Solvability Of Problem Research Articles

The solvability of indefinite nonlinear Choquard type problem with steep potential

The solvability of indefinite nonlinear Choquard type problem with steep potential

Computability of Classification and Deep Learning: From Theoretical Limits to Practical Feasibility Through Quantization

The unwavering success of deep learning in the past decade led to the increasing prevalence of deep learning methods in various application fields. However, the downsides of deep learning, most prominently its lack of trustworthiness, may not be compatible with safety-critical or high-responsibility applications requiring stricter performance guarantees. Recently, several instances of deep learning applications have been shown to be subject to theoretical limitations of computability, undermining the feasibility of performance guarantees when employed on real-world computers. We extend the findings by studying computability in the deep learning framework from two perspectives: From an application viewpoint in the context of classification problems and a general limitation viewpoint in the context of training neural networks. In particular, we show restrictions on the algorithmic solvability of classification problems that also render the algorithmic detection of failure in computations in a general setting infeasible. Subsequently, we prove algorithmic limitations in training deep neural networks even in cases where the underlying problem is well-behaved. Finally, we end with a positive observation, showing that in quantized versions of classification and deep network training, computability restrictions do not arise or can be overcome to a certain degree.

Inverse initial problem for fractional wave equation with the Hadamard fractional derivative

This paper investigates an inverse initial problem for a time-fractional wave equation involving the Hadamard fractional derivative. Unlike the more widely studied Caputo and Riemann–Liouville derivatives, the Hadamard derivative is defined via a logarithmic kernel and exhibits distinct analytical features, making it suitable for modeling processes with slow memory decay and multiplicative structures. The study is motivated by recent advances in fractional calculus and the growing interest in fractional models across applied sciences. Building on prior work concerning the extremum principle and solvability of boundary value problems with Hadamard-type operators, we establish sufficient conditions for the unique solvability of the inverse problem. The analysis is carried out in terms of eigenfunction expansions and leverages properties of the two-parameter Mittag–Leffler function. The findings contribute to the theory of inverse problems for fractional wave equations and highlight the role of Hadamard derivatives in capturing complex temporal dynamics in mathematical models.

On a mixed problem with integral conditions for third-order equation

The solvability of mixed problems with integral conditions for third-order partial differential equations is an important area of research in the theory of differential equations and their applications. Such problems arise in various fields of physics, continuum mechanics, oscillation theory, and other processes. In this paper, we consider a non-local problem with an integral condition for a third-order partial differential equation with the thermal conductivity operator in the main part. The theorems on the existence and uniqueness of the solution of the studied non-local problem are proved here. Methods of the theory of differential equations, the Green function and the theory of integral equations are used to the prove the solvability of the problem. The problem under study is reduced to an equivalent second kind Volterra integral equation which is certainly solvable.

A State-Dependent Dynamic Nonlinear Problems on Time Scales

Considering the phenomena that depend on their past state or past history, it was noted that they were given more importance. The mathematical models of these phenomena can be explained by state-dependent differential equations or a self-referred type. This article is dedicated to studying the solvability of state-dependent or self-referred dynamic nonlinear problems on time scales. Here, we got the existence of at least one solution to state-dependent dynamic nonlinear problems on time scales and a unique solution it has. Further more, we obtained results on the dependency of solutions for state-dependent dynamic nonlinear problems on time scales with respect to initial values.

A boundary integral equation formulation for transient electromagnetic transmission problems on Lipschitz domains

We propose a boundary integral formulation for the dynamic problem of electromagnetic scattering and transmission by homogeneous dielectric obstacles. In the spirit of Costabel and Stephan, we use the transmission conditions to reduce the number of unknown densities and to formulate a system of coupled boundary integral equations describing the scattered and transmitted waves. The system is transformed into the Laplace domain where it is proven to be stable and uniquely solvable. The Laplace domain stability estimates are then used to establish the stability and unique solvability of the original time domain problem. Finally, we show how the bounds obtained in both Laplace and time domains can be used to derive error estimates for semi discrete Galerkin discretizations in space and for fully discrete numerical schemes that use Convolution Quadrature for time discretization and a conforming Galerkin method for discretization of the space variables.

Algorithmic Problems for Computation Trees

In this paper, we study three algorithmic problems involving computation trees: the optimization, solvability, and satisfiability problems. The solvability problem is concerned with recognizing computation trees that solve problems. The satisfiability problem is concerned with recognizing sentences that are true in at least one structure from a given set of structures. We study how the decidability of the optimization problem depends on the decidability of the solvability and satisfiability problems. We also consider various examples with both decidable and undecidable solvability and satisfiability problems.

