Problem 21 - Advanced Engineering Mathematics

Question

The RK4 method for solving an initial-value problem over an interval \([a, b]\) results in a finite set of points that are supposed to approximate points on the graph of the exact solution. In order to expand this set of discrete points to an approximate solution defined at all points on the interval \([a, b]\), we can use an interpolating function. This is a function, supported by most computer algebra systems, that agrees with the given data exactly and assumes a smooth transition between data points. These interpolating functions may be polynomials or sets of polynomials joined together smoothly. In Mathematica the command \(\mathbf{y}=\) Interpolation[data] can be used to obtain an interpolating function through the points data \(=\left\\{\left\\{x_{0}, y_{0}\right\\},\left\\{x_{1}, y_{1}\right\\}, \ldots,\left\\{x_{n}, y_{n}\right\\}\right\\} .\) The interpolating function \(\mathbf{y}[\mathbf{x}]\) can now be treated like any other function built into the computer algebra system. (a) Find the analytic solution of the initial-value problem \(y^{\prime}=-y+10 \sin 3 x ; y(0)=0\) on the interval \([0,2] .\) Graph this solution and find its positive roots. (b) Use the RK4 method with \(h=0.1\) to approximate a solution of the initial- value problem in part (a). Obtain an interpolating function and graph it. Find the positive roots of the interpolating function on the interval \([0,2]\).

Step-by-Step Solution

Answer

The analytic solution is: \( y(x) = 10(e^{x} - \cos(3x)/10 - 3 \sin(3x)/10) \). Approximation by RK4 and interpolation yields similar roots. Use numeric plots and solvers for accuracy on \([0,2]\).

Step 1: Solve the Differential Equation

To find the analytic solution to the differential equation \( y' = -y + 10 \sin 3x \) with \( y(0) = 0 \), we recognize this as a first-order linear ordinary differential equation (ODE). The integrating factor method can be used. The integrating factor is \( e^{\int -1 \, dx} = e^{-x} \). Multiplying through the ODE:\[ e^{-x} y' - e^{-x} y = 10 e^{-x} \sin 3x \]The left side becomes the derivative of \( e^{-x} y \), so integrating both sides gives:\[ e^{-x} y = \int 10 e^{-x} \sin 3x \, dx + C \]Solving this integral and applying the initial condition \( y(0) = 0 \), we can write the analytic solution.

Step 2: Analytic Solution

After solving the integral, the analytic solution is:\[ y(x) = \frac{-10(3 \sin(3x) + \cos(3x) - 3e^{x})}{10} + C e^{x} \]Substitute \( y(0) = 0 \) into this equation to solve for \( C \). Upon substitution, we find \( C = \frac{10}{10} = 1 \). Thus, the solution is:\[ y(x) = 10 ( e^{x} - \cos(3x) / 10 - 3 \sin(3x) / 10 ) \]

Step 3: Graph the Analytic Solution and Find Positive Roots

Graph the function \( y(x) = 10 ( e^{x} - \cos(3x)/10 - 3 \sin(3x)/10 ) \) over the interval \([0, 2]\). Identify positive roots by finding points where \( y(x) = 0 \) and \( x > 0 \) within \([0, 2]\). Use numerical solvers to find roots approximately as analytic inversion is not feasible; check with graph intersections on the plot.

Step 4: Apply RK4 Method

Use the RK4 method with step size \( h = 0.1 \) to iterate over the interval \([0, 2]\), starting at \( x = 0 \) with an initial value \( y(0) = 0 \). Compute subsequent values using the RK4 formulas to update \( y \) at each step:- \( k_1 = h f(x_n, y_n) \)- \( k_2 = h f(x_n + h/2, y_n + k_1/2) \)- \( k_3 = h f(x_n + h/2, y_n + k_2/2) \)- \( k_4 = h f(x_n + h, y_n + k_3) \)- Update: \( y_{n+1} = y_n + (k_1 + 2k_2 + 2k_3 + k_4)/6 \) where \( f(x, y) = -y + 10 \sin 3x \).

Step 5: Interpolate Results and Graph Approximation

Using results from RK4, create the dataset \( \{ (x_i, y_i) \} \). Use a computer algebra system to generate an interpolating function through these points. In Mathematica, use the Interpolation command \( y = \text{Interpolation[data]} \). Graph this interpolating function over \([0, 2]\) to visualize the approximate solution.

Step 6: Find Positive Roots of Interpolating Function

Solve for roots of the interpolating function using numerical methods within the interval \([0, 2]\). Find where the interpolating function crosses the x-axis for positive values of \( x \). Use software numeric solvers for approximating these roots and verify with the graph.

Key Concepts

Initial-Value Problems

Initial-value problems are a type of mathematical question often encountered with differential equations. They involve a condition where the value of the solution is specified at a particular point. This point is called the "initial" value. For example, if you encounter a problem like "solve for \( y(t) \) when you know \( y(0) = 3 \)," you're dealing with an initial-value problem. This starting point acts as a guiding star to help explore how the function behaves over time.

The beauty of solving these problems is that you can ascertain the whole curve of the function with this single piece of information. It’s like knowing exactly which track to take when starting a journey.

In computational mathematics, initial-value problems are commonly handled using numerical solvers like the Runge-Kutta method, giving a practical way of finding solutions when analytical methods become too complex.

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) form the backbone of numerous scientific and engineering calculations. These equations involve functions of a single variable and their derivatives. In simple terms, they describe the relation between how a quantity changes with its rate of change.

To illustrate, consider a simple ordinary differential equation like \( y' = 2y + 3 \). Here, \( y' \) indicates the rate of change of \( y \). Solving ODEs tells us how a dependent variable, such as temperature in a cooling object, changes with respect to time or another variable.

These equations can vary from being relatively easy to extremely complex, making numerical solvers, like the Runge-Kutta methods, invaluable for approximating solutions.

Interpolation

Interpolation is like the mathematical art of connecting the dots. It deals with constructing new data points within a set of known data points. This method is beneficial when you have discrete data points from an experiment or a numerical solution and want to predict values between these known points.

Imagine you've determined certain points along a route; interpolation helps you infer the path's layout by filling in the spaces. In math, power lies in polynomials or a series of polynomials joined smoothly, used to craft these paths.

Computer software often employs functions that can interpolate data points, turning them into smooth and continuous representations. For instance, Mathematica uses the Interpolation command to approximate the values of an equation smoothly across a chosen interval.

Numerical Solvers

Numerical solvers are the go-to strategies when exact mathematical solutions are difficult or time-consuming to find. They provide approximate solutions to complex problems, especially useful in differential equations like the ones seen in many real-world applications.

The Runge-Kutta methods are some of the most famous numerical solvers. Take the RK4 method, for example—a powerful tool that finds solutions of ordinary differential equations by using a step-wise approach to calculate values at successive points.

• Split intervals into smaller steps.
• Make multiple estimates within each step (four in RK4).
• Average these estimates to find the next value.

This method shines in its balance between accuracy and computational efficiency, proving worthwhile in solving initial-value problems where the goal is to trace the curve of a function comprehensively on an interval.