On one nonlocal problem for the Euler--Darboux equation

M. V. Dolgopolov ; I. N. Rodionova ; V. M. Dolgopolov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 20 (2016) no. 2, pp. 259-275

Résumé

The boundary value problem with displacement is determined for the generalized Euler–Darboux equation in the field representing the first quadrant. This problem, unlike previous productions, specifies two conditions, connect integrals and fractional derivatives from the values of the sought solution in the boundary points. On the line of singularity of the coefficients of the equations the matching conditions continuous with respect to the solution and its normal derivation are considered. The authors took for the basis of solving the earlier obtained by themselves the Cauchy problem solution of the special class due to the integral representations of one of the specified functions acquired simple form both for positive and for negative values of Euler–Darboux equation parameter. The nonlocal problem set by the authors is reduced to the system of Volterra integral equations with unpacked operators, the only solution which is given explicitly in the corresponding class of functions. From the above the uniqueness of the solution of nonlocal problem follows. The existence is proved by the direct verification. This reasoning allowed us to obtain the solution of nonlocal problem in the explicit form both for the positive and for the negative values of Euler–Darboux equation parameter.

Détail

Keywords: integral equations system, boundary value problem, partial differential equation.