Inverse Boundary Value Problem for Fourth-Order Pseudo-Hyperbolic Equation with Nonlocal Integral Conditions of the Second Kind

In this paper, we consider a nonlinear inverse boundary value problem for a fourthorder pseudo hyperbolic equation with nonlocal conditions of the integral type. First, we introduce the definition of a classical solution to the problem. The purpose of this paper is to determine the unknown coefficient of the right-hand side and to solve the problem of interest. The problem is considered in a rectangular domain. To study the solvability of the inverse problem, we perform a transformation from the original problem to some auxiliary inverse problem with trivial boundary conditions. Using the principle of contraction mappings, we prove the existence and uniqueness of solutions to the auxiliary problem. Then we again perform a transformation to the problem and as a result obtain the solvability of the inverse problem.

Inverse Problem of Identifying a Time-Dependent Source Term in a Fractional Degenerate Semi-Linear Parabolic Equation

This work investigates the inverse problem of identifying a time-dependent source term in a time-fractional semi-linear degenerate parabolic equation using integral measurement data. We establish the unique solvability of the inverse problem within a suitable functional framework. The proof methodology is based on the Rothe method, where the variational formulation is discretized in time, and a priori estimates for discrete solutions are derived. These estimates are then utilized to demonstrate the convergence of Rothe approximations to a unique weak solution. Additionally, we develop a numerical scheme based on the L1-Galerkin finite element method, combined with iterative refinement, to reconstruct the unknown source term. The numerical performance of the proposed method is validated through a series of computational experiments, demonstrating its stability and robustness against noisy data.

Weak Closed-Loop Solvability of Linear Quadratic Stochastic Optimal Control Problems with Partial Information

Weak Closed-Loop Solvability of Linear Quadratic Stochastic Optimal Control Problems with Partial Information

On the Regularity Problem for Parabolic Operators and the Role of Half-Time Derivative

In this paper we present the following result on regularity of solutions of the second order parabolic equation ∂tu-div(A∇u)+B·∇u=0 on cylindrical domains of the form Ω=O×R where O⊂Rn is a uniform domain (it satisfies both interior corkscrew and Harnack chain conditions) and has a boundary that is n-1-Ahlfors regular. Let u be a solution of such PDE in Ω and the non-tangential maximal function of its gradient in spatial directions N~(∇u) belongs to Lp(∂Ω) for some p>11$$\\end{document}]]>. Furthermore, assume that for u|∂Ω=f we have that Dt1/2f∈Lp(∂Ω). Then both N~(Dt1/2u) and N~(Dt1/2Htu) also belong to Lp(∂Ω), where Dt1/2 and Ht are the half-derivative and the Hilbert transform in the time variable, respectively. We expect this result will spur new developments in the study of solvability of the Lp parabolic Regularity problem as thanks to it it is now possible to formulate the parabolic Regularity problem on a large class of time-varying domains.

INVERSE PROBLEM OF DETERMINING THE KERNEL IN THE INTEGRO-DIFFERENTIAL BEAM VIBRATION EQUATION

This paper addresses the problem of identifying the kernel that represents the memory of the medium in the integro-differential equation governing the forced vibrations of a beam. The problem is reduced to integral equations of the second kind of Volterra type with respect to the solution of the direct problem and the unknown kernel of the inverse problem. To solve these equations, we apply the method of compressive mappings in the space of continuous functions with exponential weight norm. The global solvability of the inverse problem and the conditional stability of the solution are established. In addition, the paper introduces an efficient numerical approach for solving the inverse problem associated with the beam equation. The method leverages the finite difference method to obtain solutions for both the displacement field u(x, t) and the media viscosity coefficient k(t), with the latter expressed in terms of an integral. Rigorous algorithmic development ensures accurate numerical solutions, which are validated through comparison with analytical solutions, demonstrating a high level of agreement. Furthermore, the proposed scheme is evaluated under noisy conditions, where it exhibits robustness in reducing numerical fluctuations over time. This comprehensive study highlights the reliability and effectiveness of the developed numerical approach, positioning it as a valuable tool for tackling inverse problems in structural mechanics and related fields.

On the existence of solutions to nonlinear boundary value problems for non-flat isotropic shells of Timoshenko type in arbitrary curvilinear coordinates

We study the solvability of a boundary value problem for a system of five nonlinear second-order partial differential equations under given nonlinear boundary conditions, which describes the equilibrium state of elastic non-flat inhomogeneous isotropic shells with loose edges in the framework of the Timoshenko shear model, assigned to arbitrary curvilinear coordinates. The boundary value problem is reduced to a nonlinear operator equation for generalized displacements in Sobolev space, the solvability of which is established using the contraction mapping principle.

