{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "#2: Transform IVP $dy/dt=1-y^3$; $y(-1)=3$ into an equivalent problem with the initial point at the origin.\n", "\n", "Sol'n: Replace $t\\leftarrow t+1$ and $y\\leftarrow y-3$ and the IVP becomes $dy/dt=1-(y-3)^3$; $y(0)=0$\n", "\n", "This ode is separable so we can solve it: $\\int \\dfrac{dy}{1-y^3}dy=\\int dt$" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "#4Suppose $y'=-y/2+t$; $y(0)=0$. Let $\\phi_0(t)=0$ and define $\\{\\phi_n(t)\\}$ by the method of successive approximations: $\\phi_{n+1}(t)=\\displaystyle\\int\\limits_0^tf(s,\\phi_n(s))ds$\n", "\n", "a. Determine $\\phi_n(t)$ for an arbitrary value of $n$.\n", "\n", "The plan is to compute successive approximations and try to suss out a pattern.\n", "\n", "Sol'n: $\\phi_1(t)=\\displaystyle\\int\\limits_0^tf(s,0)ds=\\int\\limits_0^tsds=\\dfrac{s^2}{2}\\Big|_0^t=\\dfrac{t^2}{2}$
\n", "$\\phi_2(t)=\\displaystyle\\int\\limits_0^tf(s,s^2/2)ds=\\int\\limits_0^t-\\dfrac{s^2}{4}+s\\,ds=\\dfrac{-s^3}{12}+\\dfrac{s^2}{2}\\Big|_0^t=-\\dfrac{t^3}{12}+\\dfrac{t^2}{2}=-\\dfrac{t^3}{2\\cdot(3\\cdot 2)}+\\dfrac{t^2}{2}$
\n", "\n", "$\\phi_3(t)=\\displaystyle\\int\\limits_0^tf(s,s^3/(2(3\\cdot 2))+s^2/2)ds=\\int\\limits_0^t\\dfrac{s^3}{2^2(3\\cdot 2)}-\\dfrac{s^2}{2\\cdot 2}+s\\,ds=\\dfrac{s^4}{2^2*4!}-\\dfrac{s^3}{2(3\\cdot 2)}+\\dfrac{s^2}{2}\\Big|_0^t=\\dfrac{t^4}{2^2*4!}-\\dfrac{t^3}{2(3\\cdot 2)}+\\dfrac{t^2}{2}=\\sum\\limits_{k=2}^4\\dfrac{(-1)^kt^k}{2^{k-2}k!}=\\sum\\limits_{k=0}^4\\dfrac{4(-t/2)^k}{k!}+2t-4$\n", "\n", "The pattern will persist, so that $\\phi_n=\\sum\\limits_{k=0}^{n+1}\\dfrac{4(-t/2)^k}{k!}-2t+4\\rightarrow 4e^{-t/2}+2t-4$ as $n\\rightarrow\\infty$\n", "\n", "Well, that's enough hard pencil and paper work. Here's a little script for cranking these things out...and graphing them!" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ] }, "execution_count": 4, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "" ] }, "execution_count": 4, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "" ] }, "execution_count": 4, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "" ] }, "execution_count": 4, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "" ] }, "execution_count": 4, "metadata": { }, "output_type": "execute_result" }, { "data": { "image/png": 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" }, "execution_count": 4, "metadata": { }, "output_type": "execute_result" } ], "source": [ "s,t = var('s t')\n", "phi = function('phi')(s)\n", "def f(t,y):\n", " return -y/2+t\n", "phi=integral(f(s,0),(s,0,t))\n", "show(phi)\n", "p1 = plot(phi,(t,0,6))\n", "for i in xrange(1,5):\n", " phi = integral(-phi(s)/2+s,(s,0,t))\n", " show(phi)\n", " p1 += plot(phi,(t,0,6))\n", "\n", "p1+=plot(4*e^(-t/2)+2*t-4,(t,0,6),color='red')\n", "show(p1)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "b. Plot $\\phi_n(t)$ for $n=1,\\dots,4$. Observe whether the iterates appear to be converging.\n", "\n", "SOLN: See above. Looks good!\n", "\n", "c. Express $\\lim\\limits_{n\\rightarrow\\infty}\\phi_n(t)=\\phi(t)$ in terms of elementary functions; that is, solve the given initial value problem.\n", "\n", "SOLN: This was worked out above by sussing out the pattern of the Taylor series for $e^{-t/2}$. Shall we give it a go the other way?
\n", "The ode is linear: $\\dfrac{dy}{dt}+\\dfrac{1}{2}y=t$ has the integrating factor $e^{t/2}$, so we have $\\dfrac{d}{dt}\\left(e^{t/2}y\\right)=te^{t/2}$ whence $e^{t/2}y=2(t-2)e^{t/2}+c$
\n", "The initial conditions require $0=-4+c$, so $c=4$ and the solution is $y=2(t-2)+4e^{t/2}$" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "#8. Consider $y'=1−y^3, y(0)=0$
\n", "let $\\phi_0(t)=0$ and use the method of successive approximations to approximate the solution of the initial value problem.
