Numerical methods for differential equations

GUIDING QUESTION: How can I compute a solution to a differential equation?

Guiding philosophy: Translate a differential equation into a system of algebraic equations.

Motivating example

When solving a differential equation, we seek a certain function satisfying some conditions.
For instance, we might seek a function $v: \mathbb{R} \to \mathbb{R}$ such that $$v''(x) = -m^2 v(x)$$
whenever $0 < x < \pi$, for some integer $m$, and such that $v(0) = v(\pi) = 0$.

Motivating example

$$v''(x) = -m^2 v(x), \quad 0 < x < \pi, \quad v(0) = v(\pi) = 0$$ You may think of this data an infinite collection of equations, one for each point in the interval $[0, \pi]$.
We can't solve this infinite system numerically, but we can discretize the domain to obtain a finitely many algebraic equations.

Motivating example

$$v''(x) = -m^2 v(x), \quad 0 < x < \pi, \quad v(0) = v(\pi) = 0$$ You may think of this data an infinite collection of equations, one for each point in the interval $[0, \pi]$.
Fix $n$ and let $x_j = 0 + jh$, with $h = {\pi - 0 \over n}$.
Define $v_j = v(x_j)$. We'll set up a system of algebraic equations to find the unkown $v_j$, by approximating the derivative term $v''(x_j)$ as a linear combination of $v_j$.

Motivating example

$$v''(x) = -m^2 v(x), \quad 0 < x < \pi, \quad v(0) = v(\pi) = 0$$ Define $v_j = v(x_j)$. We'll set up a system of algebraic equations to find the unkown $v_j$, by approximating the derivative term $v''(x_j)$ as a linear combination of $v_j$.
We can do this in at least two ways.
First, we may approximate $v$ using its interpolating polynomial at the $x_j$ and use the differentiation matrix.

Via interpolant

$$v''(x) = -m^2 v(x), \quad 0 < x < \pi, \quad v(0) = v(\pi) = 0$$ We may approximate $v$ using its interpolating polynomial at the $x_j$ and use the differentiation matrix.
The system $$v''(x_j) + m^2 v_j = 0, \quad j = 1, \ldots, n-1$$ becomes $$D^2 \mathbf{v} + m^2 \mathbf{v} = 0,$$ with $D_{ij} = L_i'(x_j)$ and $L_i$ denoting the $i$th Lagrange basis polynomial corresponding to the nodes $x_0, x_1, \ldots, x_n$.

Via interpolant

Therefore, we can obtain an approximation for the values $v_j = v(x_j)$ by solving the algebraic system $$\begin{bmatrix} v_0 \\ D^2 \mathbf{v} + m^2 \mathbf{v} \\ v_n \end{bmatrix} = 0. $$
In this case, we obtain a linear system!

Finite difference schemes

Alternatively, we may approximate the derivative terms $v''(x_j)$ using a finite difference scheme.
If we choose the central difference approximation, the system $$v''(x_j) + m^2 v_j = 0, \quad j = 1, \ldots, n-1$$ becomes $${v_{j-1} - 2v_{j} + v_{j+1} \over h^2} + m^2 v_j = 0, \quad j = 1, \ldots, n-1.$$

Finite difference schemes

Alternatively, we may approximate the derivative terms $v''(x_j)$ using a finite difference scheme.
If we choose the central difference approximation, the system $$v''(x_j) + m^2 v_j = 0, \quad j = 1, \ldots, n-1$$ becomes $$v_{j-1} + (h^2 m - 2) v_{j} + v_{j+1} = 0, \quad j = 1, \ldots, n-1.$$

Finite difference schemes

Alternatively, we may approximate the derivative terms $v''(x_j)$ using a finite difference scheme.
Incorporating the boundary data, we obtain the tridiagonal system \begin{align} v_0 \qquad \qquad \qquad \qquad \quad &= 0 \\ v_0 + (h^2m^2 - 2) v_1 + v_2 \qquad &= 0 \\ \ddots \qquad \qquad \\ v_{n-2} + (h^2m^2 - 2) v_{n-2} + v_n &= 0 \\ v_n &= 0 \end{align}
We may re-write the last system as $$(A - \lambda I) \tilde{\mathbf{v}} = 0,$$ with $$A = \begin{bmatrix} -2 & 1 & & & \\ 1 & -2 & 1 & \\ & \ddots & \ddots & \ddots \\ && 1 & -2 & 1 \\ & & & 1 & -2 \end{bmatrix}, \quad \lambda = (mh)^2,$$ and $\tilde{\mathbf{v}} = \begin{bmatrix} v_1 & \cdots & v_{n-1} \end{bmatrix}^T$.
We've seen this system before, in Problem 1 of HW4!

Finite difference method

Given a differential equation, the finite difference method consists of
  1. Discretize the domain.
    That is, choose a finite set of points in the domain at which you will approximate the solution.
  2. Choose a finite difference approximation for all derivative terms.
  3. Incorporate boundary conditions and associated system!
We may also apply this finite difference method to partial differential equations.
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