TITLE

Riemann-Hadamard function for Darboux-Goursat problem and its multidimensional application

AUTHOR(S)
Nikolov, Aleksey; Popivanov, Nedyu
PUB. DATE
November 2014
SOURCE
AIP Conference Proceedings;2014, Vol. 1631, p175
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
We consider a planar Darboux-Goursat problem with singular coefficients. It is known that its unique solution (which is not a solution in the classical sense) may have strong power type singularity isolated at one boundary point even for very smooth functions in the right-hand side of the equation. In the present work we derive an exact asymptotic formula giving the dependence between the right-hand side function of the equation and the possible singularity of the solution of the problem. To obtain our results, we use Riemann-Hadamard function for this problem and firstly we find an integral representation of the solution of the problem. Further, we apply these results to study a four-dimensional boundary value problem for weakly hyperbolic equation, introduced by Protter in 1952. Its adjoint homogeneous problem has infinitely many nontrivial classical solutions and by this reason the Protter's problem is not well posed, but it is solvable in a generalized sense. We clarify the exact asymptotic behavior of singular generalized solutions of this problem, using the close connection between the Protter's problem and the planar Darboux-Goursat problem, which we considered.
ACCESSION #
99577174

 

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