Convergence of Euler's method Suppose Euler's method is applied to the initial value problem y′(t) = ay, y(0) = 1, which has the exact solution y(t) = eᵃᵗ. For this exercise, let h denote the time step (rather than Δt). The grid points are then given by tₖ = kh. We let uₖ be the Euler approximation to the exact solution y(tₖ), for k = 0, 1, 2, ...
b. Show by substitution that uₖ = (1 + ah)ᵏ is a solution of the equations in part (a), for k = 0, 1, 2, ...

Verified step by step guidance

1. Recall the Euler's method update formula for the initial value problem \(y'(t) = ay\) with step size \(h\): \[u_{k+1} = u_k + h f(t_k, u_k)\] Since \(f(t, y) = ay\), this becomes: \[u_{k+1} = u_k + h a u_k = u_k (1 + a h)\]
2. Given the initial condition \(u_0 = y(0) = 1\), we want to verify that the proposed solution \[u_k = (1 + a h)^k\] satisfies the Euler update formula for all \(k = 0, 1, 2, \ldots\)
3. Substitute \(u_k = (1 + a h)^k\) into the right-hand side of the Euler update formula: \[u_k (1 + a h) = (1 + a h)^k (1 + a h) = (1 + a h)^{k+1}\]
4. Compare this with the left-hand side \(u_{k+1}\), which according to the proposed solution is: \[u_{k+1} = (1 + a h)^{k+1}\]
5. Since both sides are equal, this confirms by substitution that \(u_k = (1 + a h)^k\) satisfies the Euler method recurrence relation for all \(k\).

Key Concepts

Euler's Method

Euler's method is a numerical technique for approximating solutions to initial value problems of differential equations. It uses a stepwise approach, updating the solution by moving along the slope given by the differential equation at each grid point. The formula is u_{k+1} = u_k + h f(t_k, u_k), where h is the step size.

Initial Value Problem (IVP) and Exact Solution

An initial value problem specifies a differential equation along with a starting value, such as y'(t) = ay with y(0) = 1. The exact solution to this IVP is y(t) = e^{at}, which provides a benchmark to compare numerical approximations like Euler's method.

Substitution to Verify Solutions

Substitution involves plugging a proposed solution into the given equation to verify its validity. Here, substituting u_k = (1 + ah)^k into the Euler update formula confirms it satisfies the recurrence relation, demonstrating that this expression correctly represents the numerical solution at each step.