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Chapter 5: Problem 3

Use the Modified Euler method to approximate the solutions to each of the following initial-value problems, and compare the results to the actual values. a. \(\quad y^{\prime}=y / t-(y / t)^{2}, \quad 1 \leq t \leq 2, \quad y(1)=1\), with \(h=0.1\); actual solution \(y(t)=t /(1+\ln t)\). b. \(\quad y^{\prime}=1+y / t+(y / t)^{2}, \quad 1 \leq t \leq 3, \quad y(1)=0\), with \(h=0.2\); actual solution \(y(t)=t \tan (\ln t)\). c. \(\quad y^{\prime}=-(y+1)(y+3), \quad 0 \leq t \leq 2, \quad y(0)=-2\), with \(h=0.2 ;\) actual solution \(y(t)=\) \(-3+2\left(1+e^{-2 t}\right)^{-1}\) d. \(\quad y^{\prime}=-5 y+5 t^{2}+2 t, \quad 0 \leq t \leq 1, \quad y(0)=\frac{1}{3}\), with \(h=0.1\); actual solution \(y(t)=t^{2}+\frac{1}{3} e^{-5 t}\).

Short Answer

Expert verified
The short answer involves the application of Modified Euler Method to solve given differential equations and comparing the results with the actual values. Actual comparisons would need numerical outputs from the series of calculations and therefore cannot be succinctly provided in this space.

Step by step solution

01

Identify Variables and Constants

Recognize the format of the problem and identify variables, functions and intervals. Identify the differential equation, the provided initial condition (y(a) = b), the interval, and the step size \(h\). For instance, for problem (a), the differential equation \(y' = y / t - ( y / t )^2\), initial condition y(1) = 1, the interval \(1 \leq t \leq 2\), and the step size \(h = 0.1\).
02

Implement Modified Euler's Method

Implement the Modified Euler's method to approximate the solution of the initial value problem. The formula for Modified Euler method is \(y_{n+1} = y_n + h/2 * ( f(t_n, y_n ) + f(t_{n+1}, y'_n ) )\), where \(y'_n = y_n + h * f(t_n, y_n)\). f(t,y) is the differential equation written in the form \(y' = f(t,y)\). The variables \(t_n\) and \(y_n\) represent the current time and position, \(y'_n\) is the predicted next position (Euler鈥檚 method) and \(y_{n+1}\) is the corrected next position (Modified Euler鈥檚 method). Repeat this calculation for all values of \(t\) in the interval. For example, in problem (b), calculate \(y_n\) for all \(t\) in interval \(1 <= t <= 3\) with the increment of \(h = 0.2 \).
03

Compare with Actual Values

Substitute the numerical approximations obtained into the provided solutions to compare the numerical and actual values. Calculate the actual values by plugging in the same \(t\) values into the analytical solution provided. For example, in problem (c), substitute \(t\) values obtained into the actual solution \(y(t)= -3+2 / (1+e^{-2*t})\) and evaluate.
04

Analyze Error

Analyze the differences between the actual and numerical solutions. This can be done by taking the absolute difference between the numerical and actual values or by calculating the percentage error. All solutions may not perfectly match, especially those at the end of the interval because the error in the numerical methods such as the Euler method accumulates as more steps are taken.

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Key Concepts

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

Numerical Analysis
Numerical analysis involves the study of algorithms that use numerical approximation to solve problems in mathematics and physics that cannot easily be solved analytically. When confronted with differential equations that describe complex systems, such as those in engineering or physics, it often becomes necessary to resort to numerical methods for solutions because exact solutions may be hard to obtain or non-existent.

Methods like the Modified Euler method, a cornerstone in numerical analysis, offer a way to approximate the solutions to differential equations. This iterative technique incrementally advances the solution by computing slopes at both the beginning and the estimated end of a step, which improves accuracy compared to the original Euler's method. This approach strikes a balance between the need for accuracy and the constraints of computational resources.
Initial-Value Problems
Initial-value problems (IVPs) are a type of differential equation problem format where one seeks the solution of a differential equation subject to a given initial condition. An IVP typically takes the form of \( y' = f(t, y) \) with an accompanying initial state such as \( y(t_0) = y_0 \).

