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Chapter 7: Problem 14

In each exercise, the given iteration is the result of applying Euler's method, Heun's method, or the modified Euler's method to an initial value problem of the form $$ y^{\prime}=f(t, y), \quad y\left(t_{0}\right)=y_{0}, \quad t_{0} \leq t \leq t_{0}+T . $$ Identify the numerical method, and determine \(t_{0}, T\), and \(f(t, y)\). \(y_{n+1}=y_{n}+h\left[\sin \left(t_{n}+\frac{h}{2}+y_{n}+\frac{h}{2} \sin \left(t_{n}+y_{n}\right)\right)\right], \quad y_{0}=1\) \(t_{n}=-1+n h, \quad h=0.05, \quad n=0,1,2, \ldots, 199\)

Short Answer

Expert verified
Based on the provided information and calculations, the numerical method used in this exercise is the Modified Euler's method. The function \(f(t,y)\) is given by \(f(t,y) = \sin(t+y)\). The initial value of \(t_0\) is -1 and the total interval \(T\) is 9.95.

Step by step solution

01

Identify the numerical method

Given equation: $$y_{n+1} = y_n + h\left[ \sin \left(t_n + \frac{h}{2} + y_n + \frac{h}{2} \sin(t_n + y_n) \right) \right]$$ By comparing with each method's iterative equations, we can see that the given equation is in the form of the Modified Euler's method. So, the numerical method used in this exercise is the Modified Euler's method.
02

Determine \(f(t,y)\)

From the Modified Euler's formula: $$y_{n+1} = y_n + h\cdot f(t_n + \frac{h}{2} , y_n + \frac{h}{2} f(t_n, y_n))$$ Comparing this with the given equation, we can deduce that: $$f(t, y) = \sin(t+y)$$
03

Determine \(t_0\)

The provided information says that: $$t_n = -1 + nh,\quad h = 0.05,\quad n=0,1,2,...,199$$ For \(n=0\), we have: $$t_0 = -1 + (0)(0.05) = -1$$
04

Determine \(T\)

We can find \(T\) by calculating the last value of \(t_n\), when \(n=199\): $$t_{199} = -1 + (199)(0.05) = -1 + 9.95 = 8.95$$ Now, we can find \(T\): $$T = t_{199} - t_0 = 8.95 - (-1) = 9.95$$ Finally, we can summarize the values we have found: Numerical Method: Modified Euler's Method $$f(t,y) = \sin(t+y)$$ $$t_0 = -1$$ $$T = 9.95$$

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Key Concepts

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

Euler's Method
Euler's Method is a simple yet foundational numerical technique to solve initial value problems related to ordinary differential equations (ODEs). Imagine you have an equation that ties a function and its derivative, and you want to predict the function's behavior step by step.
Euler's method provides a straightforward mechanism to approximate this.

In essence, Euler鈥檚 method uses basic knowledge from calculus, where the derivative tells how a function changes. It employs the following formula:
  • Start with an initial value: \( y(t_0) = y_0 \).
  • Update the value using the slope: \( y_{n+1} = y_n + hf(t_n, y_n) \).
Here, \( h \) is the step size, which dictates how big each step in the calculation is.
The smaller the \( h \), the more accurate our approximation tends to be.

The method iterates over a specified interval, using the slope (obtained via the derivative) at a point to estimate the function's next value.
This basic approach makes Euler鈥檚 method intuitive but not very accurate for complex problems with high variability or nonlinearities.
Modified Euler's Method
Modified Euler's Method builds on the simplicity of Euler鈥檚 Method but enhances accuracy to tackle more complex situations.
This method is useful when solving initial value problems in differential equations where more precision is needed.
The basic idea involves taking an initial guess at the next point using Euler's formula and then refining it.The formula generally used for this method is:

  • Predict Step (Euler): \( y_{ ext{predict}} = y_n + hf(t_n,y_n) \).
  • Correct Step: \( y_{n+1} = y_n + \frac{h}{2} \left( f(t_n, y_n) + f(t_n + h, y_{ ext{predict}}) \right) \).
The modification you see occurs in the second formula above where you use predictions, average, and then arrive at a more precise next step.

This process is repeated for each interval, making it far superior in accuracy when compared to the basic Euler鈥檚 approach.

In the problem provided, the step of using the sinusoidal function in Modified Euler's formula helps capture more complicated dynamics that a simple time-stepping wouldn鈥檛 handle as well.
Initial Value Problem
An Initial Value Problem (IVP) is a form of differential equation that requires the solution to pass through a specific point on its domain.
These problems typically follow a format like \( y'(t) = f(t, y), \) where you're given an initial condition \( y(t_0) = y_0 \). The goal is to find the function \( y \) that satisfies this differential equation.

Initial value problems are common in real-world applications where you start from a known initial condition and seek to understand how a system evolves over time.
To handle these problems numerically, methods like Euler's or its modifications (like Modified Euler's used in the exercise) come into play, where iteratively computed approximations can give insights into the solution's behavior.

Solving an IVP involves determining how a function changes, meaning it comprises more than just solving the equation algebraically.
It demands an understanding of the system dynamics, initial conditions, and often a computational approach to get practical solutions.

