Mixed type integro-differential equation with fractional order Caputo operators and spectral parameters

The issues of unique solvability of a boundary value problem for a mixed type integro-differential equation with two Caputo time-fractional operators and spectral parameters are considered. A mixed type integro-differential equation is a partial integro-differential equation of fractional order in both positive and negative parts of multidimensional rectangular domain under consideration. The fractional Caputo operator's order is less in the positive part of the domain, than the order of Caputo operator in the negative part of the domain. Using the method of Fourier series, two systems of countable systems of ordinary fractional integro-differential equations with degenerate kernels are obtained. Further, a method of degenerate kernels is used. To determine arbitrary integration constants, a system of algebraic equations is obtained. From this system, regular and irregular values of spectral parameters are calculated. The solution of the 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. To prove the convergence of Fourier series, the properties of the Mittag-Leffler function, Cauchy-Schwarz inequality and Bessel inequality are used. The continuous dependence of the problem solution on a small parameter for regular values of spectral parameters is also studied. The results are formulated as a theorem.

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

Axioms ◽

10.3390/axioms9040121 ◽

2020 ◽

Vol 9 (4) ◽

pp. 121

Author(s):

Tursun K. Yuldashev ◽

Erkinjon T. Karimov

Keyword(s):

Differential Equation ◽

Inverse Problem ◽

Fourier Series ◽

Fractional Order ◽

Mixed Type ◽

Algebraic Equations ◽

Negative Part ◽

Spectral Parameters ◽

Fredholm Type ◽

Integro Differential Equation

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.

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

Axioms ◽

10.3390/axioms9020068 ◽

2020 ◽

Vol 9 (2) ◽

pp. 68 ◽

Cited By ~ 1

Author(s):

Tursun K. Yuldashev ◽

Bakhtiyor J. Kadirkulov

Keyword(s):

Differential Equation ◽

Fourier Series ◽

Boundary Value Problem ◽

Spectral Parameter ◽

Mixed Type ◽

Boundary Value ◽

Nonlinear Partial Differential Equation ◽

Rectangular Domain ◽

Partial Differential ◽

Negative Part

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.

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Nonlocal problem for a nonlinear fractional mixed type integro-differential equation with spectral parameters

10.1063/5.0057147 ◽

2021 ◽

Author(s):

Tursun K. Yuldashev ◽

Farhod D. Rakhmonov

Keyword(s):

Differential Equation ◽

Mixed Type ◽

Nonlocal Problem ◽

Spectral Parameters ◽

Integro Differential Equation

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On Boundary Value Problems for a Mixed Type Fractional Differential Equation with Caputo Operator

BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS ◽

10.31489/2021m1/127-137 ◽

2021 ◽

Vol 101 (1) ◽

pp. 127-137

Author(s):

T.K. Yuldashev ◽

◽

B.I. Islomov ◽

U.Sh. Ubaydullaev ◽

◽

...

Keyword(s):

Differential Equation ◽

Fourier Series ◽

Boundary Value Problems ◽

Unique Solvability ◽

Mixed Type ◽

Boundary Value ◽

Rectangular Domain ◽

Hyperbolic Type ◽

Mixed Equation ◽

Fractional Differential

This article is devoted to study the boundary value problems of the first and second kind with respect to the spatial variable for a mixed inhomogeneous differential equation of parabolic-hyperbolic type with a fractional Caputo operator in a rectangular domain. In the study of such boundary value problems, we abandoned the boundary value condition with respect to the first argument and instead it is used additional gluing condition. In this case, in the justification of the unique solvability of the problems, the conditions on the boundary domain are removed. This allowed us to weaken the criterion for the unique solvability of boundary value problems under consideration. The solution is constructed in the form of Fourier series with eigenfunctions corresponding to homogeneous spectral problems. Estimates for the convergence of Fourier series are obtained as a regular solution of this mixed equation.

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