Uniquely solvable problems for abstract Legendre equation

For loaded abstract Legendre equation we find sufficient conditions of solvability of the Cauchy problem and the boundary control problem. We also consider nonlocal problem that contains fractional integral of a function with respect to another functio

Solvability of degenerating hyperbolic differential equations with unbounded operator coefficients

We consider initial value problems for a number of hyperbolic equations with a power-law degeneracy and with operator coefficients in a Banach space and establish sufficient conditions for the unique solvability of these problems in terms of the coefficients of the equation, the degeneracy order, and the initial element

On the solvability of boundary value problems for an abstract singular equations on a finite interval

Sufficient conditions for the unique solvability of boundary value problems for a number of abstract singular equations that are formulated in terms of the zeros of the modified function Bessel and the resolvent of the operator coefficient of the considered equation

Dirichlet problem on the half-line for an abstract Euler-Poisson-Darboux equation containing powers of an unbounded operator

We consider an abstract Euler-Poisson-Darboux equation containing powers of an unbounded operator that is the generator of a Bessel operator function. Sufficient conditions for the unique solvability of the Dirichlet problem on the half-line are obtained. The question concerning the convergence of the solution to zero at infinity is investigate

Uniqueness criterion for solutions of nonlocal problems on a finite interval for abstract singular equations

For abstract singular equations, nonlocal problems belonging to the class of ill-posed problems are considered. A uniqueness criterion for solutions is establishe

On the unique solvability of boundary value problems for an abstract Euler-Poisson-Darboux equations on a finite interval

Sufficient conditions for the unique solvability of boundary value problems for a number of abstract singular equations that are formulated in terms of the zeros of the modified function Bessel and the resolvent of the operator coefficient of the considered equation

On the solvability of initial problems for abstract singular equations containing fractional derivatives

With the help of integral representations of the Poisson type, it is established that the Cauchy problem for a number of abstract singular equations with fractional derivatives reduces to a simpler problem for a non-singular equatio

A family of singular differential equations

A family of singular differential equations with variable coefficients and parameter k ∈ R is introduced into the consideration. The properties inherent in all differential equations of this family are investigated and theorems on the solvability of a number of initial problems for the considered family are formulate

Transmutation operators as a solvability concept of abstract singular equations

One of the methods of studying differential equations is the transmutation operators method. Detailed study of the theory of transmutation operators with applications may be found in the literature. Application of transmutation operators establishes many important results for different classes of differential equations including singular equations with Bessel operato

Solution of a boundary value problem for velocity-linearized Navier-Stokes equations in the case of a heated spherical solid particle settling in fluid

Assuming that the fluid viscosity is an exponential-power function of temperature, a boundary value problem for the Navier-Stokes equations linearized with respect to velocity is solved and the uniqueness of the solution is proved. The problem of a nonuniformly heated spherical solid particle settling in fluid is considered as an applicatio