Question

Convergence of Euler's method Suppose Euler's method is applied to the initial value problem \(y^{\prime}(t)=a y, y(0)=1,\) which has the exact solution \(y(t)=e^{a t} .\) For this exercise, let \(h\) denote the time step (rather than \(\Delta t\) ). The grid points are then given by \(t_{k}=k h .\) We let \(u_{k}\) be the Euler approximation to the exact solution \(y\left(t_{k}\right),\) for \(k=0,1,2, \ldots\). a. Show that Euler's method applied to this problem can be written \(u_{0}=1, u_{k+1}=(1+a h) u_{k},\) for \(k=0,1,2, \ldots\). b. Show by substitution that \(u_{k}=(1+a h)^{k}\) is a solution of the equations in part (a), for \(k=0,1,2, \ldots\). c. Recall from Section 4.7 that \(\lim _{h \rightarrow 0}(1+a h)^{1 / h}=e^{a} .\) Use this fact to show that as the time step goes to zero \((h \rightarrow 0,\) with \(\left.t_{k}=k h \text { fixed }\right),\) the approximations given by Euler's method approach the exact solution of the initial value problem; that is, \(\lim _{h \rightarrow 0} u_{k}=\lim _{h \rightarrow 0}(1+a h)^{k}=y\left(t_{k}\right)=e^{a t_{k}}\).

Short answer

Question: Show that Euler's method applied to the initial value problem \(y'(t) = ay, y(0)=1\) can be written as \(u_0 = 1\), \(u_{k+1} = (1 + ah)u_k\), and that the given solution holds. Also, demonstrate that the convergence of Euler's method to the exact solution as the time step approaches zero. Solution: By applying Euler's method to the initial value problem and following the steps outlined, we showed that the method can be written as \(u_0 = 1\), \(u_{k+1} = (1 + ah)u_k\). By substitution, we verified that the given solution \(u_k = (1+ah)^k\) holds true. Furthermore, we proved that as the time step \(h\) goes to zero, the approximation given by Euler's method converges to the exact solution, that is, \(\lim_{h\rightarrow 0} u_k = \lim_{h\rightarrow 0} (1 + ah)^k = y(t_k) = e^{at_k}\).

Step-by-step solution

Apply Euler's method to the initial value problem

Euler's method is a numerical integration method to solve the initial value problem. It approximates the solution of the differential equation at each step. The method is applied as follows: \(u_{k+1} = u_k + hy'(t_k)\). The differential equation given is \(y'(t) = ay\), and we are given the initial value \(y(0)=1\). Thus, our update rule for Euler's method is: \(u_{k+1} = u_k + h(au_k)\), where \(u_0 = 1\) as given.

Rewrite the Euler's method in the desired form

We rewrite the update rule found in Step 1: \(u_{k+1} = u_k + ah u_k = (1 + ah)u_k\). Thus, we have shown that Euler's method applied to this problem can be written as \(u_0 = 1\), \(u_{k+1} = (1 + ah)u_k\), for \(k=0,1,2,\dots\).

Verify the solution by substitution

The given solution is \(u_k = (1+ah)^k\). We substitute this into the equation we derived in Step 2 and check if it holds true for all values of \(k\). By substitution, we get \(u_{k+1} = (1+ah)^{k+1}\) and \(u_k = (1+ah)^k\). Using the update rule we derived in Step 2, we write: \((1+ah)^{k+1} = (1+ah)(1+ah)^k\). Since this equation holds true, we have confirmed that the given solution is indeed a solution for Euler's method applied to this problem.

Show convergence of Euler's method to the exact solution

We are given that \(\lim_{h\rightarrow 0}(1+ah)^{1/h} = e^a\). Now, we want to show that as the time step \(h\) goes to zero, the approximation \(u_k\) converges to the exact solution \(y(t_k) = e^{a t_k}\). In order to do this, we analyze the limit as \(h \rightarrow 0\) of \(u_k = (1+ah)^k\). Since \(t_k = kh\), we need to find the limit of \((1+ah)^{t_k/h}\) as \(h \rightarrow 0\). Using the fact given that \(\lim_{h\rightarrow 0}(1+ah)^{1/h} = e^a\), we have \(\lim_{h\rightarrow 0}(1+ah)^{t_k/h} = e^{at_k}\). Therefore, as the time step goes to zero, the approximation given by Euler's method converges to the exact solution, that is, \(\lim_{h\rightarrow 0} u_k = \lim_{h\rightarrow 0} (1 + ah)^k = y(t_k) = e^{at_k}\).

Key concepts

Numerical Integration

Numerical integration refers to methods used to approximate the solutions of integrals or differential equations when analytical solutions are too complex or unknown. In contexts like Euler's method, it allows us to approximate the behavior of systems described by differential equations. This is particularly useful when we need to find how a system evolves over time or to solve complex problems in physics, engineering, and biology.

• Popular choice for its simplicity and straightforward implementation.
• Involves discretizing the time or space domain into steps.
• Step-by-step update rule is used to approximate the solution.

Numerical integration is essential in applied mathematics, offering a toolkit for dealing with practical problems where exact solutions are impractical.

Differential Equations

Differential equations are mathematical equations that involve derivatives of a function. They describe the rates of change and are fundamental in modeling real-world systems. For instance, the initial value problem in our exercise is a differential equation given by \(y'(t) = ay\). Here, the solution of this differential equation tells us how \(y\), a quantity of interest, changes over time.

• Describe natural phenomena like motion, heat, and sound.
• Can be ordinary (ODE) or partial (PDE) depending on the number of variables.
• Initial conditions are crucial for determining unique solutions.

Understanding differential equations is key to applying methods like Euler's to solve complex dynamic systems.

Initial Value Problem

An initial value problem is a specific type of differential equation problem that provides conditions at the beginning of the observation or calculation time. In our case, the initial condition is \(y(0)=1\). This sets a specific starting point from which the approximate solution begins.

• Gives a clear scenario and starting point.
• Vital for solving differential equations accurately.
• Euler's method uses these initial values to iteratively predict future values.

Initial value problems help to ensure that solutions to differential equations not only exist but are also uniquely defined by their starting conditions.

Convergence Analysis

Convergence analysis involves studying how numerical methods approximate the exact solution as the step size becomes finer. For Euler's method applied to our problem, it is important to show that as \(h \to 0\), the numerical solution converges to the exact solution \(y(t_k)=e^{at_k}\).

• Shows reliability and accuracy of the method.
• Ensures that small step sizes result in solutions close to the exact answer.
• Uses limits to compare numerical and exact solutions.

Through convergence analysis, we ascertain that Euler's method is reliable for approximating solutions provided the step size is sufficiently small, giving confidence in its practical applications.