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Most popular questions from this chapter

In each exercise, (a) Find the exact solution of the given initial value problem. (b) As in Example 1, use a step size of \(h=0.05\) for the given initial value problem. Compute 20 steps of Euler's method, Heun's method, and the modified Euler's method. Compare the numerical values obtained at \(t=1\) by calculating the error \(\left|y(1)-y_{20}\right|\). \(y^{\prime}+2 y=4, \quad y(0)=3\)

In each exercise, (a) Verify that the given function is the solution of the initial value problem posed. If the initial value problem involves a higher order scalar differential equation, rewrite it as an equivalent initial value problem for a first order system. (b) Execute the fourth order Runge-Kutta method (16) over the specified \(t\)-interval, using step size \(h=0.1\), to obtain a numerical approximation of the exact solution. Tabulate the components of the numerical solution with their exact solution counterparts at the endpoint of the specified interval. \(t^{2} y^{\prime \prime}-t y^{\prime}+y=t^{2}, \quad y(1)=2, \quad y^{\prime}(1)=2 ; \quad y(t)=t(t+1-\ln t) ; \quad 1 \leq t \leq 2\)

In most applications of numerical methods, as in Exercises 1619 , an exact solution is unavailable to use as a benchmark. Therefore, it is natural to ask, "How accurate is our numerical solution?" For example, how accurate are the solutions obtained in Exercises 16-19 using the step size \(h=0.05\) ? This exercise provides some insight. Suppose we apply Heun's method or the modified Euler's method to the initial value problem \(y^{\prime}=f(t, y), y\left(t_{0}\right)=y_{0}\) and we use a step size \(h\). It can be shown, for most initial value problems and for \(h\) sufficiently small, that the error at a fixed point \(t=t^{*}\) is proportional to \(h^{2}\). That is, let \(n\) be a positive integer, let \(h=\left(t^{*}-t_{0}\right) / n\), and let \(y_{n}\) denote the method's approximation to \(y\left(t^{*}\right)\) using step size \(h\). Then $$ \lim _{h \rightarrow 0 \atop t^{*} \text { fixed }} \frac{y\left(t^{*}\right)-y_{n}}{h^{2}}=C, \quad C \neq 0 $$ As a consequence of this limit, reducing a sufficiently small step size by \(\frac{1}{2}\) will reduce the error by approximately \(\frac{1}{4}\). In particular, let \(\hat{y}_{2 n}\) denote the method's approximation to \(y\left(t^{*}\right)\) using step size \(h / 2\). Then, for most initial value problems, we expect that \(y\left(t^{*}\right)-\hat{y}_{2 n} \approx\left[y\left(t^{*}\right)-y_{n}\right] / 4 .\) Rework Example 1 , using Heun's method and step sizes of \(h=0.05, h=0.025\), and \(h=0.0125 .\) (a) Compare the three numerical solutions at \(t=0.05,0.10,0.15, \ldots, 0.95 .\) Are the errors reduced by about \(\frac{1}{4}\) when the step size is reduced by \(\frac{1}{2}\) ? (Since the solution becomes unbounded as \(t\) approaches 1 from the left, the expected error reduction may not materialize near \(t=1\).) (b) Suppose the exact solution is not available. How can the Heun's method solutions obtained using different step sizes be used to estimate the error? [Hint: Assuming that $$ y\left(t^{*}\right)-\hat{y}_{2 n} \approx \frac{\left[y\left(t^{*}\right)-y_{n}\right]}{4} $$ derive an expression for \(y\left(t^{*}\right)-\hat{y}_{2 n}\) that involves only \(\hat{y}_{2 n}\) and \(\left.y_{n} .\right]\) (c) Test the error monitor derived in part (b) on the initial value problem in Example \(1 .\)

\(m x^{\prime \prime}+\frac{2 k \delta}{\pi} \tan \left(\frac{\pi x}{2 \delta}\right)=F(t), \quad x(0)=0, \quad x^{\prime}(0)=0 ; \quad 0 \leq t \leq 15\) This problem was used to model a nonlinear spring-mass system (see Exercise 18 in Section 6.1). The motion is assumed to occur on a frictionless horizontal surface. In this equation, \(m\) is the mass of the object attached to the spring, \(x(t)\) is the horizontal displacement of the mass from the unstretched equilibrium position, and \(\delta\) is the length that the spring can contract or elongate. The spring restoring force has vertical asymptotes at \(x=\pm \delta\). Time \(t\) is in seconds. Let \(m=100 \mathrm{~kg}, \delta=0.15 \mathrm{~m}\), and \(k=100 \mathrm{~N} / \mathrm{m}\). Assume that the spring-mass system is initially at rest with the spring at its unstretched length. At time \(t=0\), a force of large amplitude but short duration is applied: $$ F(t)=\left\\{\begin{array}{ll} F_{0} \sin \pi t, & 0 \leq t \leq 1 \\ 0, & 1

\(y^{\prime \prime}+4(1+3\) tanh \(t) y=0, \quad y(0)=1, \quad y^{\prime}(0)=0 ; \quad 0 \leq t \leq 10\). This problem might model the motion of a spring-mass system in which the mass is released from rest with a unit initial displacement at \(t=0\) and with the spring stiffening as the motion progresses in time. Plot the numerical solutions for \(y(t)\) and \(y^{\prime}(t)\). Since tanh \(t\) approaches 1 for large values of \(t\), we might expect the solution to approximate a solution of \(y^{\prime \prime}+16 y=0\) for time \(t\) sufficiently large. Do your graphs support this conjecture?
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