Submitted Abstracts

Inverse problem with periodic conditions for the Burgers equation

Madi Yergaliyev

In the work, in Sobolev spaces, we establish the unique solvability of the inverse problem with periodic conditions for the Burgers equation and the related initial-boundary value problem for the loaded Burgers equation. The methods of functional analysis, a priori estimates, and Faedo-Galerkin are used.

Direct and inverse problems for a two-dimensional parabolic equation with involution

Batirkhan Turmetov and Maira Koshanova

The work is devoted to the study of solvability of direct and inverse problems for a two-dimensional parabolic equation with involution. The studied problems are solved by reducing them to direct and inverse problems for classical two-dimensional differential equations of parabolic type. On the basis of well-known theorems obtained for auxiliary problems, theorems on the existence and uniqueness of the solution of the studied problems are proved. The explicit form of solutions of the studied problems is constructed in the form of a series.

An analog of Nazarov-Podkorytov lemma in non-commutative Lp space

Dostilek Dauitbek

We obtain a non-commutative analog of Nazarov-Podkorytov lemma in non-commutative Lp space.

Coercive estimates and compactness of a resolvent of a class of singular parabolic operators

Mussakan B. Muratbekov, Madi M. Muratbekov and Sabit Igisinov

We study the singular parabolic operator $$ Lu=\frac{\partial u}{\partial t}-\frac{\partial^2 u}{\partial x^2}+q(x)u $$ initially defined on $C^{\infty}_{0,\pi}(\bar{\Omega})$ where $\bar{\Omega}=\{(t,x):-\pi\leq t\leq \pi, -\infty<x<\infty\}$. $C^{\infty}_{0,\pi}(\bar{\Omega})$ is the set, which consist of infinitely differentiable finite functions with respect to the x and satisfying the condition $$ u(-\pi,x)=u(\pi,x). $$ We assume that the coefficients of L are continuous functions in $\mathds{R}=(-\infty, \infty)$ and a strongly growing functions at infinity. The operator L admits closure in $L_2(\Omega)$ and the closure we also denote by L. In the paper, we have proved that there exists a bounded inverse operator and found a condition on $ q (x) $ that ensures the existence of the estimate $$ \| \frac{\partial u}{\partial t}\|_{L_2(\Omega)}+ \| \frac{\partial^2 u}{\partial x^2}\|_{L_2(\Omega)}+\|q(x)u\|_{L_2(\Omega)}\leq c (\|Lu\|_{L_2(\Omega)}+\|u\|_{L_2(\Omega)}), $$ with some restrictions on the coefficients, in addition to the above conditions; the compactness of the resolvent is proved; two-sided estimates of singular numbers (s-numbers) are obtained and an example is given of how above estimates will make it possible to find estimates for the eigenvalues of the operator under consideration.

On effective methods of regularization with discretization of integral equations

Temirbekov Nurlan, Temirbekova Laura and Nurmangaliyeva Maya

The classical methods of regularization of Tikhonov A.N. are well known in the scientific literature. [1]. Many works [5, 6, 15-20] are devoted to the study of mathematical issues of the regularization method for numerical solving the first kind of Fredholm integral equation. In recent years, there has been an increased interest in projection methods for solving Fredholm integral equations with bases in the form of Legendre, Haar, Chebyshev wavelets [2-14]. These methods allow solving Fredholm integral equations of the first kind with fairly good accuracy [2-14]. In this paper, we consider the issues of stability and convergence of the projection Galerkin-Bubnov method with Legendre wavelets as the basis function.

On a nonlocal boundary value problem for a three-dimensional Tricomi equation in a prismatic unbounded domain

S.Z.Dzhamalov, R.R.Ashurov, M.A.Sultanov, Kh.Sh. Turakulov

The article investigates the unique solvability of a generalized solution of one nonlocal boundary value problem for the three-dimensional Tricomi equation in a prismatic unbounded domain by the method of “regularization” and a priori estimates using the Fourier transform.

