In mathematical theory of differential equations the Chaplygin Theorem states about the existence and uniqueness of the solution to an initial value problem for the first order explicit ordinary differential equation. This theorem was stated by Sergey Chaplygin in 1919. It is one of many comparison theorems. The Chaplygin Method is a method of solving an ordinary differential equation that fits the criteria of his theorem.
Contents
Chaplygin's Theorem
Chaplygin's Theorem is focused on differential inequalities and how certain characteristics can be used.
Given the aforementioned initial value problem and respective upper boundary solution
z
¯
(
t
)
{\displaystyle {\overline {z}}\left(t\right)}
and lower boundary solution
z
_
(
t
)
{\displaystyle {\underline {z}}\left(t\right)}
for
t
∈
[
t
0
;
α
]
{\displaystyle t\in \left[t_{0};\alpha \right]}
. If the right part
f
(
t
,
y
(
Remarks
Weakening inequalities
Inside inequalities within both of definitions of the upper boundary solution and the lower boundary solution signs of inequalities (all at once) can be altered to unstrict. As a result, inequalities signs at Chaplygin's theorem conclusion would change to unstrict by
is already known to be an existent solution for the initial value problem in
t
∈
[
t
0
;
α
]
{\displaystyle t\in \left[t_{0};\alpha \right]}
, the Lipschitz condition requirement can be omitted entirely for proving the resulting inequality. There exists applications for this method while researching whether the solution is stable or not ( pp. 7–9). This is often called "Differential inequality method" in literature and, for example, Grönwall's inequality can be proven using this technique.
Continuation of the solution towards positive infinity
Chaplygin's theorem answers the question about existence and uniqueness of the solution in
t
∈
[
t
0
;
α
]
{\displaystyle t\in \left[t_{0};\alpha \right]}
and the constant
K
{\displaystyle K}
from the Lipschitz condition is, generally speaking, dependent on
α
{\displaystyle \alpha }
:
K
=
K
(
α
)
{\displaystyle K=K\left(\alpha \right)}
. If for
t
∈
[
t
0
Example of Applying Chaplygin's Method
Given:
y
′
(
x
)
=
e
x
2
{\displaystyle y'(x)=e^{x^{2}}}
,
y
(
x
0
)
=
y
0
{\displaystyle y(x_{0})=y_{0}}
, find the first 2 iterations of approximations.
L
e
t
:
f
(
x
,
y
)
=
e
x
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(1). Chaplygin's Method of iterated approximations can be applied here. After confirming that the differential equations satisfy Chaplygin's Theorem mentioned above, one can apply his method.
Let y(x) be a solution of the initial value Cauchy problem.
. Then, when any plane x=constant intersects with the surface
z
=
f
(
x
,
y
)
{\displaystyle z=f(x,y)}
, the curve of the intersection is convex from below. This results in that any arc of that curve lies below the chord and above the tangent through any of its points.
Now, let us suppose that the equation of the tangent line for the intersection of the plane x=constant and curve
z
=
f
(
x
,
y
)
{\displaystyle z=f(x,y)}
at the point
y
=
u
0
(
x
)
{\displaystyle y=u_{0}(x)}
has 2 parts, the curve of the intersection and the chord.
also satisfy the inequality condition (3) for all x.
However, the rate of convergence is less than that given by (5). So, although the second method has a more straightforward formula, many more iterations of approximations are needed to provide as an accurate result as the first method.
The main difficulty of Chaplygin's Method lies in the construction of the initial approximations
u
0
(
x
)
,
v
0
(
x
)
{\displaystyle u_{0}(x),v_{0}(x)}
.
A little reminder that can be used here is studying the concavity of
y
′
{\displaystyle y'}
. There are 2 situations.
1) When
y
′
{\displaystyle y'}
is concave up (convex): the lower bound approximation can be the tangent line or the first terms from the Taylor Series expansion of
y
′
{\displaystyle y'}
. The upper bound approximation can be taken by finding the secant line.
2) When
y
′
{\displaystyle y'}
is concave down: the lower bound approximation can be the secant line. The upper bound approximation can be the tangent line or the first terms of the Taylor Series.
Essentially, the basis of approximations is about tangent and secant lines, and Taylor Series expansions.
y
(
t
0
)
=
y
0
{\displaystyle y\left(t_{0}\right)=y_{0}}
.
For the initial value problem described above the upper boundary solution and the lower boundary solution are the functions
At this point, it is time to use the Chaplygin Integrals listed in the method section. One should determine the Lipschitz constant and plug everything in correctly to the integral formula. Thus yielding an approximation of the differential equation.
