Beyond the hypothesis of boundedness for the random coefficient of the Legendre differential equation with uncertainties. (English) Zbl 1474.34396
Summary: In this paper, we aim at relaxing the boundedness condition for the input coefficient \(A\) of the Legendre random differential equation, to permit important unbounded probability distributions for \(A\). We demonstrate that the formal solution constructed using the Fröbenius approach is indeed the mean square solution on the domain \((- 1, 1) \), under mean fourth integrability of the initial conditions \(X_0, X_1\) and sublinear growth of the \(8n\)-th norm of \(A\). Under linear growth of the \(8n\)-th norm of \(A\), the mean square solution is only defined on a neighborhood of zero contained in \((- 1, 1)\). These conditions are closely related to the finiteness of the moment-generating function of \(A\). Numerical experiments on the approximation of the solution statistics for unbounded equation coefficients \(A\) illustrate the theoretical findings.
MSC:
| 34F05 | Ordinary differential equations and systems with randomness |
| 34A30 | Linear ordinary differential equations and systems |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
Keywords:
random differential equation; Frobenius method; mean square calculus; mean fourth calculus; moment-generating functionReferences:
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