Abstract
The jump problem and problems with defects on the type change line for model mixed-type equations in the mixed domains are investigated. The explicit solutions of the jump problem are obtained by the method of integral equations and by the Fourier transformation method. The problems with defects are reduced to singular integral equations. Some results for the solution of the equation under consideration are discussed concerning the existence and uniqueness for the solution of the suggested problem.
Highlights
Consider the jump problem and problems with defects on the type change line for the mixedtype equation of the first kind sgn y y m ∂2u ∂x2 ∂2u ∂y2 0, m ≥ 0.This equation is a model equation among mixed-type equations of the first kind
In the formulation of the boundary value problems in the mixed domain, it is usually required that the unknown solution u x, y and its normal derivative should be continuous on the type change line y 0, that is, the Boundary Value Problems u x, 0 0 − u x, 0 − 0 0
It is shown that explicit solutions of the jump problem can be used as potentials under researching boundary value problems with defects
Summary
Consider the jump problem and problems with defects on the type change line for the mixedtype equation of the first kind sgn y y m ∂2u ∂x2. This equation is a model equation among mixed-type equations of the first kind. In the formulation of the boundary value problems in the mixed domain, it is usually required that the unknown solution u x, y and its normal derivative should be continuous on the type change line y 0, that is, the 2 conditions. Problems with defects on the type change line will form special class of boundary value problems for the mixed-type equations with discontinuous coefficients in the conjugation conditions. It is shown that explicit solutions of the jump problem can be used as potentials under researching boundary value problems with defects
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