View source for Differentiability of solutions (of differential equations)

A property of solutions of differential equations, viz. that the solutions posses a specific number of continuous derivatives with respect to the independent variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d0317701.png" /> and the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d0317702.png" /> appearing in the equation. In the theory of differential equations the problem is posed as follows: What are the properties which the right-hand side of the equation must have for the solution to have a given number of continuous derivatives with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d0317703.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d0317704.png" />? This problem has been most thoroughly investigated for ordinary differential equations [[#References|[1]]], [[#References|[4]]].

Consider an equation of the type (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d0317705.png" /> may also be a vector):

<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/d/d031/d031770/d0317706.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d0317707.png" /> is a parameter (usually also a vector), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d0317708.png" /> be a solution of (1) defined by the initial condition

<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/d/d031/d031770/d0317709.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

First differentiability of the solution with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177010.png" /> is considered. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177011.png" /> is continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177013.png" />, the theorem on the existence of a continuous solution of the problem (1)–(2) is applicable in some domain, and then it follows from the identity which is obtained after substitution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177014.png" /> in (1) that the continuous derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177015.png" /> also exists. The presence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177016.png" /> continuous derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177017.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177019.png" /> means that there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177020.png" /> continuous derivatives of the solution with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177021.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177022.png" /> may be found (expressed in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177023.png" />) by successive differentiation of the identity obtained by substituting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177024.png" /> in (1).

In several problems, e.g. in constructing the asymptotics of the solution in the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177025.png" />, it is necessary to study derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177026.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177027.png" />. In order to be specific, the existence of derivatives with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177028.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177029.png" /> will be considered. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177030.png" /> is continuous and has continuous partial derivatives with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177032.png" /> in some domain, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177033.png" /> exists and is defined from the so-called variational equation (equation in variations, which is linear in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177034.png" />), obtained from (1) by differentiating both parts with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177035.png" /> and putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177036.png" />:

<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/d/d031/d031770/d03177037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

and with the aid of the initial condition

<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/d/d031/d031770/d03177038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177039.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177040.png" />; if, however, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177041.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177042.png" />.

The derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177043.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177044.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177045.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177046.png" /> (under the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177047.png" /> has continuous partial derivatives up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177048.png" />) is defined by the variational equation of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177049.png" />-th order, which differs from (3) only in its inhomogeneity, and it depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177050.png" />. In the presence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177051.png" /> continuous derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177052.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177053.png" />, Taylor's formula may be used as the asymptotic formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177054.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177055.png" />:

<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/d/d031/d031770/d03177056.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

This is very important, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177058.png" /> can then be found from equations simpler than (1).

If the right-hand side depends analytically on its arguments, the solution is an analytic function of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177059.png" /> (see, for example, [[#References|[2]]]).

The problem of differentiability of solutions with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177060.png" /> is still meaningful in several cases when the right-hand side does not depend regularly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177061.png" />. In one such case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177062.png" /> appears as the coefficient in front of the derivative:

<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/d/d031/d031770/d03177063.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</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/d/d031/d031770/d03177064.png" /></td> </tr></table>

If (6) is rewritten in the form (1), i.e. is solved with respect to the derivatives, a pole-type singularity appears on the right-hand side as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177065.png" />. It is found that, in the presence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177066.png" /> continuous derivatives of the right-hand sides and under certain special conditions (the so-called stability conditions), expansion (5) is valid, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177067.png" /> are the limit values of the derivatives with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177068.png" /> of the solution of (6) as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177069.png" />, which are defined by the variational equation constructed according to the same rule: (6) is differentiated with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177071.png" /> is set equal to zero. However, as distinct from the regular case, the system of variational equations will be of a lower order than (6), and the initial values for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031770/d03177072.png" /> will no longer be zero — but will be equal to (usually non-zero) constants, obtained by a definite rule [[#References|[3]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.G. Petrovskii,   "Ordinary differential equations" , Prentice-Hall  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Tikhonov,   "On the dependence of solutions of differential equations on a small parameter"  ''Mat. Sb.'' , '''22''' :  2  (1948)  pp. 193–204  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.B. Vasil'eva,   V.F. Butuzov,   "Asymptotic expansions of solutions of singularly perturbed equations" , Moscow  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.N. Tikhonov,   A.B. Vasil'eva,   A.G. Sveshnikov,   "Differential equations" , Springer  (1985)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.A. Coddington,   N. Levinson,   "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.R. Smith,   "Singular perturbation theory" , Cambridge Univ. Press  (1985)</TD></TR></table>