O C++, language so keen,

With classes, objects and templates so sheen.

Inheritance and polymorphism galore,

Abstracting the complexity to the core.

Support for low and high level alike,

Helping programmers day and night.

Pointers, references and RAII,

OOP and generic programming, my oh my!

Exception handling and lambda's too,

There's so much you help me do!

A powerful language, sometimes complex,

But in the end, truly adept.

That's my poem about C++,

A language with power, through and through!

Copy code console . log ( "hello world" );

If that's the question you're looking for, here's the bottom line for New York: $2.13 per day.

It's not surprising then, because as you can see

2 is the number of times a point in a circle is square. Thus, one is squared twice: 2 x 2 = 19 x 18 = 51 / 2 x 2 = 49, and I was

O Captain, my Captain, you have led us through the fray
Through the storm and the turmoil, through the darkest of days
With your unwavering courage, your steadfast heart ablaze
You have lifted us up where the eagles soar and play

Your voice is as thunder, your soul a raging fire
You guide us through the tempest, you lift us ever higher
With you at our helm, we shall never falter or tire
For you are the Captain, and we are your loyal choir

Together we shall march, to the beat of our own drum
We shall conquer every fear, we shall overcome
With our eyes on the prize, and our hearts full of song
We shall claim victory, as we journey ever on

So here's to you, O Captain, our light in the dark
May your courage never falter, may your fire never spark
For in your hands we trust, our fate and our ark
O Captain, my Captain, you are the beating of our heart.

The above code will print the numbers 0 through 4. It will not print the number 5 because the range function includes the first value (i.e. 0) but not the last value (i.e. 5).

The 2D Laplace equation is: $$ \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0 $$ where $f$ is a scalar function.

It is used to describe steady-state heat flow, electrostatics, fluid flow, and other phenomena in two dimensions under the assumption of no sources. The Laplace equation arises from simplifying the Poisson equation by setting the charge density to 0. It is a key partial differential equation used in modeling many physical systems.

The Laplace equation implies that the sum of second derivatives of $f$ with respect to $x$ and $y$ is zero, which means the curvature of the solution surface is constant. So, the solutions have a minimal surface with constant negative Gaussian curvature.

Does this help explain the 2D Laplace equation? Let me know if you have any other questions!

Here's the Python code to solve the 2D Laplace equation in Cartesian coordinates:

import numpy as np
import matplotlib.pyplot as plt

# Set up grid
nx = 101
ny = 101
nt = 100

dx = 1. / (nx - 1)
dy = 1. / (ny - 1)

x = np.linspace(0, 1, nx)
y = np.linspace(0, 1, ny)

# Initialize solution
u = np.zeros((nx, ny))

# Set boundary conditions
u[0,:] = 0
u[-1,:] = 0
u[:,0] = 0
u[:,-1] = 0

# Set up source term
f = np.zeros((nx, ny))
f[int(nx/2), int(ny/2)] = 1 / (dx * dy)

# Set up iteration parameters
tolerance = 1e-5
omega = 1.8

# Iterative solver
for n in range(nt):

    for i in range(1, nx-1):
        for j in range(1, ny-1):
            u_old = u[i,j]
            u[i,j] = (1-omega)*u[i,j] + omega*(dx**2*(u[i+1,j]+u[i-1,j]) + dy**2*(u[i,j+1]+u[i,j-1]) - dx**2*dy**2*f[i,j]) / (2*(dx**2+dy**2))
            if abs(u[i,j] - u_old) < tolerance:
                break

# Plot solution
X, Y = np.meshgrid(x, y)
plt.contourf(X, Y, u.T, 100, cmap='viridis')
plt.colorbar()
plt.xlabel('x')
plt.ylabel('y')
plt.title('2D Laplace equation solution')
plt.show()

Explanation:

This code uses finite difference methods to solve the 2D Laplace equation, which is given by:

∇²u(x,y) = 0

on the square domain x=(0,1) and y=(0,1) with vanishing boundary conditions.

We set up a grid with nx x ny points in the x and y directions, respectively. We initialize a solution vector u with zeros, and then set the boundary conditions to zero. We also set up a source term f, which is a delta function located at the center of the domain. The source term is used to enforce the vanishing boundary conditions.

We then use an iterative solver to solve the Laplace equation. The solver updates the values of the solution vector at each point on the grid until the difference between consecutive iterations is below a certain tolerance.

Finally, we plot the solution using Matplotlib, which creates a contour plot of the solution on the domain. The contour plot is a visualization of the solution u(x,y) as a height map, which provides a 2D picture of the solution.