Solving Differential Equations with ode45: A Lateral Comparison to Help You Choose the Optimal Solver

# Solving Differential Equations with ode45: A Comparative Analysis to Help You Choose the Optimal Solver

## 1. An Overview of Differential Equation Solving**

A differential equation is an equation that describes the relationship between an unknown function and its derivatives. They are widely used in physics, engineering, biology, and other fields to model various dynamic systems. Differential equation solvers are numerical methods designed to solve differential equations by discretizing them into algebraic equations and using iterative methods.

ode45 is a common differential equation solver in MATLAB, employing the Dormand-Prince method, an explicit Runge-Kutta method known for its accuracy and efficiency. The ode45 solver offers a wealth of parameters and options, allowing users to control the solution accuracy, step size, and tolerance to meet the needs of various applications.

## 2.1 The Principle and Algorithm of the ode45 Solver

### 2.1.1 An Introduction to the Runge-Kutta Method

The Runge-Kutta method is a numerical technique for solving ordinary differential equations by converting them into a set of integral equations and then approximating the solution by iteratively solving these equations.

The specific steps of the Runge-Kutta method are as follows:

1. Given the ordinary differential equation:

y' = f(x, y)

2. With the given initial condition:

y(x0) = y0

3. Iterative computation:

for i = 1 to n
     k1 = h * f(x(i-1), y(i-1))
     k2 = h * f(x(i-1) + h/2, y(i-1) + k1/2)
     k3 = h * f(x(i-1) + h/2, y(i-1) + k2/2)
     k4 = h * f(x(i-1) + h, y(i-1) + k3)
     y(i) = y(i-1) + (k1 + 2*k2 + 2*k3 + k4) / 6
     x(i) = x(i-1) + h
end for

Where:
* `h` is the step size
* `n` is the number of iterations
* `k1`, `k2`, `k3`, `k4` are the Runge-Kutta coefficients

### 2.1.2 The Implementation of the Dormand-Prince Method

The Dormand-Prince method is a variant of the Runge-Kutta method that uses a 7th-order Runge-Kutta formula to solve ordinary differential equations. Its specific steps are as follows:

1. Given the ordinary differential equation:

y' = f(x, y)

2. With the given initial condition:

y(x0) = y0

3. Iterative computation:

for i = 1 to n
     k1 = h * f(x(i-1), y(i-1))
     k2 = h * f(x(i-1) + h/5, y(i-1) + k1/5)
     k3 = h * f(x(i-1) + 3*h/10, y(i-1) + 3*k1/10 - k2/10)
     k4 = h * f(x(i-1) + h/2, y(i-1) + k1/2 - k2/2 + k3/2)
     k5 = h * f(x(i-1) + 2*h/5, y(i-1) + k1/5 - k2/5 + 3*k3/5 - k4/5)
     k6 = h * f(x(i-1) + h, y(i-1) + k1/7 - k2/7 + 2*k3/7 - k4/7 + 6*k5/7)
     k7 = h * f(x(i-1) + h, y(i-1) + k1/8 - k2/8 + k3/8 - k4/8 + k5/8 + 2*k6/8)
     y(i) = y(i-1) + (7*k1 + 32*k3 + 12*k4 + 32*k5 + 7*k6 + 8*k7) / 90
     x(i) = x(i-1) + h
end for

Where:
* `h` is the step size
*