Discrete Processes Research Articles

Topology Optimization of Continuum Structures Based on Binary Hunter-Prey Optimization Algorithm

According to BESO’s principle of binarizing continuous design variables and the excellent performance of the standard HPO algorithm in terms of solving continuous optimization problems, a discrete binary Hunter-prey optimization algorithm is introduced to construct an efficient topology optimization model. It was used to solve the problems that the BESO method of topology optimization has, such as easily falling into the local optimal value and being unable to obtain the optimal topology configuration; the metaheuristic algorithm was able to solve the topology optimization model’s low computational efficiency and could easily produce intermediate elements and unclear boundaries. Firstly, the BHPO algorithm was constructed by discrete binary processing using the s-shape transformation function. Secondly, BHPO-BESO topology optimization theory was established by combining the BHPO algorithm with BESO topology optimization. Using the sensitivity information of the objective function and the updated principle of the meta-heuristic of the BHPO algorithm, a semi-random search for the optimal topology configuration was carried out. Finally, numerical simulation experiments were conducted by using the three typical examples of the cantilever beam, simply supported beam, and clamping beam as optimization objects and the results were compared with the solution results of BESO topology optimization. The experimental results showed that compared with BESO, BHPO-BESO could find the optimal topology configuration with lower compliance and maximum stiffness, and it has higher computational efficiency, which can solve the above problems.

An Analysis of a Fractional-Order Model of Colorectal Cancer and the Chemo-Immunotherapeutic Treatments with Monoclonal Antibody

The growth of colorectal cancer tumors and their reactions to chemo-immunotherapeutic treatment with monoclonal antibodies (mAb) are discussed in this paper using a system of fractional order differential equations (FDEs). mAb medications are still at the research stage; however, this research takes into account the mAbs that are already in use. The major goal is to demonstrate the effectiveness of the mAb medication Cetuximab and the significance of IL-2 levels in immune system support. The created model is broken down into four sub-systems: cell populations, irinotecan (CPT11) concentration for treatment, IL-2 concentration for immune system support, and monoclonal antibody Cetuximab. We show the existence and uniqueness of the initial value problem (IVP). After that, we analyze the stability of the equilibrium points (disease-free and co-existing) using the Routh–Hurwitz criteria. In addition, in applying the discretization process, we demonstrate the global stability of the constructed system around the equilibrium points based on specific conditions. In the end, simulation results were carried out to support the theory of the manuscript.

МАНЕВР ПОВЕРНЕННЯ СУДНА НА ЗАДАНУ ТРАЄКТОРІЮ РУХУ ПІСЛЯ РОЗХОДЖЕННЯ З ЦІЛЛЮ

This paper proposes a method of decomposing a set of initial bearings into subsets, each of which corresponds to a type of vessel passing strategy based on their initial parameters and the nature of the close-quarters situation. The method aims to perform a vessel evasive action and return it to the programmed (given) course. The paper investigates the problem of collision avoidance between vessels during their passing using methods of the theory of optimal discrete processes. The paper determines the most hazardous initial vessel bearing in a situation of approaching a target (another vessel, hereafter referred to as the target vessel or TV) by decomposing the set of initial bearings into subsets based on vessel movement and distance between them in the case of a close-quarters situation. The paper examines in detail the process of vessel evasive action in a close-quarters situation with a target vessel using computer simulation models in various initial conditions to choose the most optimal safe vessel passing strategy. The paper discusses the peculiarity of choosing possible safe vessel passing strategies. The paper examines different types of vessel return courses to the programmed course, depending on the situation of close-quarters situation with a target vessel, using computer simulation models. The paper determines possible risks and hazards that arise when returning the vessel to the given course during the safe vessel passing process with a target vessel. The paper presents examples of typical vessel passing situations with a target vessel at short distances using computer simulation models, including the circling maneuver of a vessel during passing with another vessel and overtaking maneuver of one vessel by another. Keywords: safety of navigation, prevention of collision of vessels, passing of vessels by evasion, the maneuver of returning the vessel to the programmed track.

Integrable equations associated with the finite‐temperature deformation of the discrete Bessel point process

AbstractWe study the finite‐temperature deformation of the discrete Bessel point process. We show that its largest particle distribution satisfies a reduction of the 2D Toda equation, as well as a discrete version of the integro‐differential Painlevé II equation of Amir–Corwin–Quastel, and we compute initial conditions for the Poissonization parameter equal to 0. As proved by Betea and Bouttier, in a suitable continuum limit the last particle distribution converges to that of the finite‐temperature Airy point process. We show that the reduction of the 2D Toda equation reduces to the Korteweg–de Vries equation, as well as the discrete integro‐differential Painlevé II equation reduces to its continuous version. Our approach is based on the discrete analogue of Its–Izergin–Korepin–Slavnov theory of integrable operators developed by Borodin and Deift.

