On a Boundary-value Problem for Boussinesq type Nonlinear Integro-Differential Equation with Reflecting Argument

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: 99%

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

Kadirkulovich

2020

UMJ

“…Nonlocal boundary value problems for different type of equations were studied in the works of many authors, in particular, in [29][30][31][32][33][34][35][36]. Nonlinear differential and integro-differential equations without mixing of the type of equations were studied in [37][38][39][40][41][42] by the Fourier series method.…”

Section: Problem Statementmentioning

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.

“…We rewrite Formulas (39) and (40) in the form of countable systems of nonlinear integral equations (CSNIE)…”

Section: Redefinition Functionsmentioning

confidence: 99%

“…So, we determined the redefinition functions as a Fourier series (57). Using representations (39) and (40),we can present Fourier series (33) and (34) of the main unknown functions as…”

Section: Convergence Of Fourier Series (57)mentioning

confidence: 99%

“…The presence of an integral term in differential equations of the first and second order has an important role in the theory of dynamical systems of automatic control [36,37]. Boundary value problems for integro-differential equations with spectral parameters have singularities in studying the questions of one-value solvability [38,39]. Mixed type integer order integro-differential equations with degenerate kernels and spectral parameters are studied in [40,41].…”

Section: Introductionmentioning

confidence: 99%

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Inverse Problem for a Mixed Type Integro-Differential Equation with Fractional Order Caputo Operators and Spectral Parameters

Karimov

2020

Axioms

The questions of the one-value solvability of an inverse boundary value problem for a mixed type integro-differential equation with Caputo operators of different fractional orders and spectral parameters are considered. The mixed type integro-differential equation with respect to the main unknown function is an inhomogeneous partial integro-differential equation of fractional order in both positive and negative parts of the multidimensional rectangular domain under consideration. This mixed type of equation, with respect to redefinition functions, is a nonlinear Fredholm type integral equation. The fractional Caputo operators’ orders are smaller in the positive part of the domain than the orders of Caputo operators in the negative part of the domain under consideration. Using the method of Fourier series, two systems of countable systems of ordinary fractional integro-differential equations with degenerate kernels and different orders of integro-differentation are obtained. Furthermore, a method of degenerate kernels is used. In order to determine arbitrary integration constants, a linear system of functional algebraic equations is obtained. From the solvability condition of this system are calculated the regular and irregular values of the spectral parameters. The solution of the inverse problem under consideration is obtained in the form of Fourier series. The unique solvability of the problem for regular values of spectral parameters is proved. During the proof of the convergence of the Fourier series, certain properties of the Mittag–Leffler function of two variables, the Cauchy–Schwarz inequality and Bessel inequality, are used. We also studied the continuous dependence of the solution of the problem on small parameters for regular values of spectral parameters. The existence and uniqueness of redefined functions have been justified by solving the systems of two countable systems of nonlinear integral equations. The results are formulated as a theorem.