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Implicit Euler's Method Application

  1. Nov 19, 2014 #1
    1. The problem statement, all variables and given/known data
    dx/dt= -x2-2x(1+t+t2)
    x(1)=2
    estimate x(1.2) with h=0.2
    2. Relevant equations
    Implicit Euler:
    6895b3f6ac1d4e5a575bd8c2e6489195.png
    I was taught that we must solve for yk+1 using Newton's method:
    af2d6f780d8673d64e8cc328ae52631d.png
    This doesn't seem like it will work because newton's method assumes a function of only one variable.
    According to wikipedia though: The backward Euler method is an implicit method: the new approximation 6675fbec7c9571df0f0c413acd3fcab8.png appears on both sides of the equation, and thus the method needs to solve an algebraic equation for the unknown [PLAIN]http://upload.wikimedia.org/math/6/6/7/6675fbec7c9571df0f0c413acd3fcab8.png. [Broken]
    3. The attempt at a solution
    dx/dt= -x2-2x(1+t+t2)
    let x(1)=2 be denoted by x0, x(1.2) is denoted by x1
    using implicit euler formula:
    x1=2+0.2[-x12-2x1(1+1.2+1.22)]
    0=2-0.2x12-2.456x1
    ==> proceed with quadratic formula to solve for x1

    My question is whether this method is valid to solve for x1?

    Edit: it appears that it is possible to solve for x1 using multivariable newton's method. This would require finding the inverse matrix of the Jacobian of F.
     
    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. Nov 19, 2014 #2

    BvU

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    Solve the quadratic equation for x1 and you have the implicit euler prediction for x(1.2), completely according to the recipe.

    Solving the implicit euler eqn using Newtons method may be possible, but it doesn't seem logical to me: implicit euler replaces the derivative by the function value at the end of the interval (i.e. a constant).
    Newton's method is to solve equations of the type f(x) = 0. Here you are solving a differential equation.
     
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