Question

In each exercise, assume that a numerical solution is desired on the interval \(t_{0} \leq t \leq t_{0}+T\), using a uniform step size \(h\). (a) As in equation (8), write the Euler's method algorithm in explicit form for the given initial value problem. Specify the starting values \(t_{0}\) and \(\mathbf{y}_{0}\). (b) Give a formula for the \(k\) th \(t\)-value, \(t_{k}\). What is the range of the index \(k\) if we choose \(h=0.01\) ? (c) Use a calculator to carry out two steps of Euler's method, finding \(\mathbf{y}_{1}\) and \(\mathbf{y}_{2}\). Use a step size of \(h=0.01\) for the given initial value problem. Hand calculations such as these are used to check the coding of a numerical algorithm.\(\mathbf{y}^{\prime}=\left[\begin{array}{cc}\frac{1}{t} & \sin t \\\ 1-t & 1\end{array}\right] \mathbf{y}+\left[\begin{array}{l}0 \\\ t^{2}\end{array}\right], \quad \mathbf{y}(1)=\left[\begin{array}{l}0 \\\ 0\end{array}\right], \quad 1 \leq t \leq 6\)

Short answer

Answer: \(\mathbf{y}_{1}=\left[\begin{array}{l}0 \\\ 0.01\end{array}\right]\) and \(\mathbf{y}_{2}=\left[\begin{array}{l}0.000099 \\\ 0.020301\end{array}\right]\).

Step-by-step solution

Explicit form of Euler's Method

Euler's Method for a system of first-order linear differential equations is given by: \(\mathbf{y}_{k+1} = \mathbf{y}_{k} + h\mathbf{F}(t_{k}, \mathbf{y}_{k})\) where \(\mathbf{F}(t_{k}, \mathbf{y}_{k})\) is the system of equations given by: \(\mathbf{y}^{\prime}=\left[\begin{array}{cc}\frac{1}{t} & \sin t \\\ 1-t & 1\end{array}\right] \mathbf{y}+\left[\begin{array}{l}0 \\ t^{2}\end{array}\right]\) For the given problem, the starting values are: \(t_{0} = 1\) and \(\mathbf{y}_{0}=\left[\begin{array}{l}0 \\\ 0\end{array}\right]\)

Formula for kth t value and range of k

The kth t value can be expressed as: \(t_{k} = t_{0} + kh\) In our case, \(t_{0} = 1\) and \(h = 0.01\), so: \(t_{k} = 1 + 0.01k\) Since we need to solve for the interval \(1 \leq t \leq 6\), we can find the minimum and maximum values of k: For \(t = 1\), \(k = \frac{1-1}{0.01} = 0\) (minimum value) For \(t = 6\), \(k = \frac{6-1}{0.01} = 500\) (maximum value) So, the range of k is from \(0 \leq k \leq 500\).

Calculate y1 and y2 using Euler's Method

Now we proceed with two steps of Euler's method. First, we need to compute \(\mathbf{F}(t_0, \mathbf{y}_0)\): \(\mathbf{F}(t_0, \mathbf{y}_0) = \left[\begin{array}{cc}\frac{1}{1} & \sin 1 \\\ 1-1 & 1\end{array}\right] \left[\begin{array}{l}0 \\\ 0\end{array}\right]+\left[\begin{array}{l}0 \\\ 1^{2}\end{array}\right] = \left[\begin{array}{l}0 \\\ 1\end{array}\right]\) Now calculate \(\mathbf{y}_{1}\): \(\mathbf{y}_{1} = \mathbf{y}_{0} + h \mathbf{F}(t_0, \mathbf{y}_0) = \left[\begin{array}{l}0 \\\ 0\end{array}\right] + 0.01 \left[\begin{array}{l}0 \\\ 1\end{array}\right] = \left[\begin{array}{l}0 \\\ 0.01\end{array}\right]\) Next, we need to compute \(\mathbf{F}(t_1, \mathbf{y}_1)\): \(\mathbf{F}(t_1, \mathbf{y}_1) = \left[\begin{array}{cc}\frac{1}{1.01} & \sin 1.01 \\\ 1-1.01 & 1\end{array}\right] \left[\begin{array}{l}0 \\\ 0.01\end{array}\right]+\left[\begin{array}{l}0 \\\ (1.01)^{2}\end{array}\right] = \left[\begin{array}{l}0.0099 \\\ 1.0301\end{array}\right]\) Finally, calculate \(\mathbf{y}_{2}\): \(\mathbf{y}_{2} = \mathbf{y}_{1} + h \mathbf{F}(t_1, \mathbf{y}_1) = \left[\begin{array}{l}0 \\\ 0.01\end{array}\right] + 0.01 \left[\begin{array}{l}0.0099 \\\ 1.0301\end{array}\right] = \left[\begin{array}{l}0.000099 \\\ 0.020301\end{array}\right]\) Thus, using Euler's method with a step size of \(h=0.01\), the approximate values of \(\mathbf{y}_1\) and \(\mathbf{y}_2\) are: \(\mathbf{y}_{1}=\left[\begin{array}{l}0 \\\ 0.01\end{array}\right]\) \(\mathbf{y}_{2}=\left[\begin{array}{l}0.000099 \\\ 0.020301\end{array}\right]\)

Key concepts

Numerical Solution

When we encounter complex mathematical problems, especially those involving differential equations, finding an exact analytical solution can be quite challenging or sometimes impossible. This is where numerical solutions step in as a powerful tool. A numerical solution involves creating an approximation that closely matches the true solution of the mathematical problem. One of the most common methods for approximating solutions is called Euler's method, which is particularly useful for solving initial value problems related to differential equations.

• Numerical solutions allow us to calculate approximate values of a function at discrete points.
• Euler's method is a straightforward iterative process starting from an initial value and proceeding stepwise, using a chosen step size.
• Each iteration provides a new, approximate solution by considering the slope of the function from the previous step.

Understanding the fundamental idea behind numerical solutions is crucial for students as it broadens their ability to tackle a wide range of mathematical challenges beyond the scope of textbook examples.

Differential Equations

Differential equations are a type of equation that involves an unknown function and its derivatives. These equations describe the relationship between a function and the rate at which it changes. Real-world applications of differential equations are vast and include areas such as physics, engineering, economics, and biology.

Why Are They Challenging?

They often represent complex systems and require sophisticated methods to solve, particularly when they can't be expressed in a simple formulaic way. Euler's method breaks down these complex relationships into more manageable calculations, producing an approximation of the function's behavior over an interval.

• Differential equations express physical laws, such as motion, growth, or decay.
• The solution to a differential equation is a function, not a single value, making the problem multidimensional.
• Euler's method provides a practical approach when exact solutions are unattainable.

Initial Value Problem

An initial value problem in the context of differential equations is a problem where the solution to the differential equation is required to satisfy a specific initial condition. This is akin to knowing the position of a moving object at a certain time and asking where it will be at some later time, given its continuous motion.

Significance of Initial Conditions

Initial conditions provide a starting point for the solution, allowing us to anchor our numerical approximation. When applying Euler's method, we always begin with this known initial value and then employ a formula to predict the following points, hence developing the solution progressively with each step.

• The initial value essentially sets the stage for the entire problem-solving process.
• Knowing the initial conditions, we can apply Euler's method to approximate the subsequent values of the solution.
• The accuracy of the numerical method will depend on the step size and the nature of the differential equation itself.

An in-depth grasp of initial value problems empowers students to approach real-world problems that start from known conditions and evolve over time according to governing equations.