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

Karimov

2020

Axioms

Self Cite

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.

Section: Expansion Of the Solution Of The Direct Problem (1)-(4) Intomentioning

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%

See 1 more Smart Citation

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

Karimov

2020

Axioms

Self Cite

“…In [14], a hyperbolicparabolic system arising in pulse combustion is investigated. Mixed type fractional differential equations are studied in many works of scientists, in particular in [15][16][17][18][19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

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

Izvestiya Instituta Matematiki I Informatiki Udmurtskogo Gosudarstvennogo Universiteta

Karimov

2021

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.

“…More detailed information related to the theory of fractional integro-dierentiation, including the Hilfer fractional derivative, one can nd in [16]. In [17] by analytical method is studied the unique solvability of boundary value problem for weak nonlinear partial dierential equations of mixed type with fractional Hilfer operator. In [18] is studied the solvability of nonlocal problem for a mixed type fourth-order dierential equation with Hilfer fractional operator.…”

Section: Introductionmentioning

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.