Question

For the given initial value problem, an exact solution in terms of familiar functions is not available for comparison. If necessary, rewrite the problem as an initial value problem for a first order system. Implement one step of the fourth order Runge-Kutta method (14), using a step size \(h=0.1\), to obtain a numerical approximation of the exact solution at \(t=0.1\). \(y^{\prime \prime}+z+t y=0\) \(z^{\prime}-y=t, \quad y(0)=1, \quad y^{\prime}(0)=2, \quad z(0)=0\)

Short answer

Question: Apply one step of the fourth-order Runge-Kutta method with step size \(h=0.1\) to the given initial-value problem for second-order differential equations, and find the numerical approximation of the exact solution at \(t=0.1\). Given equations: $$y^{\prime \prime}+z+t y=0,$$ $$z^{\prime}-y=t,$$ with initial conditions: $$y(0)=1,$$ $$y^{\prime}(0)=2,$$ $$z(0)=0.$$ The numerical approximation of the exact solution at \(t=0.1\) using the fourth-order Runge-Kutta method is: $$u(0.1) \approx 1.1,$$ $$v(0.1) \approx 1.9835,$$ $$z(0.1) \approx 0.10815.$$

Step-by-step solution

Convert the second-order ODE into a first-order system

Introduce two new functions \(u(t) = y(t\)) and \(v(t) = y'(t)\) to transform the given second-order ODE into a system of two first-order ODEs. The derivatives of the two new functions are: $$u'(t) = v(t),$$ $$v'(t) = -z(t) - t u(t).$$ Now we have a first-order system of the form: $$\mathbf{w}'(t) = \mathbf{F}(t, \mathbf{w}(t)),$$ where \(\mathbf{w} = \begin{pmatrix} u \\ v \\ z \end{pmatrix}\) and \(\mathbf{F} = \begin{pmatrix} v \\ -z-tu \\ u-t \end{pmatrix}\). The given initial conditions are \(u(0)=1\), \(v(0)=2\), and \(z(0)=0\).

Prepare the fourth-order Runge-Kutta method

The fourth-order Runge-Kutta method is a numerical method to approximate the solution of a first-order ODE system. It can be written as follows: 1. Calculate \(k_1 = h\mathbf{F}(t_n, \mathbf{w}_n)\), 2. Calculate \(k_2 = h\mathbf{F}(t_n + \frac{h}{2}, \mathbf{w}_n + \frac{k_1}{2})\), 3. Calculate \(k_3 = h\mathbf{F}(t_n + \frac{h}{2}, \mathbf{w}_n + \frac{k_2}{2})\), 4. Calculate \(k_4 = h\mathbf{F}(t_n + h, \mathbf{w}_n + k_3)\), 5. Update \(\mathbf{w}_{n+1} = \mathbf{w}_n + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)\). We will use this method to calculate the numerical approximation of the exact solution at \(t=0.1\).

Apply the fourth-order Runge-Kutta method

We have to implement one step of the Runge-Kutta method with step size \(h=0.1\). We start with the initial conditions \(t_0=0\), \(u_0=1\), \(v_0=2\), and \(z_0=0\). Then, using the given functions and relations, we carry out the Runge-Kutta method step by step: 1. \(k_1 = 0.1\begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0.2 \\ 0 \\ 0.1 \end{pmatrix}\), 2. \(k_2 = 0.1\begin{pmatrix} 2 \\ -0.05 \\ 1.05 \end{pmatrix} = \begin{pmatrix} 0.2 \\ -0.005 \\ 0.105 \end{pmatrix}\), 3. \(k_3 = 0.1\begin{pmatrix} 2 \\ -0.0475 \\ 1.0475 \end{pmatrix} = \begin{pmatrix} 0.2 \\ -0.00475 \\ 0.10475 \end{pmatrix}\), 4. \(k_4 = 0.1\begin{pmatrix} 2 \\ -0.10975 \\ 1.12975 \end{pmatrix} = \begin{pmatrix} 0.2 \\ -0.010975 \\ 0.112975 \end{pmatrix}\), 5. \(\mathbf{w}_1 = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix} + \frac{1}{6}(\begin{pmatrix} 0.2 \\ 0 \\ 0.1 \end{pmatrix} + 2\begin{pmatrix} 0.2 \\ -0.005 \\ 0.105 \end{pmatrix} + 2\begin{pmatrix} 0.2 \\ -0.00475 \\ 0.10475 \end{pmatrix} + \begin{pmatrix} 0.2 \\ -0.010975 \\ 0.112975 \end{pmatrix}) = \begin{pmatrix} 1.1 \\ 1.9835 \\ 0.10815 \end{pmatrix}\). So, the numerical approximation of the exact solution at \(t=0.1\) obtained using one step of the fourth-order Runge-Kutta method is: $$u(0.1) \approx 1.1,$$ $$v(0.1) \approx 1.9835,$$ $$z(0.1) \approx 0.10815.$$

Key concepts

Initial Value Problem

An initial value problem (IVP) is a fundamental concept in differential equations where the solution must satisfy a given set of conditions at the starting point, typically time zero. This involves finding a function, usually denoted as 'y(t)' that not only satisfies a differential equation but also meets specific values, known as initial conditions, at a particular point. For instance, in the context of the exercise, we are presented with a second-order ordinary differential equation (ODE) with the initial conditions provided at time 't = 0'.

With these conditions, the goal is to determine the unique trajectory of the system from that starting point forward. Given the inherent complexity of many differential equations, analytical solutions are not always obtainable. In such cases, numerical methods, like the Runge-Kutta method, become invaluable tools for approximating the behavior of the system beyond the initial state.

System of Differential Equations

A system of differential equations consists of multiple equations involving derivatives of several dependent variables with respect to one or more independent variables. In the given exercise, we transition from a single second-order ODE to a system of first-order ODEs for simplification and to facilitate the application of the Runge-Kutta method.

Breaking down a higher-order ODE into a system of first-order ODEs is a common strategy, as first-order systems can be more systematically approached through numerical methods. The system expresses how each variable changes in relation to others, capturing the interdependencies within complex dynamic systems. When addressing a system, the Runge-Kutta method, or similar numerical solutions, can yield approximate values for each variable at desired points, thus building a comprehensive picture of how the system evolves over time.

Numerical Approximation

Numerical approximation involves estimating solutions to problems where exact answers might be impossible to find due to complexity or imprecision in the model. In the field of differential equations, numerical methods enable us to approximate solutions by creating sequences that converge towards the true solution. The Runge-Kutta method employed in the exercise is a prime example of such a technique. It offers a precise approximation of the solution at discrete points along the interval of interest, in this case at t = 0.1.

This process is not only crucial for cases lacking closed-form solutions but is also invaluable in various applied fields like physics, engineering, and finance where understanding the behavior of systems through simulation is more practical than solving equations analytically.

Second-Order ODE

A second-order ODE is a differential equation that contains the second derivative of a function. In our exercise, the function 'y' depends on variable 't' and includes its second derivative y''. The complexity of second-order ODEs typically means they are harder to solve analytically, making numerical methods such as the Runge-Kutta approach more significant.

Second-order ODEs often arise in physics, particularly in mechanics and wave theory, as they can describe systems with acceleration or curvature. By converting the second-order ODE into a system of first-order ODEs, as done in the step-by-step solution, we utilize powerful numerical techniques to obtain a useful approximation even when an exact solution remains out of reach.