Representation Of Solution Research Articles

Integral Representations of Solution in the Problem on Skew Incidence of a Surface Wave on the Straight Shoreline Water Wedge

In the linear approximation of the surface gravitational waves of small amplitude a classical model problem about the incursion of a surface wave under some angle on the shoreline is solved. The problem is formulated for the harmonic potential of velocity of the fluid in the 3D water wedge with the Robin-Steklov boundary condition on the free surface and with the no-flow condition along the normal on the bed of the water domain. Some critical comments about a known in the literature solution having a “non-physical” singularity of the logarithmic type on the coastal line are given. The asymptotics with respect to distance from the shoreline of the obtained solution, bounded on the edge, is constructed. The reflection coefficient of the wave reflected from the shoreline is calculated.

A Q-learning-based multi-population algorithm for multi-objective distributed heterogeneous assembly no-idle flowshop scheduling with batch delivery

Distributed heterogeneous factory environments have become the mainstream in real-world manufacturing enterprises. Scheduling the assembly no-idle flowshops in such distributed heterogeneous environments is significant for practitioners. This problem takes into account the heterogeneity between different factories and the batch delivery process. To minimize inventory and tardiness costs of this new problem, a novel Q-learning-based multi-population multi-objective evolutionary algorithm is proposed. The algorithm incorporates five problem-specific properties to design the solution representation and calculate objectives. To obtain high-quality initial dual populations, a collaborative initialization technique combining four NEH heuristics and a completely random approach is proposed. To enhance algorithm’s search efficiency, a Q-learning-based selection method is developed to coordinate four job-related and four product-related operators. The dual-cooperation strategy is introduced to improve the global search capability. Experimental results demonstrate the effectiveness of all the designed components, indicating that the proposed algorithm outperforms five existing algorithms. It is well-suited for addressing real-world distributed heterogeneous scheduling scenarios by the proposed algorithm.

A tri-chromosome-based evolutionary algorithm for energy-efficient workflow scheduling in clouds

Cloud computing is increasingly attracting workflow applications, where workflows need to satisfy execution deadlines and energy consumption is to be minimized. So far, numerous studies have adopted evolutionary algorithms to optimize the energy consumption of workflow execution. Dynamic voltage and frequency scaling (DVFS) has been widely employed to save energy on computing devices running workflow tasks. However, most existing evolutionary algorithms focus on evolving task execution order or mapping from tasks to resources, while neglecting the evolution of task runtime to leverage the dynamic voltage and frequency scaling (DVFS) technology for further energy saving. To compensate for that deficiency, this paper designs a tri-chromosome-based evolutionary algorithm, namely TCEA, to evolve three types of decision vectors (i.e., task order, task and resource mapping, and task runtime) simultaneously using three problem-specific mechanisms. Firstly, we construct a search space by using the tasks’ minimum and optimal runtime, and propose a solution representation mechanism to simplify the decision vector for task runtime between 0 and 1. Secondly, we design a deadline constraint handling mechanism to distribute those durations exceeding the deadline to each task based on their extension of the minimum runtime. Thirdly, we exploit the workflow structure to cluster decision variables without direct constraints into the same group. During each iteration, only the order of tasks within a group evolves to avoid precedence constraints, thus performing searches within the feasible space. At last, we conduct comparison experiments on five types of real-world workflows with 30 to 1000 tasks. The energy consumed by TCEA is much less than those consumed by the state-of-the-art workflow scheduling algorithms, demonstrating the superior performance of TCEA in energy saving.

Packing squares independently

Given a set of squares and a strip with bounded width and infinite height, we consider a square strip packaging problem, which we call the square independent packing problem (SIPP), to minimize the strip height so that all the squares are packed into independent cells separated by horizontal and vertical partitions. For the SIPP, we first investigate efficient solution representations and propose a compact representation that reduces the search space from Ω(n!) to O(2n), with n the number of given squares, while guaranteeing that there exists a solution representation that corresponds to an optimal solution. Based on the solution representation, we show that the problem is NP-hard. To solve the SIPP, we propose a dynamic programming method that can be extended to a fully polynomial-time approximation scheme (FPTAS). We also propose three mathematical programming formulations based on different solution representations and confirm their performance through computational experiments with a mathematical programming solver. Finally, we discuss several extensions that are relevant to practical applications.

