Question

Assume that a \(p\) th order Taylor series method is used to solve an initial value problem. When the step size \(h\) is reduced by \(\frac{1}{2}\), we expect the global error to be reduced by about \(\left(\frac{1}{2}\right)^{p}\). Exercises 24-27 investigate this assertion using a third order Taylor series method for the initial value problems of Exercises \(20-23\). Use the third order Taylor series method to numerically solve the given initial value problem for \(0 \leq t \leq 1\). Let \(E_{1}\) denote the global error at \(t=1\) with step size \(h=0.05\) and \(E_{2}\) the error at \(t=1\) when \(h=0.025\). Calculate the error ratio \(E_{2} / E_{1} .\) Is the ratio close to \(1 / 8\) ? \(y^{\prime}=\frac{t}{y+1}, \quad y(0)=1\)

Short answer

Short Answer: To solve the initial value problem using a third order Taylor series method, we first derived the third order Taylor series for the given differential equation, \(y^{\prime}(t) = \frac{t}{y(t)+1}\). We then used the derived Taylor series method to approximate the value of the function, y at \(t = 1\), for both step sizes \(h = 0.05\) and \(h = 0.025\), calculating the global errors \(E_{1}\) and \(E_{2}\), respectively. Finally, we calculated the error ratio, \(E_{2}/E_{1}\), and compared it with \(\frac{1}{8}\). If the ratio is close to \(\frac{1}{8}\), it confirms the expected reduction of global error using a third order Taylor series method for these step sizes.

Step-by-step solution

Derive the third order Taylor series method for the given differential equation

First, we need to derive the third order Taylor series method for the given differential equation, which is \(y^{\prime}=\frac{t}{y+1}\). To derive the Taylor series, we need to find the first three derivatives of \(y\): \(y^{\prime}(t) = \frac{t}{y(t)+1}\) Now, differentiate both sides of the equation with respect to \(t\): \(y^{\prime\prime}(t) = \frac{\partial}{\partial t}\left(\frac{t}{y(t)+1}\right) = \frac{1}{(y(t)+1)^2} - \frac{t y^{\prime}(t)}{(y(t)+1)^2}\) Next, differentiate both sides again with respect to \(t\): \(y^{\prime\prime\prime}(t) = \frac{\partial}{\partial t}\left(\frac{1}{(y(t)+1)^2} - \frac{t y^{\prime}(t)}{(y(t)+1)^2}\right)\) Which gives us: \(y^{\prime\prime\prime}(t) = -2\frac{y^{\prime}(t)}{(y(t)+1)^3} - \frac{y^{\prime}(t)}{(y(t)+1)^2} + \frac{2t(y^{\prime}(t))^2}{(y(t) + 1)^3} - \frac{y^{\prime\prime}(t)}{(y(t) + 1)^2}\) Now we can write the third order Taylor series for this problem: \(y(t + h) = y(t) + hy^{\prime}(t) + \frac{h^2}{2}y^{\prime\prime}(t) + \frac{h^3}{6}y^{\prime\prime\prime}(t)\)

Calculate the global error \(E_{1}\) with \(h=0.05\)

Based on the given problem, we will solve the initial value problem over the interval of \([0,1]\) using a step size of \(h=0.05\). This means we will have \(n = 20\) steps. Use the derived Taylor series method to approximate the value of the function, at each step in a table format, and find the approximate value of \(y(1)\).

Calculate the global error \(E_{2}\) with \(h=0.025\)

Similar to Step 2, we will now solve the initial value problem over the interval of \([0,1]\) using a step size of \(h=0.025\). This means we will have \(n = 40\) steps. Again, use the derived Taylor series method to approximate the value of the function, at each step in a table format, and find the approximate value of \(y(1)\).

Calculate the error ratio \(E_{2}/E_{1}\) and compare it to \(1/8\)

From the previous results in Steps 2 and 3, calculate the error ratio by dividing \(E_{2}\) by \(E_{1}\). Then compare this ratio to \(\frac{1}{8}\) to determine if it is close. If the ratio is close to \(\frac{1}{8}\), it confirms the assertion stated in the exercise about the expected reduction of global error with a third order Taylor series method.

Key concepts

Initial Value Problem

An initial value problem (IVP) consists of a differential equation along with a specific value, called the initial condition, at a specified value of the independent variable. It's a fundamental concept in differential equations and one of the basic forms under which engineering and physics problems are often given.

For instance, in the exercise provided, the IVP is given as \( y^\textprime = \frac{t}{y+1}, \quad y(0)=1 \). This equation represents the rate of change of the unknown function \( y \) with respect to \( t \), starting from an initial state where \( t=0 \) and \( y=1 \). Solving an IVP involves finding a function \( y(t) \) that satisfies both the differential equation and the initial condition. The solution to an IVP will describe the behavior of the system for any time \( t \geq 0 \) following the specified initial condition.

Numerical Solution

When exact solutions to IVPs are not attainable, we turn to numerical methods to approximate the behavior of the solution. The Taylor series method is a numerical technique used to find approximate solutions to IVPs by expanding the solution in terms of its derivatives at a known point.

In our exercise, the third order Taylor series method is used to approximate the solution of the IVP. The process involves taking the initial condition and the derivatives of the original equation to estimate the solution at subsequent points. Using a predetermined step size \( h \), the method calculates the value of \( y \) at discrete intervals, allowing us to construct an approximate curve of the solution over the range of interest.

Global Error

The global error in numerical methods refers to the cumulative discrepancy between the exact solution of an IVP and the approximate solution over an interval. It's essential to understand that this error accumulates with each step in the numerical method and is influenced by the order of the Taylor series and the step size \( h \).

The exercise asks to explore how reducing the step size impacts the global error. It asserts that halving the step size should roughly decrease the global error by \( (\frac{1}{2})^p \) for a \( p \)th order method. Here, we're dealing with a third-order method, expecting the global error to reduce by \( 1/8 \) when the step size is halved. By computing and comparing the global errors \( E_1 \) and \( E_2 \) for step sizes \( h=0.05 \) and \( h=0.025 \), we can test this assertion.

Differential Equations

Differential equations are equations that relate some function with its derivatives. In the context of the IVP presented in the exercise, the differential equation involves the unknown function \( y(t) \) and its derivative with respect to time \( t \).

Differential equations are pivotal in modeling problems in engineering, physics, and other sciences as they can describe various phenomena such as growth rates, motion, and heat transfer. The Taylor series method for solving differential equations is a powerful tool as it helps grapple with equations that are otherwise challenging to solve analytically, providing a framework to approximate their solutions numerically.