The Linear Cauchy Problem for a Class of Differential Equations with Distributional Coefficients

C.O.R. Sarrico

Abstract: We consider the problem $X^{(n)}=\sum_{i=1}^{n}U_{i}X^{(n-i)}+V$, $X^{(n-i)}(t_{0})=a_{i}$ in dimension 1 ($X\in\calc{D}'$ is unknown, $n$ is a positive integer, $V\in\calc{D}'$, $U_{1},...,U_{n}\in C^{\infty}\oplus \calc{D}_{m}^{\prime p}$, $\calc{D}_{m}^{\prime p}=\calc{D}^{\prime p}\cap \calc{D}_{m}'$, $\calc{D}^{\prime p}$ is the space of distributions of order $\le p$ in the sense of Schwartz, $\calc{D}_{m}'$ is the space of distributions with nowhere-dense support, $a_{1},...,a_{n}\in\C$ and $t_{0}\in\R$).
Necessary and sufficient conditions for existence and uniqueness of this problem in $C^{q}\oplus\calc{D}_{m}'$ where $q=\max(n,n-1+p)$ are given and also the way of getting an explicit solution when it exists.
The solutions are considered in a generalized sense defined with the help of the distributional product we introduced in [2] and they are consistent with the usual solutions.
As an example we take $X'(t)=i\,g\,\delta'(t)\,X(t)$, $X(t_{0})=1$ for a certain $t_{0}<0$ ($i=\sqrt{-1}$, $g\in\R$ and $\delta$ is the Dirac measure) and we prove that in our sense, its unique solution in $C^{1}\oplus\calc{D}_{m}'$ is $X(t)=1+i\,g\,\delta(t)$ (Colombeau [1] also considers this problem with another approach). More examples are presented.

Keywords: Ordinary differential equations; products of distributions; distributions, generalized functions.

Classification (MSC2000): 34A30; 46F10

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