Bottom SciencePython Programming
Basics
Numerical Methods
- Bisection Method
- Secant Method
- Regular Falsi (False Position) Method
- Newton Raphson Method
- Gauss Elimination Method
- Gauss Jordan Method
- Gauss-Seidel Method
- Lagrange Interpolation Method
- Newton Divided Difference Interpolation
- Newton Forward Difference Interpolation
- Newton Backward Difference Interpolation
- Trapezoidal Rule
- Simpson 1/3rd Rule
- Simpson 3/8 Rule
- Euler’s Method
- Euler’s Modified Method
- Runge-Kutta 2nd Order Method
- Runge-Kutta 4th Order Method
- Cubic Spline Method
- Bilinear Interpolation Method
- Milne’s Method
- More topics coming soon…
Advance Numerical Methods
Euler’s method is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.
It is a simple and widely used method for approximating the solution of a first-order ODE at discrete time steps.
The idea behind Euler’s method is to approximate the solution of the ODE at discrete time steps by using the derivative at the current time step to estimate the solution at the next time step.
Here’s a simple example of how to use Euler’s method to solve the ODE dy/dx = -y with the initial value y(0) = 1:
Euler’s Method in Python
import numpy as np
def euler_method(f, y0, x0, x_end, h):
# Initialize the solution array
x = np.arange(x0, x_end+h, h)
y = np.zeros(len(x))
y[0] = y0
#Euler's Method [By Bottom Science]
# Iterate over the steps
for i in range(1, len(x)):
y[i] = y[i-1] + h*f(x[i-1], y[i-1])
return x, y
# Define the ODE function
def f(x, y):
return -y
# Set the initial condition and the step size
y0 = 1
x0 = 0
x_end = 10
h = 0.1
# Solve the ODE
x, y = euler_method(f, y0, x0, x_end, h)
for xx,yy in zip(x,y):
print("x = ",xx," y = ",yy)
