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

Tursun K Yuldashev,Bakhtiyor J Kadirkulov

doi:10.3390/axioms9020068

Abstract

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.

Highlights

One of the most striking areas of mathematical analysis is the invention of fractional-order integro-differential operators
We study the solvability of problem (1)–(5) for various values of the spectral parameter
We considered a nonlocal boundary value problem T ω for a weak nonlinear partial differential equation of mixed type with fractional Hilfer operator D α,γ in a positive rectangular domain Ω 1 = {0 < t < b, 0 < x, y < l } and with spectral parameter ω in a negative rectangular domain Ω 2 = {− a < t < 0, 0 < x, y < l }

Summary

Introduction

One of the most striking areas of mathematical analysis is the invention of fractional-order integro-differential operators. Hilfer solved a Cauchy type problem for a fractional order equation with the same operator, applying in this case the Laplace transforms. Using the integral Fourier, Laplace, and Mellin transforms, he investigated the Cauchy problem for the generalized diffusion equation, the solution of which is presented in the form of the Fox H-function. It is applied in [9,10], the generalized fractional integro-differentiation operator in studying the dielectric relaxation in glass-forming liquids with different chemical compositions. The study of nonlinear differential and functional-differential equations of fractional order is relevant

Problem Statement

Nonhomogeneous Ordinary Differential Equation With Hilfer Operator

Convergence of Fourier Series

Irregular Value of Spectral Parameter ω

Conclusions