Question

find approximate values of the solution of the given initial value problem at \(t=0.1,0.2,0.3,\) and 0.4 (a) Use the Euler method with \(h=0.05\) (b) Use the Euler method with \(h=0.025\). (c) Use the backward Euler method with \(h=0.05\) (d) Use the backward Euler method with \(h=0.025\) $$ y^{\prime}=2 t+e^{-t y}, \quad y(0)=1 $$

Short answer

Question: Approximate the values of the solution of the given initial value problem (IVP) at t=0.1, 0.2, 0.3, and 0.4 using the Euler and backward Euler methods with step sizes h=0.05 and h=0.025. IVP: $\frac{dy}{dt} = 2t + e^{-ty}$ with initial condition y(0)=1.

Step-by-step solution

Understand and Apply the Euler Method

The Euler method is a first-order numerical method for approximating the solution of an ordinary differential equation (ODE). It can be written as: $$ y_{n+1} = y_n + h f(t_n, y_n) $$ where \(f(t_n, y_n)\) is the derivative of the function \(y(t)\) at point \((t_n, y_n)\). In our problem, we have: $$ f(t, y) = 2t + e^{-ty} $$ Now we will apply the Euler method. (a) Euler method with \(h=0.05\): We will approximate the solution at the following points: \(t = 0.1, 0.2, 0.3, 0.4\) using the given stepsize h=0.05. Start with the initial condition \(y(0)=1\). Therefore, \((t_0, y_0) = (0, 1)\) and apply the Euler method using the stepsize 0.05.

Understand and Apply the Backward Euler Method

The backward Euler method is an implicit first-order numerical method used to approximate the solution of an ODE. The backward Euler method can be written as: $$ y_{n+1} = y_n + h f(t_{n+1}, y_{n+1}) $$ To apply the backward Euler method, we need to solve for \(y_{n+1}\) at each step, which can be challenging depending on the equation. However, we can generally use an iterative method such as the Newton-Raphson method. Now we will apply the backward Euler method. (c) Backward Euler method with \(h=0.05\): We will approximate the solution, following the same procedure mentioned above and using an iterative method to solve for \(y_{n+1}\).

Approximate the Values of the Solution

Use Euler's method and the backward Euler method for the given function with the stated step sizes, and compute the approximations for y(t) at t=0.1, 0.2, 0.3, and 0.4. (a) Approximate solution using Euler method with h=0.05: After applying the Euler method, perform calculations to approximate the values of y(t) at the required points. (b) Approximate solution using Euler method with h=0.025: Repeat the Euler method using a smaller step size (h=0.025) and compute the approximated values of y(t). (c) Approximate solution using backward Euler method with h=0.05: Apply the backward Euler method using h=0.05, and calculate the approximations for y(t) at the required points. (d) Approximate solution using backward Euler method with h=0.025: Use the backward Euler method with h=0.025, and compute the approximations for y(t) at the given points. After completing these steps, you will have the approximate values of the solution at the required points using both Euler and backward Euler methods with different step sizes.

Key concepts

Euler Method

The Euler method is a straightforward numerical technique used for solving ordinary differential equations (ODEs) with given initial conditions, often referred to as an initial value problem. This method allows us to approximate the solution by using a series of straight-line segments between the points on the curve of the solution.
To apply the Euler method, follow these steps: start with an initial point \((t_0, y_0)\), then approximate the next point using \((t_{n+1}, y_{n+1}) = (t_n + h, y_n + h f(t_n, y_n))\). Here, \((t_n, y_n)\) are the current point coordinates, \(h\) is the step size, and \(f(t, y)\) represents the derivative or slope at \(t_n\).
Although simple, the Euler Method's accuracy largely depends on the step size \(h\). Smaller step sizes yield more accurate results but require more calculations. This method forms the backbone for understanding numerical approximations in more complex methods.

Backward Euler Method

The Backward Euler method is a popular choice for solving stiff ODEs, where rapid changes require a stable numerical method to avoid inaccurate results. Unlike the explicit Euler method, the Backward Euler is implicit and requires solving an equation at each step, which can be accomplished using techniques like the Newton-Raphson method.
The formula utilized in this method is: \((t_{n+1}, y_{n+1}) = (t_n + h, y_n + h f(t_{n+1}, y_{n+1}))\). The implicit nature of the equation means \(y_{n+1}\) is on both sides, and hence, solving it involves more computation.
The Backward Euler method is robust, especially useful for problems where maintaining stability is critical even with a larger step size. This makes it the go-to method for many practical applications where traditional methods struggle.

Initial Value Problem

An Initial Value Problem (IVP) in the context of differential equations is a problem where the solution is required to satisfy not only the differential equation but also an initial condition. The initial condition is given as \(y(t_0) = y_0\), where \(t_0\) is a known point, and \(y_0\) is the value of the function at this point.
The purpose of solving an IVP is to predict future behavior of the function based on its current state. Understanding and solving IVPs is crucial in fields such as physics, engineering, and finance, where predicting dynamic systems' behavior is essential.
By utilizing numerical methods like Euler or Backward Euler, we can approximate solutions to these problems even when an analytical solution is difficult or impossible to obtain.

Ordinary Differential Equation

An Ordinary Differential Equation (ODE) involves a function and its derivatives, usually with respect to a single variable. They are called "ordinary" to contrast them with partial differential equations, which involve multiple variables. ODEs appear in various scientific contexts when modeling the rate of change of processes.
In general form, an ODE is represented as \((y' = f(t, y))\), where \(y'\) denotes the derivative of \(y\) with respect to \(t\), and \(f(t, y)\) is a given function. Solving an ODE means finding \(y\) as a function of \(t\) that satisfies both the equation and any given initial conditions.
The challenge with many ODEs is that they don't have closed-form solutions, making numerical methods crucial for finding approximate solutions. Understanding ODEs enables us to simulate real-world systems mathematically, forecasting phenomena like population dynamics or mechanical vibrations.