On a nonlocal problem for a fourth-order parabolic equation with the fractional Dzhrbashyan–Nersesyan operator

Kadirkulovich

2020

UMJ

In this paper, we consider a non-self-adjoint boundary value problem for a fourth-order 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. The mixed type differential equation under consideration is a fourth order differential equation with respect to the second variable. Regarding the first variable, this equation is a fractional differential equation in the positive part of the segment, and is a second-order differential equation with spectral parameter in the negative part of this segment. A rational method of solving a nonlocal problem with respect to the Hilfer operator is proposed. Using the spectral method of separation of variables, the solution of the problem is constructed in the form of Fourier series. Theorems on the existence and uniqueness of the problem are proved for regular values of the spectral parameter. For sufficiently large positive integers in unique determination of the integration constants in solving countable systems of differential equations, the problem of small denominators arises. Therefore, to justify the unique solvability of this problem, it is necessary to show the existence of values of the spectral parameter such that the quantity we need is separated from zero for sufficiently large \(n\). For irregular values of the spectral parameter, an infinite number of solutions in the form of Fourier series are constructed. Illustrative examples are provided.

Section: Problem Statementmentioning

confidence: 68%

Nonlocal Problem for a Mixed Type Fourth-Order Differential Equation With Hilfer Fractional Operator

Kadirkulovich

2020

UMJ

“…We study the solvability of problem (1)-( 5) for various values of the spectral parameter. This work is a further development of the results of [35,[38][39][40][42][43][44][45].…”

Section: Problem Statementmentioning

confidence: 78%

“…Similarly, by virtue of the conditions of the theorem and applying Cauchy-Schwarz inequality and properties (43) and (44) for the first difference of approximations (45), we derive:…”

Section: Solvability Of Scsnie (30) and (31)mentioning

confidence: 99%

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

Kadirkulov

2020

Axioms

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.

“…Now we show the uniqueness of this solution. Assuming that the integral equation (13) has two dierent solutions x(t, ω) and y(t, ω) on the interval [t 0 ; T ], we obtain the following integral inequality (22) Applying GronwallBellman inequality to estimate (22), we obtain that there holds identity x (t, ω) − y (t, ω ≡ 0 for all t ∈ [t 0 ; T ]. Therefore, the Cauchy type problem (1), (2) has a unique solution on the interval (t 0 ; T ).…”

Section: Unique Solvability Of Problem (1) (2)mentioning

confidence: 99%

On Solvability of an Initial Value Problem for Hilfer type Fractional Differential Equation with Nonlinear Maxima

Kadirkulov

2020

DEMR

In this article we consider the questions of one-valued solvability of initial value problem for a nonlinear Hilfer type fractional differential equation with nonlinear maxima. By the aid of uncomplicated integral transformation based on Dirichlet formula, this initial value problem is reduced to the nonlinear Volterra type fractional integral equation with nonlinear maxima. It is proved the theorem of existence and uniqueness of the solution of given initial value problem in an interval under consideration. It is proved also the stability of the desired solution with respect to given parameter.