. Solving of initial-boundary value problems for the wave equation using retarded surface potential and Laguerre transform, Mat. Stud. 44 (2015), 185-203. Approach for solving of initial-boundary value problems for the homogeneous wave equation is described and proved. It is based on the Laguerre transform in the time domain and the boundary integral equations. Retarded potentials are used for representation of generalized solutions of such problems. The densities of retarded potentials are expanded in Fourier-Laguerre series which coefficients have special convolution form. As a result, initial-boundary value problems are reduced to a sequence of boundary integral equations. Для решения смешанных задач для однородного волнового уравнения описан и обосно-ван подход, который базируется на интегральном преобразовании Лагерра по временной переменной и граничных интегральных уравнениях. Для представления обобщенных ре-шений таких задач используются запаздывающие поверхностные потенциалы, плотности которых ищут в виде ряда Фурье-Лагерра. Для коэффициентов разложения получены аналитические формулы и исходные задачи сведены к последовательности граничных ин-тегральных уравнений.1. Вступ. Запiзнюючi потенцiали використовують для iнтегрального зображення кла-сичних ([28]) i узагальнених ([2, 3, 9]) розв'язкiв мiшаних задач для хвильового рiвняння в областях загального вигляду. Вони дають змогу замiнити мiшанi задачi для хвильово-го рiвняння еквiвалентними залежними вiд часової змiнної граничними iнтегральними рiвняннями (ЧГIР), у яких невiдомi величини -густини потенцiалiв -визначаються в кожен момент часу лише на межi областi ([7, 11, 21, 17, 27]).Вiдзначимо, що iснування та єдинiсть розв'язкiв ЧГIР дослiджено в працях [2, 3] з використанням перетворення Лапласа за часовою змiнною у широких функцiйних про-сторах. Там же обгрунтовано метод Гальоркiна для чисельного розв'язування таких iнтегральних рiвнянь. Разом з тим, в працях [11,4] вiдзначено складнiсть алгоритмiв

We consider a numerical solution of the initial-boundary value problem for the homogeneous wave equation with the Neumann condition using the retarded double layer potential. For solving an equivalent time-dependent integral equation we combine the Laguerre transform (LT) in the time domain with the boundary elements method. After LT we obtain a sequence of boundary integral equations with the same integral operator and functions in right-hand side which are determined recurrently. An error analysis for the numerical solution in accordance with the parameter of boundary discretization is performed. The proposed approach is demonstrated on the numerical solution of the model problem in unbounded three-dimensional spatial domain.

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The Laguerre transform is applied to the convolution product of functions of a real argument (over the time axis) with values in Hilbert spaces. The main results have been obtained by establishing a relationship between the Laguerre and Laplace transforms over the time variable with respect to the elements of Lebesgue weight  spaces. This relationship is built using a special generating function. The obtained dependence makes it  possible to extend the known properties of the Laplace transform to the case of the Laguerre transform. In particular, this approach concerns the transform of a convolution of functions. The Laguerre transform is determined by a system of Laguerre functions, which forms an orthonormal basis in the weighted Lebesgue space. The inverse Laguerre transform is constructed as a Laguerre series. It is proven that the direct and the inverse Laguerre transforms are mutually inverse operators that implement an isomorphism of square-integrable functions and infinite squares-summable sequences. The concept of a q-convolution in spaces of sequences is introduced as a discrete analogue of the convolution products of functions. Sufficient conditions for the existence of convolutions in the weighted Lebesgue spaces and in the corresponding spaces of sequences are investigated. For this purpose, analogues of Young’s inequality for such spaces are proven. The obtained results can be used to construct solutions of evolutionary problems and time-dependent boundary integral equations.