Representation Of Solution Research Articles

Physical significance and periodic solutions of the high-order good Jaulent-Miodek model in fluid dynamics

<p>Using Whitham modulation theory, this paper examined periodic solutions and the problem of discontinuous initial values for the higher-order good Jaulent-Miodek (JM) equation. The physical significance of the JM equations was discussed by considering the reduction of Euler's equation. Next, the zero- and one-phase periodic solutions of the JM equation, along with the associated Whitham equations, were derived. The analysis included the degeneration of the one-phase periodic solution and the genus-one Whitham equation by examining the limits of the modulus $ m $ of the Jacobi elliptic functions. Additionally, analytical and graphical representations of rarefaction wave solutions and periodic wave patterns were provided, and a solution for discontinuous initial values in the JM equation was presented. The results of this study offer a theoretical foundation for analyzing discontinuous initial values in nonlinear dispersion equations.</p>

Solutions and local stability of the Jacobsthal system of difference equations

<abstract><p>We presented a comprehensive theory for deriving closed-form expressions and representations of the general solutions for a specific case of systems involving Riccati difference equations of order $ m+1 $, as discussed in the literature. However, our focus was on coefficients dependent on the Jacobsthal sequence. Importantly, this system of difference equations represents a natural extension of the corresponding one-dimensional difference equation, uniquely characterized by its theoretical solvability in a closed form. Our primary objective was to demonstrate a direct linkage between the solutions of this system and Jacobsthal and Lucas-Jacobsthal numbers. The system's capacity for theoretical solvability in a closed form enhances its distinctiveness and potential applications. To accomplish this, we detailed offer theoretical explanations and proofs, establishing the relationship between the solutions and the Jacobsthal sequence. Subsequently, our exploration addressed key aspects of the Jacobsthal system, placing particular emphasis on the local stability of positive solutions. Additionally, we employed mathematical software to validate the theoretical results of this novel system in our research.</p></abstract>

On the Complexiton Solutions to the Conformable Fractional Hirota–Satsuma–Ito Equation

This study analyzes the Hirota–Satsuma–Ito equation, which discusses the propagation of unidirectional shallow‐water waves and the interactions between two long waves with different dispersion forms. For the proposed equation, the sine‐Gordon expansion method has been considered. This method is derived from the sine‐Gordon equation. Different types of solutions, namely, bright, periodic, and dark‐bright soliton solutions, are derived. When these solutions are compared to other previously published research, to our knowledge, the study concludes that they are innovative, and this method was not applied to this equation. The validation of the obtained solutions is verified and plotted as three‐dimensional figures to comprehend physical phenomena. With the proper parameter values, distinct graphs are created to convey the physical representation of specific solutions. The results of this paper show that the method effectively improves a system’s nonlinear dynamical behavior. This study will be useful to a wide range of engineers who specialize in engineering models. The findings show that the computational approach is successful, simple, and even applicable to complex systems.

Numerical Approximation of Spatially Loaded Time-Fractional Diffusion Equation

This article is dedicated to numerically solving the spatially loaded time-fractional diffusion equation with initial and Dirichlet-type boundary conditions. We apply finite difference approximation to the considered problem and use the well-known L1 method to approximate the Caputo fractional derivative. The numerical approximation of the problem yields a loaded implicit difference scheme. To obtain the solution, we employ a special parametric representation of solutions of auxiliary linear systems using the superposition property. In conclusion, we showcase several numerical tests, validate the outcomes, and observe the convergence of errors.

Bi-Trajectory Hybrid Search to Solve Bottleneck-Minimized Colored Traveling Salesman Problems

A bottleneck-minimized colored traveling salesman problem is an important variant of colored traveling salesman problems. It is useful in handling the planning problems with partially overlapped workspace such as the scheduling transportation resources for timely delivery of goods. In this work, we propose an efficient bi-trajectory hybrid search method for it. The proposed method integrates a route-based crossover operator to generate promising offspring solutions, a multi-neighborhood simulated annealing to perform local optimization, and a stagnation-detect-escape mechanism to help the search escape from local optima. We also propose a bidirectional adjacency solution representation method to encode a solution, which extends the traditional adjacency representation method by additionally employing an array to represent its reverse counterpart. Extensive evaluations on two sets of 58 widely used benchmark instances demonstrate that the proposed method significantly outperforms state-of-the-art algorithms. Investigations on key algorithm modules are performed to confirm the novelty and effectiveness of the proposed ideas and strategies. Finally, we verify the generalization of the proposed method via its use to solve a colored traveling salesman problem with its aim to balance workload among salesmen. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This work is motivated by the problems of scheduling transportation resources for timely delivery of goods. It proposes an effective bi-trajectory hybrid search to tackle bottleneck-minimized colored traveling salesman problem. The proposed approach performs bi-trajectory hybrid evolutionary search by maintaining only two individuals during the whole search process. Extensive numerical experiments and comparisons show that our proposed algorithm significantly outperforms state-of-the-art algorithms and can help decision-makers in designing best-routing solutions.