An approximation to the solution of the source identification telegraph problem with nonlocal condition

Haitham Al hazaimeh

In this talk, the source identification problem for the telegraph equation is studied. We propose The first order of accuracy absolute stable difference scheme to find the numerical solution of the one-dimensional identification problem for the telegraph equation with the nonlocal condition. The advantage of the resulting difference scheme is that the algorithm is very simple so it is very easy to implement. The results of numerical experiments are presented and are compared with the exact solution to verify the accurate nature of our method.

Homeomorphic maps of finite-dimensional Euclidean spaces

Firudin Muradov

The purpose of the present note is to characterize open and bounded closed sets of finite-dimensional Euclidean spaces by ternary semigroups of homeomorphic maps.

Variational methods for constructing iterative algorithms

Almas Temirbekov, Bakhytzhan Zhumagulov

Fictitious domain methods for partial differential equations are the most effective for solving complex problems of science and technology. The main reason for the popularity of these methods is that they allow the use of fairly simple structured grids on the auxiliary domain containing the actual domain, which allows building uniform difference schemes. The use of differential methods of end-to-end counting based on the method of fictitious domains is very convenient for the development of programming modules for solving applied problems. This approach significantly increases the degree of programming automation. The method of a fictitious domain allows satisfying boundary conditions on the actual boundary using variational principles. To solve by the finite element method, two grids are constructed in the extended domain: a triangulation over the entire domain and a curved grid on the actual boundary. Then the searched variables are fitted on the curved boundary at each iteration of the conjugate gradient method. In some cases, this process may even behave worse than solving the original problem on an uneven grid fitted with a curved boundary. The results of numerical calculations show convergence to the solution in the ”average” due to the use of variational principles, i.e. the functional containing the basic equation and the boundary condition is minimized. In this talk, the study of the extreme problem of the method of fictitious domains based on the use of the Lagrange functional, with a multiplier defined on the actual boundary and associated with genuine boundary conditions, is considered.

A new finite difference algorithm for boundary value problems involving transmission conditions

Semih Çavuşoğlu and Oktay Sh. Mukhtarov

Finite difference methods are numerical methods for approximating the solution to various types of differential equations using finite difference equations to approximate derivatives. The idea is to replace ordinary or partial derivatives appearing in the boundary-value problem with finite differences that approximate them. There is extensive literature on this topic but, as a rule, ordinary differential equations or partial differential equations were studied without an internal singular point and without corresponding transmission conditions.

It is our main goal here to develop finite difference method to deal with an boundary value problem involving additional transmission conditions at the interior singular point. To check the reliability and efficiency of the proposed new finite-difference algorithm, the following example of a boundary value problem with additional transmission conditions was solved

y'' -2(2x+1) y'+ ( (2x+1)^{2}-2) y =0,  x in [-1,0) u (0,1],

y (-1)=0,  y (1) = 3,

y(0^{-})=2y(0^{+}), y'(0^{-})=3y'(0^{+}) .

Source identification problems for the neutron transport differential and difference equations

Abdulgafur Taskin

In this study, a source identification problem for the two-dimensional neutron transport equation is studied. For the approximate solution of this problem, a first order of accuracy difference scheme is presented. Stability inequalities for the solution of these differential and difference problems are established. Numerical results are given.

Numerical solution to elliptic source identification problem with non-local integral condition

Charyyar Ashyralyyev

Stability estimates for solutions of elliptic source identification problem with an integral condition for derivative and difference schemes for its solution were studied in [1. Ashyralyyev C., Cay A., Well-posedness of Neumann-type elliptic overdetermined problem with integral condition, AIP Conference Proceedings 2018, Vol. 1997 020026; 2. Ashyralyyev C., Cay A., Numerical solution to elliptic inverse problem with Neumann-type integral condition and overdetermination, Bulletin of the Karaganda University-Mathematics. 2020, Vol. 99, no 3, 5-17].

