Question

Let \(h\) be a fixed positive step size, and let \(\lambda\) be a nonzero constant. Suppose we apply Heun's method or the modified Euler's method to the initial value problem \(y^{\prime}=\lambda y, y\left(t_{0}\right)=y_{0}\), using this step size \(h\). Show, in either case, that \(y_{k}=\left(1+h \lambda+\frac{(h \lambda)^{2}}{2 !}\right) y_{k-1}\) and hence \(y_{k}=\left(1+h \lambda+\frac{(h \lambda)^{2}}{2 !}\right)^{k} y_{0}, \quad k=1,2, \ldots\)

Short answer

In this exercise, we applied Heun's method to the given differential equation \(y' = \lambda y\) with initial condition \(y(t_0) = y_0\) and step size \(h\). We derived the expression \(y_k = \left(1 + h\lambda + \frac{(h\lambda)^2}{2!}\right)y_{k-1}\) and further derived the expression for \(y_k\) in terms of \(y_0\) and \(k\) as \(y_k = \left(1 + h\lambda + \frac{(h\lambda)^2}{2!}\right)^k y_0\). These expressions give us the numerical solution to the initial value problem after \(k\) steps using Heun's method.

Step-by-step solution

Recall Heun's method

Heun's method, also known as the modified Euler's method, is a numerical method used for solving ordinary differential equations (ODEs) given an initial value problem. The method is an improved version of Euler's method, using an average of the slope at the initial point and the slope at the endpoint of the interval to compute the next value of \(y\). Given a differential equation of the form \(y'(t) = f(t, y)\) and an initial condition \(y(t_0) = y_0\), the method proceeds in the following manner: 1. Set \(k = 1\). 2. Compute the slope at the initial point: \(k_1 = f(t_{k-1}, y_{k-1})\). 3. Compute the endpoint: \(t_k = t_{k-1} + h\). 4. Compute the slope at the endpoint: \(k_2 = f(t_k, y_{k-1} + hk_1)\). 5. Update the solution: \(y_k = y_{k-1} + \frac{h}{2}(k_1 + k_2)\). 6. Increment \(k\) and repeat steps 2-5 until the desired number of steps is reached. Now, let's apply Heun's method to the given ODE \(y' = \lambda y\) with initial condition \(y(t_0) = y_0\).

Applying Heun's method to the given ODE

In our case, we have that \(f(t, y) = \lambda y\). We will apply Heun's method to obtain the general form for \(y_k\). 1. Compute the slope at the initial point: \(k_1 = f(t_{k-1}, y_{k-1}) = \lambda y_{k-1}\). 2. Compute the endpoint: \(t_k = t_{k-1} + h\). 3. Compute the slope at the endpoint: \(k_2 = f(t_k, y_{k-1} + hk_1) = \lambda(y_{k-1} + h\lambda y_{k-1})\). 4. Update the solution: \(y_k = y_{k-1} + \frac{h}{2}(k_1 + k_2) = y_{k-1} + \frac{h}{2}\left(\lambda y_{k-1} + \lambda(y_{k-1} + h\lambda y_{k-1})\right)\). Now, let's simplify this expression for \(y_k\).

Simplifying the expression for \(y_k\)

We can simplify the expression for \(y_k\) as follows: \begin{align*} y_k &= y_{k-1} + \frac{h}{2}\left(\lambda y_{k-1} + \lambda(y_{k-1} + h\lambda y_{k-1})\right) \\ &= y_{k-1} + \frac{h}{2}\left(\lambda y_{k-1} + \lambda y_{k-1} + h\lambda^2 y_{k-1}\right) \\ &= y_{k-1} + h\lambda y_{k-1} + \frac{h^2\lambda^2}{2}y_{k-1} \\ &= \left(1 + h\lambda + \frac{(h\lambda)^2}{2!}\right)y_{k-1}. \end{align*} Now, we have arrived at the desired expression for \(y_k\): \(y_k = \left(1 + h\lambda + \frac{(h\lambda)^2}{2!}\right)y_{k-1}\).

Expressing \(y_k\) in terms of \(y_0\) and \(k\)

Next, we will express \(y_k\) in terms of \(y_0\) and \(k\). We can do this by iterating the expression we found in step 3 for \(y_k\), from \(k=1\) to \(k\). For \(k=1\), we have: \(y_1 = \left(1 + h\lambda + \frac{(h\lambda)^2}{2!}\right)y_0\). For \(k=2\), we have: \(y_2 = \left(1 + h\lambda + \frac{(h\lambda)^2}{2!}\right)y_1 = \left(1 + h\lambda + \frac{(h\lambda)^2}{2!}\right)^2y_0\). By continuing this pattern, we can see that for any \(k\), we have: \(y_k = \left(1 + h\lambda + \frac{(h\lambda)^2}{2!}\right)^k y_0\). This is the final expression we have been asked to derive.

Key concepts

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are fundamental in the mathematical modeling of natural systems, capturing how various quantities evolve over time under the influence of numerous factors. An ODE is an equation that involves functions of only one independent variable and its derivatives. The simplest form of an ODE is a first-order equation, which can be written in general form as \( y'(t) = f(t, y) \), where \( y'(t) \) represents the derivative of \( y \) with respect to \( t \) and \( f(t, y) \) is a given function.

Such equations are prevalent in various branches of science, including physics, biology, and engineering, describing phenomena like radioactive decay, population dynamics, and the motion of a pendulum. The initial condition, typically denoted as \( y(t_0) = y_0 \), specifies the state of the system at the beginning of the period of interest, anchoring the solution to a definite starting point. Solving an ODE entails determining the function \( y(t) \) that satisfies both the differential equation and the initial condition.

Numerical Methods

Numerical methods are essential tools for solving mathematical problems that cannot be addressed analytically or require a considerable amount of simplification for an analytical approach to be feasible. They provide algorithms for approximating solutions to various mathematical problems, including ODEs. Numerical methods allow the approximation of solutions at discrete points, making it possible to handle complex equations that describe real-world phenomena where an exact solution is unattainable.

One such method is Heun's method, which improves upon the basic Euler's method by taking the average of two slopes: one at the start and another at the projected end of the interval. It is an explicit technique, meaning it calculates the next point's value based solely on previously calculated values without solving any algebraic equations at each step. A key advantage of Heun's method over Euler's method is increased accuracy for slightly more computational complexity.

Initial Value Problem

The initial value problem (IVP) for ODEs is a task where one seeks the function \( y(t) \) that not only solves the differential equation \( y'(t) = f(t, y) \) but also satisfies an initial condition \( y(t_0) = y_0 \). This problem can be viewed as tracking the future behavior of a system given a known starting position.

When applying numerical methods like Heun's method, the IVP is approached iteratively, starting from the initial condition and advancing step-by-step in a controlled manner over the domain. The step size \( h \) determines the fineness of the approximation and thus influences the accuracy of the solution. Small \( h \) values typically lead to more precise results but require more computation. The challenge is to choose a step size that provides an adequate trade-off between accuracy and computational efficiency. The output of Heun's method is a set of points \( (t_k, y_k) \) that approximate the function \( y(t) \) at discrete intervals, forming a polygonal curve that can be used to visualize the solution of the IVP.