Vol. 31 No. 2 (2024)

Open Access

Articles

Article ID: 2431

A NUMERICAL METHOD TO SOLVE THE VISCOSITY PROBLEM OF THE BURGERS EQUATION

by Gérard ZONGO, Ousséni SO, Geneviève BARRO

Advances in Differential Equations and Control Processes, Vol.31, No.2, 2024;

Considering the viscosity problem of the Burgers equation, we give a numerical solution using the Cole-Hopf transformation.

Open Access

Articles

Article ID: 2432

ON THE EXISTENCE OF CLASSICAL SOLUTION TO ONE-DIMENSIONAL FOURTH ORDER SEMILINEAR EQUATIONS

by Samed J. Aliyev, Maftun N. Heydarova, Arzu G. Aliyeva

Advances in Differential Equations and Control Processes, Vol.31, No.2, 2024;

In this paper, we prove the existence in small of classical solution of one-dimensional mixed problem for one class of fourth order semilinear Sobolev type equations by combining the generalized contracted mapping principle with Schauder’s fixed point principle.

Open Access

Articles

Article ID: 2433

ON FINITE CHARACTER GEOMETRICAL PROPERTY OF THE DIFFERENTIAL REALIZATION OF NONSTATIONARY HYPERBOLIC SYSTEMS

by A. V. Lakeyev, V. A. Rusanov, A. V. Banshchikov, R. A. Daneev

Advances in Differential Equations and Control Processes, Vol.31, No.2, 2024;

Topological-algebraic investigation of the problem of existence of realization of finite-dimensional continuous dynamic processes in the class of second-order ordinary differential equations in a separable Hilbert space has been conducted. Simultaneously, analytical-geometric conditions of continuity of the process of constructing projections for the Rayleigh-Ritz nonlinear functional operator together with computation of the fundamental group of its image have been determined. The results may be applied to a posteriori modeling nonstationary hyperbolic systems.

Open Access

Articles

Article ID: 2434

THE LIMIT OF BLOW-UP DYNAMICS SOLUTIONS FOR A CLASS OF NONLINEAR CRITICAL SCHRÖDINGER EQUATIONS

by N’takpe Jean-Jacques, L. Boua Sobo Blin, Nachid Halima, Kambire D. Gnowille

Advances in Differential Equations and Control Processes, Vol.31, No.2, 2024;

This paper considers the asymptotic behavior of solutions of equations of evolutions, and concentrates on the analysis of the critical blow-up solutions for a class of evolutions for nonlinear Schrödinger equations in a bounded domain. More precisely, the numerical approximation of the blow-up rate below the one of the known explicit explosive solutions is studied, which has strictly positive energy for the following initial-boundary value problem: $$ (P) \begin{cases}u_t(t, x)-i \alpha \Delta u(t, x)-i \beta f(t, x)=0, & x \in \mathbb{R}^d, t \in(0, T), \\ u(x, 0)=u_0(x), & x \in \mathbb{R}^d,\end{cases} $$ where $i=\sqrt{-1}, \quad \alpha \in \mathbb{R}, \quad \beta \in \mathbb{R}, \quad d \geq 1, \quad u$ is a complex-valued function of the variable $x \in \mathbb{R}^d, \Delta$ is the Laplace operator in $\mathbb{R}^d$ and the time $t \geq 0$. The paper proposes a general setting to study and understand the behavior of the blow-up solutions in a finite time as a function of the parameters $\alpha, \beta$, with initial condition $u(0, x)=u_0$, in the energy space $H^1 \in \mathbb{R}^d$, also in the case where $\mathbb{R}^d$ is large enough and its size $d$ is taken as parameter. Some assumptions are found under which the solution of the above problem blows-up in a finite time, study the dynamics of blow-up solutions and estimate its blow-up time. Finally, some numerical experiments to illustrate the analysis have been provided.

Open Access

Articles

Article ID: 2435

SOLVABILITY FOR CONTINUOUS CLASSICAL BOUNDARY OPTIMAL CONTROL OF COUPLE FOURTH ORDER LINEAR ELLIPTIC EQUATIONS

by Eman Hussain Mukhalf Al-Rawdhanee

Advances in Differential Equations and Control Processes, Vol.31, No.2, 2024;

In this paper, we study continuous classical boundary optimal control problem for the couple fourth order of linear elliptic system with variable coefficients. The existence theorem of a unique couple vector state solution of the weak form obtaining from the couple fourth order of linear elliptic system with Neumann conditions (NCs) is demonstrated for fixed continuous classical couple boundary control vector (CCCPBCTV) utilizing Hermite finite element method. The existence theorem of a couple continuous classical boundary optimal control vector dominated with the considered problem is proved. The existence and uniqueness of the solution of the couple adjoint equations (CPAEs) is discussed, when the classical couple optimal boundary control is given. Finally, the Fréchet derivative (FrD) of the Hamiltonian is obtained to establish the theorem of the necessary condition for optimality.

Open Access

Articles

Article ID: 2436

APPROXIMATED SOLUTIONS OF THE HOMOGENEOUS LINEAR FRACTIONAL DIFFUSION-CONVECTION-REACTION EQUATION

by Bamogo Hamadou, Nebié AbdoulWassiha, Francis Bassono, Minoungou Youssouf, Bagayogo Moussa

Advances in Differential Equations and Control Processes, Vol.31, No.2, 2024;

Our work focused on solving a homogeneous linear fractional diffusion, diffusion-convection and diffusion-convection-reaction model with various initial conditions and appropriate parameters. We used the Adomian decomposition method (ADM) to find exact or approximate solutions.

Open Access

Articles

Article ID: 2437

DECOUPLING OF NONLINEAR CONTROL SYSTEM BASED ON LIE SYMMETRY METHOD

by Zheng Mingliang, Nie Wenyan, Cheng Daguang, Yu Liang

Advances in Differential Equations and Control Processes, Vol.31, No.2, 2024;

The Lie symmetry method is introduced for the decoupling problem of nonlinear control system. Firstly, the key technology of Lie symmetry theory for differential equations is introduced; secondly, a kind of nonlinear control system model is established, and the conditions and properties of Lie symmetry are given in detail. Finally, the decoupling global and local forms of the system are given through the derived distribution of infinitesimal generators. The numerical results show the effectiveness of Lie symmetry method. As long as the infinitesimal generator is constructed, the cascade decoupling form of nonlinear control system could be obtained.

Open Access

Articles

Article ID: 2438

PERIODIC SOLUTIONS OF A SECOND-ORDER NONLINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION

by A. T. Alymbaev, A. Bapa Kyzy, F. K. Sharshembieva

Advances in Differential Equations and Control Processes, Vol.31, No.2, 2024;

The article considers the problem of constructing a 2π-periodic solution of a quasilinear second-order integro-differential equation. Using the Green's function of bounded solutions on the number line, the integro-differential equation is reduced to an integral equation.A 2π-periodic solution to the integral equation is found using the projection-iteration method.A 2π- periodic solution is sought as the limit of successive 2π-periodic functions representable as a Fourier series. An estimate of the error of the difference between the exact and approximate solutions is obtained.