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

Abstract

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.

Inverse Problem for a Partial Differential Equation with Gerasimov–Caputo-Type Operator and Degeneration

“…Substituting the presentation of Fourier coefficients (33) of the main unknown function into Fourier series (25), we obtain…”

“…6-8), [24]. In [25], an inverse problem to determine the right-hand side for a mixed type integro-differential equation with fractional order Gerasimov-Caputo operators is considered. The problem of determining the source function for a degenerate parabolic equation with the Gerasimov-Caputo operator was investigated [26].…”

Unique Solvability of a Boundary Value Problem for a Loaded Fractional Parabolic-Hyperbolic Equation with Nonlinear Terms

“…The theory of boundary value problems for equations of mixed type of fractional order is also one of the intensively developing directions in the general theory of partial differential equations. It should be noted that local and nonlocal boundary value problems for equations of parabolic-hyperbolic and elliptichyperbolic types, including various integro-differential operators of fractional order, have been studied by many authors (see works [16][17][18][19][20][21]).…”

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

“…In [18] is studied the solvability of nonlocal problem for a mixed type fourth-order dierential equation with Hilfer fractional operator. In [19] it is considered an inverse problem for a mixed type integro-dierential equation with fractional order Caputo operators.…”