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diff --git a/Higher_Engineering_Mathematics_by_B._S._Grewal/chapter27_2.ipynb b/Higher_Engineering_Mathematics_by_B._S._Grewal/chapter27_2.ipynb new file mode 100644 index 00000000..6b7c3c32 --- /dev/null +++ b/Higher_Engineering_Mathematics_by_B._S._Grewal/chapter27_2.ipynb @@ -0,0 +1,1084 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Chapter 27 : Numerical Solution Of Ordinary Differential Equations" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 27.1, page no. 686" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Solution through picards method \n", + "The no of iterations required3\n", + "y(0)=1 and y(x)=x+y \n", + "Y = x**4/24 + x**3/3 + x**2 + x + 1\n" + ] + } + ], + "source": [ + "import numpy,sympy,math\n", + "\n", + "x = sympy.Symbol('x')\n", + "print \"Solution through picards method \"\n", + "n = int(raw_input(\"The no of iterations required\")) \n", + "print \"y(0)=1 and y(x)=x+y \"\n", + "yo = 1\n", + "yn = 1\n", + "for i in range(1,n+1):\n", + " yn = yo+sympy.integrate(yn+x,(x,0,x))\n", + "print \"Y = \",yn" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 27.2, page no. 687" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Solution through picards method \n", + "The no of iterations required : 2\n", + "y(0)=1 and y(x)=x+y \n", + "Y = -Integral(2*x/(2*log(x + 1) + 1), x) + Integral(2*x/(2*log(x + 1) + 1), (x, 0)) - Integral(-2*log(x + 1)/(2*log(x + 1) + 1), x) + Integral(-2*log(x + 1)/(2*log(x + 1) + 1), (x, 0)) - Integral(-1/(2*log(x + 1) + 1), x) + Integral(-1/(2*log(x + 1) + 1), (x, 0)) + 1.0\n" + ] + } + ], + "source": [ + "import numpy,sympy,math\n", + "\n", + "x = sympy.Symbol('x')\n", + "print \"Solution through picards method \"\n", + "n = int(raw_input(\"The no of iterations required : \")) \n", + "print \"y(0)=1 and y(x)=x+y \"\n", + "yo = 1\n", + "y = 1\n", + "for i in range(1,n+1):\n", + " f = (y-x)/(y+x) \n", + " y = yo+sympy.integrate(f,(x,0,x))\n", + "x = 0.1\n", + "print \"Y = \",y.evalf()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 27.5, page no. 690" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Solution using Eulers Method \n", + "0.3\n", + "1.362\n", + "Input the number of iteration :− 4\n", + "The value of y is :− 1.5282\n" + ] + } + ], + "source": [ + "print \"Solution using Eulers Method \"\n", + "print x\n", + "print y\n", + "n = int(raw_input(\"Input the number of iteration :− \"))\n", + "x = 0\n", + "y = 1\n", + "for i in range(1,n+1):\n", + " y1 = x+y\n", + " y = y+0.1*y1\n", + " x = x+0.1\n", + "print \"The value of y is :− \",y" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 27.6, page no. 691" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Solution using Eulers Method \n", + "0.4\n", + "1.5282\n", + "Input the number of iteration :− 2\n", + "1.02\n", + "1.03642857143\n", + "The value of y is :− 1.03642857143\n" + ] + } + ], + "source": [ + "print \"Solution using Eulers Method \"\n", + "print x\n", + "print y\n", + "n = int(raw_input(\"Input the number of iteration :− \"))\n", + "x = 0\n", + "y = 1\n", + "for i in range(1,n+1):\n", + " y1 = (y-x)/(y+x)\n", + " y = y+0.02*y1\n", + " x = x+0.1\n", + " print y\n", + "print \"The value of y is :− \",y" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 27.7, page no. 692" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Solution using Eulers Method \n", + "0.2\n", + "1.03642857143\n", + "Input the number of iteration :− 3\n", + "1.1\n", + "1.11\n", + "1.1105\n", + "1.110525\n", + "−−−−−−−−−−−−−−−−−−−−−−− \n", + "1.2315775\n", + "1.2315775\n", + "1.242630125\n", + "1.24318275625\n", + "1.24321038781\n", + "−−−−−−−−−−−−−−−−−−−−−−− \n", + "1.38753142659\n", + "1.38753142659\n", + "1.39974747853\n", + "1.40035828113\n", + "1.40038882126\n", + "−−−−−−−−−−−−−−−−−−−−−−− \n", + "1.57042770339\n", + "The value of y is :− 1.40038882126\n" + ] + } + ], + "source": [ + "print \"Solution using Eulers Method \"\n", + "print x\n", + "print y\n", + "n = int(raw_input(\"Input the number of iteration :− \"))\n", + "x = 0.1\n", + "m = 1\n", + "y = 1\n", + "yn = 1\n", + "y1 = 1\n", + "k = 1\n", + "for i in range(1,n+1):\n", + " yn = y \n", + " for i in range(1,5):\n", + " m = (k+y1)/2\n", + " yn = y+0.1*m\n", + " y1 = (yn+x)\n", + " print yn\n", + " print \"−−−−−−−−−−−−−−−−−−−−−−− \"\n", + " y = yn \n", + " m = y1\n", + " yn = yn+0.1*m \n", + " print yn\n", + " x = x+0.1\n", + " yn = y \n", + " k = m\n", + "print \"The value of y is :− \",y" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 27.8, page no. 694" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Solution using Eulers Method \n", + "0.2\n", + "2\n", + "Input the number of iteration :− 3\n", + "2.06020299957\n", + "2.06551474469\n", + "2.06561668931\n", + "2.06561864352\n", + "−−−−−−−−−−−−−−−−−−−−−−−\n", + "2.13665600548\n", + "2.13665600548\n", + "2.14156348217\n", + "2.14164742068\n", + "2.14164885497\n", + "−−−−−−−−−−−−−−−−−−−−−−−\n", + "2.22267196493\n", + "2.22267196493\n", + "2.22722645093\n", + "2.22729646948\n", + "2.22729754504\n", + "−−−−−−−−−−−−−−−−−−−−−−−\n", + "2.31757184825\n", + "The value of y is :− 2.22729754504\n" + ] + } + ], + "source": [ + "import math\n", + "\n", + "print \"Solution using Eulers Method \"\n", + "print x\n", + "print y\n", + "n = int(raw_input(\"Input the number of iteration :− \"))\n", + "x = 0.2\n", + "m = 0.301\n", + "y = 2\n", + "yn = 2\n", + "y1 = math.log10(2)\n", + "k = 0.301\n", + "for i in range(1,n+1):\n", + " yn = y\n", + " for i in range(1,5):\n", + " m = (k+y1)/2\n", + " yn = y+0.2*m\n", + " y1 = math.log10(yn+x)\n", + " print yn\n", + " print \"−−−−−−−−−−−−−−−−−−−−−−−\"\n", + " y = yn\n", + " m = y1\n", + " yn = yn+0.2*m \n", + " print yn\n", + " x = x+0.2\n", + " yn = y\n", + " k = m\n", + "print \"The value of y is :− \",y" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 27.9, page no. 696" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Solution using Eulers Method \n", + "0.8\n", + "2.22729754504\n", + "Input the number of iteration :− 2\n", + "1.2\n", + "1.2295445115\n", + "1.23088482816\n", + "1.23094524903\n", + "−−−−−−−−−−−−−−−−−−−−−−− \n", + "1.49284119302\n", + "1.49284119302\n", + "1.52407510156\n", + "1.52534665765\n", + "1.52539814634\n", + "−−−−−−−−−−−−−−−−−−−−−−− \n", + "1.85241216588\n", + "The value of y is :− 1.52539814634\n" + ] + } + ], + "source": [ + "import math\n", + "\n", + "print \"Solution using Eulers Method \"\n", + "print x\n", + "print y\n", + "n = int(raw_input(\"Input the number of iteration :− \"))\n", + "x = 0.2 \n", + "m = 1\n", + "y = 1\n", + "yn = 1\n", + "y1 = 1\n", + "k = 1\n", + "for i in range(1,n+1):\n", + " yn = y\n", + " for i in range(1,5):\n", + " m = (k+y1)/2\n", + " yn = y+0.2*m \n", + " y1 = math.sqrt(yn)+x\n", + " print yn\n", + " print \"−−−−−−−−−−−−−−−−−−−−−−− \"\n", + " y = yn\n", + " m = y1\n", + " yn = yn+0.2*m\n", + " print yn\n", + " x = x+0.2\n", + " yn = y\n", + " k = m\n", + "print \"The value of y is :− \",y" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "## Example 27.10, page no. 697" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Runges method\n", + "The required approximate value is :− \n", + "1.0\n" + ] + } + ], + "source": [ + "print \"Runges method\"\n", + "def f(x,y):\n", + " y = x+y\n", + " return y\n", + "x = 0\n", + "y = 1\n", + "h = 0.2\n", + "k1 = h*f(x,y)\n", + "k2 = h*f(x+1/2*h,y+1/2*k1)\n", + "kk = h*f(x+h,y+k1)\n", + "k3 = h*f(x+h,y+kk)\n", + "k = 1/6*(k1+4*k2+k3)\n", + "print \"The required approximate value is :− \"\n", + "y = y+k\n", + "print y" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 27.11, page no. 697" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Runges method\n", + "The required approximate value is :− \n", + "1.0\n" + ] + } + ], + "source": [ + "print \"Runges method\"\n", + "def f(x,y):\n", + " y = x+y\n", + " return y\n", + "x = 0\n", + "y = 1\n", + "h = 0.2\n", + "k1 = h*f(x,y)\n", + "k2 = h*f(x+1/2*h,y+1/2*k1)\n", + "k3 = h*f(x+1/2*h,y+1/2*k2)\n", + "k4 = h*f(x+h,y+k3)\n", + "k = 1/6*(k1+2*k2+2*k3+k4)\n", + "print \"The required approximate value is :− \"\n", + "y = y+k\n", + "print y" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 27.12, page no. 698" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Runga kutta method\n", + "The required approximate value is :− \n", + "1.0\n", + "To find y(0.4) put x=0.2 y=above value ie 1.196 h=0.2 \n", + "The required approximate value is :− \n", + "1.0\n" + ] + } + ], + "source": [ + "print \"Runga kutta method\"\n", + "def f(x,y):\n", + " y = (y**2-x**2)/(x**2+y**2)\n", + " return y\n", + "x = 0\n", + "y = 1\n", + "h = 0.2\n", + "k1 = h*f(x,y)\n", + "k2 = h*f(x+1/2*h,y+1/2*k1)\n", + "k3 = h*f(x+1/2*h,y+1/2*k2)\n", + "k4 = h*f(x+h,y+k3)\n", + "k = 1/6*(k1+2*k2+2*k3+k4)\n", + "print \"The required approximate value is :− \"\n", + "y = y+k\n", + "print y\n", + "print \"To find y(0.4) put x=0.2 y=above value ie 1.196 h=0.2 \"\n", + "x = 0.2\n", + "h = 0.2\n", + "k1 = h*f(x,y)\n", + "k2 = h*f(x+1/2*h,y+1/2*k1)\n", + "k3 = h*f(x+1/2*h,y+1/2*k2)\n", + "k4 = h*f(x+h,y+k3)\n", + "k = 1/6*(k1+2*k2+2*k3+k4)\n", + "print \"The required approximate value is :− \"\n", + "y = y+k\n", + "print y" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 27.13, page no. 699" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Runga kutta method\n", + "The required approximate value is :− \n", + "1.0\n", + "To find y(0.4) put x=0.2 y=above value ie 1.196 h=0.2 \n", + "The required approximate value is :− \n", + "1.0\n" + ] + } + ], + "source": [ + "print \"Runga kutta method\"\n", + "def f(x,y):\n", + " yy = x+y**2\n", + " return yy\n", + "x = 0\n", + "y = 1\n", + "h = 0.1\n", + "k1 = h*f(x,y)\n", + "k2 = h*f(x+1/2*h,y+1/2*k1)\n", + "k3 = h*f(x+1/2*h,y+1/2*k2) \n", + "k4 = h*f(x+h,y+k3)\n", + "k = 1/6*(k1+2*k2+2*k3+k4)\n", + "print \"The required approximate value is :− \"\n", + "y = y+k\n", + "print y\n", + "print \"To find y(0.4) put x=0.2 y=above value ie 1.196 h=0.2 \"\n", + "x = 0.1\n", + "h = 0.1\n", + "k1 = h*f(x,y)\n", + "k2 = h*f(x+1/2*h,y+1/2*k1)\n", + "k3 = h*f(x+1/2*h,y+1/2*k2)\n", + "k4 = h*f(x+h,y+k3)\n", + "k = 1/6*(k1+2*k2+2*k3+k4)\n", + "print \"The required approximate value is :− \"\n", + "y = y+k\n", + "print y" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 27.14, page no. 700" + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "y1= x**2/2\n", + "y2= -x**5/20 + x**2/2\n", + "y3= -x**5/20 + x**2/2\n", + "Determining the initial values for milnes method using y3 \n", + "x=0.0 y0=0.0 f0=0 \n", + "x=0.2 y1= x**2/2\n", + "f1= -x**4/4 + 0.2\n", + "x=0.4 y2= -x**5/20 + x**2/2\n", + "f2= -(-x**5/20 + x**2/2)**2 + 0.4\n", + "x=0.6 y3= -x**5/20 + x**2/2\n", + "f3= -(-x**5/20 + x**2/2)**2 + 0.6\n", + "Using predictor method to find y4\n", + "y4= -0.1*x**4 - 0.2*(-x**5/20 + x**2/2)**2 + 0.24\n", + "f4= -(-x**5/20 + x**2/2)**2 + 0.8\n", + "Using predictor method to find y5 \n", + "[-1.53353211362738, 2.59188514369719, -1.51825864909117 - 1.96576535664013*I, -1.51825864909117 + 1.96576535664013*I, -0.611593620665791 - 2.14551655586226*I, -0.611593620665791 + 2.14551655586226*I, 0.00495892544779777 - 0.780181689871346*I, 0.00495892544779777 + 0.780181689871346*I, 1.59571682927425 - 0.8271211188914*I, 1.59571682927425 + 0.8271211188914*I]\n", + "f5 = [-1.28459666867223, 2.38248090853886, -1.33416685318757 - 1.68618279990398*I, -1.33416685318757 + 1.68618279990398*I, -1.00716479884721 - 2.10036734766602*I, -1.00716479884721 + 2.10036734766602*I, 0.142926398918135 - 1.33989714259572*I, 0.142926398918135 + 1.33989714259572*I, 1.64946313318333 - 0.385565643189579*I, 1.64946313318333 + 0.385565643189579*I]\n", + "Hence y(1)= [-1.53353211362738, 2.59188514369719, -1.51825864909117 - 1.96576535664013*I, -1.51825864909117 + 1.96576535664013*I, -0.611593620665791 - 2.14551655586226*I, -0.611593620665791 + 2.14551655586226*I, 0.00495892544779777 - 0.780181689871346*I, 0.00495892544779777 + 0.780181689871346*I, 1.59571682927425 - 0.8271211188914*I, 1.59571682927425 + 0.8271211188914*I]\n" + ] + } + ], + "source": [ + "import sympy\n", + "\n", + "x = sympy.Symbol('x')\n", + "yo = 0\n", + "y = 0\n", + "h = 0.2\n", + "f = x-y**2\n", + "y = sympy.integrate(f,(x,0,x))\n", + "y1 = yo+y\n", + "print \"y1= \",y1\n", + "f = x-y**2\n", + "y = sympy.integrate(f,(x,0,x))\n", + "y2 = yo+y\n", + "print \"y2= \",y2\n", + "y = x-y**2\n", + "y = sympy.integrate(f,(x,0,x))\n", + "y3 = yo+y\n", + "print \"y3= \",y3\n", + "print \"Determining the initial values for milnes method using y3 \"\n", + "print \"x=0.0 y0=0.0 f0=0 \"\n", + "print \"x=0.2 y1= \",\n", + "x = 0.2\n", + "print y1\n", + "#y1 = sympy.solve(y1)\n", + "print \"f1=\",\n", + "f1 = x-y1**2\n", + "print f1\n", + "print \"x=0.4 y2= \",\n", + "x = 0.4\n", + "print y2\n", + "print \"f2= \",\n", + "f2 = x-y2**2\n", + "print f2\n", + "print \"x=0.6 y3= \",\n", + "x = 0.6\n", + "print y3\n", + "print \"f3=\",\n", + "f3 = x-y3**2\n", + "print f3\n", + "print \"Using predictor method to find y4\"\n", + "x = 0.8\n", + "y4 = yo+4/3*h*(2*f1-f2+2*f3)\n", + "print \"y4=\",y4\n", + "f4 = x-y**2\n", + "print \"f4= \",f4\n", + "print \"Using predictor method to find y5 \"\n", + "x = 1.0\n", + "y5 = sympy.solve(y1+4/3*h*(2*f2-f3+2*f4))\n", + "print y5\n", + "f5 = sympy.solve(x-y**2)\n", + "print \"f5 =\",f5\n", + "print \"Hence y(1)= \",y5" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 27.15, page no. 704" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Runga kutta method\n", + "The required approximate value is :− \n", + "1.0\n", + "To find y(0.4) put x=0.2 y=above value ie 1.196 h=0.2 \n", + "The required approximate value is :− \n", + "1.0\n", + "To find y(0.4) put x=0.2 y=above value ie 1.196 h=0.2 \n", + "The required approximate value is :− \n", + "1.0\n", + "y0 y1 y2 y3 are respectively: \n", + "1.0 1.0 1.0 1\n", + "f0 f1 f2 f3 are respectively: \n", + "1.3 1.2 1.1 1\n", + "finding y4 using predictors milne method x=0.4\n", + "y4 = 1.48\n", + "f4 = 2.7824\n", + "Using corrector method : \n", + "y4 = -0.0333333333333333*(-x**5/20 + x**2/2)**2 + 1.24\n", + "f4 = -0.0133333333333333*(-x**5/20 + x**2/2)**2 + (-0.0333333333333333*(-x**5/20 + x**2/2)**2 + 1.24)**2 + 0.496\n" + ] + } + ], + "source": [ + "print \"Runga kutta method\"\n", + "def f(x,y):\n", + " yy = x*y+y**2\n", + " return yy\n", + "y0 = 1\n", + "x = 0\n", + "y = 1\n", + "h = 0.1\n", + "k1 = h*f(x,y)\n", + "k2 = h*f(x+1/2*h,y+1/2*k1)\n", + "k3 = h*f(x+1/2*h,y+1/2*k2)\n", + "k4 = h*f(x+h,y+k3)\n", + "ka = 1/6*(k1+2*k2+2*k3+k4)\n", + "print \"The required approximate value is :− \"\n", + "y1 = y+ka \n", + "y = y+ka \n", + "print y\n", + "print \"To find y(0.4) put x=0.2 y=above value ie 1.196 h=0.2 \"\n", + "x = 0.1\n", + "h = 0.1\n", + "k1 = h*f(x,y)\n", + "k2 = h*f(x+1/2*h,y+1/2*k1)\n", + "k3 = h*f(x+1/2*h,y+1/2*k2)\n", + "k4 = h*f(x+h,y+k3)\n", + "kb = 1/6*(k1+2*k2+2*k3+k4)\n", + "print \"The required approximate value is :− \"\n", + "y2 = y+kb \n", + "y = y+kb\n", + "print y\n", + "print \"To find y(0.4) put x=0.2 y=above value ie 1.196 h=0.2 \"\n", + "x = 0.2\n", + "h = 0.1\n", + "k1 = h*f(x,y)\n", + "k2 = h*f(x+1/2*h,y+1/2*k1)\n", + "k3 = h*f(x+1/2*h,y+1/2*k2)\n", + "k4 = h*f(x+h,y+k3)\n", + "kc = 1/6*(k1+2*k2+2*k3+k4)\n", + "print \"The required approximate value is :− \"\n", + "y3 = y+kc\n", + "y = y+kc \n", + "print y\n", + "f0 = f(0,y0)\n", + "f1 = f(0.1,y1)\n", + "f2 = f(0.2,y2)\n", + "f3 = f(0.3,y3)\n", + "print \"y0 y1 y2 y3 are respectively: \"\n", + "print y3,y2,y1,y0\n", + "print \"f0 f1 f2 f3 are respectively: \"\n", + "print f3,f2,f1,f0\n", + "print \"finding y4 using predictors milne method x=0.4\"\n", + "h = 0.1\n", + "y4 = y0+4*h/3*(2*f1-f2+2*f3)\n", + "print \"y4 = \",y4\n", + "print \"f4 = \",f(0.4,y4)\n", + "print \"Using corrector method : \"\n", + "y4 = y2+h/3*(f2+4*f3+f4) \n", + "print \"y4 = \",y4\n", + "print \"f4 = \",f(0.4,y4) " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 