Question

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 \(k=0,1,2, \dots\) c. \(\lim _{h \rightarrow 0}(1+a h)^{1 / h}=e^{a} .\) Use this fact to show that as the time step goes to zero \(\left(h \rightarrow 0, \text { with } t_{k}=k h \text { fixed }\right),\) 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: Write Euler's method for the given initial value problem (IVP) \(y'(t) = ay, y(0)=1\) and verify the solution obtained using Euler's method. Show that as the time step goes to zero, the approximations given by Euler's method approach the exact solution of the given initial value problem. Answer: Euler's method for the given IVP can be written as \(u_0 = 1\) and \(u_{k+1} = (1 + ah)u_k\) for \(k = 0, 1, 2, \dots\). The solution \(u_k=(1+ah)^k\) satisfies the equations for Euler's method. As the time step goes to zero, the approximations given by Euler's method approach the exact solution: \(\lim_{h \to 0} u_k = \lim_{h \to 0}(1+ah)^k = y(t_k) = e^{at_k}\).

Step-by-step solution

(a) Write Euler's method for the given problem

Euler's method is written as follows: $$ u_{k+1} = u_k + hf(t_k, u_k) $$ For the given problem, we have \(f(t_k, u_k) = ay_k = ay\). Thus, the Euler's method is given by: $$ u_{k+1} = u_k + h(au_k) = u_k + ahu_k = (1 + ah)u_k $$ The initial condition is \(u_0 = y(0) = 1\). So, Euler's method for the given problem can be written as: $$ u_0 = 1, \quad u_{k+1} = (1 + ah)u_k, \quad k = 0, 1, 2, \dots $$

(b) Verify the solution \(u_k=(1+ah)^k\) by substitution

To show that \(u_k=(1+ah)^k\) is a solution of the equations obtained in part (a), we will substitute and verify if the equation is satisfied. For \(k=0\), we have \(u_0 = (1+ah)^0 = 1\), which satisfies the initial condition. Now let's substitute the solution \(u_k=(1+ah)^k\) into the equation \(u_{k+1} = (1+ah)u_k\): $$ u_{k+1} = (1+ah)(1+ah)^k = (1+ah)^{k+1} $$ Thus, the solution \(u_k=(1+ah)^k\) satisfies the equations for \(k=0, 1, 2, \dots\)

(c) Show that as the time step goes to zero, the approximations given by Euler's method approach the exact solution

We have established that \(u_k=(1+ah)^k\) is a solution of our Euler's method. We are given the limit: $$ \lim_{h \to 0} (1+ah)^{1/h}=e^a $$ To show that the approximations given by the Euler's method approach the exact solution as the time step goes to zero, we will take the limit of \(u_k\) as \(h\to 0\). Recall that \(t_k = kh\): $$ \lim_{h \to 0} u_k = \lim_{h \to 0} (1+ah)^k = \lim_{h \to 0} ((1+ah)^{1/h})^k = \lim_{h \to 0} (e^a)^k = e^{ak} $$ Now we can compare this limit with the exact solution \(y(t_k) = e^{at_k} = e^{akh}\). Since \(kh = t_k\) and \(e^{ak} = e^{at_k}\), the approximations given by Euler's method approach the exact solution as the time step goes to zero. This shows that: $$ \lim_{h \to 0} u_k = \lim_{h \to 0}(1+ah)^k = y(t_k) = e^{at_k} $$

Key concepts

Initial Value Problem

An initial value problem in mathematics involves finding a function that satisfies a differential equation and meets a specified value at a given point, called the initial condition. For example, in our problem, we have a differential equation: \( y'(t) = ay \) with the initial condition \( y(0) = 1 \). The initial condition provides a starting point for the function we're looking to determine.

Initial value problems are critical in modeling real-world phenomena where the condition at a specific time needs to dictate future behavior. Whether in physics, engineering, or finance, they help in predicting future outcomes based on known starting conditions, improving the effectiveness of simulations and analyses.

By applying Euler’s method to our initial value problem, we can approximate the solution numerically. This leads us to our next concept, where numerical techniques are key in tackling problems that don’t have straightforward analytical solutions.

Numerical Approximation

Numerical approximation plays a crucial role in solving initial value problems where exact analytical solutions are challenging to find. Euler's method provides a straightforward approach to generate these approximations, offering iterative steps to approximate the values of the unknown function.

In our example, Euler's method approximates the solution of the differential equation by taking a small step \( h \) in time to estimate subsequent values using the formula:

• \( u_{k+1} = (1 + ah)u_k \).

The choice of the step size \( h \) affects the approximation's accuracy. Smaller steps often lead to better accuracy, but require more computations.

Here's why numerical approximation is valuable:

• It provides solutions in a form suitable for computational methods.
• It helps when dealing with complex systems where analytical expressions are impractical.

By converting the problem into a series of simpler iterative computations, numerical methods like Euler's allow us to model dynamic systems effectively.

Convergence

Convergence in numerical methods refers to how the approximation provided by a method approaches the exact solution as the steps or iterations increase. It is crucial in ensuring that the numerical approximation becomes more accurate as more points are calculated or as the step size diminishes.

In the context of Euler's method for our initial value problem, we are interested in how \( u_k = (1 + ah)^k \) behaves as the step size, \( h \), approaches zero. The key result here is:

• \( \lim_{h \to 0}(1+ah)^{1/h} = e^a \).

This implies that, as \( h \) diminishes and the number of steps increases, the approximation converges to the exact solution \( y(t_k) = e^{at_k} \).

Understanding convergence ensures:

• The accuracy of the numerical solution in practice.
• The reliability of predictions made based on the model.

Convergence is a critical property that confirms the efficacy of using Euler’s method for approximations.

Differential Equations

Differential equations form the backbone of many scientific models, describing relationships involving rates of change over time. They are essential in fields such as physics, biology, and economics, linking derivative terms with function expressions.

For instance, in our exercise, the differential equation \( y'(t) = ay \) symbolizes exponential growth or decay, a common phenomenon in natural processes and finance. The solution to such equations provides insights into future behavior based on present conditions.

Differential equations can be:

• Ordinary Differential Equations (ODEs): Involving a single independent variable.
• Partial Differential Equations (PDEs): Involving multiple independent variables and partial derivatives.

Solving these equations often involves techniques like separation of variables, integration, or numerical methods.

Understanding and solving differential equations through methods like Euler’s is crucial in translating mathematical models into practical, predictive tools.