Chaplygin method

A method of approximately solving the initial value (Cauchy) problem for systems of ordinary differential equations of the first order, consisting in the simultaneous construction of two families of approximations to the solution. For example, in the case of the initial value (Cauchy) problem for a single equation of the first order

$$ \tag{1 } y ^ \prime = f ( x , y ) ,\ \ ( x , y ) \in R ,\ \ y ( x _ {0} ) = y _ {0} , $$

$$ R = \{ ( x , y ) : | x - x _ {0} | \leq a , | y - y _ {0} | \leq b \} , $$

one of these families approaches the solution from below, and the other from above.

At the basis of the method lies the Chaplygin theorem on differential inequalities. Let $ y ( x) $ be a solution of (1) and suppose that the curves $ y = u ( x) $ and $ y = v ( x) $ lie entirely inside the rectangle $ R $, pass through the point $ ( x _ {0} , y _ {0} ) $, and for $ x > x _ {0} $ satisfy the inequalities

$$ u ^ \prime ( x) - f ( x , u ( x) ) < 0 ,\ \ v ^ \prime ( x) - f ( x , v ( x) ) > 0 . $$

Then for $ x > x _ {0} $ the following inequalities hold:

$$ \tag{2 } u ( x) < y ( x) < v ( x) . $$

Functions $ u ( x) $ and $ v ( x) $ that satisfy the hypotheses of Chaplygin's theorem give upper and lower bounds for the solution to (1).

Given a pair of initial approximations $ u _ {0} ( x) $ and $ v _ {0} ( x) $ satisfying (2), Chaplygin's method enables one to construct a pair $ u _ {1} ( x) , v _ {1} ( x) $ of closer approximations, satisfying

$$ \tag{3 } u _ {0} ( x) < u _ {1} ( x) < y ( x) < \ v _ {1} ( x) < v _ {0} ( x) . $$

In the case where $ \partial ^ {2} f / \partial y ^ {2} $ is of fixed sign throughout $ R $, the pair $ u _ {1} ( x) , v _ {1} ( x) $ can be obtained as the solution of the pair of linear differential equations with initial condition $ y ( x _ {0} ) = y _ {0} $. If, for example, $ \partial ^ {2} f / \partial y ^ {2} > 0 $ in $ R $, then the curve of intersection of any plane $ x = \textrm{ const } $ with the surface $ z = f ( x , y ) $ is convex from below, and any arc of that curve lies below the chord and above the tangent through any of its points. Suppose that for some $ x = \textrm{ const } $ the equation of the tangent to the curve $ z = f ( x , y ) $ at the point $ y = u _ {0} ( x) $ is

$$ z = k ( x) y + p ( x) , $$

where

$$ k ( x) = f _ {y} ^ { \prime } ( x , u _ {0} ( x) ) ,\ \ p ( x) = f ( x , u _ {0} ( x) ) - u _ {0} ( x) k ( x) , $$

and that the equation of the chord of the same curve joining the points $ y = u _ {0} ( x) $ and $ y = v _ {0} ( x) $ is

$$ z = l ( x) y + q ( x) , $$

where

$$ l ( x) = \frac{f ( x , v _ {0} ( x) ) - f ( x , u _ {0} ( x) ) }{v _ {0} ( x) - u _ {0} ( x) } , $$

$$ q ( x) = f ( x , u _ {0} ( x) ) - u _ {0} ( x) l ( x) . $$

Then for that value of $ x $ the inequalities

$$ \tag{4 } k ( x) y + p ( x) < f ( x , y ) < \ l ( x) y + q ( x) $$

hold. Condition (4) is satisfied uniformly for $ x $ in $ R $; the solution $ y = u _ {1} ( x) $ to the initial value (Cauchy) problem $ y ^ \prime = k ( x) y + p ( x) $, $ y ( x _ {0} ) = y _ {0} $, and the solution $ y = v _ {1} ( x) $ to the initial value (Cauchy) problem $ y ^ \prime = l ( x) y + p ( x) $, $ y ( x _ {0} ) = y _ {0} $, satisfy condition (2). It can be shown that they also satisfy condition (3). Given the pair $ u _ {1} ( x) , v _ {1} ( x) $, one can construct in the same way a pair $ u _ {2} ( x) , v _ {2} ( x) $, etc. The process converges very quickly:

$$ \tag{5 } v _ {n} - u _ {n} \leq \frac{c}{2 ^ {2 ^ {n} } } , $$

where the constant $ c $ depends neither on $ x $ nor on $ n $.

A second way of constructing closer approximations $ u _ {n} ( x) , v _ {n} ( x) $ from given approximations $ u _ {n-} 1 ( x) , v _ {n-} 1 ( x) $ does not require the sign of $ \partial ^ {2} f / \partial y ^ {2} $ to be fixed in $ R $. In this method

$$ u _ {n} ( x) = u _ {n-} 1 ( x) + \int\limits _ {x _ {0} } ^ { x } e ^ {- k ( x - t ) } [ f ( t , u _ {n-} 1 ( t) ) - u _ {n-} 1 ^ \prime ( t) ] d t , $$

$$ v _ {n} ( x) = v _ {n-} 1 ( x) + \int\limits _ {x _ {0} } ^ { x } e ^ {- k ( x - t ) } [ v _ {n-} 1 ^ \prime ( t) - f ( t , v _ {n-} 1 ( t) ) ] d t , $$

where $ k $ is the Lipschitz constant of $ f ( x , y ) $ in $ R $. In this case the pairs $ u _ {n} ( x) , v _ {n} ( x) $ and $ u _ {n-} 1 ( x) , v _ {n-} 1 ( x) $ also satisfy condition (3) for all $ x $, but the rate of convergence is less than that given by (5).

The main difficulty in the application of Chaplygin's method lies in the construction of initial approximations $ u _ {0} ( x) , v _ {0} ( x) $.

The method was proposed by S.A. Chaplygin in 1919.

References

[1]	S.A. Chaplygin, "A new method of approximate integration of differential equations" , Moscow-Leningrad (1950) (In Russian)
[2]	N.N. Luzin, "On Academician S.A. Chaplygin's method of approximate integration" Trudy Ts.A.G.I. , 141 (1932) pp. 1–32 (In Russian)
[3]	S.G. Mikhlin, Kh.L. Smolitskii, "Approximate method for solution of differential and integral equations" , American Elsevier (1967) (Translated from Russian)

References

[a1]	L. Collatz, "The numerical treatment of differential equations" , Springer (1966) (Translated from German)