Class Of Problems Research Articles

On ellipticity of operators with shear mappings

The nonlocal boundary value problems are considered, in which the main operator and the operators in the boundary conditions include the differential operators and twisting operators. The de nition of the trajectory symbols for this class of problems is given. We show that the elliptic problems de ne the Fredholm operators in the corresponding Sobolev spaces. The ellipticity condition of such nonlocal boundary value problem is given.

K voprosu o chislennom reshenii nekonservativnykh giperbolicheskikh sistem uravneniy

Issues related to the lack of convergence in the application of formally path-conservative difference schemes for solving nonconservative hyperbolic systems of equations are numerically investigated. This problem is central in constructing well-posed difference schemes for solving this class of problems. The basic concepts of the theory of nonconservative hyperbolic systems of equations and the corresponding problems of constructing difference schemes for their solution are outlined. A variant of the HLL method is proposed that allows using an arbitrary explicitly specified path. For a model system of Burgers equations, the shock adiabates and paths corresponding to the viscous regularization of a system of a given form are explicitly calculated. The reasons for the lack of convergence of numerical solutions of exact ones in the case of incorrect application of the corresponding algorithms are analyzed. It is shown that, at least in the particular case considered, a variant of the HLL method that is formally conservative along the way gives the correct solution of the problem.

Fredholm Integral Equation for Problems of Acoustic Scattering by Three-Dimensional Transparent Structures

We consider the differential and integral statements of problems of acoustic scattering by three-dimensional bounded transparent structures described by an integral equation. The results of the numerical solution of the integral equation describing the class of problems under consideration are presented. A theorem on existence and uniqueness of the solution is proved.

On a New Class of Impulsive η-Hilfer Fractional Volterra-Fredholm Integro-Differential Equations

This work addresses the idea of the uniqueness and existence results for a class of boundary value problems (BVPs) for implicit Volterra-Fredholm integro-differential equations (V-FIDEs) with fractional η-Hilfer nonlinear equations and multi-point fractional boundary non-instantaneous conditions. The conclusions are confirmed by the fixed point of Krasnoselskii's theorem and the Banach contraction principle. Finally, a concrete example is given to illustrate our main conclusions.

Convexificators for nonconvex multiobjective optimization problems with uncertain data: robust optimality and duality

In this paper, we investigate robust optimality conditions and duality for a class of nonconvex multiobjective optimization problems with uncertain data in the worst case by the upper semi-regular convexificator. The Fermat principle for a locally Lipschitz function is presented in terms of the upper semi-regular convexificator. We establish robust necessary optimality conditions of the Fritz-John type and KKT type for the uncertain nonconvex multiobjective optimization problems. In addition, robust sufficient optimality conditions as well as saddle point conditions are derived under the generalized ∂ ˆ ∗ -pseudoquasiconvexity and generalized convexity, respectively. The robust duality relations between the original problem and its mixed robust dual problem are obtained under a generalized pseudoconvexity assumption.

Numerical Solution of the Fokker-Planck Equation Using Orthogonal Collocation

This study aims to propose a numerical methodology to solve the Fokker-Planck Equation using the Orthogonal Collocation Method. In this approach, the original problem, represented by a partial differential equation, is rewritten as a system of initial value equations via discretization of spatial variable. The resulting system is integrated considering the Runge-Kutta-Fehlberg Method. The proposed methodology is applied in three case studies that presented an analytical solution. The obtained results demonstrate that the proposed approach is a good alternative for solving this class of problems. Keywords: Fokker-Planck Equation, Orthogonal Collocation, Numerical Method.

