Аңдатпа
This paper investigates the Goursat problem for a second-order loaded hyperbolic differential equation in a rectangular domain. The loading term depends on the value of the solution at a fixed spatial point, which complicates the analytical treatment of the problem. A parameterization-based approach is developed by introducing an additional unknown function, reducing the original problem to an equivalent parameter-dependent Cauchy problem. An explicit integral representation of the solution is derived using an integrating-factor technique. Based on this representation, an iterative algorithm is constructed for solving the equivalent problem. Sufficient conditions ensuring the existence and uniqueness of a classical solution are established, and a priori estimates are obtained for the unknown functions. The convergence of the proposed iterative procedure is analyzed. The proposed approach provides a rigorous framework for studying loaded hyperbolic differential equations and can be applied to a wider class of boundary value problems with spatial loading.