Boundary Value Problem for Weak Nonlinear Partial Differential Equations of Mixed Type with Fractional Hilfer Operator

In this paper, we consider a boundary value problem for a nonlinear partial differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. With respect to the first variable...

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Main Authors:	Tursun K. Yuldashev, Bakhtiyor J. Kadirkulov
Format:	Article
Language:	English
Published:	MDPI AG 2020-06-01
Series:	Axioms
Subjects:	mixed type nonlinear equation boundary value problem hilfer operator mittag–leffler function spectral parameter solvability
Online Access:	https://www.mdpi.com/2075-1680/9/2/68

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spelling	doaj-eca561743db548b1a412cbfa30b48e852020-11-25T03:07:16ZengMDPI AGAxioms2075-16802020-06-019686810.3390/axioms9020068Boundary Value Problem for Weak Nonlinear Partial Differential Equations of Mixed Type with Fractional Hilfer OperatorTursun K. Yuldashev0Bakhtiyor J. Kadirkulov1Uzbek-Israel Joint Faculty of High Technology and Engineering Mathematics, National University of Uzbekistan, Tashkent 100174, UzbekistanTashkent State Institute of Oriental Studies, Tashkent 100060, UzbekistanIn this paper, we consider a boundary value problem for a nonlinear partial differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. With respect to the first variable, this equation is a nonlinear fractional differential equation in the positive part of the considering segment and is a second-order nonlinear differential equation with spectral parameter in the negative part of this segment. Using the Fourier series method, the solutions of nonlinear boundary value problems are constructed in the form of a Fourier series. Theorems on the existence and uniqueness of the classical solution of the problem are proved for regular values of the spectral parameter. For irregular values of the spectral parameter, an infinite number of solutions of the mixed equation in the form of a Fourier series are constructed.https://www.mdpi.com/2075-1680/9/2/68mixed type nonlinear equationboundary value problemhilfer operatormittag–leffler functionspectral parametersolvability
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Tursun K. Yuldashev Bakhtiyor J. Kadirkulov
spellingShingle	Tursun K. Yuldashev Bakhtiyor J. Kadirkulov Boundary Value Problem for Weak Nonlinear Partial Differential Equations of Mixed Type with Fractional Hilfer Operator Axioms mixed type nonlinear equation boundary value problem hilfer operator mittag–leffler function spectral parameter solvability
author_facet	Tursun K. Yuldashev Bakhtiyor J. Kadirkulov
author_sort	Tursun K. Yuldashev
title	Boundary Value Problem for Weak Nonlinear Partial Differential Equations of Mixed Type with Fractional Hilfer Operator
title_short	Boundary Value Problem for Weak Nonlinear Partial Differential Equations of Mixed Type with Fractional Hilfer Operator
title_full	Boundary Value Problem for Weak Nonlinear Partial Differential Equations of Mixed Type with Fractional Hilfer Operator
title_fullStr	Boundary Value Problem for Weak Nonlinear Partial Differential Equations of Mixed Type with Fractional Hilfer Operator
title_full_unstemmed	Boundary Value Problem for Weak Nonlinear Partial Differential Equations of Mixed Type with Fractional Hilfer Operator
title_sort	boundary value problem for weak nonlinear partial differential equations of mixed type with fractional hilfer operator
publisher	MDPI AG
series	Axioms
issn	2075-1680
publishDate	2020-06-01
description	In this paper, we consider a boundary value problem for a nonlinear partial differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. With respect to the first variable, this equation is a nonlinear fractional differential equation in the positive part of the considering segment and is a second-order nonlinear differential equation with spectral parameter in the negative part of this segment. Using the Fourier series method, the solutions of nonlinear boundary value problems are constructed in the form of a Fourier series. Theorems on the existence and uniqueness of the classical solution of the problem are proved for regular values of the spectral parameter. For irregular values of the spectral parameter, an infinite number of solutions of the mixed equation in the form of a Fourier series are constructed.
topic	mixed type nonlinear equation boundary value problem hilfer operator mittag–leffler function spectral parameter solvability
url	https://www.mdpi.com/2075-1680/9/2/68
work_keys_str_mv	AT tursunkyuldashev boundaryvalueproblemforweaknonlinearpartialdifferentialequationsofmixedtypewithfractionalhilferoperator AT bakhtiyorjkadirkulov boundaryvalueproblemforweaknonlinearpartialdifferentialequationsofmixedtypewithfractionalhilferoperator
_version_	1724671491467902976

Boundary Value Problem for Weak Nonlinear Partial Differential Equations of Mixed Type with Fractional Hilfer Operator

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