On Existence of Multiple Normalized Solutions to a Class of Elliptic Problems in Whole $$\mathbb {R}^N$$ Via Penalization Method

On Existence of Multiple Normalized Solutions to a Class of Elliptic Problems in Whole $$\mathbb {R}^N$$ Via Penalization Method

Design Space Exploration and Evaluation Using Margin-Based Trade-Offs

Abstract Design space exploration and margin analysis can inform critical decisions early in engineering design, helping to handle the uncertainties of early design while ensuring design performance. In practice, the complexity of many products makes such decision-making challenging. This paper addresses the challenge with a new design framework that relies on the margin value method to evaluate sets of concepts that are combinatorially generated from an enhanced function-means tree. The basis for concept comparison is the margin value in each design alternative. The margin value method is expanded to address a broad class of design problems by using surrogate models and novel metrics for evaluating different conceptual alternatives. Visualization tools are introduced to support the evaluations. The efficacy of the framework is demonstrated using the design of a structural aero-engine component involving simulation models and uncertain load specifications. Overall, this paper shows how design concepts can be compared objectively and distilled to a set of alternatives that would retain their values throughout product development.

Inertial Proximal Point Algorithms for Solving a Class of Split Feasibility Problems

Inertial Proximal Point Algorithms for Solving a Class of Split Feasibility Problems

Location Planning of Emergency Medical Facilities Using the p-Dispersed-Median Modeling Approach

This research employs a spatial optimization approach customized for addressing equitable emergency medical facility location problems through the p-dispersed-median problem (p-DIME). The p-DIME integrates two conflicting classes of spatial optimization problems, dispersion and median problems, aiming to identify the optimal locations for emergency medical facilities to achieve an equitable spatial distribution of emergency medical services (EMS) while effectively serving demand. To demonstrate the utility of the p-DIME model, we selected Gyeongsangbuk-do in South Korea, recognized as one of the most challenging areas for providing EMS to the elderly population (aged 65 and over). This challenge arises from the significant spatial disparity in the distribution of emergency medical facilities. The results of the model assessment gauge the spatial disparity of EMS, provide significantly enhanced solutions for a more equitable EMS distribution in terms of service coverage, and offer policy implications for future EMS location planning. In addition, to address the computational challenges posed by p-DIME’s inherent complexity, involving mixed-integer programming, this study introduces a solution technique through constraint formulations aimed at tightening the lower bounds of the problem’s solution space. The computational results confirm the effectiveness of this approach in ensuring reliable computational performance, with significant reductions in solution times, while still producing optimal solutions.

Linear and non-linear thermal system identification based on the integral of non-integer order — Application to solve inverse heat conduction linear and non-linear problems

The identification of thermal systems is an approach particularly well suited to solve inverse heat conduction problems (IHCP) in the case of complex systems where many parameters are unknown and where even the sensors must be modeled in order to obtain accurate results. In this paper, we propose significant advances in the formulation of the mathematical model structure by basing it exclusively on the non-integer order integration operator. We show that this approach is consistent with the heat diffusion process and that it allows to get rid of the step of pre-filtering the experimental signals to the sensors which was necessary when using the derivation operator. Demonstrations are made on the simulated configurations through simple analytical solutions and also a complex real configuration. I generalize the use of the non-integer integration operator in the treatment of non-linear heat conduction problems based on the Volterra series decomposition. We show in particular that the use of a generalized non-integer transfer function gives encouraging and generalizable results for a large class of related problems.

A co‐rotational virtual element method for 2D elasticity and plasticity

AbstractThe virtual element method (VEM) allows discretization of the problem domain with polygons in 2D. The polygons can have an arbitrary number of sides and can be concave or convex. These features, among others, are attractive for meshing complex geometries. VEM applied to linear elasticity problems is now well established. Nonlinear problems involving plasticity and hyperelasticity have also been explored by researchers using VEM. Clearly, techniques for extending the method to nonlinear problems are attractive. In this work, a novel first‐order consistent VEM is applied within a static co‐rotational framework. To the author's knowledge, this has not appeared before in the literature with virtual elements. The formulation allows for large displacements and large rotations in a small strain setting. For some problems avoiding the complexity of finite strains, and alternative stress measures, is warranted. Furthermore, small strain plasticity is easily incorporated. The basic method, VEM specific implementation details for co‐rotation, and representative benchmark problems are illustrated. Consequently, this research demonstrates that the co‐rotational VEM formulation successfully solves certain classes of nonlinear solid mechanics problems. The work concludes with a discussion of results for the current formulation and future research directions.

