В роботі приводяться диференціальні оператори, які переводять довільні функції в регулярні розв"язки рівняння гіперболічного типу другого порядку та ітеративного рівняння. В обох випадках розв"язана задача Коші в замкнутій формі.
Partial differential equations and their iterations play an important role in the approximation theory, mapping theory, in mathematical physics problems that are associated with vibration phenomena as well as in many other practical problems of Applied Mathematics. Building of differential operators, that determine the hyperbolic and elliptic type equations solutions, as well as the hyperbolic and elliptic type equations systems solutions, is, according to the authors, one of the most attractive methods for solving such problems. A method for obtaining solutions of the second and higher-order hyperbolic equation by the differential operator is proposed, i.e. any function of one variable and its derivatives is reflected by differential operator in a regular solution of the equation. Also a differential operator, that converts arbitrary function in a regular solution of the iterative equation, is built. Necessary and sufficient conditions are established, when the differential operator uniquely determine the regular solut&ions of the second-order hyperbolic equation and iterative equation. In both cases, as an example, Cauchy problem is solved in a closed form.
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