Hyperbolic Equation with Piecewise-Constant Argument of Generalized Type and Solving Boundary Value Problems for It
Assanova A.T.
December 2021Pleiades journals
Lobachevskii Journal of Mathematics
2021#42Issue 153584 - 3593 pp.
Abstract: We consider problem with conditions on the characteristics forsecond order system of hyperbolic equation withpiecewise-constant argument of generalized type. By a new unknownfunctions this problem is transformed to a family of problems forsystem of differential equations with piecewise-constant argumentof generalized type and unknown functions. Further, weinvestigate the family of differential equations withpiecewise-constant argument of generalized type in the rectangulardomain. This domain is divided into N parts, the values of asolution at the interior lines of the subdomains are considered asadditional functional parameters, and the family of differentialequations with piecewise-constant argument of generalized type isreduced to a family of Cauchy problems for system of differentialequations with parameters on the subdomains. Using the solutionsto these problems, new general solutions to the family ofdifferential equations with piecewise-constant argument ofgeneralized type are introduced and their properties areestablished. Based on the general solution, boundary condition,and continuity conditions of a solution at the interior lines ofthe partition, the system of linear functional equations withrespect to parameters is composed. Its coefficients and right handsides are found by solving the family of Cauchy problems forlinear differential equations on the subdomains. It is shown thatthe solvability of family boundary value problems is equivalent tothe solvability of systems composed. Methods for solving problemwith conditions on the characteristics for second order system ofhyperbolic equation with piecewise-constant argument ofgeneralized type are proposed, which are based on the solving offamily boundary value problems.
family of boundary value problems , new general solution , problem with conditions on the characteristics , solvability , system of hyperbolic equation with piecewise-constant argument of generalized type
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Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Institute of Mathematics and Mathematical Modeling
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