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 $ t $ and the parameter $ \mu $ 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 $ t $ and $ \mu $? This problem has been most thoroughly investigated for ordinary differential equations [1], [4].

Consider an equation of the type ( $ x $ may also be a vector):

$$ \tag{1 } \frac{dx}{dt} = f ( t , x , \mu ) , $$

where $ \mu $ is a parameter (usually also a vector), and let $ x ( t , \mu ) $ be a solution of (1) defined by the initial condition

$$ \tag{2 } \left . x \right | _ {t = t _ {0} } = x _ {0} . $$

First differentiability of the solution with respect to $ t $ is considered. If $ f $ is continuous with respect to $ t $ and $ x $, 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 $ x ( t , \mu ) $ in (1) that the continuous derivative $ x _ {t} $ also exists. The presence of $ n $ continuous derivatives of $ f $ with respect to $ t $ and $ x $ means that there exist $ n+ 1 $ continuous derivatives of the solution with respect to $ t $; $ x _ {t} ^ {(} n) $ may be found (expressed in terms of $ x ( t , \mu ) $) by successive differentiation of the identity obtained by substituting $ x ( t , \mu ) $ in (1).

In several problems, e.g. in constructing the asymptotics of the solution in the parameter $ \mu $, it is necessary to study derivatives of $ x ( t , \mu ) $ with respect to $ \mu $. In order to be specific, the existence of derivatives with respect to $ \mu $ for $ \mu = 0 $ will be considered. If $ f ( t , x , \mu ) $ is continuous and has continuous partial derivatives with respect to $ x $ and $ \mu $ in some domain, $ \eta _ {1} = x _ \mu $ exists and is defined from the so-called variational equation (equation in variations, which is linear in $ \eta _ {1} $), obtained from (1) by differentiating both parts with respect to $ \mu $ and putting $ \mu = 0 $:

$$ \tag{3 } \frac{d \eta _ {1} }{dt} = f _ {x} ( t , x ( t , 0 ) , 0 ) \eta _ {1} + f _ \mu ( t , x ( t , 0 ) , 0 ) , $$

and with the aid of the initial condition

$$ \tag{4 } \left . \eta _ {1} \right | _ {t = t _ {0} } = 0 $$

if $ x _ {0} $ is independent of $ \mu $; if, however, $ x _ {0} = x _ {0} ( \mu ) $, then $ \eta _ {1} \mid _ {t = t _ {0} } = x _ {0} ( 0) $.

The derivative $ \eta _ {k} $ of $ x ( t , \mu ) $ with respect to $ \mu $ of order $ k $( under the condition that $ f $ has continuous partial derivatives up to order $ k $) is defined by the variational equation of the $ k $- th order, which differs from (3) only in its inhomogeneity, and it depends on $ t , x ( t , 0 ) , \eta _ {1} \dots \eta _ {k-} 1 $. In the presence of $ k+ 1 $ continuous derivatives of $ x ( t , \mu ) $ with respect to $ \mu $, Taylor's formula may be used as the asymptotic formula for $ x ( t , \mu ) $ with respect to $ \mu $:

$$ \tag{5 } x ( t , \mu ) = x ( t , 0 ) + \mu \eta _ {1} ( t) + \dots + \mu ^ {k} \eta _ {k} ( t) + O ( \mu ^ {k+} 1 ) . $$

This is very important, since $ x ( t , 0 ) $ and $ \eta _ {i} $ 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 $ \mu $( see, for example, [2]).

The problem of differentiability of solutions with respect to $ \mu $ is still meaningful in several cases when the right-hand side does not depend regularly on $ \mu $. In one such case $ \mu $ appears as the coefficient in front of the derivative:

$$ \tag{6 } \mu \frac{dy}{dt} = F ( y , x , t ) ,\ \left . y \right | _ {t = t _ {0} } = y _ {0} , $$

$$ \frac{dx}{dt} = f ( y , x , t ) ,\ \left . x \right | _ {t = t _ {0} } = x _ {0.} $$

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 $ \mu \rightarrow 0 $. It is found that, in the presence of $ k+ 1 $ continuous derivatives of the right-hand sides and under certain special conditions (the so-called stability conditions), expansion (5) is valid, where $ \eta _ {i} $ are the limit values of the derivatives with respect to $ \mu $ of the solution of (6) as $ \mu \rightarrow 0 $, which are defined by the variational equation constructed according to the same rule: (6) is differentiated with respect to $ \mu $ and $ \mu $ 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 $ \eta _ {i} $ will no longer be zero — but will be equal to (usually non-zero) constants, obtained by a definite rule [3].

References

[1]	I.G. Petrovskii, "Ordinary differential equations" , Prentice-Hall (1966) (Translated from Russian)
[2]	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)
[3]	A.B. Vasil'eva, V.F. Butuzov, "Asymptotic expansions of solutions of singularly perturbed equations" , Moscow (1973) (In Russian)
[4]	A.N. Tikhonov, A.B. Vasil'eva, A.G. Sveshnikov, "Differential equations" , Springer (1985) (Translated from Russian)

References

[a1]	E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17
[a2]	D.R. Smith, "Singular perturbation theory" , Cambridge Univ. Press (1985)