Math Problem Statement

Consider the IVP with solution y(t) ( y′(t) = f (t, y), a < t < b y(a) = y0 Using a similar strategy as in the derivation of Explicit Euler, derive Implicit Euler’s method: w0 = y0 wi+1 = wi + hf (ti+1, wi+1) Hint: use Taylor expansion centered at ti+1 and evaluated at ti Note: Implicit Euler is more computationally expensive then Explicit Euler, but it is more stable and gives better results for certain ODEs (stiff ODEs). We will see this later in the class.

Solution

To derive the Implicit Euler method for solving the initial value problem (IVP)

y(t)=f(t,y),a<t<b,y(a)=y0,y'(t) = f(t, y), \quad a < t < b, \quad y(a) = y_0,

we use a similar approach to the derivation of the Explicit Euler method, but with a key difference in evaluating the derivative. Here's the step-by-step derivation:


Step 1: Taylor Expansion Centered at ti+1t_{i+1}

The Taylor expansion of y(t)y(t) about ti+1t_{i+1} is:

y(ti)=y(ti+1)(ti+1ti)y(ti+1)+(ti+1ti)22!y(c),y(t_i) = y(t_{i+1}) - (t_{i+1} - t_i)y'(t_{i+1}) + \frac{(t_{i+1} - t_i)^2}{2!}y''(c),

where cc is some point in [ti,ti+1][t_i, t_{i+1}].

Rearranging to solve for y(ti+1)y(t_{i+1}), and neglecting the higher-order term (ti+1ti)22!y(c)\frac{(t_{i+1} - t_i)^2}{2!}y''(c), we get:

y(ti+1)y(ti)+(ti+1ti)y(ti+1).y(t_{i+1}) \approx y(t_i) + (t_{i+1} - t_i)y'(t_{i+1}).

For simplicity, let the step size h=ti+1tih = t_{i+1} - t_i. This gives:

y(ti+1)y(ti)+hy(ti+1).y(t_{i+1}) \approx y(t_i) + h y'(t_{i+1}).


Step 2: Approximation Using the IVP

From the IVP, y(t)=f(t,y)y'(t) = f(t, y). Substituting y(ti+1)y'(t_{i+1}) into the equation, we have:

y(ti+1)y(ti)+hf(ti+1,y(ti+1)).y(t_{i+1}) \approx y(t_i) + h f(t_{i+1}, y(t_{i+1})).

This is the Implicit Euler method in terms of yy. Since the exact solution y(ti+1)y(t_{i+1}) is generally unknown, we approximate it numerically using wiw_i, the numerical approximation of y(ti)y(t_i). Thus, the recurrence relation becomes:


Step 3: Final Implicit Euler Formula

wi+1=wi+hf(ti+1,wi+1).w_{i+1} = w_i + h f(t_{i+1}, w_{i+1}).


Key Characteristics of Implicit Euler

  • Implicit nature: The formula requires solving for wi+1w_{i+1} at each step, often requiring iterative numerical techniques (e.g., Newton's method).
  • Stability: This method is more stable than Explicit Euler, particularly for stiff ODEs, making it suitable for such problems.

Would you like a worked example to illustrate this derivation or more details about its implementation?


5 Follow-Up Questions

  1. How does the stability of the Implicit Euler method compare to the Explicit Euler method for stiff ODEs?
  2. What numerical methods can be used to solve for wi+1w_{i+1} in the Implicit Euler formula?
  3. Can you derive the Explicit Euler method for comparison with the Implicit Euler method?
  4. What are the advantages and disadvantages of using Implicit Euler in practice?
  5. How does the choice of step size hh affect the accuracy and stability of Implicit Euler?

Tip

When implementing the Implicit Euler method, ensure that your numerical solver for wi+1w_{i+1} is efficient, as this step is computationally intensive but critical for accurate results.

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Math Problem Analysis

Mathematical Concepts

Ordinary Differential Equations (ODEs)
Numerical Methods
Taylor Series Expansion
Implicit Methods

Formulas

y(t_{i+1}) ≈ y(t_i) + h * y'(t_{i+1})
y'(t) = f(t, y)
w_{i+1} = w_i + h * f(t_{i+1}, w_{i+1})

Theorems

Taylor Series Approximation
Stability of Numerical Methods

Suitable Grade Level

Undergraduate (Numerical Analysis or Differential Equations)