Examination of Sturm-Liouville problem with proportional derivative in control theory

The current study is intended to provide a comprehensive application of Sturm-Liouville (S-L) problem by benefiting from the proportional derivative which is a crucial mathematical tool in control theory. This advantageous derivative, which has been presented to the literature with an interesting approach and a strong theoretical background, is defined by two tuning parameters in control theory and a proportional-derivative controller. Accordingly, this research is presented mainly to introduce the beneficial properties of the proportional derivative for analyzing the S-L initial value problem. In addition, we reach a novel representation of solutions for the S-L problem having an importing place in physics, supported by various graphs including different values of arbitrary order and eigenvalues under a specific potential function.

Solution Representations for Poisson’s Equation, Martingale Structure, and the Markov Chain Central Limit Theorem

The solution of Poisson’s equation plays a key role in constructing the martingale through which sums of Markov correlated random variables can be analyzed. In this paper, we study three different representations for the solution for countable state space irreducible Markov chains, two based on entry time expectations, and the other based on a potential kernel. Our consideration of null recurrent chains allows us to extend our theory to positive recurrent nonexplosive Markov jump processes. We also develop the martingale structure induced by these solutions to Poisson’s equation, under minimal conditions, and establish verifiable Lyapunov conditions to support our theory. Finally, we provide a central limit theorem for Markov dependent sums, under conditions weaker than have previously appeared in the literature.

A two-stage cross-neighborhood search algorithm bridging different solution representation spaces for solving the hybrid flow shop scheduling problem

In this paper, we address the classical hybrid flow shop scheduling problem(HFSP) to minimize makespan. This problem is known to be NP-hard and widely exists in many industrial systems such as electronics, paper, textile, and manufacturing industries. The existing work has shown that searching in multiple solution representation spaces is beneficial to improving the performance of the algorithm. However, the connection of the solutions in different solution representation spaces was ignored. To address this issue, this paper proposes a two-stage cross-neighborhood search algorithm (TSCNSA) for solving the hybrid flow shop scheduling problem. TSCNSA contains two individuals, each of which is responsible for searching in a solution representation space. The algorithm is divided into two stages: exploration and exploitation. In the exploration stage, the two individuals search in the solution representation space generated by the job-based encoding method and two decoding methods respectively to locate the potential area. In the exploitation stage, the two individuals search in the solution representation space generated by the operation-based encoding and two decoding methods respectively. The information obtained in the exploration stage will be transferred in the exploitation stage. Besides, each individual searches in its neighborhood and then generates an expert solution in the other solution representation space to guide the search for the other individual. Finally, the performance of the TSCNSA is proved by testing the well-known 586 benchmark instances. 151 new upper bounds are obtained, including 144 instances in Jose's benchmark instances, 2 instances in Liao's benchmark instances, and 5 instances in Carlier's benchmark instances.

Neumann series of Bessel functions for inverse coefficient problems

Consider the Sturm–Liouville equation with a real‐valued potential . Let be its solution satisfying certain initial conditions for a number of , where , and are some complex numbers. Denote , where . We propose a method for solving the inverse problem of the approximate recovery of the potential and number from the following data . In general, the problem is ill‐posed; however, it finds numerous practical applications. Such inverse problems as the recovery of the potential from a Weyl function or the inverse two‐spectra Sturm–Liouville problem are its special cases. Moreover, the inverse problem of determining the shape of a human vocal tract also reduces to the considered inverse problem. The proposed method is based on special Neumann series of Bessel functions representations for solutions of Sturm–Liouville equations. With their aid the problem is reduced to the classical inverse Sturm–Liouville problem of recovering from two spectra, which is solved again with the help of the same representations. The overall approach leads to an efficient numerical algorithm for solving the inverse problem. Its numerical efficiency is illustrated by several examples.

A self-learning knowledge-based MOEA/D for distributed heterogeneous assembly permutation flowshop scheduling with batch delivery

With the requirements of just-in-time and make-to-order, it is necessary to consider the batch delivery in distributed heterogeneous assembly flow shop scheduling as well as the minimization of inventory and tardiness costs. Hence, this work proposes a new mixed integer linear programming model and a self-learning knowledge-based multi-objective evolutionary algorithm based on decomposition. In this algorithm, 4 problem-specific properties are extracted and embedded in the designed solution representation, forward-reverse objective calculation and local search operators. A new hybrid initialization with six knowledge-based NEH heuristics and a random generation is developed to generate better initial solutions. A self-learning selection strategy is developed to coordinate the above knowledge-based local search operators. The experimental study, based on 900 small and 540 large instances, demonstrates that the designed components such as hybrid initialization and self-learning selection strategy are effective, and the proposed algorithm is superior to 6 existing algorithms in the literature in solving this new problem.

