{"id":4667,"date":"2014-01-17T06:05:54","date_gmt":"2014-01-16T21:05:54","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=4667"},"modified":"2014-08-01T18:59:39","modified_gmt":"2014-08-01T09:59:39","slug":"numerical-methods-for-solving-differential-equations","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4667","title":{"rendered":"Numerical methods for solving differential equations"},"content":{"rendered":"
\n

Given the boudary-value problem<\/p>\n

\\displaystyle dy\/dx = f(x, y)\\ \\ \\ y(x_0) = y_0\\ \\ \\ (1)<\/p>\n

it may not be possible to obtain an exact solution. In such case various methods are available for obtaining an approximate or numerical solution. In the following we list several methods. <\/p>\n

    \n
  1. Step by step or Euler method<\/li>\n
  2. Taylor series method<\/li>\n
  3. Picard’s method<\/li>\n
  4. Runge-Kutta method<\/li>\n<\/ol>\n

    1. Step by step or Euler method<\/h2>\n

    In this method we replace the differential equation of (1) by the approximation<\/p>\n

    \\displaystyle \\frac{y(x_0 + h)-y(x_0)}{h} = f(x_0, y_0)\\ \\ \\ (2)<\/p>\n

    so that<\/p>\n

    \\displaystyle y(x_0 + h) = y(x_0) + hf(x_0, y_0)\\ \\ \\ (3)<\/p>\n

    By continuity in this manner we can then find y(x_0 + 2h),\\ y(x_0 + 3h), etc. We choose h sufficiently small so as to obtain good approximations. <\/p>\n

    A modified procedure of this method can also be used. <\/p>\n

    2. Taylor series method<\/h2>\n

    By successive differentiation of the differential equation in (1) we can find y'(x_0),\\ y''(x_0),\\ y'''(x_0),\\cdots. Then the solution is given by the Taylor series<\/p>\n

    \\displaystyle y(x) = y(x_0) + y'(x_0)(x - x_0) + \\frac{y''(x_0)(x - x_0)^2}{2!} + \\cdots\\ \\ \\ (4)<\/p>\n

    assuming that the series converges. If it does we can obtain y(x_0 + h) to any desired accuracy. <\/p>\n

    3. Picard’s method<\/h2>\n

    By integrating the differential equation in (1) and using the boundary condition, we find <\/p>\n

    \\displaystyle y(x) = y_0 + \\int^{x}_{x_0}\\! f(u, y)du\\ \\ \\ (5)<\/p>\n

    Assuming the approximation y_1(x) = y_0, we obtain from (5) a new approximation. <\/p>\n

    \\displaystyle y_2(x) = y_0 + \\int^{x}_{x_0}\\! f(u, y_1)du\\ \\ \\ (6)<\/p>\n

    Using this in (5) we obtain another approximation. <\/p>\n

    \\displaystyle y_3(x) = y_0 + \\int^{x}_{x_0}\\! f(u, y_2)du\\ \\ \\ (7)<\/p>\n

    Continuing in this manner we obtain a sequence of approximations y_1, y_2, y_3,\\cdots. The limit of this sequence, if it exists, is the required solution. However, by carrying out the procedure a few times, good approximations can be obtained. <\/p>\n

    4. Runge-Kutta method<\/h2>\n

    This method consists of computing<\/p>\n

    \\displaystyle \\left.\\begin{array}{rcl}k_1 & = & hf(x_0, y_0) \\\\ k_2 & = & hf(x_0 + \\frac{1}{2}h, y_0 + \\frac{1}{2}k_1) \\\\ k_3 & = & hf(x_0 + \\frac{1}{2}h, y_0 + \\frac{1}{2}k_2 \\\\ k_4 & = & hf(x_0 + \\frac{1}{2}h, y_0 + \\frac{1}{2}k_3) \\end{array} \\right\\}\\ \\ \\ (8)<\/p>\n

    Then<\/p>\n

    \\displaystyle y(x_0 + h) = y_0 + \\frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)\\ \\ \\ (9)<\/p>\n

    These methods can also be adapted for higher order differential equations by writing them as several first order equations. <\/p>\n<\/blockquote>\n