\n", "a. Calculate $\\phi_1(t),\\dots,\\phi_3(t)$.\n", "\n", "SOL'N: Here, $y$ is autonomous, so we can write $y'=f(y)$
\n", "$\\phi_1(t)=\\displaystyle\\int\\limits_0^tf(0)ds=\\int\\limits_0^t1ds=s\\Big|_0^t=t$
\n", "$\\phi_2(t)=\\displaystyle\\int\\limits_0^tf(s)ds=\\int\\limits_0^t1-s^3\\,ds=s-\\dfrac{s^4}{4}\\Big|_0^t=t-\\dfrac{t^4}{4}$
\n", "\n", "$\\phi_3(t)=\\displaystyle\\int\\limits_0^tf(s-s^4/4)ds=\\int\\limits_0^t1-\\left(s-\\dfrac{s^4}{4}\\right)^3\\,ds=t-\\dfrac{t^4}{4}+\\dfrac{3t^7}{7\\cdot 4}-\\dfrac{3t^{10}}{10\\cdot 4^2}+\\dfrac{t^{13}}{13\\cdot 4^3}$\n", "\n", "b. Plot $\\phi_1(t),\\dots,\\phi_3(t)$ Observe whether the iterates appear to be converging." ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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" }, "execution_count": 5, "metadata": { }, "output_type": "execute_result" } ], "source": [ "t = var('t')\n", "p2 = plot(t,(t,0,2))\n", "p2 += plot(t-t^4,(t,0,2))\n", "p2 += plot(t-t^4/4+3*t^7/28-3*t^10/160+t^13/(13*64),(t,0,2),color='red')\n", "show(p2)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "It doesn't look great...what's going on? Well, the equation is separable so we can solve it in explicit, closed form:
\n", "$\\displaystyle\\int\\dfrac{dy}{1-y^3}=\\int dt$\n", "\n", "Being lazy, I'll use Python's numpy to get the antiderivative:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ] }, "execution_count": 7, "metadata": { }, "output_type": "execute_result" } ], "source": [ "y=var('y')\n", "show(integral(1/(1-y^3),y))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "...and...moving right along!\n", "\n", "#10 Consider $y'=\\dfrac{3t^2+4t+2}{2(y-1)},y(0)=0$
\n", "Let $\\phi_0(t)=0$ and use the method of successive approximations to approximate the solution of the initial value problem.\n", "\n", "a. Calculate $\\phi_1(t),\\dots, \\phi_4(t)$, or (if necessary) Taylor approximations to these iterates. Keep terms up to order six.\n", "\n", "SOLN: " ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "$\\phi_1(t)=\\displaystyle\\int\\limits_0^tf(s,0)ds=\\int\\limits_0^t\\dfrac{-3s^2}{2}-2s-1\\,ds=\\dfrac{-s^3}{2}-s^2-s\\Big|_0^t=\\dfrac{-t^3}{2}-t^2-t$
\n", "$\\phi_2(t)=\\displaystyle\\int\\limits_0^tf(s,\\dfrac{-s^3}{2}-s^2-s)ds=-\\int\\limits_0^t\\dfrac{3s^2+4s+2}{s^3+2s^2+2s+2}\\,ds=-\\ln|t^3+2t^2+2t+2|+\\ln(2)$
\n" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "data": { "text/html": [ "" ] }, "execution_count": 14, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "" ] }, "execution_count": 14, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "" ] }, "execution_count": 14, "metadata": { }, "output_type": "execute_result" }, { "name": "stderr", "output_type": "stream", "text": [ "WARNING: Some output was deleted.\n" ] } ], "source": [ "s,t = var('s t')\n", "assume(t>0)phi = function('phi')(s)\n", "def f(t,y):\n", " return (3*t^2+4*t+2)/(2*(y-1))\n", "\n", "phi=integral(f(s,0),(s,0,t))\n", "show(phi)\n", "p1 = plot(phi,(t,0,6))\n", "for i in xrange(1,3):\n", " #phi = -phi(s)/2+s\n", " phi(t)=integral(f(s,phi(s)),(s,0,t))\n", " show(phi)\n", " p1 += plot(phi,(t,0,6))\n", "\n", "p1+=plot(1+sqrt(t^3+2*t^2+2*t+1),(t,0,6),color='red')\n", "p1+=plot(1-sqrt(t^3+2*t^2+2*t+1),(t,0,6),color='red')\n", "\n", "show(p1)" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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" }, "execution_count": 17, "metadata": { }, "output_type": "execute_result" } ], "source": [ "p1+=plot(1+sqrt(t^3+2*t^2+2*t+1),(t,0,6),color='red')\n", "p1+=plot(1-sqrt(t^3+2*t^2+2*t+1),(t,0,6),color='red')\n", "show(p1)" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ] }, "execution_count": 3, "metadata": { }, "output_type": "execute_result" } ], "source": [ "t=var('t')\n", "y = function('y')(t)\n", "de = 2*(y-1)*diff(y,t)-3*t^2-4*t-2\n", "h = desolve(de, y,ics=[0,0])\n", "show(h)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "#14 If $\\partial f/\\partial y$ is continuous in the rectangle $D$, show that there is a
\n", "positive constant $K$ such that $|f(t,y_1)−f(t,y_2)|\\leq K|y_1−y_2|$, (31)
\n", "where $(t,y_1)$ and $(t,y_2)$ are any two points in $D$ having the same $t$ coordinate.
