Abstract The article considers a Dirichlet-type boundary value problem for generalized harmonic functions of arbitrary order in simply connected domains with smooth boundaries. Using the general representation of generalized harmonic functions via two analytic functions of a complex variable and based on the method of conjugation of analytic functions, explicit solutions are obtained for both the homogeneous Dirichlet problem and the nonhomogeneous Dirichlet problem in the classes of first-order generalized harmonic functions in circular domains. The main logical scheme of the obtained method for solving the studied boundary value problem is as follows: first, taking into account that the circle is an analytic curve and introducing two auxiliary analytic functions, the solution of the considered boundary value problem is reduced to solving the simplest classical conjugation problem for the auxiliary analytic functions; then, by solving two linear first-order Euler differential equations in the classes of analytic functions of a complex variable, we obtain the general solution of the original boundary value problem.
Nagornaya et al. (Sun,) studied this question.