SOLUTION OF THE MIXED PROBLEMS BY THE METHOD OF CHARACTERISTICS FOR THE WAVE EQUATION WITH THE INTEGRAL CONDITION
Abstract
The mixed problem for the wave equation with one integral condition and one Dirichlet’s condition on the right boundary of the domain is considered in the one-dimensional case. It is proved that the fulfillment of the matching conditions is necessary and sufficient for existence and uniqueness of the classical solution of the given mixed problem under certain smoothness conditions for the given functions. The method of characteristics is used for analysis of the problem. This method is reduced to partitioning the original domain by characteristics line in sub-domains where the solution of the given problem is constructed with the help of initial, boundary and integral conditions. However, in some sub-domains the solution of the problem is reduced to Volterra’s second-type equation. For this equation, the theorems of correct solvability are fulfilled. Matching conditions are obtained by equating the values of the solution and its derivatives up to the second-order, including on characteristics. The obtained results allow building either the analytical solution of the given problem if Volterra’s equation solution can be constructed in explicit form, or the approximate solution with the help of numerical methods. However, in building the approximate solution, the additional conjugation conditions for solution and its derivatives should be introduced on characteristics.
About the Author
I. I. Stolyarchuk
Belarusian State University, Minsk
Belarus
Master of Physics and Mathematics, Postgraduate of the Mathematical Cybernetics Department, Faculty of Mechanics and Mathematics
Belarus
Master of Physics and Mathematics, Postgraduate of the Mathematical Cybernetics Department, Faculty of Mechanics and Mathematics
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