ODE and PDE, Numerical Approximations I

February 3, 2021

Content

• 1. Direction Field
• 2. Euler’s Method

1. Direction Field

Direction fields are valuable tools in studying the solutions of differential equations of the form:

where $f$ is a given function of the two variables $t$ and $y$, sometimes referred to as the rate function. A direction field for equations (1) can be constructed by evaluating $f$ at each point of a rectangular grid. At each point of the grid, a short line segment is drawn whose slope is the value of $f$ at that point. Each line segment is tangent to the graph of the solution passing through that point. The plots give you a general intuition about what does the solution look like.

Like example on B&D book, page 5, the direction field of equation: $$\frac{d p}{d t}=\frac{p}{2}-450$$

2. Euler’s Method

For a first-order IVP given $f$ and $\partial f /\partial y$ are continuous:

there are two facts driving the numerical approximations approaches:from existence and uniqueness theorem

1. There is a unique solution $y=\phi(t)$ in some interval around $t=t_0$

2. It is usually not possible to find the $\phi$ by symbolic manipulations.

From the diretive fied plotted above, we could imagine that if we have a fine enough grid, it presents a reasonable approximation of the solution, which motivates the Euler’s method as numerical approximation to equation (2)’s solutionThe formalized approximation here is $f'(t) = \frac{f(t+\Delta t)-f(t)}{\Delta t}$. There are many forms of this truncation approximation, with different levels of error. :

1. Start from the initial point $(t_0, y_0)$

2. Move a small step further $t_1 = t_0 + \Delta t$:
   $y_1 = y_0+f(t_0,y_0)\Delta t$

3. Iteratively move the point further like above:
   $y_n = y_{n-1}+f(t_{n-1},y_{n-1})\Delta t$

Now you have the a bunch of points $(t_i,y_i), i = 0, \dots,n$, starting from initial value $(t_0,y_0)$. Connect them with segemental lines, then you have one numerical approximation. The local truncation error level is $O(h^2)$.

Example from B&D book section 2.7, example 1, page 78. Use Euler's method to solve: $$\frac{d y}{d t}=3-2 t-0.5 y, \quad y(0)=1$$ Evaluate at five points $(0,0.2,0.4,0.6,0.8,1.0)$: You can see the error gap grows larger as $t$ increases, because the truncation error is accumulating.

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