# Assanova A. T.

### Numerical method for the solution of linear boundary-value problem for integrodifferential equations based on spline approximations

Assanova A. T., Bakirova E. A., Iskakova N. B.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 9. - pp. 1176-1191

UDC 517.642

We propose a numerical method for the solution of linear boundary-value problem for system of integrodifferential equations.
This method is based on the approximation of the integral term by a cubic spline and reduction of the original problem to
a linear boundary-value problem for a system of loaded differential equations. We also propose new algorithms for finding
the numerical solution and a method for the construction of approximate solution to the approximating boundary-value
problem.

### To the theory of nonlocal problems with integral conditions for systems of equations of hyperbolic type

Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1313-1323

We consider a nonlocal problem with integral conditions for a system of hyperbolic equations of the second order. By method of introduction of functional parameters, the investigated problem is reduced to an equivalent problem formed by the Goursat problem for a system of hyperbolic equations with parameters and integral relations. Algorithms for finding approximate solutions of this problem are constructed and their convergence to the exact solution is demonstrated. Sufficient conditions for the unique solvability of the equivalent problem are obtained in terms of the initial data. Moreover, the conditions of unique solvability of the nonlocal problem with integral conditions for system of hyperbolic equations are established in terms of the coefficients of the system and kernels in the integral conditions.

### On the unique solvability of a nonlocal boundary-value problem for systems of loaded hyperbolic equations with impulsive actions

Assanova A. T., Bakirova E. A., Kadirbayeva Zh. M.

Ukr. Mat. Zh. - 2017. - 69, № 8. - pp. 1011-1029

We consider a nonlocal boundary-value problem with impulsive actions for a system of loaded hyperbolic equations and establish the relationship between the unique solvability of this problem and the unique solvability of a family of two-point boundary-value problems with impulse actions for the system of the loaded ordinary differential equations by method of introduction of additional functions. Sufficient conditions are obtained for the existence of a unique solution to a family two-point boundary-value problems with impulsive effects for the system of loaded ordinary differential equations by using method of parametrization. The algorithms of finding the solutions are constructed. The conditions of unique solvability of the nonlocal boundary-value problem for a system of loaded hyperbolic equations with impulsive actions are established. The numerical realization of the algorithms of the method of parametrization is proposed for the solution of the family of two-point boundary-value problems with impulsive actions for the system of the loaded ordinary differential equations. The results are illustrated by specific examples.

### Well-Posed Solvability of a Nonlocal Boundary-Value Problem for the Systems of Hyperbolic Equations with Impulsive Effects

Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 291-303

We consider a nonlocal boundary-value problem for a system of hyperbolic equations with impulsive effects. The relationship is established between the well-posed solvability of the nonlocal boundary-value problem for a system of hyperbolic equations with impulsive effects and the well-posed solvability of a family of two-point boundary-value problems for a system of ordinary differential equations with impulsive effects. Sufficient conditions for the existence of a unique solution of the family of two-point boundary-value problems for a system of ordinary differential equations with impulsive effects are obtained by method of introduction of functional parameters. The algorithms are proposed for finding the solutions. The necessary and sufficient conditions of the well-posed solvability of a nonlocal boundary-value problem for a system of hyperbolic equations with impulsive effects are established in the terms of the initial data.

### On a Nonlocal Boundary-Value Problem for Systems of Impulsive Hyperbolic Equations

Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 315-328

We consider a nonlocal boundary-value problem for a system of impulsive hyperbolic equations. Conditions for the existence of a unique solution of the problem are established by the method of functional parameters, and an algorithm for its determination is proposed.

### Periodic solutions of systems of hyperbolic equations bounded on a plane

Assanova A. T., Dzhumabaev D. S.

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 562-572

For a linear system of hyperbolic equations of the second order with two independent variables, we investigate the problem of the existence and uniqueness of a solution periodic in both variables and a solution periodic in one of the variables and bounded on a plane. By using the method of introduction of functional parameters, we obtain sufficient conditions for the unique solvability of the problems under consideration.

### On a bounded almost periodic solution of a semilinear parabolic equation

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 828–830

We obtain sufficient conditions for the existence and uniqueness of a bounded almost periodic solution of a semilinear parabolic equation.