On divergence type linear and quasilinear equations in the complex plane

This is a survay of our recent results concerning divergence type linear and quasilinear equations in the complex plane. It contains a number of existence, representation and regularity theorems for solutions of fundamental boundary value problems for such equations. The degeneration case of uniform ellipticity is also covered by means of BMO, VMO, Calderon-Zigmund, Lehto and Orlicz technic.

Improved ant colony algorithm for the mixed-model parallel two-sided assembly lines balancing problem

ABSTRACT In response to the increasing demand for individualized products in the ‘small-lot, multi-variety’ market, there is an urgent need to enhance the efficiency and intelligence level of assembly lines. To address this, a novel solution method for the mixed-model parallel two-sided assembly lines balancing problem (MMPTSALBP) is proposed. The first step is to define the type and layout of the parallel two-sided assembly lines. Next, a mathematical model is constructed with the objective of minimizing the number of workstations required for the MMPTSALBP, proposing an improved ant colony algorithm that utilizes double string representation for the initial solution encoding method and a double pheromone matrix update to solve the model. Its efficiency is tested on benchmark datasets according to some performance measures and classifications. A study validates its effectiveness on 25 classical arithmetic cases by comparison with the solutions of an heuristic algorithm and an artificial fish swarm algorithm that had been reported to perform well.

Numerical computational approach for 6th order boundary value problems

This study introduces numerical computational methods that employ fourth-kind Chebyshev polynomials as basis functions to solve sixth-order boundary value problems. The approach transforms the BVPs into a system of linear algebraic equations, expressed as unknown Chebyshev coefficients, which are subsequently solved through matrix inversion. Numerical experiments were conducted to validate the accuracy and efficiency of the technique, demonstrating its simplicity and superiority over existing solutions. The graphical representation of the method's solution is also presented.

Theoretical analysis of a SIRD model with constant amount of alive population and COVID-19 applications

This study deals with a theoretical analysis of the integrability properties and analytical solutions of an initial-value problem for a SIRD model with a constant amount of alive population (SIRD-CAAP), which is in the form of a fourth-dimensional and first-order coupled system of nonlinear ordinary differential equations, by using the partial Hamiltonian method. This research represents a COVID-19 study as a real-world problem by using the analytical results obtained in the study. The first integrals and the associated exact analytical solutions are investigated of the model with respect to algebraic relations among the model parameters. Then, for both cases, the dynamical behaviors of the model based on the analytical solutions are analyzed, and the graphical representations of the closed-form solutions are demonstrated and compared. In addition, it is shown that the SIRD-CAAP model can be decoupled based on its first integrals for all cases from the mathematical perspective point of view. Furthermore, the periodicity properties and the classification of the regimes of the solutions with respect to the model parameter constraints are discussed. Finally, the COVID-19 applications are given using the data related to the different countries.

A comparative analysis of fractional model of second grade fluid subject to exponential heating: application of novel hybrid fractional derivative operator

In this article, a new approach to study the fractionalized second grade fluid flow is described by the different fractional derivative operators near an exponentially accelerated vertical plate together with exponentially variable velocity, energy and mass diffusion through a porous media is critically examined. The phenomenon has been expressed in terms of partial differential equations, then transformed the governing equations in non-dimentional form. For the sake of better rheology of second grade fluid, developed a fractional model by applying the new definition of Constant Proportional-Caputo hybrid derivative (CPC), Atangana Baleanu in Caputo sense (ABC) and Caputo Fabrizio (CF) fractional derivative operators that describe the generalized memory effects. For seeking exact solutions in terms of Mittag-Leffler and G-functions for velocity, temperature and concentration equations, Laplace integral transformation technique is applied. For physical significance of various system parameters on fluid velocity, concentration and temperature distributions are demonstrated through various graphs by using graphical software. Furthermore, for being validated the acquired solutions, accomplished a comparative analysis with some published work. It is also analyzed that for exponential heating and non-uniform velocity conditions, the CPC fractional operator is the finest fractional model to describe the memory effect of velocity, energy and concentration profile. Moreover, the graphical representations of the analytical solutions illustrated the main results of the present work. Also, in the literature, it is observed that to derived analytical results from fractional fluid models developed by the various fractional operators, is difficult and this article contributing to answer the open problem of obtaining analytical solutions the fractionalized fluid models.

Solution to delayed linear discrete system with constant coefficients and second‐order differences and application to iterative learning control

SummaryIn this article, we give an alternative representation and prior estimation of solution to delayed linear discrete systems with a second‐order difference with the help of the notation of delayed discrete matrix functions via their norm estimations. As an application, this solution is used to study iterative learning control under the suitable updating laws and sufficient conditions to guarantee that asymptotic convergence tracking results are presented. Finally, two numerical examples are given to verify the theoretical results.