Let  \Omega =(0,L)^{n} be open cube in the n-dimensional vector space R_{n , \overline{\Omega }=\Omega \cup \partial \Omega  and

a_{1}(x),...,a_{n}(x),\vartheta (x),\eta (x),\xi (x),\ (x\in \Omega ),\ f(t,x)~(x\in \Omega ,t\in (0,1)\ )

be given sufficiently smooth functions. In addition,  a_{s}(x)\geq \delta >0,s=1,...,n, for all x in \Omega $. In paper [ 3. Ashyralyyev C., On the source identification elliptic problem with Dirichlet and integral conditions in Springer Proceedings in Mathematics & Statistics "Functional Analysis in Interdisciplinary Applications, II ". 2021. Vol. 351, 63-73], well-posedness of the following identification multidimensional elliptic problem with integral and first kind boundary conditions:

\begin{equation} \left\{ \begin{array}{l} -u_{tt}(x,t)~-~\sum\limits_{r=1}^{n}(a_{r}(x)u_{x_{r}}(x,t))_{x_{r}}+\sigma u(x,t)~=~f(x,t)~+~p(x), \\ ~x\in \Omega ,t\in (0,1), \\ {u}\left( x,0\right) =\vartheta \left( x\right) ,~{u}\left( x,1\right) =\int\limits_{0}^{1}\rho \left( \lambda \right) {u}\left( \lambda ,x\right) d\lambda +\eta (x),~ \\ u\left( {\gamma },x\right) =\xi (x),x\in \overline{\Omega }\ (0<{\gamma }<1), \\ \ u(x,t)~=~0,~x\in S,~t\in \lbrack 0,1].% \end{array}% \right. \label{ap1} \end{equation}

was established. In this paper, we propose the second order of accuracy difference scheme for approximate solution of source identification problem for multi dimensional elliptic partial differentional equation. Numerical results are given in test examples.

A note on stability of parabolic difference equations on n-torus

Allaberen Ashyralyev, Fatih Hezenci and Yaşar Sozen

In the present talk, we consider nonlocal boundary value problems for parabolic equations of reverse type on N-Torus {T}^N. We set up the first order of accuracy difference scheme for the numerical solution of nonlocal boundary value problems for parabolic equations on circle {T}^1 and  2-torus {T}^2. For the solutions of the difference scheme, we establish the stability estimates and coercivity estimates in various Holder norms for the solutions of such boundary value problems. Furthermore, theoretical results are supported by numerical experiments.

A numerical algorithm for the third order partial delay differential equation with involution and Neumann condition

Allaberen Ashyralyev, Suleiman Ibrahim, Evren Hincal

In the present talk, the initial value problem for the third-order partial delay differential equation with involution and Neumann boundary condition is studied. The first order of accuracy difference schemes for the numerical solution of the third-order partial delay differential equation with involution and Neumann boundary condition is presented. The illustrative numerical results are provided.

Time-dependent source identification problems for delay hyperbolic differential and difference equations

Bishar Chato Haso

In the present work, a time-dependent source of identification problem for a one-dimensional delay hyperbolic equation is studied. The first order of accuracy difference scheme for this source identification problem is presented. The theorem on the stability estimates for the solution of this difference scheme is established. Numerical analysis and discussions are presented.

Global weak solutions for a shallow water wave equation

Hatice Taskesen, Necat Polat

In this work, we consider the Cauchy problem of a shallow water wave equation which is a natural approximation to the Euler equation. Utilizing the potential well method, we discuss the existence of global weak solutions in two different initial energy levels.

Global well-posedness for a stochastic nonlinear wave equation

Hatice Taskesen, Sıddık Polat

In this study, we obtain conditions guaranteeing the local and global existence of solutions for a stochastic wave equation with multiplicative noise. We also investigate the dependence of initial data on solutions.

