View source for 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c0214901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c0214902.png" /></td> </tr></table>

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

At the basis of the method lies the [[Chaplygin theorem|Chaplygin theorem]] on differential inequalities. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c0214903.png" /> be a solution of (1) and suppose that the curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c0214904.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c0214905.png" /> lie entirely inside the rectangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c0214906.png" />, pass through the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c0214907.png" />, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c0214908.png" /> satisfy the inequalities

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c0214909.png" /></td> </tr></table>

Then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149010.png" /> the following inequalities hold:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

Functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149013.png" /> that satisfy the hypotheses of Chaplygin's theorem give upper and lower bounds for the solution to (1).

Given a pair of initial approximations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149015.png" /> satisfying (2), Chaplygin's method enables one to construct a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149016.png" /> of closer approximations, satisfying

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

In the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149018.png" /> is of fixed sign throughout <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149019.png" />, the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149020.png" /> can be obtained as the solution of the pair of linear differential equations with initial condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149021.png" />. If, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149022.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149023.png" />, then the curve of intersection of any plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149024.png" /> with the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149025.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149026.png" /> the equation of the tangent to the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149027.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149028.png" /> is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149029.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149030.png" /></td> </tr></table>

and that the equation of the chord of the same curve joining the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149032.png" /> is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149033.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149034.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149035.png" /></td> </tr></table>

Then for that value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149036.png" /> the inequalities

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

hold. Condition (4) is satisfied uniformly for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149038.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149039.png" />; the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149040.png" /> to the initial value (Cauchy) problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149042.png" />, and the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149043.png" /> to the initial value (Cauchy) problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149045.png" />, satisfy condition (2). It can be shown that they also satisfy condition (3). Given the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149046.png" />, one can construct in the same way a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149047.png" />, etc. The process converges very quickly:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149048.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

where the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149049.png" /> depends neither on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149050.png" /> nor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149051.png" />.

A second way of constructing closer approximations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149052.png" /> from given approximations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149053.png" /> does not require the sign of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149054.png" /> to be fixed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149055.png" />. In this method

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149056.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149057.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149058.png" /> is the [[Lipschitz constant|Lipschitz constant]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149059.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149060.png" />. In this case the pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149062.png" /> also satisfy condition (3) for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149063.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021490/c02149064.png" />.

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.A. Chaplygin,   "A new method of approximate integration of differential equations" , Moscow-Leningrad  (1950)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.G. Mikhlin,   Kh.L. Smolitskii,   "Approximate method for solution of differential and integral equations" , American Elsevier  (1967)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Collatz,   "The numerical treatment of differential equations" , Springer  (1966)  (Translated from German)</TD></TR></table>