Representation of solutions and finite‐time stability for fractional delay oscillation difference equations

In this article, an explicit solution of the homogeneous fractional delay oscillation difference equation of order is given by constructing discrete sine‐ and cosine‐type delayed Mittag‐Leffler functions. Then, the discrete Laplace transform technique as an effective tool for solving the nonhomogeneous term, which is utilized to explore the solution of corresponding nonhomogeneous equation. Next, finite‐time stability of the homogeneous equation is studied based on the representation of the solution. Furthermore, we show a numerical example to elaborate the correctness of stability theory. Finally, an exact solution for the nonhomogeneous fractional difference equation with is further presented via using the discrete two‐parameter delayed Mittag‐Leffler function.

Hadamard integrator for time-dependent wave equations: Lagrangian formulation via ray tracing

Starting from the time-domain Kirchhoff-Huygens representation of wave solutions, we propose a novel Hadamard integrator for the self-adjoint time-dependent wave equation in an inhomogeneous medium. First, we create a new asymptotic series based on the Gelfand-Shilov function, dubbed Hadamard's ansatz, to approximate the Green's function of the time-dependent wave equation. Accordingly, the governing equations and related initializations for the eikonal and Hadamard coefficients are derived using the properties of the Gelfand-Shilov generalized function. Second, incorporating the leading term of Hadamard's ansatz into the Kirchhoff-Huygens representation, we develop an original Hadamard integrator for the Cauchy problem of the time-dependent wave equation and derive the corresponding Lagrangian formulation in geodesic polar coordinates. Third, to construct the Hadamard integrator in the Lagrangian formulation efficiently, we use a short-time ray tracing method to obtain equal-time wavefront locations accurately, and we further develop fast algorithms to compute Chebyshev-polynomial based low-rank representations of both wavefront locations and variants of Hadamard coefficients. Fourth, equipped with these low-rank representations, we apply the Hadamard integrator to efficiently solve time-dependent wave equations with highly oscillatory initial conditions, where the time step size is independent of the initial conditions. By judiciously choosing the medium-dependent time step, our new Hadamard integrator can propagate wave field beyond caustics implicitly and advance spatially overturning waves in time naturally. Moreover, since the integrator is independent of initial conditions, the Hadamard integrator can be applied to many different initial conditions once it is constructed. Both two-dimensional and three-dimensional numerical examples illustrate the accuracy and performance of the proposed method.

The Airy equation with nonlocal conditions

AbstractWe study a third‐order dispersive linear evolution equation on the finite interval subject to an initial condition and inhomogeneous boundary conditions but, in place of one of the three boundary conditions that would typically be imposed, we use a nonlocal condition, which specifies a weighted integral of the solution over the spatial interval. Via adaptations of the Fokas transform method (or unified transform method), we obtain a solution representation for this problem. We also study the time periodic analog of this problem, and thereby obtain long time asymptotics for the original problem with time periodic boundary and nonlocal data.

Interpretable self-organizing map assisted interactive multi-criteria decision-making following Pareto-Race

The problem-solving task of the multi-criteria decision-making (MCDM) approach involves decision makers’ (DMs’) interaction by incorporating their preferences to arrive at one or more preferred near Pareto-optimal solution(s). The Pareto-Race is an interactive MCDM approach that relies on finding preferred solutions as the DM navigates on the Pareto surface by selecting the preferred reference direction and uniform/varying speed using the concept of Achievement Scalarization Function (ASF) or its augmented version (AASF). Selecting a reference direction or speed is possible, algorithmically, but the effectiveness of the Pareto-Race concept relies on suitable visualization methods that present the series of past solutions and convey the current objective trade-off information to assist the DM’s in decision-making task. Traditional multi-dimensional Pareto-optimal front visualization methods like parallel coordinate plots, radial visualization, and heat maps are deemed insufficient for this purpose. Therefore, the paper proposes integrating the concept of the Pareto-Race MCDM approach with a recently developed interpretable self-organizing map (iSOM) based visualization method. The iSOM maps high-dimensional data into lower-dimensional space, facilitating visual convenience for the DM. The proposed method requires a finite-size representation of non-dominated solutions near the complete Pareto Front to generate iSOM plots of objectives. We also propose visualizing the metrics such as closeness to constraint boundaries, trade-off value, and robustness using iSOM to check the quality of the preferred solutions, which is rarely considered in existing MCDM approaches. iSOM plots of objective functions, closeness to constraint, trade-off, and robustness metrics assist the DM in effectively choosing new reference direction and step size in each iteration to arrive at the most preferred solution. The proposed iSOM-enabled Pareto-Race approach is demonstrated on benchmark analytical and real-world engineering examples with 3 to 10 objectives.