Stochastic representation for solutions of a system of coupled HJB-Isaacs equations with integral–differential operators

In this paper, we focus on the stochastic representation of a system of coupled Hamilton–Jacobi–Bellman–Isaacs (HJB–Isaacs (HJBI), for short) equations which is in fact a system of coupled Isaacs’ type integral-partial differential equation. For this, we introduce an associated zero-sum stochastic differential game, where the state process is described by a classical stochastic differential equation (SDE, for short) with jumps, and the cost functional of recursive type is defined by a new type of backward stochastic differential equation (BSDE, for short) with two Poisson random measures, whose wellposedness and a prior estimate as well as the comparison theorem are investigated for the first time. One of the Poisson random measures μ appearing in the SDE and the BSDE stems from the integral term of the HJBI equations; the other random measure in BSDE is introduced to link the coupling factor of the HJBI equations. We show through an extension of the dynamic programming principle that the lower value function of this game problem is the viscosity solution of the system of our coupled HJBI equations. The uniqueness of the viscosity solution is also obtained in a space of continuous functions satisfying certain growth condition. In addition, also the upper value function of the game is shown to be the solution of the associated system of coupled Isaacs’ type of integral-partial differential equations. As a byproduct, we obtain the existence of the value for the game problem under the well-known Isaacs’ condition.

Generalized Brenier Principle and the Closure Problem of Landgren–Monin–Novikov Hierarchy for Vorticity Field

Brenier’s concept – a representation of solutions to the equations of ideal incompressible fluids in terms of probability measures on the set of Lagrangian trajectories in the case of their stochasticity, is a generalization of Arnold’s principle of least action of finding smooth solutions of Euler’s equations. In this work, the variational generalized Brenier principle (Brenier, J. Am. Math. Soc. 1989) is used to close the infinite chain of Landgren–Monin–Novikov equations for the n-point probability density functions fn of the vortex field of two-dimensional turbulence. In addition, within the framework of the statistical approach, an approximation of the variational problem with conditions at the ends posed by Shnirelman (Mat. Sat. 1985) for the Euler equation is proposed.

Exploring the inverse line-source scattering problem in dielectric cylinders with deep neural networks

Abstract This study presents a novel approach utilizing deep neural networks to
address the inverse line-source scattering problem in dielectric cylinders. By employing
Multi-layer Perceptron models, we intend to identify the number, positions, and
strengths of hidden internal sources. This is performed by using single-frequency
phased data, from limited measurements of real electric and real magnetic surface fields.
Training data are generated by solving corresponding direct problems, using an exact
solution representation. Through extended numerical experiments, we demonstrate
the efficiency of our approach, including scenarios involving noise, reduced sample
sizes, and fewer measurements. Additionally, we examine the empirical scaling laws
governing model performance and conduct a local analysis to explore how our neural
networks handle the inherent ill-posedness of the considered inverse problems.

A two-phase adaptive large neighborhood search algorithm for the electric location routing problem with range anxiety

