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Chapter 3: Problem 13

Use the improved Euler's method with tolerance to approximate the solution to $$ y^{\prime}=1-y+y^{3}, \quad y(0)=0 $$ at \( x=1 \). For a tolerance of \( \varepsilon=0.003 \), use a stopping procedure based on the absolute error.

Short Answer

Expert verified
The final solution is the approximate solution of the ODE at \(x=1\) obtained using the improved Euler's method, ensuring the absolute difference between two successive iterations is less than the tolerance \(\varepsilon=0.003\). The exact value will depend on the chosen step size \(h\) and the number of iterations performed.

Step by step solution

01

Set up the initial conditions

First, label the given ODE and the initial condition: \(y'=f(x,y)=1-y+y^{3}\) and \(y(0)=0\). Initialize \(x = 0\), \(y = 0\), and the step size, \(h\) (which can be experimented with to get improved precision).
02

Implement the improved Euler's method

The improved Euler's method uses a two-step process. First, use the Euler's method to forecast, then correct with an average slope. Follow these equations: \[k_1 = hf(x,y)\] \[k_2 = hf(x+h, y+k_1)\] \[y_{n+1}=y_n + 0.5*(k_1+k_2)\]. Repeat these steps until the absolute error is less than the given tolerance.
03

Calculate until the stopping condition is met

Compute \(y_{n+1}\) from \(y_n\), and compare the absolute difference \(|y_{n+1}-y_n|\) with the tolerance \(\varepsilon=0.003\). If the absolute difference is greater, Update \(y = y_{n+1}\), \(x = x + h\) and compute the next \(y_{n+1}\). Repeat until the absolute difference is less than the tolerance.
04

Check the value at x=1

Once the absolute difference is less than the tolerance, check the value of \(y\) at \(x=1\). This is the approximate solution of the ODE at \(x=1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ordinary Differential Equations (ODE)
Ordinary Differential Equations (ODEs) are equations that involve a function of one independent variable and its derivatives. An ODE provides a description of how this function changes over time or space. In the exercise, the given ODE is \(y' = 1 - y + y^3\) with the initial condition \(y(0) = 0\). The term \(y'\) represents the derivative of \(y\) with respect to \(x\), describing the rate at which \(y\) changes. By solving this ODE, we determine how \(y\) behaves as \(x\) changes.
The specific goal of the exercise is to find the approximate value of \(y\) at \(x=1\) using a numerical method. Due to the complexity of most ODEs, analytical solutions (i.e., exact formulas for the function) are not always possible. This is where numerical methods come into play, allowing us to generate approximate solutions that are usually good enough for practical applications.
Numerical Methods
Numerical methods are techniques used to find approximate solutions to mathematical problems that cannot be solved analytically. The improved Euler's method is one such numerical method, particularly useful for solving ODEs. It's an enhancement of the basic Euler's method and provides more accuracy by considering the average slope over an interval.
  • In the first step of the improved Euler's method, we compute an initial estimate of \(k_1\) using the formula \(k_1 = hf(x, y)\), where \(h\) is the step size.
  • Next, we calculate a second slope \(k_2 = hf(x + h, y + k_1)\) based on the updated values.
  • The approximated next value of \(y\), denoted as \(y_{n+1}\), is then determined by averaging the two slopes: \(y_{n+1} = y_n + 0.5 * (k_1 + k_2)\).
This process is repeated iteratively until a specified condition is met, like achieving a desired level of accuracy. By adjusting the step size \(h\), one can trade off between computational effort and the accuracy of the approximation.
Tolerance in Numerical Computations
Tolerance in numerical computations refers to the permissible level of error in approximating solutions. It's a measure of the precision required for the solution to be considered acceptable. In the context of the improved Euler's method, tolerance determines when to stop the iterative process.

For this exercise, a tolerance of \(\varepsilon = 0.003\) sets the bar for how close the approximate solution should be to the actual value. The iteration continues until the absolute difference between successive approximations, \(|y_{n+1} - y_n|\), is less than the specified tolerance. This ensures that the solution is stable and reliable.
When dealing with numerical computations:
  • A smaller tolerance value may lead to a more precise solution but requires more iterations, increasing computational time.
  • A larger tolerance can save time but may sacrifice accuracy.
Choosing the right tolerance balances accuracy with computational efficiency.

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Most popular questions from this chapter

Use the fourth-order Runge-Kutta subroutine with h = 0.25 to approximate the solution to the initial value problem $$ y^{\prime}=2 y-6, \quad y(0)=1 $$ at x = 1. (Thus, input N = 4.) Compare this approximation to the actual solution \( y=3-2 e^{2 x} \) evaluated at x = 1.

In 1980 the population of alligators on the Kennedy Space Center grounds was estimated to be 1500. In 2006 the population had grown to an estimated 6000. Using the Malthusian law for population growth, estimate the alligator population on the Kennedy Space Center grounds in the year 2020.

Fluid Flow. In the study of the nonisothermal flow of a Newtonian fluid between parallel plates, the equation $$ \frac{d^{2} y}{d x^{2}}+x^{2} e^{y}=0, \quad x>0 $$ was encountered. By a series of substitutions, this equation can be transformed into the first-order equation $$ \frac{d v}{d u}=u\left(\frac{u}{2}+1\right) v^{3}+\left(u+\frac{5}{2}\right) v^{2} $$ Use the fourth-order Runge-Kutta algorithm to approximate \( v(3) \text { if } v(u) \text { satisfies } v(2)=0.1 \). For a tolerance of e = 0.0001, use a stopping procedure based on the relative error.

A sailboat has been running (on a straight course) under a light wind at 1 m/sec. Suddenly the wind picks up, blowing hard enough to apply a constant force of 600 N to the sailboat. The only other force acting on the boat is water resistance that is proportional to the velocity of the boat. If the proportionality constant for water resis- tance is $$b=100 \mathrm{N}-\sec / \mathrm{m}$$ and the mass of the sailboat is 50 kg, find the equation of motion of the sailboat. What is the limiting velocity of the sailboat under this wind?

Building Temperature. In Section 3.3 we modeled the temperature inside a building by the initial value problem $$ \begin{aligned} \frac{d T}{d t} &=K[M(t)-T(t)]+H(t)+U(t) \\\ T\left(t_{0}\right) &=T_{0} \end{aligned} $$ where \( M \) is the temperature outside the building, \( T \) is the temperature inside the building, \( H \) is the additional heating rate, \( U \) is the furnace heating or air conditioner cooling rate, \( K \) is a positive constant, and \( T_{0} \) is the initial temperature at time \( t_{0} \). In a typical model, \( t_{0}=0 \) (midnight), \( T_{0}=65^{\circ} F, H(t)=0.1 \), \( U(t)=1.5[70-T(t)] \), and $$ M(t)=75-20 \cos (\pi t / 12) $$ The constant \( K \) is usually between 1>4 and 1>2, depending on such things as insulation. To study the effect of insulating this building, consider the typical building described above and use the improved Euler's method subroutine with \( h=2 / 3 \) to approximate the solution to (13) on the interval \( 0 \leq t \leq 24 \) (1 day) for \( K=0.2,0.4 \) and 0.6.
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