In the three-dimensional domain a Boussinesq type nonlinear partial integro-differential equation of the fourth order with a degenerate kernel, integral form conditions, spectral parameters and ...reflecting argument is considered. The solution of this partial integro-differential equation is studied in the class of generality functions. The method of separation of variables and the method of a degenerate kernels are used. Using these methods, the nonlocal boundary value problem is integrated as a countable system of ordinary differential equations. When we define the arbitrary integration constants there are possible five cases with respect to the first spectral parameter. Calculated values of the spectral parameter for each case. Further, the problem is reduced to solving countable system of linear algebraic equations. Irregular values of the second spectral parameter are determined. At irregular values of the second spectral parameter the Fredholm determinant is degenerate. Other values of the second spectral parameter, for which the Fredholm determinant does not degenerate, are called regular values. Taking the values of the first spectral parameter into account for regular values of the second spectral parameter the corresponding solutions were constructed and we obtained the countable system of nonlinear integral equations for each of five cases. To establish the unique solvability of this countable system of nonlinear integral equations we use the method of successive approximations and the method of compressing mappings. Using the Cauchy–Schwarz inequality and the Bessel inequality, we proved the absolute and uniform convergence of the obtained Fourier series. The stability of the solution of the boundary value problem with respect to given functions in integral conditions is proved. The conditions under which the solution of the boundary value problem will be small are studied. For the irregular values of the second spectral parameter each of the five cases is checked separately. The orthogonality conditions are used. Cases are determined in which the problem has an infinite number of solutions and these solutions are constructed as Fourier series. For other cases, the absence of nontrivial solutions of the problem is proved. The corresponding theorems are formulated.
The questions of weakly generalized solvability of a nonlinear inverse problem in nonlinear optimal control of thermal processes for a parabolic differential equation are studied. The parabolic ...equation is considered under initial and boundary conditions. To determine the recovery function, a nonlocal integral condition is specified. Moreover, the recovery function nonlinearly enters into the differential equation. Is applied the method of variable separation based on the search for a solution to the mixed inverse problem in the form of a Fourier series. It is assumed that the recovery function and nonlinear term of the given differential equation are also expressed as a Fourier series. For fixed values of the control function, the unique solvability of the inverse problem is proved by the method of compressive mappings. The quality functional has a nonlinear form. The necessary optimality conditions for nonlinear control are formulated. The determination of the optimal control function is reduced to a complicated functional-integral equation, the process of solving which consists of solving separately taken two nonlinear functional and nonlinear integral equations. Nonlinear functional and integral equations are solved by the method of successive approximations. Formulas are obtained for the approximate calculation of the state function of the controlled process, the recovery function, and the optimal control function. Is proved the absolutely and uniformly convergence of the obtained Fourier series.
In this paper are considered the questions of unique solvability and redefinitions of a nonlocal inverse problem for the Fredholm integro-differential equation of the second order with degenerate ...kernel, integral condition, and spectral parameter. Calculations of the value of the spectral parameter are reduced to the solve of trigonometric equations. Systems of algebraic equations are obtained. The singularities that arose in determining arbitrary constants are studied. A criterion for unique solvability of the problem is established and the corresponding theorem is proved.
In this paper, we consider the problems on the solvability and constructing solutions of one nonlocal boundary-value problem for the multidimensional, fourth-order, integro-differential Benney–Luke ...equation with degenerate kernel and spectral parameters. For various values of spectral parameters, necessary and sufficient conditions of the existence of a solution are obtained. The Fourier series for solutions of the problem corresponding to various sets of spectral parameters are obtained. For regular values of spectral parameters, the absolute and uniform convergence of the series and the possibility of their termwise differentiation with respect to all variables are proved. The problem is also examined studied for cases of irregular values of spectral parameters.
The problems of solvability and construction of solutions of a nonlocal boundary value problem for the second-order Fredholm integro-differential equation with degenerate kernel, integral conditions, ...spectral parameters and reflecting deviation are considered. Using the method of the degenerate kernel, the boundary value problem is integrated as an ordinary differential equation. When we define the arbitrary integration constants there are possible five cases with respect to the first spectral parameter. Calculated values of the spectral parameter for each case. Further, the problem is reduced to solving systems of linear algebraic equations. Irregular values of the second spectral parameter are determined. At irregular values of the second spectral parameter the Fredholm determinant is degenerate. Other values of the second spectral parameter, for which the Fredholm determinant does not degenerate, are called regular values. Taking the values of the first spectral parameter into account for regular values of the second spectral parameter the corresponding solutions were constructed for each of five cases. The stability of the solution of the boundary value problem for given values in integral conditions is proved. The conditions under which the solution of the boundary value problem will be small are studied. For the irregular values of the second spectral parameter each of the five cases is checked separately. The orthogonality conditions are used. Cases are determined in which the problem has an infinite number of solutions and these solutions are constructed. For other cases, the absence of nontrivial solutions of the problem is proved.
Using the Fourier method of separation of variables, we examine the classical solvability and construct solutions of a nonlocal inverse boundary-value problem for the fourth-order Benney–Luke ...integro-differential equation with degenerate kernel. We prove the criterion of the unique solvability of the inverse boundary-value problem and examine the stability of solutions with respect to the recovery function.
We study the solvability and the construction of the solution of a boundary value problem with a nonlocal integral boundary condition for a three-dimensional analog of the fourth-order homogeneous ...Boussinesq type differential equation. Separation of variables is used to derive a criterion for the unique solvability of this nonlocal problem. The problem is also considered in the case of violation of the unique solvability criterion.
In this paper, we consider a nonlinear parabolic differential equation with involution. With respect to spatial variable is used Dirichlet boundary value conditions and spectral problem with ...involution is obtained. Eigenvalues and eigenfunctions of the spectral problems are found. The Fourier series method of separation of variables is applied. The countable system of nonlinear integral equations is obtained. Theorem on a unique solvability of the countable system of nonlinear integral equations is proved. The method of successive approximations is used in combination with the method of contraction mapping. The generalized solution of the nonlinear mixed problem is obtained in the form of Fourier series. Absolutely and uniformly convergence of Fourier series is proved.
In this paper, the classical solvability of a nonlocal boundary-value problem for a three dimensional, homogeneous, fourth-order, pseudoelliptic integro-differential equation with degenerate kernel ...is proved. The spectral Fourier method based on the separation of variables is used and a countable system of algebraic equations is obtained. A solution is constructed explicitly in the form of a Fourier series. The absolute and uniform convergence of the series obtained and the possibility of termwise differentiation of the solution with respect to all variables are justified. A criterion of the unique solvability of the problem considered is ascertained.