In response to environmental pollution and global warming, electric vehicles (EVs) have been widely applied in supply chain, and lots of researches have been focused on the electric location routing problem (ELRP) to optimize the location of battery-swapping stations (BSSs). However, few studies have simultaneously considered the EV’s collaborative behavior and range anxiety in ELRP. Therefore, this study focuses on formulating an ELRP with battery-swapping stations considering both collaborative behavior and range anxiety. The model takes into account the collaboration between the supply and demand sides to share supply chain resources effectively. The detour probability function is employed to handle the uncertainty caused by drivers’ range anxiety during the trip, which can be evaluated through a proposed range anxiety function. In addition, a new two-dimensional matrix-based solution representation is proposed to explore ELRP optimal solutions intuitively and efficiently. To solve the proposed model effectively, a two-phase adaptive large neighborhood search (TALNS) algorithm that integrates the adaptive large neighborhood search (ALNS) algorithm and the extended binary particle swarm optimization (EBPSO) algorithm is proposed. In the EBPSO algorithm, a new position update mechanism and local search strategy are used to strengthen the local search ability. In the ALNS algorithm, multiple destroy and repair operators are developed for the proposed model. Further, to validate the effectiveness and performance of the proposed algorithm, comparison experiments with the Gurobi optimizer and other baseline heuristic algorithms are conducted.

Sinc method in spectrum completion and inverse Sturm–Liouville problems

Cardinal series representations for solutions of the Sturm–Liouville equation with a complex‐valued potential are obtained, by using the corresponding transmutation operator. Consequently, partial sums of the series approximate the solutions uniformly with respect to in any strip of the complex plane. This property of the obtained series representations leads to their applications in a variety of spectral problems. In particular, we show their applicability to the spectrum completion problem, consisting in computing large sets of the eigenvalues from a reduced finite set of known eigenvalues, without any information on the potential as well as on the constants from boundary conditions. Among other applications this leads to an efficient numerical method for computing a Weyl function from two finite sets of the eigenvalues. This possibility is explored in the present work and illustrated by numerical tests. Finally, based on the cardinal series representations obtained, we develop a method for the numerical solution of the inverse two‐spectra Sturm–Liouville problem and show its numerical efficiency.

Correctness of the mixed problem for degenerate multidimensional elliptic-parabolic equations

The initial-boundary value problem (Dirichlet problem) for general elliptic-parabolic equations of second order was first posed by G. Fichera. Further investigation of this problem was carried out in the monograph by O.A. Oleinik and E.V. Radkevich and the works by V.N. Vragov. In these works, the authors examined mixed problems for degenerate multidimensional elliptic equations. The articles by S.A. Aldashev focused on the correctness (in the sense of uniqueness of solvability) of the Dirichlet problem in a cylindrical domain for multidimensional elliptic-parabolic equations. A mixed problem for these equations has not been studied. In this paper, the authors demonstrate the uniqueness of solvability and obtain an explicit representation of the classical solution of the mixed problem for degenerate multidimensional elliptic-parabolic equations. The proposed method allows reducing the problem under study to a mixed problem for a degenerate multidimensional elliptic equation examined by S.A. Aldashev.

Representation of solutions of M-spectral problem for diffusion operator

In this article, we offer the M-Sturm Liouville problem for the diffusion operator depending on initial condition. We obtain the representation of the solution of the M-Sturm Liouville problem for diffusion operator through the M-Laplace transform. The purpose of this article is to advantage demonstrate a more generalized version of the representation of the solution of the Sturm Liouville problem for diffusion operator in classical analysis.

Trace formulas revisited and a new representation of KdV solutions with short-range initial data

Trace formulas revisited and a new representation of KdV solutions with short-range initial data

Green’s Tensor of BIOT’s Equations by Stationary Oscillations

Various mathematical models of deformable solids mechanics are used in the study of seismic processes in the earth's crust. The processes of waves propagation are most studied in elastic media. But these models do not take into account many real properties of the ambient array. These are, for example, the presence of groundwater, which complicates the construction and operation of surface and underground structures, affect the magnitude and distribution of stresses. Models, which take into account the water saturation form the earth's crust structures, the presence of gas bubbles, etc., are multi-component medium. A variety of multicomponent media, the complexity of the processes associated with their deformation, lead to a large difference in the methods of analysis and modelling used in the solution of wave problems. In this paper the processes of wave propagation in a two-component Biot medium under the action of periodic forces of various forms are considered. Using the Fourier transform of generalized functions, fundamental solutions are constructed - the Green's tensor of the Biot equations and its properties are studied. This tensor describes the process of propagation of harmonic waves of a fixed frequency in spaces of dimension N = 1,2,3 under the action of power sources concentrated at the origin of coordinates, described by a singular delta function. On its basis, generalized solutions of these equations are constructed under the action of various sources of periodic disturbances, which are described by both regular and singular generalized functions. For regularly acting forces, integral representations of solutions are given, which can be used to calculate the stress-strain state of a porous water-saturated medium

