Transmutation Operators Based on Homotope Analysis for Nonlocal Boundary Value Problems

Transmutation operators nonlocal boundary value problems in terms of the homotopic analysis method is used to solve and research nonlocal boundary value problems for the Laplace equation in domains with plane symmetry. The idea of parametric homotopic analysis in solving multipoint boundary value problems is implemented to determine the transformation operators. The solution of a multipoint boundary value problem is the sum of a series whose terms are solutions of Dirichlet problems taken with certain weights. Similarly, the solution of the boundary value problem for the Laplace equation in the right half-plane with internal conjugation conditions is expressed in terms of solutions of a series of successive Dirichlet problems. A non-local boundary value problem is immersed in a one-parameter or multi-parameter collection of boundary value problems. This collection, in limiting cases, contains the considered boundary value problem and the model Dirichlet problem. The solution of non-local multipoint boundary value problems for the Laplace equation in the right half-plane, the solution of mixed multipoint boundary value problems for the one-dimensional heat equation are found. Formulas have been obtained for solving a number of model problems with integral boundary conditions. The Homotopic Analysis method is applied to study the multipoint Sturm-Liouville problem for the Fourier operator on the real semiaxis. The adjoint Sturm-Liouville problem is stated. To extend the sine transform and its inverse are extended to the case of a multipoint boundary value problem.