FDM Discretization: Solving Y'' + Xy = 1 With N=4

by Aria Freeman 50 views

Hey guys! Today, we're diving deep into the fascinating world of finite difference methods (FDM). Specifically, we'll tackle the ordinary differential equation (ODE) problem: y'' + xy = 1, with boundary conditions y(0) = 1 and y(1) = 0. Our mission? To discretize this bad boy using FDM with N = 4, ultimately forming the matrices A and b that represent our system of equations. Buckle up, it's gonna be a fun ride!

Problem Setup: Laying the Foundation

Before we jump into the discretization process, let's clearly define our problem. We have the following second-order ODE:

y''(x) + x * y(x) = 1

with boundary conditions:

  • y(0) = 1
  • y(1) = 0

Our goal is to approximate the solution y(x) numerically. To do this, we'll use the finite difference method, which involves dividing the interval [0, 1] into N equal subintervals. In our case, N = 4, meaning we'll have 5 grid points. Let's define these grid points as:

xi = i * h, i = 0, 1, 2, 3, 4

where h is the step size, calculated as:

h = (1 - 0) / N = 1 / 4 = 0.25

So, our grid points are:

  • x0 = 0
  • x1 = 0.25
  • x2 = 0.5
  • x3 = 0.75
  • x4 = 1

Now that we have our grid, we can move on to approximating the derivatives using finite differences.

Finite Difference Approximations: Cracking the Code

The heart of the finite difference method lies in approximating derivatives using difference quotients. We'll use the central difference approximation for the second derivative, which is the most common and generally provides better accuracy compared to forward or backward differences. The central difference approximation for the second derivative at a point xi is given by:

y''(xi) ≈ (yi-1 - 2yi + yi+1) / h2

where yi represents the approximate value of y(x) at xi. This formula essentially says that the second derivative at a point is proportional to the difference between the values of the function at the neighboring points. It's a powerful and intuitive way to approximate curvature.

Now, let's substitute this approximation into our ODE. At each interior grid point (i = 1, 2, 3), we have:

(yi-1 - 2yi + yi+1) / h2 + xi * yi = 1

This equation represents a discrete approximation of our ODE at grid point xi. We'll apply this equation at each interior grid point to generate a system of linear equations.

Forming the System of Equations: The Matrix Dance

Let's write out the equations for i = 1, 2, and 3, keeping in mind that h = 0.25:

  • i = 1: (y0 - 2y1 + y2) / (0.25)2 + 0.25 * y1 = 1
  • i = 2: (y1 - 2y2 + y3) / (0.25)2 + 0.5 * y2 = 1
  • i = 3: (y2 - 2y3 + y4) / (0.25)2 + 0.75 * y3 = 1

We also have our boundary conditions:

  • y0 = 1
  • y4 = 0

Now, let's simplify these equations and rearrange them to a more matrix-friendly form. Multiplying each equation by h2 = (0.25)2 = 0.0625, we get:

  • y0 - 2y1 + y2 + 0.0625 * 0.25 * y1 = 0.0625
  • y1 - 2y2 + y3 + 0.0625 * 0.5 * y2 = 0.0625
  • y2 - 2y3 + y4 + 0.0625 * 0.75 * y3 = 0.0625

Substituting y0 = 1 and y4 = 0, and further simplifying, we have:

  • -1.984375y1 + y2 = -0.9375
  • y1 - 1.96875y2 + y3 = 0.0625
  • y2 - 1.953125y3 = 0.0625

These three equations can now be expressed in matrix form as A * y = b, where:

A = [[-1.984375, 1, 0], [1, -1.96875, 1], [0, 1, -1.953125]]

y = [[y1], [y2], [y3]]

b = [[-0.9375], [0.0625], [0.0625]]

The Matrices A and b: Our Final Form

We've successfully formed the matrices A and b! Matrix A is a 3x3 tridiagonal matrix, which is a common structure in finite difference discretizations of second-order ODEs. The tridiagonal structure arises from the central difference approximation, where each equation involves the values of y at three neighboring grid points. This structure is advantageous for computational efficiency, as specialized algorithms can be used to solve tridiagonal systems quickly.

Matrix b is a 3x1 column vector representing the right-hand side of our system of equations. It incorporates the constant term from the ODE and the boundary conditions.

So, to recap, we have:

A = [[-1.984375, 1, 0], [1, -1.96875, 1], [0, 1, -1.953125]]

b = [[-0.9375], [0.0625], [0.0625]]

These matrices represent the discretized form of our original ODE problem. Solving this system of linear equations (A * y = b) will give us the approximate values of y at the interior grid points, y1, y2, and y3.

Solving the System: The Next Step

Now that we have our matrices A and b, the next step is to solve the linear system A * y = b for the vector y. There are several methods we could use, such as Gaussian elimination, LU decomposition, or iterative methods like the Gauss-Seidel method. For a tridiagonal system like this, the Thomas algorithm (also known as the tridiagonal matrix algorithm or TDMA) is a particularly efficient and commonly used method. This algorithm is a simplified form of Gaussian elimination that takes advantage of the tridiagonal structure to reduce the computational cost.

Once we solve for y, we'll have the approximate values of the solution at the interior grid points. Combined with our boundary conditions, we'll have a complete approximation of the solution to the original ODE.

Conclusion: A Finite Difference Triumph

We've successfully discretized the ODE y'' + xy = 1 using the finite difference method with N = 4. We derived the central difference approximation for the second derivative, set up the system of equations, and formed the matrices A and b. This process is a fundamental step in numerically solving differential equations and is used extensively in various fields of engineering and science. We've seen how a seemingly continuous problem can be transformed into a discrete one, which we can then solve using linear algebra techniques. Understanding this process opens the door to solving a wide range of complex problems that don't have analytical solutions. Keep exploring, keep learning, and you'll be amazed at what you can achieve with numerical methods!

Remember, this is just the beginning. We can further improve the accuracy of our solution by increasing N (using more grid points) or using higher-order finite difference approximations. The world of numerical methods is vast and exciting, so keep your curiosity alive and happy coding!