Isogeometric analysis in computation of complex-geometry flow problems with moving boundaries and interfaces

Flows with moving boundaries and interfaces (MBI) are a large class of problems that include fluid–particle and fluid–structure interactions, and in broader terms, moving solid surfaces. They also include multi-fluid flows, and as a special case of that, free-surface flows, sometimes in combination with moving solid surfaces. In some classes of MBI problems the solid surfaces could be in fast, linear or rotational relative motion or could come into contact. In almost all real-world applications, the solid surfaces would have complex geometries. All these problems are frequently encountered in engineering analysis and design, pose some of the most formidable computational challenges, and have a common core computational technology need. Bringing solution and analysis to them motivated the development of a good number of core computational methods and special methods targeting specific classes of MBI problems. This paper is an overview of some of those core and special methods. The focus is on isogeometric analysis, complex geometries, incompressible-flow Space–Time Variational Multiscale (ST-VMS) and Arbitrary Lagrangian–Eulerian VMS (ALE-VMS) methods, compressible-flow ST Streamline-Upwind/Petrov–Galerkin (ST-SUPG) and ALE-SUPG methods, and some of the special methods developed in connection with these core ST and ALE methods. The incompressible-flow ST-VMS and ALE-VMS and compressible-flow ST-SUPG and ALE-SUPG are moving-mesh methods, where the mesh moves to have mesh-resolution control near the fluid–solid interfaces, enabling high-resolution boundary-layer representation, an essential feature when the accuracy in representing the boundary layer is a priority. The computational examples presented are car and tire aerodynamics with road contact and tire deformation, ventricle–valve–aorta flow, and gas turbine flow.

Multi-level neural networks for accurate solutions of boundary-value problems

The solution to partial differential equations using deep learning approaches has shown promising results for several classes of initial and boundary-value problems. However, their ability to surpass, particularly in terms of accuracy, classical discretization methods such as the finite element methods, remains a significant challenge. Deep learning methods usually struggle to reliably decrease the error in their approximate solution. A new methodology to better control the error for deep learning methods is presented here. The main idea consists in computing an initial approximation to the problem using a simple neural network and in estimating, in an iterative manner, a correction by solving the problem for the residual error with a new network of increasing complexity. This sequential reduction of the residual of the partial differential equation allows one to decrease the solution error, which, in some cases, can be reduced to machine precision. The underlying explanation is that the method is able to capture at each level smaller scales of the solution using a new network. Numerical examples in 1D and 2D dealing with linear and non-linear problems are presented to demonstrate the effectiveness of the proposed approach.

Fourier-Gegenbauer pseudospectral method for solving time-dependent one-dimensional fractional partial differential equations with variable coefficients and periodic solutions

In this paper, we present a novel pseudospectral (PS) method for solving a new class of initial-value problems (IVPs) of time-dependent one-dimensional fractional partial differential equations (FPDEs) with variable coefficients and periodic solutions. A main ingredient of our work is the use of the recently developed periodic RL/Caputo fractional derivative (FD) operators with sliding positive fixed memory length of Bourafaet al. (2021) or their reduced forms obtained by Elgindy (2023) as the natural FD operators to accurately model FPDEs with periodic solutions. The proposed method converts the IVP into a well-conditioned linear system of equations using the PS method based on Fourier collocations and Gegenbauer quadratures. The reduced linear system has a simple special structure and can be solved accurately and rapidly by using standard linear system solvers. A rigorous study of the computational storage requirements as well as the error and convergence of the proposed method is presented. The idea and results presented in this paper are expected to be useful in the future to address more general problems involving FPDEs with periodic solutions.

On Caputo-Hadamard fractional pantograph problem of two different orders with Dirichlet boundary conditions

This manuscript aims to study the effectiveness of two different levels of fractional orders in the frame of Caputo-Hadamard (CH)-derivatives on a special type class of delay problem supplemented by Dirichlet boundary conditions. The corresponding Hadamard fractional integral equation is derived for a proposed CH-fractional pantograph system. The Banach, Schaefer, and Krasnoselskii fixed point theorems (FPTs), are used to investigate sufficient conditions of the existence and uniqueness theorems for the proposed system. Furthermore, the Green functions properties are investigated and used to discuss the Ulam-Hyers (UH) stability and its generalized by utilizing nonlinear analysis topics. Finally, three mathematical examples are provided with numerical results and figures by using Matlab software to illustrate the validity of our findings.

