Differential Equations/Successive Approximations

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y'=f(x,y) has a solution $y$ satisfying the initial condition y(x₀)=y₀, then it must satisfy the following integral equation:

y=y₀+ $\int_{x_0}^x f(t, y(t))dt$

Now we will solve this equation by the method of successive approximations.

Define $y 1$ as:

$y_1=y_0+\int_{x_0}^x f(t,y_0)dt$

And define $y n$ as

$y_n=y_0+\int_{x_0}^x f(t,y_{n-1})dt$

We will now prove that:

If f(x,y) is bounded and the Lipschitz condition is satisfied, then the sequence of functions converges to a continuous function
This function satisfies the differential equation
This is the unique solution to this differential equation with the given initial condition.

[edit] Proof

First, we prove that $y n$ lies in the box, meaning that $|y_n(x)-y_0|<\frac{1}{2}h$ . We prove this by induction. First, it is obvious that $|y_1(x)-y_0|\le\frac{1}{2}h$ . Now suppose that $|y_{n-1}(x)-y_0|\le\frac{1}{2}h$ . Then $|f(t,y_{n-1}(t))|\le M$ so that

$|y_n(x)-y_0|\le\int_{x_0}^x |f(t,y_{n-1}(t))|dt\le M(x-x_0)\le \frac{1}{2}Mw\le \frac{1}{2}h$ . This proves the case when $x 0 < x$ , and the case when $x < x 0$ is proven similarily.

We will now prove by induction that $|y_n(x)-y_{n-1}(x)|<\frac{MK^{n-1}}{n!}(x-x_0)^n$ . First, it is obvious that $| y 1 (x) - y 0 | < M (x - x 0)$ . Now suppose that it is true up to n-1. Then

$|y_n(x)-y_{n-1}(x)|\le\int_{x_0}^x |f(t,y_{n-1}(t))-f(t,y_{n-2}(t))|dt<\int_{x_0}^x K|y_{n-1}(t)-y_{n-2}(t)|dt$ due to the Lipschitz condition.

Now,

$|y_n(x)-y_{n-1}(x)|<\frac{MK^{n-1}}{(n-1)!}\int_{x_0}^x ||u-x_0|^{n-1}du=\frac{MK^{n-1}}{n!}|x-x_0|^n$ .

Therefore, the series of series $y_0+\sum_{n=1}^\infty (y_n(x)-y_{n-1}(x))$ is absolutely and uniformly convergent for $|x-x_0|\le\frac{1}{2}w$ because it is less than the exponential function.

Therefore, the limit function $y(x)=y_0+\sum_{n=1}^\infty (y_n(x)-y_{n-1}(x))=\lim_{n\rightarrow\infty}y_n(x)$ exists and is a continuous function for $|x-x_0|\le\frac{1}{2}w$ .

Now we will prove that this limit function satisfies the differential equation.

Differential Equations/Successive Approximations

From Wikibooks, the open-content textbooks collection

[edit] Proof

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