Question

Use the technique discussed in Problem 20 to show that the approximation obtained by the Euler method converges to the exact solution at any fixed point as \(h \rightarrow 0 .\) $$ y^{\prime}=\frac{1}{2}-t+2 y, \quad y(0)=1 \quad \text { Hint: } y_{1}=(1+2 h)+t_{1} / 2 $$

Short answer

Question: Show that the approximation obtained by the Euler method converges to the exact solution as the step size, h, approaches 0 for the given first-order ordinary differential equation and initial value problem. Given: \(y'(t) = \frac{1}{2} - t + 2y(t)\) and \(y(0) = 1\) Answer: The Euler method's approximation, \(y_1 = (1+2h) + \frac{ht}{2}\) converges to the exact solution as \(h \rightarrow 0\), which can be seen when comparing it with the Taylor series expansion of the exact solution, \(y(t) = 1 + \frac{5}{2}t + 2t^2 + \cdots\).

Step-by-step solution

Taylor series expansion of the exact solution

Find the Taylor series expansion of the exact solution y(t) about the point t=0: $$ y(t) = y(0) + y'(0)t + \frac{1}{2}y''(0)t^2 + \cdots $$ We have y(0)=1 and we need to find y'(0) and y''(0).

Compute derivatives

Compute the first and second derivatives of y(t) using the given ODE: $$ y'(t) = \frac{1}{2} - t + 2y(t) $$ Now we can find y'(0): $$ y'(0) = \frac{1}{2} - 0 + 2\cdot 1 = \frac{5}{2} $$ Now differentiate y'(t) with respect to t to find y''(t): $$ y''(t) = -1 + 2y'(t) $$ Now we can find y''(0): $$ y''(0) = -1 + 2\cdot \frac{5}{2} = 4 $$

Exact solution Taylor series

Plug y(0), y'(0), and y''(0) into the Taylor series expansion of y(t): $$ y(t) = 1 + \frac{5}{2}t + \frac{1}{2}(4)t^2 + \cdots $$

Euler method's first step

Write the Euler method approximation y1 using the given hint: $$ y_1 = (1+2h) + \frac{t_1}{2} $$ where \(t_1 = h\).

Compare Taylor series expansions

Compare the Taylor series expansion for the exact solution y(t) with the Euler method approximation y1, as h approaches 0: $$ y(t) = 1 + \frac{5}{2}t + 2t^2 + \cdots $$ $$ y_1 = (1+2h) + \frac{ht}{2} $$ Plugging in \(t=h\) into the Taylor series of y(t): $$ y(h) = 1 + \frac{5}{2}h + 2h^2 + \cdots $$ As \(h \rightarrow 0,\) we have: $$ y_1 = (1 + 2h) + \frac{h^2}{2} \approx 1 + \frac{5}{2}h + 2h^2 $$ which is the same as the Taylor series expansion of y(t) up to the second order term. Thus, the approximation obtained by the Euler method converges to the exact solution at any fixed point as \(h \rightarrow 0\).

Key concepts

Taylor Series Expansion

Taylor Series Expansion is a powerful tool for approximating functions using polynomials. Imagine you want to predict how a function behaves near a particular point, say zero. The Taylor series helps you achieve just that by creating a polynomial that closely matches your function around that point.

• The Taylor series of a function like \( y(t) \) is constructed using the function's derivatives at that point.
• The expansion is a sum of terms involving these derivatives and powers of \( t \), starting from \( t^0 \).

It’s like building a skyscraper using layers of information obtained from each derivative.

In our example, the Taylor series starts as \( y(t) = y(0) + y'(0)t + \frac{1}{2}y''(0)t^2 + \cdots \). Here, you substitute values you calculate (\( y(0), y'(0), y''(0), \,\) etc.) to get the approximate polynomial. This series gives a good approximation to how \( y(t) \) behaves near zero, especially when higher derivatives are included.

Convergence of Numerical Methods

When we talk about the convergence of numerical methods, we refer to how closely the approximation methods, like Euler’s Method, approach the exact solution of a function as a certain parameter gets smaller.

• In Euler’s Method, \( h \), the step size, is this parameter. The smaller \( h \) is, the closer our approximation is expected to be to the true solution.
• Convergence is important because it assures us that our numerical approximations are reliable.

As we showed, when \( h \rightarrow 0 \), the Euler approximation aligns with the exact Taylor expansion. Essentially, by decreasing \( h \), Euler’s steps become more precise, imitating the behavior of the real function. This property is crucial for validating numerical methods as they replace complex calculations in practical applications.

Differential Equations

Differential Equations are mathematical equations involving derivatives, which represent rates of change. They allow us to describe various phenomena such as motion, growth, or decay.

• In our case, we're working with a simple first-order differential equation, \( y'(t) = \frac{1}{2} - t + 2y(t) \).
• These equations require specific techniques for solving them, like Euler’s Method or integrating factors for exact solutions.

Understanding how to manipulate these equations is essential for predicting system behaviors over time.

In applications, knowing the initial condition, such as \( y(0) = 1 \), is important for solving these equations. This initial condition is your starting point and together with the differential equation, it uniquely determines the solution. The goal is often to find what happens to \( y(t) \) as \( t \) changes, which is why methods like Euler’s are developed—to approximate \( y(t) \) for various \( t \), especially when a clean analytical solution is not possible.