Abstract:
An $n$-dimensional system of equations with dominant partial derivatives of the $n$th order is being studied. The distinguishing feature of the considered system compared to other systems with partial derivatives is the presence of a first term in the equations on the right side of the system, representing a dominant derivative, while all other derivatives appearing in the system equations are obtained from it by discarding at least one differentiation with respect to any of the independent variables. The aim of the study is to find conditions for the unique solvability of the Goursat problem for the considered system.
The main problem is reduced to a system of integral equations, the solution of which exists and is unique when the requirements of continuity of the kernels and right sides of this system are satisfied in the corresponding closed parallelepipeds of variable ranges.
Conditions under which the main problem is uniquely solvable have been obtained.
The final result in terms of the coefficients of the original system is formulated as a theorem.
Keywords:system of differential equations, Goursat problem, existence and uniqueness theorem.
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