Boundary value problems for a mixed type equation of the 3rd order with variable coefficients
DOI:
https://doi.org/10.52754/16948645_2025_4(1)_30Keywords:
existence; uniqueness; Green's function; Riemann's function; integral equation; order reduction methodAbstract
The existence and uniqueness of the solution to the boundary value problem for a third-order mixed-type equation with variable coefficients in the lower terms is proven, with conditions for the gluing of the function itself and its first- and second-order derivatives on the line y=0, where the type of equation changes when a second-order mixed parabolic-hyperbolic operator is applied to a first-order linear differential operator with constant coefficients. By reducing the order of the equation, the problem was reduced to the Tricomi problem for a second-order mixed parabolic-hyperbolic equation with continuous conditions for the function itself and its first-order derivative with respect to y on the line of change of the equation type. By the method of elimination of the system of equations obtained from the parabolic and hyperbolic parts of the domains, the solvability of the problem was reduced to the solvability of the Fredholm integral equation of the second kind. A sufficient condition for the solvability of the integral equation was obtained through estimates of the kernel of the equation. The solution of the problem was split into two problems in the regions under consideration: in the parabolic part of the region, the first boundary value problem for the heat conduction equation was solved using the Green's function method, and in the hyperbolic part of the region, bounded by the characteristics of the equation and the line y=0, the solution of the problem using the Riemann function construction method was determined as the solution of the Cauchy problem. By applying a curvilinear integral, the solution to the problem in the areas under consideration was found. The necessity of the requirement of continuity of the function itself and its first two derivatives with respect to y on the line of change of the equation type was justified. Sufficient conditions for the unique classical solvability of the boundary value problem were established. The obtained conditions for the solvability of the boundary value problem provided a theoretical basis for the development of numerical methods for solving applied problems in aerohydrodynamics, geophysics, and engineering thermodynamics
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