Wave equation driven by fractional generalized stochastic processes. (English) Zbl 1288.26006
The author uses the approach of fractional derivatives of generalized stochastic processes to solve equations driven by fractional derivatives of singular noises and initial data. The perturbation of the wave equation by fractional time and space derivatives of generalized processes are also examined using a Wiener process and a nonlinear term. It is shown that the solutions of the perturbed wave equation for \(n \leq 3\) belong to the extended Colombeau space \({\mathcal G}^{\Omega_ e}_{{\mathcal C_ k}}(\operatorname{Re}^ n), k\in N^{k}\), when the perturbed initial data and additive term are from \({\mathcal G}^{\Omega}(\operatorname{Re}^ n)\).
Reviewer: James Adedayo Oguntuase (Abeokuta)
MSC:
| 26A33 | Fractional derivatives and integrals |
| 46F30 | Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.) |
| 60G20 | Generalized stochastic processes |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
Keywords:
nonlinear stochastic wave equations; fractional derivatives of Colombeau generalized stochastic processes; influence of the fractional singular additive term and/or the singular initial data; existence-uniqueness result in extended Colombeau settingsReferences:
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