Differential Equations with Variable Coefficients in the Mechanics of Inhomogeneous Bodies

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The paper considers differential equations in partial derivatives of an elliptic type with variables, piecewise-smooth coefficients, depending on the coordinates (initial equations). It is shown that the solution of the original equation can be represented as an integral equation through the solution of the accompanying equation with constant coefficients of the same type. This representation includes a fundamental solution to the original equation. Under the assumption that the accompanying solution is smooth, the integral equation implies the representation of the original solution in the form of a series with respect to all possible derivatives of the related solution. The coefficients of the series are called structural functions, since they are determined by the functional dependence of the coefficients of the initial equations either on coordinates, or on time, or on coordinates and time. Structural functions are identically equal to zero in the case when the initial coefficients coincide with the corresponding constant coefficients of the accompanying equation. For structural functions, systems of recurrence equations are obtained. It is shown that in the case of a plate with non-uniform thickness, the structural functions depend only on the coordinate in the thickness of the plate, and structural equations become ordinary differential equations that integrate in a general way. A scheme for solving the plate problem is considered.