[Numerical Solutions to Ordinary Differential Equations: A Basic Introduction and Implementation in MATLAB]

# Advanced篇: Numerical Solutions to Ordinary Differential Equations: An Introductory Guide with MATLAB Implementation

## 1. Theoretical Foundations of Numerical Methods**

**2.1 Basic Concepts of Numerical Methods**

Numerical methods are approaches used to solve ordinary differential equations with the aid of computers. These methods convert the differential equation into a system of algebraic equations and then iteratively solve the system to approximate the solution of the original differential equation.

**2.1.1 Accuracy and Stability**

The accuracy of a numerical method refers to the closeness of the approximation to the exact solution, while stability pertains to the convergence of the solution. The accuracy of a numerical method depends on its order - the higher the order, the greater the accuracy. Stability is determined by the convergence factor, with smaller convergence factors indicating better stability.

**2.1.2 Explicit and Implicit Methods**

Explicit methods compute the solution at the next time step based on the known solution at the current time step, whereas implicit methods compute it based on the unknown solution at the current time step and the known solution at the next time step. Explicit methods are less computationally intensive but less stable, while implicit methods are more computationally intensive but more stable.

# 2. Theoretical Foundations of Numerical Methods

### 2.1 Basic Concepts of Numerical Methods

#### 2.1.1 Accuracy and Stability

In numerical methods, accuracy and stability are two critical concepts.

***Accuracy** measures how closely the numerical solution approximates the exact solution. It is influenced by the step size, method order, and rounding errors.
***Stability** measures whether the numerical method produces stable results during the calculation process. If a method is stable for small step sizes, it is called A-stable; if it is stable for all step sizes, it is called L-stable.

#### 2.1.2 Explicit and Implicit Methods

Numerical methods can be classified into explicit and implicit methods.

***Explicit methods** directly use the current solution to calculate the next time step's solution, with a simple calculation process. However, explicit methods have strict step size limitations; otherwise, unstable results may occur.
***Implicit methods** use the current and next time step's solutions to calculate the next time step's solution, with a more complex calculation process. But, implicit methods allow for looser step size restrictions and better stability.

### 2.2 Common Numerical Methods

#### 2.2.1 Euler's Method

Euler's method is the simplest explicit method, with the formula:

y_{n+1} = y_n + h * f(t_n, y_n)

where h is the step size, and f(t, y) is the ordinary differential equation.

**Logical Analysis:** Euler's method approximates the derivative at the next time step by using the derivative at the current time step to obtain the solution at the next time step.

**Parameter Explanation:**

* h: Step size
* y_n: Solution at the current time step
* f(t_n, y_n): Derivative at the current time step

#### 2.2.2 Improved Euler's Method

The Improved Euler's Method, also known as the midpoint method, has the formula:

y_{n+1} = y_n + h * f(t_n + h/2, y_n + h/2 * f(t_n, y_n))

**Logical Analysis:** The Improved Euler's Method approximates the derivative at the next time step by using the average of the current and next time step's derivatives to obtain the solution at the next time step.

**Parameter Ex