Solving Differential Equations with ode45: A Powerful Tool for Prediction and Analysis, Resolving 5 Practical Problems

# 1. Overview of Solving Differential Equations with ode45

ode45 is a powerful function in MATLAB for solving ordinary differential equations (ODEs). It employs a numerical integration technique known as the Runge-Kutta method, which is renowned for its accuracy and stability. ode45 is particularly useful for solving a variety of ODEs, including initial value problems and boundary value problems.

The syntax for the ode45 function is as follows:

[t,y] = ode45(@odefun,tspan,y0)

Where:

- `@odefun` is a function handle that defines the differential equation to be solved.
- `tspan` is a vector specifying the range of time over which to solve the problem.
- `y0` is a vector specifying the initial conditions of the differential equation.

# 2. Theoretical Basis of Solving Differential Equations with ode45

### 2.1 Overview of Numerical Integration Methods

Numerical integration methods are techniques for approximating the value of integrals, widely used in solving differential equations. The basic idea is to divide the integration interval into several subintervals and calculate the integral values over each subinterval using some approximation method, ***

***monly used numerical integration methods include:

- Trapezoidal Rule: Divides the integration interval into equal subintervals and approximates the integral values using the trapezoidal formula over each subinterval.
- Simpson's Rule: Divides the integration interval into equal subintervals and approximates the integral values using the Simpson formula over each subinterval.
- Gaussian Quadrature: Uses the Gaussian integration formula, selecting specific nodes over the integration interval and approximating the integral values using the weights of these nodes.

### 2.2 Runge-Kutta Methods

The Runge-Kutta method is an explicit numerical integration method for solving ordinary differential equations. The basic idea is to convert the differential equation into a first-order integral equation, ***

***mon Runge-Kutta methods include:

#### 2.2.1 RK4 Method

The RK4 method (also known as the Runge-Kutta fourth-order method) is a fourth-order Runge-Kutta method, with the formula as follows:

y_{n+1} = y_n + h/6 * (k_1 + 2k_2 + 2k_3 + k_4)

Where:

- `y_n` is the solution at step `n`
- `y_{n+1}` is the solution at step `n+1`
- `h` is the step size
- `k_1 = f(x_n, y_n)`
- `k_2 = f(x_n + h/2, y_n + h/2 * k_1)`
- `k_3 = f(x_n + h/2, y_n + h/2 * k_2)`
- `k_4 = f(x_n + h, y_n + h * k_3)`

**Parameter Explanation:**

- `x_n`: The value of the independent variable at step `n`
- `y_n`: The value of the dependent variable at step `n`
- `f(x, y)`: The right-hand side function of the differential equation

**Logical Analysis:**

The RK4 method approximates the integral value by calculating four slopes `k_1`, `k_2`, `k_3`, and `k_4`. These slopes represent the derivative values at different moments. Then, a weighted average is used to calculate the solution for the next moment `y_{n+1}`.

#### 2.2.2 RK5 Method

The RK5 method (also known as the fifth-order Runge-Kutta method) is a fifth-order Runge-Kutta method, with the formula as follows:

y_{n+1} = y_n + h/90 * (7k_1 + 32k_3 + 12k_4 + 32k_5 + 7k_6)

Where:

- `y_n` is the solution at step `n`
- `y_{n+1}` is the solution at step `n+1`
- `h` is the step size
- `k_1 = f(x_n, y_n)`
- `k_2 = f(x_n + h/4, y_n + h/4 * k_1)`
- `k_3 = f(x_n + 3h/8, y_n + 3h/32 * k_1 + 9h/32 * k_2)`
- `k_4 = f(x_n + 12h/13, y_n + 1932h/2197 * k_1 - 7200h/2197 * k_2 + 7296h/2197 * k_3)`
- `k_5 = f(x_n + h, y_n + 439h/216 * k_1 - 8 * k_2 + 3680h/513 * k_3 - 845h/4104 * k_4)`
- `k_6 = f(x_n + h/2, y_n - 8h/27 * k_1 + 2 * k_2 - 3544h/2565 * k_3 + 1859h/4104 * k_4 - 11h/40 * k_5)`

**Parameter Explanation:**

- `x_n`: The value of the independent variable at step `n`
- `y_n`: The value of the dependent variable at step `n`
- `f(x, y)`: The right-hand side function of the differential equation

**Logical Analysis:**

The RK5 method approximates the integral value by calculating six slopes `k_1`, `k_2`, `k_3`, `k_4`, `k_5`, and `k_6`. These slopes represent the derivative values at different moments. Then, a weighted average is used to calculate the solution for the next moment `y_{n+1}`.

### 2.3 Principles of the ode45 Algorithm

The ode45 algorithm is a MATLAB function for solving ordinary differential equations, which internally uses the RK45 method (a fourth-order Runge-Kutta method). The RK45 method combines the advantages of both RK4 and RK5 methods, offe