Initial-boundary value problem for a parabolic pseudodifferential equation with a third boundary condition on the hyperplane
DOI:
https://doi.org/10.31471/2304-7399-2026-22(83)-95-104Keywords:
initial-boundary value problem, pseudo-differential equation, fundamental solution, single-layer potential, stable stochastic procesAbstract
The aim of the article is to study the third initial-boundary value problem for a parabolic pseudo-differential equation associated with an isotropic stable with an exponent greater than 1 and less than 2 stochastic process in a multidimensional Euclidean space. The equation is linear with constant coefficients with respect to the partial derivative of the unknown function in time and its fractional Laplacian of order equal to the exponent of the stochastic process. The boundary condition is formulated on the hyperplane. It equates a linear combination with variable coefficients of one-sided limits of the pseudo-derivative of order less than the number 1 less than the power of the process in the direction of the normal to the hyperplane and the value of the unknown function itself with some variable non-negative factor. The existence and uniqueness in a certain class of functions of the fundamental solution of the problem under consideration are proved.
References
1. Osypchuk M.M., Portenko M.I. Symmetric α-stable stochastic process and the third initial-boundary-value problem for the corresponding pseudo-differential equation. Ukr. Math. J. – 2018. – V. 69, №10. P. 1631–1650. https://doi.org/10.1007/s11253-018-1459-2
2. Osypchuk M.M., Portenko M.I. On simple-layer potentials for one class of pseudodifferential equations. Ukr. Math. J. – 2016. – V. 67, №11. P. 1704–1720. https://doi.org/10.1007/s11253-016-1184-7
3. Eidelman S.D., Ivasyshen S.D., Kochubei A.N. Analytic Methods in the Theory of Differential and Pseudo-differential Equations of Parabolic Type, Operator Theory Advances and Applications, vol. 152. Birkhäuser Verlag, 2004. https://doi.org/10.1007/978-3-0348-7844-9
4. Mamalyha K., Osypchuk M. On single-layer potentials, pseudo-gradients and jump theorem for an isotropic stable stochastic process. J. Pseudo-Differ. Oper. Appl. – 2024. – V. 15, №4. https://doi.org/10.1007/s11868-023-00574-y
5. Didyk S. V., Osypchuk M. M. On an initial-boundary value problem for a parabolic pseudodifferential equation. Matematychni Studii – 2025. – V. 64, №2. P. 153–160. https://doi.org/10.30970/ms.64.2.153-160
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2026 Mykhailo Osypchuk, Sergii Didyk
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