On the Cauchy problem for the modified Camassa-Holm equation with weak dissipation

Nurhan Dündar, Necat Polat

In this study, we consider the Cauchy problem of the modified Camassa-Holm equation with weak dissipation. Firstly, by using Kato’s theory, we show that the Cauchy problem for the weakly dissipative modified Camassa-Holm equation is locally well-posed in Sobolev spaces. Then we obtain the global existence result.

Asymptotic formulas for the solution of hyperbolic perturbation problems with nonlocal conditions

Ozgur Yildirim

In the present work, the abstract second-order hyperbolic problem with nonlocal conditions and an arbitrary  \varepsilon \in  (0,\infty) parameter in a Hilbert space $H$ with the self-adjoint positive definite operator A is considered. The asymptotic formula for the solution of this problem is obtained.

Solution of some nonlocal problems of Elliptic-hyperbolic equation

Pirmyrat Gurbanov

It is considered elliptic-hyperbolic equation {█(u_xx+u_yy=0, y>0,@u_xx-u_yy=0, y<0) ┤ in the area which is bounded by characteristics x+y=0, x-y=1 and straight lines x=0 ,x=1. We solve this equation subject to nonlocal boundary condition: {█(u(0,y)=φ_1 (y), u(1,y)=φ_2 (y) @u(x/8;-x/8)+u((-x+7)/8;-(x+1)/8)+u((x+2)/8;-(x+2)/8)+u((-x+5)/8;-(x+3)/8)=φ(x) )┤ Using nonlocal conditions we solve functional equation at y<0 and then find a general equation for the given parabola-hyperbolic equation. After that we prove that the solution is unique and sustainable.

Methods of differential geometry in stochastic analysis

Mamadsho Ilolov

Models of the "white noise" type were the subject of research in a number of works by the American mathematician A.V. Balakrishnan, published in the 70-80's of the last century. This author proposed a new concept of the stochastic integral, different from the well-studied integrals of Ito and Stratanovich. In the standard Ito calculus, the stochastic Wiener process or the integral of the Gaussian process is used to describe white noise, the exact definition of which is very difficult. The main motivation behind the white noise theory developed by Balakrishnan is to develop an alternative for the Ito calculus, which leads to solutions that are potentially more convenient for engineering applications. The theoretical substantiation of this method lies in the following property of functions from the Wiener space. Contour integrals of functions of bounded variation and finite energy (norms) belong to the Wiener set of zero measure. At the same time, unfortunately, physical signals are precisely functions of limited variation and finite energy. Thus, the results obtained in the framework of the Ito or Stratanovich calculus are true on Wiener sets of measure 1 and cannot be adequate when applied to various physical data.

On certain non-standard elliptic problems

Vladimir Vasilyev

The limiting behavior of the solution of a model elliptic pseudo-differential equation is studied. First, the equation is considered in a flat sector with an additional integral condition. In this case, using the formula for the general solution, the limiting behavior is studied assuming that the sector angle tends to zero. It is established that the function in the boundary condition cannot be arbitrary, but must satisfy a certain functional singular integral equation. Then the case of a 4-wedge conical canonical 3D singular domain with two parameters is studied. It is shown that the solution of such boundary value problem can have a limit with respect to endpoint values of the parameters in appropriate Sobolev – Slobodetskii space if the boundary function is a solution of a special functional singular integral equation.

A numerical study of the interaction of waves for 2D Riemann problem

Bahaddin Sinsoysal

We propose a new finite difference method to solve the Cauchy problem with a three-piece constant initial function for the Guckenheimer equation. To this end, we introduce an auxiliary problem that has advantages over the original problem, and we develop an original finite difference method to solve the auxiliary problem. The solution is used as well to compute the numerical solution of the original problem that describes all of the physical properties of the original problem correctly.

On the stability of a nonlocal BVP for a third-order partial differential equation

Kheireddine Belakroum

In this talk, the nonlocal boundary value problem for third-order partial differential equations in a Hilbert space with a self-adjoint positive definite operator is studied. The main theorem on the stability of this problem is established. In practice, stability estimates for the solution of two problems for third-order partial differential equations are obtained.