Abstract:

nd. The article studies the generalized integral transforms, such as generalizedLaplace’ integral transform, generalized Stieltjes’ integral transformation. Objective. Investigation some applications of the new generalized classical integral transforms for solving integral and differential equations, for calculation integrals which are absent in reference and scientific literature. Methods. We apply the methods the theory of functional variable, the theory of mathematical physics, the theory of special function and the methods the theory applied analysis. Results. Some new forms of generalized Laplace’ integral transform are given. With help of the r-generalized confluent hypergeometric function the generalized Stieltjes’ integral transform is introduced. The inverse theorem of the generalized Stieltjes’ integral transform is proved. New properties of the r-generalized confluent hypergeometric function are explored.Conclusions. New properties of the r-generalized confluent hypergeometric function are explored. These functions are expressing in the form by the Fox–Wright functions. Some forms of generalized Laplace’ integral transform are given. With help of the r-generalized confluent hypergeometric function the generalized Stieltjes’ integral transform is introduced. Interesting examples of applications of new generalized integral transforms in the theory of differential and integral equations, for calculation of integrals, which are absent in mathematical literature are given.

Keywords: