Question

Solve the following problems using Euler's method with stepsizes of \(h=0.2,0.1\), 0.05. Compute the error and relative error using the true answer \(Y(x) .\) For selected values of \(x\), observe the ratio by which the error decreases when \(h\) is halved. \(\begin{array}{ll}\text { (a) } \quad & Y^{\prime}(x)=[\cos (Y(x))]^{2}, \quad 0 \leq x \leq 10, \quad Y(0)=0 ; \\\ Y(x)=\tan ^{-1}(x) & & \\ \text { (b) } \quad Y^{\prime}(x)=\frac{1}{1+x^{2}}-2[Y(x)]^{2}, \quad 0 \leq x \leq 10, \quad Y(0)=0 \\ Y(x)=\frac{x}{1+x^{2}} & \\ \text { (c) } \quad Y^{\prime}(x)=\frac{1}{4} Y(x)\left[1-\frac{1}{20} Y(x)\right], \quad 0 \leq x \leq 20, \quad Y(0)=1 ; \\ Y(x)= & \frac{20}{1+19 e^{-x / 4}}\end{array}\) (d) \(Y^{\prime}(x)=-[Y(x)]^{2}, \quad 1 \leq x \leq 10, \quad Y(1)=1 ; \quad Y(x)=\frac{1}{x}\) (e) \(\quad Y^{\prime}(x)=x e^{-x}-Y(x), \quad 0 \leq x \leq 10, \quad Y(0)=1 ;\) \(y(x)=\left(1=\frac{1}{2} x^{2}\right) e^{-x}\) (f) \(\quad Y^{\prime}(x)=\frac{x^{3}}{Y(x)}, \quad 0 \leq x \leq 10, \quad Y(0)=1 ; \quad Y(x)=\sqrt{\frac{1}{2} x^{4}+1}\) (g) \(\quad Y^{\prime}(x)=\left(3 x^{2}-1\right) Y(x)^{2}, \quad 0 \leq x \leq 10, \quad Y(0)=1\); \(y(x)=\sqrt{3\left(x^{3}-x\right)+1}\)

Short answer

Apply Euler's method iteratively for each step size, compute errors and relative errors, analyze error reduction when halving step size.

Step-by-step solution

Title - Understanding Euler's Method

Euler's method is a numerical technique for solving ordinary differential equations (ODEs) by approximating solutions at discrete points. It is given by the formula: \[ Y_{n+1} = Y_n + h f(x_n, Y_n) \] where \( h \) is the step size, \( Y_n \) is the current value, and \( f(x_n, Y_n) \) is the value of the derivative at the point \( (x_n, Y_n) \).

Title - Setting Up the Problem

For each problem, first determine the initial conditions and the corresponding differential equation. Then we select step sizes of \(h = 0.2, 0.1,\) and \(0.05\).

Title - Computing With Euler's Method

For each step size, compute the approximated values using Euler’s method by applying the formula iteratively from the initial condition to the desired value of \(x\). Keep track of the values at each step.

Title - Calculating Errors and Relative Errors

Calculate the true value \( Y(x) \) using the provided true solution. Calculate the error as : \[ \text{Error} = |Y_\text{true} - Y_\text{approx}| \] and the relative error as :\[ \text{Relative Error} = \frac{|Y_\text{true} - Y_\text{approx}|}{|Y_\text{true}|} \]

Title - Examining Error Reduction

For selected values of \(x\), compute the errors for different step sizes. Observe how the error changes as the step size \(h\) is halved. Use the ratio \( \frac{\text{Error}_h}{\text{Error}_{h/2}} \) to analyze the decrease in error.

Title - Example Application (Part (a))

For Part (a): Given \( Y'(x) = [\cos(Y(x))]^2, Y(0)=0, Y(x) = \tan^{-1}(x) \), follow the previous steps. Start with \( x = 0, Y(0) = 0\) and use Euler’s formula with each step size \(h\). Compare the approximation with \( Y(x) = \tan^{-1}(x) \).

Title - Repeat for Remaining Problems

Repeat the steps for each problem (b) through (g) using the corresponding differential equations and true solutions provided.

Key concepts

Numerical Techniques

Numerical techniques are methods used to solve mathematical problems that are difficult or impossible to address analytically. These techniques often involve approximations and are essential in fields like engineering, physics, and economics. Euler’s method is one such numerical technique.
Euler's method specifically addresses the need to solve ordinary differential equations (ODEs) or problems involving rates of change. These techniques are especially useful when the exact analytical solution is not feasible or straightforward to derive.

Ordinary Differential Equations

Ordinary differential equations (ODEs) involve functions and their derivatives. They are used to model various real-world phenomena, such as population growth, heat transfer, and motion of objects. An ODE relates an unknown function, typically dependent on one variable such as time, to its rates of change.
For instance, consider an ODE of the form \(Y'(x) = f(x, Y(x))\), where \(Y(x)\) is the unknown function and \(f(x, Y(x))\) is a given function. The goal is to determine \(Y(x)\) for a given initial condition such as \(Y(0) = Y_0\). Euler's method helps approximate this function by considering small increments (or steps) and iterating over the range of interest.

Step Size

The step size, denoted by \(h\), is a critical component in numerical methods like Euler's method. It represents the increment in the independent variable (such as time or distance) for each step of the computation. Choosing an appropriate step size is crucial for balancing accuracy and computational effort.
If the step size is too large, the method may miss important changes in the function's behavior, leading to significant errors. Conversely, a very small step size improves accuracy but increases the number of computations needed, impacting performance.
In our given problems, step sizes of \(h = 0.2, 0.1, \text{ and } 0.05\) are used to observe how the error changes with different increments.

Error Calculation

When using numerical methods, it's essential to quantify how approximate results differ from true values. This difference is known as the error. Two common forms of error calculations are absolute error and relative error.
The absolute error is given by:
\[\text{Error} = |Y_{\text{true}} - Y_{\text{approx}}|\]
The relative error, which helps in understanding the error magnitude in relation to the true value, is calculated as:
\[\text{Relative Error} = \frac{|Y_{\text{true}} - Y_{\text{approx}}|}{|Y_{\text{true}}|}\]
By comparing errors at different step sizes, one can analyze the effectiveness and accuracy of the numerical method. Observing how the error decreases when the step size is halved provides insights into the method's convergence properties.

Iterative Methods

Euler's method is an example of an iterative method, where the solution is approximated through a series of repeated steps. Starting from an initial condition, the method calculates successive values using a specified formula. For Euler’s method, the iterative formula is:
\[ Y_{n+1} = Y_n + h f(x_n, Y_n)\]
Here, \(h\) is the step size, \(Y_n\) is the current value, and \(f(x_n, Y_n)\) is the derivative at the current point. The next value, \(Y_{n+1}\), is then calculated, and the process repeats for the range of the independent variable.
Iterative methods are powerful as they enable solving complex problems by breaking them down into smaller, manageable steps. The accuracy of the solution depends on the step size and the number of iterations performed.