\n", "This inequality is known as a Lipschitz condition.
\n", "Hint: Hold $t$ fixed and use the mean value theorem on $f$ as a function
\n", "of $y$ only. Choose $K$ to be the maximum value of $|\\partial f/\\partial y|$ in $D$.\n", "\n", "SOLN: Fix a value of $t$ so that the vertical line segment in the $ty$ plane from $(t,y_1)$ to $(t,y_2)$ is contained in the rectangle, $D$.
\n", "Following the hint, we recall that the mean value theorem guarantees that if $y=f(t,y)$ is differentiable
\n", "on $(y_1,y_2)$ then there exists $c\\in(y_1,y_2)$ such that $(y_2-y_1)f'(t,c)=f(t,y_2)-f(t,y_1)$.
\n", "Swapping sides, and equating the absolute values, $|f(t,y_2)-f(t,y_1)|=|f'(t,c)||(y_2-y_1)|$
\n", "Since, by assumption, $f$ is continuous in the interval, $|f'(t,y)|$ is bounded by some finite value, $K$." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "#15 and 16 If $\\phi_{n−1}(t)$ and $\\phi_n(t)$ are members of the sequence $\\{\\phi_k(t)\\}$, use
\n", "the result of Problem 14 to show that $f'(t,\\phi_n(t))−f'(t,\\phi_{n−1}(t)\\leq K|\\phi_n(t)−\\phi_{n−1}(t)|$.\n", "\n", "SOLN: Proof by induction may work here. Start with $\\phi_1(t)-\\phi_0(t)|=|\\phi_1(t)|\\leq\\displaystyle\\int\\limits_0^t|f(s,0)|ds\\leq Ms\\Big|_0^t=Mt$
\n", "where $M$ is an upper bound on the partial derivative in $D$.
\n", "Further, look at $|\\phi_2(t)-\\phi_1(t)|=\\Big|\\displaystyle\\int_0^t(f(s,\\phi_1(s))-f(s,0))ds\\Big|\\leq\\int_0^t\\Big|f(s,\\phi_1(s))-f(s,0))\\Big|ds$
\n", "This was shown in problem 14, to be $\\leq\\displaystyle\\int_0^tK|\\phi_1(s)-0|ds\\leq KN\\int_0^t|s|ds=\\dfrac{1}{2}KMt^2$
\n", "Now the inductive hypothesis is that $|\\phi_k(t)-\\phi_{k-1}(t)|\\leq\\dfrac{MK^{k-1}t^k}{k!}$ for some $k$. Then
\n", "$|\\phi_k(t)-\\phi_{k-1}(t)|\\leq\\displaystyle\\int_0^t|f(s,\\phi_k(s))-f(s,\\phi_{k-1}(s))|ds\\leq\\int_0^tK|\\phi_k(s)-\\phi_{k-1}(s)|ds\\leq\\int_0^tK\\dfrac{MK^{k-1}s^k}{k!}ds=\\dfrac{MK^kt^{k+1}}{(k+1)!}$, completing the proof by induction." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "#17. Note that $\\phi_n(t)=\\phi_1(t)+(\\phi_2(t)-\\phi_1(t))+\\cdots+(\\phi_n(t)-\\phi_{n-1}(t))$
\n", "a. Then, by the triangle inequality, $|\\phi_n(t)|\\leq |\\phi_1(t)|+|\\phi_2(t)−\\phi_1(t)|+\\cdots+|\\phi_n(t)−\\phi_{n−1}(t)|$.
\n", "b. And by the results of Problem 16, if $|t|\n", "Further, each successive difference has $|\\phi_n(t)-\\phi_{n-1}(t)|\\leq\\dfrac{MK^{n-1}h^n}{n!}$.
\n", "Thus by the triangle inequality, $|\\phi_n(t)|\\leq \\dfrac{M}{K}\\left(Kh+\\dfrac{(Kh)^2}{2!}+\\cdots+\\dfrac{(Kh)^n}{n!}\\right)$.\n", "\n", "c. Show that the sum in part b converges as $n\\rightarrow\\infty$ and, hence,
\n", "the sum in part a also converges as $n\\rightarrow\\infty$. Conclude therefore
\n", "that the sequence $\\{\\phi_n(t)\\}$ converges since it is the sequence of,br>\n", "partial sums of a convergent infinite series.\n", "\n", "SOLN: The sequence of partial sums converges to $MK(e^{Kh}-1)$. Using the comparison test
\n", "(remember Chapter 11 of 1B?!) the terms of part a must also converge and so the $n$th term must\n", "go to zero: $|\\phi_n(t)-\\phi_{n-1}(t)|\\rightarrow 0\\Rightarrow \\{\\phi_n(t)\\}$ is convergent." ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] } ], "metadata": { "kernelspec": { "display_name": "SageMath (stable)", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.14" } }, "nbformat": 4, "nbformat_minor": 0 }