On representation of solutions to the heat equation

On representation of solutions to the heat equation

Non-Local Problems for the Fractional Order Diffusion Equation and the Degenerate Hyperbolic Equation

This research explores nonlocal problems associated with fractional diffusion equations and degenerate hyperbolic equations featuring singular coefficients in their lower-order terms. The uniqueness of the solution is established using the energy integral method, while the existence of the solution is equivalently reduced to solving Volterra integral equations of the second kind and a fractional differential equation. The study focuses on a mixed domain where the parabolic section aligns with the upper half-plane, and the hyperbolic section is bounded by two characteristics of the equation under consideration and a segment of the x-axis. By utilizing the solution representation of the fractional-order diffusion equation, a primary functional relationship is derived between the traces of the sought function on the x-axis segment from the parabolic part of the mixed domain. An explicit solution form for the modified Cauchy problem in the hyperbolic section of the mixed domain is presented. This solution, combined with the problem’s boundary condition, yields a fundamental functional relationship between the traces of the unknown function, mapped to the interval of the equation’s degeneration line. Through the conjugation condition of the problem, an equation with fractional derivatives is obtained by eliminating one unknown function from two functional relationships. The solution to this equation is explicitly formulated. For a specific solution of the proposed problem, visualizations are provided for various orders of the fractional derivative. The analysis demonstrates that the derivative order influences both the intensity of the diffusion (or subdiffusion) process and the shape of the wave front.

Hybrid loading situation vehicle routing problem in the context of agricultural harvesting: A reconstructed MOEA/D with parallel populations

With the increasing level of agricultural automation, the combination of agriculture and intelligent vehicle technology is propelling the development of smart agriculture. Although this technology has already been applied widely for various agricultural production tasks, inefficient vehicle scheduling still hasn't been resolved satisfactorily. Oriented towards agricultural harvesting scenarios, a hybrid loading situation vehicle routing problem (HLSVRP) model is proposed to minimize total energy consumption and maximum completion time. A reconstructed multi-objective evolutionary algorithm based on decomposition (R-MOEA/D) is developed to solve the problem. Eight solution representations tailored specifically to the problem are introduced by R-MOEA/D, allowing an extensive exploration of the solution space. A modified Clarke & Wright (MCW) heuristic is designed to generate a high-quality initial population. A novel problem-specific parallel population updating mechanism based on the four crossover and two mutation combinations is also provided to improve the exploration ability. A collaborative search strategy is employed to facilitate cooperation among parallel populations. Finally, a series of comparative experiments conducted on various task scales and vehicle scales verify the effectiveness of the proposed algorithmic components and the exceptional performance for solving HLSVRP.

