najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2026-17-2-172-178najo-1757Research ArticleMATHEMATICSМАТЕМАТИКАThe Cauchy problem for a fourth-order equation involving a fractional derivative in the Caputo senseЗадача Коши для уравнения четвертого порядка с дробной производной в смысле Капутоhttps://orcid.org/0000-0001-7204-9127ИргашевБ. Ю.IrgashevB. Yu.

Namangan 

bahromirgasev@gmail.comUniversity of Business and Science ; Institute of Mathematics named after V.I.Romanovsky of the Academy of Sciences of the Republic of UzbekistanUzbekistan202630042026172172178Copyright © Irgashev B.Y., 20262026Иргашев Б.Ю.Irgashev B.Y.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/1757

In this article, the Cauchy problem in a half-plane is studied for a fourth-order inhomogeneous equation with a fractional derivative in the Caputo sense. The uniqueness of the solution is demonstrated using the Laplace transform. In constructing the solution, partial solutions expressed in terms of Wright functions are first found. Green’s functions are then constructed using these partial solutions. The solution is constructed explicitly using the Green function. An explicit form of the fundamental solution is also obtained.

В данной статье исследуется задача Коши в полуплоскости для неоднородного уравнения четвертого порядка с дробной производной в смысле Капуто. Единственность решения показывается с помощью преобразования Лапласа. При построении решения сначала находятся частные решения, выраженные через функции Райта. Затем с помощью этих частных решений строятся функции Грина. Решение строится в явном виде с использованием функции Грина. Также получен явный вид фундаментального решения.

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The authors declare that there are no conflicts of interest present.