Mixed local and nonlocal elliptic equation with singular and critical Choquard nonlinearity

In this article, we study a class of elliptic problems involving both local and nonlocal operators with different orders and a singular nonlinearity in combination with critical Hartree type nonlinearity (see problem ( P λ ) below). Using variational methods together with the critical point theory of nonsmooth analysis, we show the existence, the regularity and the multiplicity of weak solutions with respect to the parameter λ.

Metric Optimization in Penner Coordinates

Many parametrization and mapping-related problems in geometry processing can be viewed as metric optimization problems, i.e., computing a metric minimizing a functional and satisfying a set of constraints, such as flatness. Penner coordinates are global coordinates on the space of metrics on meshes with a fixed vertex set and topology, but varying connectivity, making it homeomorphic to the Euclidean space of dimension equal to the number of edges in the mesh, without any additional constraints imposed. These coordinates play an important role in the theory of discrete conformal maps, enabling recent development of highly robust algorithms with convergence and solution existence guarantees for computing such maps. We demonstrate how Penner coordinates can be used to solve a general class of optimization problems involving metrics, including optimization and interpolation, while retaining the key solution existence guarantees available for discrete conformal maps.

Численное моделирование цунами оползневого типа

The article proposes mathematical models and stable numerical algorithms for solving problems of deformation and flow of water-saturated porous materials. This class of problems arises in many engineering and applied fields, from field development to biomechanics. Of particular interest are the problems of the collapse of the banks of rivers, lakes and bays of the seas due to their catastrophic impact on possible consequences, for example, blocking the riverbed, the occurrence of a tsunami on the water surface of a reservoir, etc. For the mathematical description of the stress-strain state of saturated porous media, the Biot theory of three-dimensional consolidation is used, supplemented by the kinetics of damage accumulation in the medium and the Drucker-Prager yield conditions. After the formation of the main yield surface (or crack) in the material at the moment of the beginning of the collapse, its rheological properties are presented in the form of a Bingham model with a mathematical description similar to the Navier-Stokes equations with effective viscosity. A characteristic hydrodynamic feature of the problem under consideration is the presence of a free surface. Numerical modeling of the considered problems is carried out by the finite element method. The evolution of the free surface and the occurrence of a tsunami generated by a landslide are investigated.

Application of two classes of inverse problems for optimization and management of the natural hydrocarbon field development process

The article considers two classes of inverse problems in the modeling of hydrocarbon field development: optimization and adaptation of the development history. They jointly form the basis for field development management in a closed loop, based on a constantly updated 3D geological and technological model. The problems of model adaptation (identification) arise due to the point-by-point measurements and the inaccuracy of determining the initial data on porosity and permeability of the host rocks and other parameters of the deposit when building the 3D model. Such problems are regarded as the inverse ones, because the consequence – field measurements – is used to determine their cause. The problems under consideration are related to the identification of both porosity and permeability of the reservoir and other key parameters that determine the dynamics of development. Geologically consistent formulation ensures that the adapted model retains the original geological principles of the distribution of reservoir properties. The second class of problems under consideration is the problems of optimization (regulation) of development. One of the sought-after formulations is related to the optimal redistribution of a given target level of production from the field to all production wells in time. In this case, the technological limitations of surface equipment are taken into account. The maximized criterion reflects the economic effect of the asset development. In this article, without going into details of mathematical realization, the authors consider practical examples of solving these problems using full-scale 3D multiphase filtration models of the real fields. The algorithms based on the current methods of optimal control theory are used, which were developed and implemented by the researchers of Oil and Gas Research Institute of the Russian Academy of Sciences in the SimMatch software package and confirmed their effectiveness on numerous synthetic and real examples. Application and further development of these methods within the framework of a single loop, taking into account continuous information from the entire set of sensors, will significantly increase the efficiency of intelligent wells and field control systems.