Variations of heat equation on the half‐line via the Fokas method

In this review paper, we discuss some of our recent results concerning the rigorous analysis of initial boundary value problems (IBVPs) and newly discovered effects for certain evolution partial differential equations (PDEs). These equations arise in the applied sciences as models of phenomena and processes pertaining, for example, to continuum mechanics, heat‐mass transfer, solid–fluid dynamics, electron physics and radiation, chemical and petroleum engineering, and nanotechnology. The mathematical problems we address include certain well‐known classical variations of the traditional heat (diffusion) equation, including (i) the Sobolev–Barenblatt pseudoparabolic PDE (or modified heat or second‐order fluid equation), (ii) a fourth‐order heat equation and the associated Cahn–Hilliard (or Kuramoto–Sivashinsky) model, and (iii) the Rubinshtein–Aifantis double‐diffusion system. Our work is based on the synergy of (i) the celebrated Fokas unified transform method (UTM) and (ii) a new approach to the rigorous analysis of this method recently introduced by one of the authors. In recent works, we considered forced versions of the aforementioned PDEs posed in a spatiotemporal quarter‐plane with arbitrary, fully non‐homogeneous initial and boundary data, and we derived formally effective solution representations, for the first time in the history of the models, justifying a posteriori their validity. This included the reconstruction of the prescribed initial and boundary conditions, which required careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula was utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. Importantly, this analysis is indispensable for proving (non)uniqueness of solution. These works extend previous investigations. The usefulness of our closed‐form solutions will be demonstrated by studying their long‐time asymptotics. Specifically, we will briefly review some asymptotic results about Barenblatt's equation.

The set team orienteering problem

We introduce the Set Team Orienteering Problem (STOP), a generalised variant of the Set Orienteering Problem (SOP), in which customer locations are split into multiple clusters (or groups). Each cluster is associated with a profit that can be gained only if at least one customer from the cluster is visited. There is a fleet of homogeneous vehicles at a depot, and each vehicle has a limited travel time. The goal of the STOP is to find a set of feasible vehicle routes to collect the maximum profit. We first formulate the problem as a Mixed Integer Linear Programming (MILP) to mathematically describe it. A branch-and-price (B&P) algorithm is then developed to solve the problem to optimality. To deal with large instances, we propose a Large Neighbourhood Search (LNS), which relies on problem-tailored solution representation, removal, and insertion operators. Multiple experiments on newly generated instances confirm the performance of our approaches. Remarkably, when tested on the SOP using benchmarks available in the literature, our B&P method achieves optimality in 61.9% of these instances. This is the first time such a large number of SOP instances are solved to optimality. Our LNS outperforms existing algorithms proposed to solve the SOP in terms of solution quality. Out of 612 considered instances, it improves 40 best-known solutions.

Machine Learning based Algorithm Selection and Genetic Algorithms for serial-batch scheduling

Whenever combinatorial optimization problems cannot be solved by exact solution methods in reasonable time, tailor-made algorithms (heuristics, meta-heuristics) are developed. Often, these heuristics exploit structural properties and perform well on selected subsets of the problem space. For example, this is how the two best-known construction heuristics solve the scheduling problem investigated in this study (i.e., the scheduling of parallel serial-batch processing machines with incompatible job families, restricted batch capacities, arbitrary batch capacity demands, and sequence-dependent setup times). However, when the properties change, the performance of one algorithm might decrease, and another algorithm might have been the better choice. To resolve this issue, we propose using Machine Learning to exploit the strengths of different algorithms and to select the probably best-performing algorithm for each problem instance individually. To that, we investigate a variety of methods from the “learning-to-rank” literature and propose several adaptations. Furthermore, because there is no algorithm for the considered scheduling problem that is capable to explore the entire solution space, we developed two Genetic Algorithms for the improvement of initial solutions computed by the selected algorithms. Here, we put special emphasis on ensuring that the solution representation (encoding) reflects the entire solution space and that the operators (e.g., for recombination and mutation) are appropriate to explore and exploit this space completely. Our computational experiments show an average increase of 39.19% in solution quality.

Ulam–Hyers–Rassias Mittag-Leffler stability of ϖ–fractional partial differential equations

This paper offers a comprehensive analysis of solution representations for ϖ-fractional partial differential equations, specifically focusing on the linear case of the Darboux problem. We exhibit a representation of the solutions for the Darboux problem of ϖ-fractional partial differential equations in the linear case in the space of continuous functions. Through the application of the generalized Gronwall inequality, we establish the Ulam–Hyers–Rassias Mittag–Leffler stability in the space of continuous functions. Three numerical examples are presented to show the effectiveness and the applicability of our results.