In the context of the Modified Euler method, solving an IVP involves using the given initial state to predict subsequent values over a designated interval. Through iterative calculations, the method generates a series of points that approximate the trajectory of the exact solution. The step size \( h \) plays a critical role in this process, as smaller steps generally lead to more accurate approximations but require more computations.
Error Analysis
Error analysis in numerical methods is focused on evaluating the accuracy of the calculated approximation compared to the actual or true values. The error consists of two main components: local truncation error, which measures the error per step, and global error, which assesses the cumulative error over all steps.

By analyzing the error in methods like the Modified Euler method, students can understand the limitations and efficiency of different numerical methods. Errors can be computed using the absolute difference between the numerical approximation and the exact solution or in relative terms as a percentage. This process of error assessment is crucial not just for checking correctness but also for informing the selection of appropriate step sizes and improving the method itself to yield more reliable results.

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Most popular questions from this chapter

Show that Heun's method can be expressed in difference form, similar to that of the Runge-Kutta method of order four, as $$ \begin{aligned} w_{0} &=\alpha \\ k_{1} &=h f\left(t_{i}, w_{i}\right), \\ k_{2} &=h f\left(t_{i}+\frac{h}{3}, w_{i}+\frac{1}{3} k_{1}\right) \\ k_{3} &=h f\left(t_{i}+\frac{2 h}{3}, w_{i}+\frac{2}{3} k_{2}\right) \\ w_{i+1} &=w_{i}+\frac{1}{4}\left(k_{1}+3 k_{3}\right) \end{aligned} $$ for each \(i=0,1, \ldots, N-1\).

Use the Taylor method of order two with \(h=0.1\) to approximate the solution to $$ y^{\prime}=1+t \sin (t y), \quad 0 \leq t \leq 2, \quad y(0)=0. $$

Picard's method for solving the initial-value problem $$ y^{\prime}=f(t, y), \quad a \leq t \leq b, \quad y(a)=\alpha $$ is described as follows: Let \(y_{0}(t)=\alpha\) for each \(t\) in \([a, b]\). Define a sequence \(\left\\{y_{k}(t)\right\\}\) of functions by $$ y_{k}(t)=\alpha+\int_{a}^{t} f\left(\tau, y_{k-1}(\tau)\right) d \tau, \quad k=1,2, \ldots $$ a. Integrate \(y^{\prime}=f(t, y(t))\), and use the initial condition to derive Picard's method. b. Generate \(y_{0}(t), y_{1}(t), y_{2}(t)\), and \(y_{3}(t)\) for the initial- value problem $$ y^{\prime}=-y+t+1, \quad 0 \leq t \leq 1, \quad y(0)=1 $$ c. Compare the result in part (b) to the Maclaurin series of the actual solution \(y(t)=t+e^{-t}\).

A projectile of mass \(m=0.11 \mathrm{~kg}\) shot vertically upward with initial velocity \(v(0)=8 \mathrm{~m} / \mathrm{s}\) is slowed due to the force of gravity, \(F_{g}=-m g\), and due to air resistance, \(F_{r}=-k v|v|\), where \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\) and \(k=0.002 \mathrm{~kg} / \mathrm{m}\). The differential equation for the velocity \(v\) is given by $$ m v^{\prime}=-m g-k v|v| $$ a. Find the velocity after \(0.1,0.2, \ldots, 1.0 \mathrm{~s}\). b. To the nearest tenth of a second, determine when the projectile reaches its maximum height and begins falling.

Use the Extrapolation Algorithm with tolerance \(T O L=10^{-6}, \operatorname{hmax}=0.5\), and \(h \min =0.05\) to approximate the solutions to the following initial-value problems. Compare the results to the actual values. a. \(\quad y^{\prime}=y / t-(y / t)^{2}, \quad 1 \leq t \leq 4, \quad y(1)=1 ;\) actual solution \(y(t)=t /(1+\ln t)\). b. \(\quad y^{\prime}=1+y / t+(y / t)^{2}, \quad 1 \leq t \leq 3, \quad y(1)=0 ;\) actual solution \(y(t)=t \tan (\ln t)\). c. \(\quad y^{\prime}=-(y+1)(y+3), \quad 0 \leq t \leq 3, \quad y(0)=-2\); actual solution \(y(t)=-3+2\left(1+e^{-2 t}\right)^{-1}\). d. \(\quad y^{\prime}=\left(t+2 t^{3}\right) y^{3}-t y, \quad 0 \leq t \leq 2, \quad y(0)=\frac{1}{3} ;\) actual solution \(y(t)=\left(3+2 t^{2}+6 e^{t^{2}}\right)^{-1 / 2}\).
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