SOLVABILITY OF A BOUNDARY VALUE PROBLEM OF THE SAMARSKY-IONKIN TYPE FOR DIFFERENTIAL EQUATIONS WITH INVOLUTION

In this paper, we investigate a non-local boundary value problem for a second-order differential equation with an involutive transformation. The aim of this work is to apply the parametrization method developed by Professor D. Dzhumabayev to study the solvability of non-local boundary value problems in the context of differential equations with involutive transformations. It is known that the Cauchy problem for such equations may not have a unique solution. Therefore, parameters μ1= y(½), μ2= y′(½) are introduced, and variables y(x)=u(x)+μ1+μ2(x-1/2) are replaced. The parameter values are determined at the midpoint of the interval, which guarantees the existence of a unique solution for the Cauchy problem of the original equation. The performed variable substitution formally divides the problem into two components: the Cauchy problem for the initial equation and a system of linear equations with respect to the introduced parameters. By solving the Cauchy problem and substituting its solution into the boundary conditions, one can obtain a system of linear equations with respect to the parameters. If the matrix of this system is invertible, then the problem has a unique solution. In the case when the matrix is non-invertible, two scenarios are possible: either the boundary value problem is unsolvable, or it has multiple solutions. For the second case, the paper defines the eigenvalues and solvability conditions of the boundary value problem.

Optimal Operation of a Tablet Pressing Machine Using Deep-Neural-Network-Embedded Mixed-Integer Linear Programming

This paper presents a deep neural network (DNN)-embedded mixed-integer linear programming (MILP) model for fault prediction and production optimization in tablet pressing machines. The DNN predicts the probability of failures during the tablet pressing process by analyzing key operational parameters such as pressure, temperature, humidity, speed, vibration, and number of maintenance cycles. The MILP model optimizes the temperature and humidity settings, production schedules, and maintenance planning to maximize total profit while minimizing penalties for fault pressing, energy consumption, and maintenance costs. To integrate DNN into the MILP framework, Big-M constraints are applied to linearize the Rectified Linear Unit (ReLU) activation functions, ensuring solvability and global optimality of the optimization problem. A case study using the Kaggle dataset demonstrates the model’s ability to dynamically adjust production and maintenance schedules, enhancing profitability and resource utilization under fluctuating electricity prices. Sensitivity analyses further highlight the model’s robustness to variations in maintenance and energy costs, striking an effective balance between cost efficiency and production quality, which makes it a promising solution for intelligent scheduling and optimization in complex manufacturing environments.

On solvability of some boundary value problems for a nonlocal Poisson equation with periodic conditions

In the present paper, a nonlocal analog of the Laplace operator is introduced by means of involution-type mappings. New classes of boundary value problems are studied for the corresponding nonlocal analog of the Poisson equation in a unit sphere. In the problems under consideration, the boundary conditions are given in the form of a relation between the value of the unknown function in the upper hemisphere and the value in the lower hemisphere. The problems under study generalize the known periodic and antiperiodic boundary value problems for circular regions. The problems are solved by reducing them to two auxiliary problems with Dirichlet and Neumann boundary conditions for the nonlocal analog of the Poisson equation. Using known statements for the obtained auxiliary problems, we prove theorems on the existence and uniqueness of solutions of the main problems. Exact conditions for the solvability of the investigated problems are found, and integral representations of the solutions are obtained. Spectral issues related to periodic problems are also studied. Eigenfunctions and eigenvalues of these problems are found. The theorems on completeness of the system of eigenfunctions in the space L2 are proved.

A problem of simple group pursuit with possible dynamical disturbance in dynamics and phase constraints

In finite-dimensional Euclidian space, we treat the problem of pursuit of one evader by a group of pursuers, which is described by a system of the form z˙i = ai(t)ui − v, ui ∈ Ui, v ∈ V, where the functions i(t) are equal to 1 for all t, except for a certain interval of a given length, on which they are equal to zero (to each pursuer there corresponds its own interval). This fact can be interpreted in such a way that each of the pursuers has a possible failure of the control device at any previously unknown moment in time, and the length of the time interval needed to fix the failure is given, while in the process of fixing the failure the pursuers have no possibility to carry out a capture. The target sets are convex compact sets, and the evader does not leave the bounds of the convex polyhedral set. We obtain sufficient conditions for solvability of the pursuit problem.

On the Strong Solvability of a Nonlocal Boundary Value Problem for the Laplace Equation in an Unbounded Domain

On the Strong Solvability of a Nonlocal Boundary Value Problem for the Laplace Equation in an Unbounded Domain

Weak solvability of the initial-boundary value problem for the second-order Kelvin-Voigt model with smoothed Jaumann derivative

Weak solvability of the initial-boundary value problem for the second-order Kelvin-Voigt model with smoothed Jaumann derivative

Solvability of Caputo fractional integral boundary value problem at resonance in Rn

In this paper, we utilize the continuation theorem due to Mawhin to investigate the solvability for a class of Caputo fractional system integral boundary value problems at resonance. To broaden the applicability of the main results, we employ the Moore–Penrose pseudoinverse matrix to construct projectors.