27.16, page no. 705" + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Using predictor method \n", + "y11 = 2.57229741667\n", + "Using corrector method \n", + "y11 = 2.57494727679\n", + "f11 = 7.00689666251\n" + ] + } + ], + "source": [ + "def f(x,y):\n", + " yy = x**2*(1+y)\n", + " return yy\n", + "y3 = 1\n", + "y2 = 1.233\n", + "y1 = 1.548\n", + "y0 = 1.979\n", + "f3 = f(1,y3)\n", + "f2 = f(1.1,y2)\n", + "f1 = f(1.2,y1)\n", + "f0 = f(1.3,y0)\n", + "print \"Using predictor method \"\n", + "h = 0.1\n", + "y11 = y0+h/24*(55*f0-59*f1+37*f2-9*f3)\n", + "print \"y11 = \",y11\n", + "x = 1.4\n", + "f11 = f(1.4,y11)\n", + "print \"Using corrector method \"\n", + "y11 = y0+h/24*(9*f11+19*f0-5*f1+f2)\n", + "print \"y11 = \",y11\n", + "f11 = f(1.4,y11)\n", + "print \"f11 = \",f11" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 27.17, page no. 706" + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Runga kutta method\n", + "The required approximate value is :− \n", + "1.0\n", + "To find y(0.4) put x=0.2 y=above value ie 1.196 h=0.2 \n", + "The required approximate value is :− \n", + "1.0\n", + "To find y(0.4) put x=0.2 y=above value ie 1.196 h=0.2 \n", + "The required approximate value is :− \n", + "1.0\n", + "y0 y1 y2 y3 are respectively :\n", + "1.0 1.0 1.0 1\n", + "f0 f1 f2 f3 are respectively :\n", + "-0.7 -0.8 -0.9 -1\n", + "Using adams method \n", + "Using the predictor\n", + "y4 = 0.935\n", + "Using corrector method \n", + "y4 = 0.9397165625\n", + "f4 = -0.483067217837\n" + ] + } + ], + "source": [ + "print \"Runga kutta method\"\n", + "def f(x,y):\n", + " yy = x-y**2\n", + " return yy\n", + "y0 = 1\n", + "x = 0\n", + "y = 1\n", + "h = 0.1\n", + "k1 = h*f(x,y)\n", + "k2 = h*f(x+1/2*h,y+1/2*k1)\n", + "k3 = h*f(x+1/2*h,y+1/2*k2)\n", + "k4 = h*f(x+h,y+k3)\n", + "ka = 1/6*(k1+2*k2+2*k3+k4) \n", + "print \"The required approximate value is :− \"\n", + "y1 = y+ka\n", + "y = y+ka\n", + "print y\n", + "print \"To find y(0.4) put x=0.2 y=above value ie 1.196 h=0.2 \"\n", + "x = 0.1\n", + "h = 0.1\n", + "k1 = h*f(x,y)\n", + "k2 = h*f(x+1/2*h,y+1/2*k1)\n", + "k3 = h*f(x+1/2*h,y+1/2*k2)\n", + "k4 = h*f(x+h,y+k3)\n", + "kb = 1/6*(k1+2*k2+2*k3+k4)\n", + "print \"The required approximate value is :− \"\n", + "y2 = y+kb\n", + "y = y+kb\n", + "print y\n", + "print \"To find y(0.4) put x=0.2 y=above value ie 1.196 h=0.2 \"\n", + "x = 0.2\n", + "h = 0.1\n", + "k1 = h*f(x,y)\n", + "k2 = h*f(x+1/2*h,y+1/2*k1)\n", + "k3 = h*f(x+1/2*h,y+1/2*k2)\n", + "k4 = h*f(x+h,y+k3)\n", + "kc = 1/6*(k1+2*k2+2*k3+k4) \n", + "print \"The required approximate value is :− \"\n", + "y3 = y+kc\n", + "y = y+kc\n", + "print y\n", + "f0 = f(0,y0)\n", + "f1 = f(0.1,y1)\n", + "f2 = f(0.2,y2)\n", + "f3 = f(0.3,y3) \n", + "print \"y0 y1 y2 y3 are respectively :\"\n", + "print y3,y2,y1,y0\n", + "print \"f0 f1 f2 f3 are respectively :\"\n", + "print f3,f2,f1,f0\n", + "print \"Using adams method \"\n", + "print \"Using the predictor\"\n", + "h = 0.1\n", + "y4 = y3+h/24*(55*f3-59*f2+37*f1-9*f0)\n", + "x = 0.4\n", + "f4 = f(0.4,y4)\n", + "print \"y4 = \",y4\n", + "print \"Using corrector method \"\n", + "y4 = y3+h/24*(9*f4+19*f3-5*f2+f1) \n", + "print \"y4 = \",y4\n", + "f4 = f(0.4,y4)\n", + "print \"f4 = \",f4" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 27.18, page no. 708" + ] + }, + { + "cell_type": "code", + "execution_count": 36, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " Picards method\n", + "First approximation\n", + "y1 = x**2/2 + x + 2\n", + "z1 = x**2/2 - 4*x + 1\n", + "Second approximation\n", + "y2 = x**3/6 - 3*x**2/2 + x + 2\n", + "z2 = -x**5/20 - x**4/4 - x**3 - 3*x**2/2 - 4*x + 1\n", + "Third approximation\n", + "y3 = -x**6/120 - x**5/20 - x**4/4 - x**3/2 - 3*x**2/2 + x + 2\n", + "z3 = -x**7/252 + x**6/12 - 31*x**5/60 + 7*x**4/12 + 5*x**3/3 - 3*x**2/2 - 4*x + 1\n", + "y(0.1) = -0.00833333333333333*x**6 - 0.05*x**5 - 0.25*x**4 - 0.5*x**3 - 1.5*x**2 + x + 2.0\n", + "z(0.1) = -0.00396825396825397*x**7 + 0.0833333333333333*x**6 - 0.516666666666667*x**5 + 0.583333333333333*x**4 + 1.66666666666667*x**3 - 1.5*x**2 - 4.0*x + 1.0\n" + ] + } + ], + "source": [ + "import sympy\n", + "\n", + "print \"Picards method\"\n", + "x0 = 0\n", + "y0 = 2\n", + "z0 = 1\n", + "x = sympy.Symbol('x')\n", + "def f(x,y,z):\n", + " yy = x+z\n", + " return yy\n", + "def g(x,y,z):\n", + " yy = x-y**2\n", + " return yy\n", + "print \"First approximation\"\n", + "y1 = y0+sympy.integrate(f(x,y0,z0),(x,x0,x))\n", + "print \"y1 = \",y1\n", + "z1 = z0+sympy.integrate(g(x,y0,z0),(x,x0,x))\n", + "print \"z1 = \",z1\n", + "print \"Second approximation\"\n", + "y2 = y0+sympy.integrate(f(x,y1,z1),(x,x0,x))\n", + "print \"y2 = \",y2\n", + "z2 = z0+sympy.integrate(g(x,y1,z1),(x,x0,x))\n", + "print \"z2 = \",z2\n", + "print \"Third approximation\"\n", + "y3 = y0+sympy.integrate(f(x,y2,z2),(x,x0,x))\n", + "print \"y3 = \",y3\n", + "z3 = z0+sympy.integrate(g(x,y2,z2),(x,x0,x))\n", + "print \"z3 = \",z3\n", + "x = 0.1\n", + "print \"y(0.1) = \",y3.evalf()\n", + "print \"z(0.1) = \",z3.evalf()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 27.19, page no. 710" + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Using k1 k2.. for f and l1 l2... for g runga kutta formula becomes \n", + "y = 1.0\n", + "y1 = 0.0\n" + ] + } + ], + "source": [ + "import sympy\n", + "\n", + "x = sympy.Symbol('x')\n", + "def f(x,y,z):\n", + " yy = z \n", + " return yy\n", + "def g(x,y,z):\n", + " yy = x*y**2-y**2\n", + " return yy\n", + "x0 = 0\n", + "y0 = 1\n", + "z0 = 0\n", + "h = 0.2\n", + "print \"Using k1 k2.. for f and l1 l2... for g runga kutta formula becomes \"\n", + "h = 0.2\n", + "k1 = h*f(x0,y0,z0)\n", + "l1 = h*g(x0,y0,z0)\n", + "k2 = h*f(x0+1/2*h,y0+1/2*k1,z0+1/2*l1)\n", + "l2 = h*g(x0+1/2*h,y0+1/2*k1,z0+1/2*l1)\n", + "k3 = h*f(x0+1/2*h,y0+1/2*k2,z0+1/2*l2)\n", + "l3 = h*g(x0+1/2*h,y0+1/2*k2,z0+1/2*l2)\n", + "k4 = h*f(x0+h,y0+k3,z0+l3)\n", + "l4 = h*g(x0+h,y0+k3,z0+l3)\n", + "k = 1/6*(k1+2*k2+2*k3+k4)\n", + "l = 1/6*(l1+2*l2+2*l3+2*l4)\n", + "x = 0.2\n", + "y = y0+k\n", + "y1 = z0+l\n", + "print \"y = \",y\n", + "print \"y1 = \",y1" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 2", + "language": "python", + "name": "python2" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 2 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython2", + "version": "2.7.11+" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +} |