t u3\\t u4\\t\\n\\n\"\n",
"for i in range(0,6):\n",
" u1=(u2/2.0)+30.0\n",
" u2=(u1+u4+150.0)/4.0\n",
" u4=(u2/2.0)+45.0\n",
" print \"%0.2f\\t %0.2f\\t %0.2f\\t %0.2f\\n\" %(u1,u2,u2,u4)\n",
"print \" from last two iterates we conclude u1=67 u2=75 u3=75 u4=83\\n\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" u1\t u2\t u3\t u4\t\n",
"\n",
"\n",
"30.00\t 45.00\t 45.00\t 67.50\n",
"\n",
"52.50\t 67.50\t 67.50\t 78.75\n",
"\n",
"63.75\t 73.12\t 73.12\t 81.56\n",
"\n",
"66.56\t 74.53\t 74.53\t 82.27\n",
"\n",
"67.27\t 74.88\t 74.88\t 82.44\n",
"\n",
"67.44\t 74.97\t 74.97\t 82.49\n",
"\n",
" from last two iterates we conclude u1=67 u2=75 u3=75 u4=83\n",
"\n"
]
}
],
"prompt_number": 59
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex9.6:pg-362"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#bender-schmidt formula\n",
"#example 9.6\n",
"#page 362\n",
"def f(x):\n",
" return (4*x)-(x*x)\n",
"#u=[f(0),f(1),f(2),f(3),f(4)]\n",
"u1=f(0);u2=f(1);u3=f(2);u4=f(3);u5=f(4);\n",
"u11=(u1+u3)/2\n",
"u12=(u2+u4)/2\n",
"u13=(u3+u5)/2\n",
"print \"u11=%0.2f\\t u12=%0.2f\\t u13=%0.2f\\t \\n\" %(u11,u12,u13)\n",
"u21=(u1+u12)/2.0\n",
"u22=(u11+u13)/2.0\n",
"u23=(u12+0)/2.0\n",
"print \"u21=%0.2f\\t u22=%0.2f\\t u23=%0.2f\\t \\n\" %(u21,u22,u23)\n",
"u31=(u1+u22)/2.0\n",
"u32=(u21+u23)/2.0\n",
"u33=(u22+u1)/2.0\n",
"print \"u31=%0.2f\\t u32=%0.2f\\t u33=%0.2f\\t \\n\" % (u31,u32,u33)\n",
"u41=(u1+u32)/2.0\n",
"u42=(u31+u33)/2.0\n",
"u43=(u32+u1)/2.0\n",
"print \"u41=%0.2f\\t u42=%0.2f\\t u43=%0.2f\\t \\n\" % (u41,u42,u43)\n",
"u51=(u1+u42)/2.0\n",
"u52=(u41+u43)/2.0\n",
"u53=(u42+u1)/2.0\n",
"print \"u51=%0.2f\\t u52=%0.2f\\t u53=%0.2f\\t \\n\" % (u51,u52,u53)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"u11=2.00\t u12=3.00\t u13=2.00\t \n",
"\n",
"u21=1.50\t u22=2.00\t u23=1.50\t \n",
"\n",
"u31=1.00\t u32=1.50\t u33=1.00\t \n",
"\n",
"u41=0.75\t u42=1.00\t u43=0.75\t \n",
"\n",
"u51=0.50\t u52=0.75\t u53=0.50\t \n",
"\n"
]
}
],
"prompt_number": 77
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex9.7:pg-363"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"#bender-schimdt's formula and crank-nicolson formula\n",
"#example 9.7\n",
"#page 363\n",
"#bender -schimdt's formula\n",
"z=math.pi\n",
"def f(x,t):\n",
" return math.exp(z*z*t*-1)*sin(z*x)\n",
"#u=[f(0,0),f(0.2,0),f(0.4,0),f(0.6,0),f(0.8,0),f(1,0)];\n",
"u1=f(0,0)\n",
"u2=f(0.2,0)\n",
"u3=f(0.4,0)\n",
"u4=f(0.6,0)\n",
"u5=f(0.8,0)\n",
"u6=f(1.0,0)\n",
"u11=u3/2;u12=(u2+u4)/2;u13=u12;u14=u11;\n",
"print \"u11=%f\\t u12=%f\\t u13=%f\\t u14=%f\\n\\n\" % (u11,u12,u13,u14)\n",
"u21=u12/2;u22=(u12+u14)/2;u23=u22;u24=u21;\n",
"print \"u21=%f\\t u22=%f\\t u23=%f\\t u24=%f\\n\\n\" % (u21,u22,u23,u24)\n",
"print \"the error in the solution is: %f\\n\\n\" % (math.fabs(u22-f(0.6,0.04)))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"u11=0.475528\t u12=0.769421\t u13=0.769421\t u14=0.475528\n",
"\n",
"\n",
"u21=0.384710\t u22=0.622475\t u23=0.622475\t u24=0.384710\n",
"\n",
"\n",
"the error in the solution is: 0.018372\n",
"\n",
"\n"
]
}
],
"prompt_number": 119
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex9.8:pg-364"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from numpy import matrix\n",
"#heat equation using crank-nicolson method\n",
"#example 9.8\n",
"#page 364\n",
"z=0.01878;\n",
"#h=1/2;l=1/8,i=1\n",
"u01=0.0;u21=1.0/8.0;\n",
"u11=(u21+u01)/6.0;\n",
"print \" u11=%f\\n\\n\" % (u11)\n",
"print \"error is %f\\n\\n\" % (math.fabs(u11-z))\n",
"#h=1/4,l=1/8,i=1,2,3\n",
"A=matrix([['-3' ,'-1' ,'0'],['1','-3','1'],['0','1','-3']])\n",
"C=matrix([['0'],['0'],['-1/8']])\n",
"#here we found inverese of A then we multipy it with C\n",
"u12=0.01786\n",
"print \"u12=%f\\n\\n\" % (u12)\n",
"print \"error is %f\\n\\n\" %(math.fabs(u12-z))\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
" u11=0.020833\n",
"\n",
"\n",
"error is 0.002053\n",
"\n",
"\n",
"u12=0.017860\n",
"\n",
"\n",
"error is 0.000920\n",
"\n",
"\n"
]
}
],
"prompt_number": 105
},
{
"cell_type": "code",
"collapsed": false,
"input": [],
"language": "python",
"metadata": {},
"outputs": []
}
],
"metadata": {}
}
]
}