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This paper is devoted to the determination of the trigonometric Fourier series by the method of four-dimensional mathematics. In this paper, a new four-dimensional method for evaluation of the sums containing trigonometric function is proposed. Current work provides a first study and finding towards Fourier’s series in four-dimensional space.

This paper is devoted to the study of the Cauchy problem for a system of differential equations describing the unsteady flow of a compressible fluid in a homogeneous and inhomogeneous porous medium with a general nonlinear filtration law in three-dimensional space. In the work using the methods of four-dimensional mathematics, a special four-dimensional space of space and time coordinates was developed, as well as a functional space of regular functions, and analytical conditions were obtained on the general form of the nonlinear filtration law for which the Cauchy problem has a solution.