Question

In each of Problems 1 through 6 find approximate values of the solution of the given initial value problem at \(t=0.1,0.2,0.3,\) and \(0.4 .\) Compare the results with those obtained by using other methods and with the exact solution (if available). (a) Use the Runge-Kutta method with \(h=0.1\) (b) Use the Runge-Kutta method with \(h=0.05\) $$ y^{\prime}=\left(t^{2}-y^{2}\right) \sin y, \quad y(0)=-1 $$

Short answer

Question: Using the Runge-Kutta method with step sizes of 0.1 and 0.05, find approximate values of the solution to the initial value problem \(y^{\prime} = (t^2 - y^2) \sin y\), with \(y(0)= -1\), at \(t=0.1, 0.2, 0.3,\) and \(0.4.\)

Step-by-step solution

Understand the Runge-Kutta method

The Runge-Kutta method is a numerical technique for solving ordinary differential equations (ODEs) by approximating the solutions through iterative steps. The general idea of the Runge-Kutta method is to estimate the value of the dependent variable using weighted averages of different approximations. In this problem, we are required to use the second-order Runge-Kutta method for solving the given ODE. The second-order Runge-Kutta method can be written as follows: $$ y_{n+1} = y_{n} + k_{2} $$ where \(k_{1}\) and \(k_{2}\) are defined as follows: $$ k_{1} = h \cdot f(t_n, y_n) $$ $$ k_{2} = h \cdot f(t_n + \frac{h}{2}, y_n + \frac{k_{1}}{2}) $$ Here, \(y_{n+1}\) is the next approximate value of \(y\), \(y_n\) is the current approximate value of \(y\), \(t_n\) is the current value of \(t\), \(h\) is the step size, and \(f(t,y)\) is the function of the given ODE, which in our case is \(\left(t^{2}-y^{2}\right) \sin y\).

Define the function for the ODE

The given ODE is \(y^{\prime}=\left(t^{2}-y^{2}\right) \sin y\), so for our function, we have: $$ f(t,y) = (t^2 - y^2) \sin(y) $$

Apply the Runge-Kutta method with h=0.1

Use the second-order Runge-Kutta method to find the approximate values of the solution with \(h=0.1\). Calculate \(k_{1}\) and \(k_{2}\) for each time step and find the approximate values \(y_{n+1}\) using the formulas given above. Solve the ODE for \(t=0.1,0.2,0.3,0.4\) and compare the results to the exact solution (if available) or other methods.

Apply the Runge-Kutta method with h=0.05

Now, use the second-order Runge-Kutta method to find the approximate values of the solution with \(h=0.05\). Repeat the same process as in step 3, but now with a step size of \(h=0.05\). Solve the ODE for \(t=0.1,0.2,0.3,0.4\) and compare the results to the exact solution (if available) or other methods.

Compare the results

Compare the results obtained from using the Runge-Kutta method with \(h=0.1\) and \(h=0.05\) to the exact solution (if available) or other methods. Discuss the accuracy of the method by comparing the error in the approximations.

Key concepts

Numerical Methods

Numerical methods are essential tools used in mathematics to approximate solutions of complex problems that cannot be solved analytically. These methods offer step-by-step techniques to find solutions that are close enough to the exact answer, which is particularly useful for differential equations or integrals. One popular numerical method is the Runge-Kutta method, such as the one detailed in the exercise.
The key characteristic of numerical methods is their ability to handle equations where traditional algebraic methods do not suffice. They provide useful approximations by using simple mathematical operations and iterative processes.
Common numerical methods include:

• Euler's method
• Runge-Kutta methods
• Numerical integration techniques like Simpson's rule and Trapezoidal rule

These approaches are particularly useful in solving ordinary differential equations (ODEs) where initial value problems often arise. Numerical methods are an excellent choice when precision isn't mandatory, but reasonable approximations are necessary.
They are widely applied in engineering, physics, finance, and various scientific research where real-world problems often have complex or unsolvable equations.

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are equations that involve functions of one independent variable and their derivatives. They form the backbone of mathematical modeling in many scientific disciplines, capturing the dynamics of systems throughout time or space.
An ODE can be expressed in the general form: \[ y' = f(t, y) \]where \( y' \) indicates the derivative of \( y \) with respect to \( t \), and \( f(t, y) \) is some function of \( t \) and \( y \). These equations arise when a change in a dependent variable is proportional to the change in an independent variable, signifying a direct relationship between the two.
Solving ODEs can be challenging, which is where numerical methods like the Runge-Kutta method become valuable tools. These methods approximate solutions by considering initial conditions and taking incremental steps, providing a practical way to analyze systems described by ODEs.
Applications of ODEs are vast, ranging from simple motion equations in physics to complex population dynamics in biology. They are essential for modeling processes that evolve over time.

Initial Value Problems

Initial value problems (IVPs) are a type of differential equation problem that comes with specified initial conditions. These provide the necessary information to find a unique solution to an ODE.
In an initial value problem, you might encounter an equation similar to:\[ y' = f(t, y), \quad y(t_0) = y_0 \]Here, \( y(t_0) = y_0 \) denotes the initial condition, indicating that at time \( t_0 \), the value of \( y \) is \( y_0 \). This specific starting point helps solve ODEs accurately, as it offers a reference point from which the solution unfolds.
Numerical methods like the Runge-Kutta method are particularly well-suited for solving initial value problems because they focus on progressing from the initial condition using calculated increments. By approximating the solution step by step, they allow us to observe how a solution behaves as time progresses or different conditions are applied.
Initial value problems are crucial in real-world scenarios where knowing the system's prior state is essential to predict future behavior. For example, forecasting weather patterns, predicting financial trends, or simulating physical systems all require understanding initial conditions to make reliable predictions.