Why are Discrete Implications Necessary? An Analysis Through the Discretization Process

Discrete implications have been studied for almost two decades as those operators needed to perform inference processes when dealing with qualitative information from a finite chain. However, it is known that by means of some adequate transformations, fuzzy logic operators defined in [0,1] can generate the corresponding discrete operators. Thus, an immediate question arises: do we need to study discrete implications or is it enough to study implications defined in [0,1] and then to discretize them? The answer must rely on the preservation of the additional properties of fuzzy implication functions through these discretization methods. In this paper, for two specific methods based on the ceiling and floor functions it is proved that most of the additional properties are not preserved in general, showing that the preservation of the additional properties depends directly on the properties of the underlying operators considered in the discretization. Thus, sufficient, and for some properties necessary, conditions to guarantee the preservation are presented.

Data-Driven Control for Continuum Robots Based on Discrete Zeroing Neural Networks

The effectiveness of continuous-time zeroing neural network (CZNN) in continuum robot control has been preliminarily verified. However, CZNN is not friendly to digital devices and hardware implementation. Although discrete ZNN (DZNN) has been investigated in other tasks, all the existing DZNNs are designed with fixed time steps, which cannot meet the needs of different tasks. This motivates us to develop a generic DZNN model with variable number of time steps and apply it to continuum robot control. In this paper, we present a data-driven scheme based on CZNNs to learn the unknown kinematics of continuum robots and solve the kinematic control problem. Further, a unified <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> -step forward discretization formula is derived to discretize the CZNN-based scheme into DZNN-based control algorithms. Finally, we take the 1-step and the 3-step algorithms as examples to show the discretization process, and verify their efficacy by simulations and experiments.

Demonstration of the Managing for Daily Improvement Quality Improvement Methodology in an Alcohol Withdrawal Protocol Audit.

The Clinical Institute Withdrawal Assessment for Alcohol, Revised (CIWA-Ar) is an assessment tool designed to standardize care and minimize the risk of complications in patients experiencing alcohol withdrawal. After discovering an increase in medication errors and late assessments under this protocol, pharmacists at a 218-bed community hospital performed an audit of protocol compliance using a performance improvement methodology known as Managing for Daily Improvement (MDI). A daily audit of CIWA-Ar protocol compliance was performed across all hospital units, followed by discussions with frontline nurses regarding barriers to compliance. The daily audit included assessments of appropriate monitoring frequency, medication administration, and medication coverage. Nurses caring for CIWA-Ar patients were interviewed to identify perceived barriers to protocol compliance. The MDI methodology provided a framework and tools to visualize audit results. The visual management tools used in the methodology include daily tracking of 1 or more discrete process measures, daily identification of barriers to perfect process performance at the patient and process level, and collaborative action plan tracking to resolve barriers. Forty-one audits were collected for 21 unique patients over 8 days. After conversations with multiple nurses across different units, the most commonly reported barrier to compliance was a lack of communication at shift handoff. The results of this audit were discussed with nurse educators, patient safety and quality leaders, and frontline nurses. Process improvement opportunities identified from this data included improved widespread nursing education, development of protocol auto-discontinuation criteria based on scores, and determination of downtime processes for the protocol. The MDI quality tool successfully assisted in identifying end-user barriers to and focused areas of improvement of compliance with a nurse-driven CIWA-Ar protocol. This tool is elegant in its simplicity and ease of use. It can be customized to cover any timeframe or monitoring frequency while providing visualization of progress over time.

Mechanism of Phase-Locked Ice Crushing against Offshore Structures

This paper addresses a detailed analysis of the ice–structure interaction process of the phase-locked ice crushing (PLC) against offshore structures. Directly measured ice load, structure response data, and in situ observation from the field measurements on the Molikpaq lighthouse and jacket platform were used in the study. This paper summarizes a new ductile damage-collapse (DDC) failure mechanism for the PLC process. The DDC mechanism shows that the ice failure is a discrete ductile crushing process rather than a ductile–brittle transition process. The analysis identifies that the ice has a failure length in PLC and this failure length plays an important role in understanding the interaction. It reveals that PLC can occur on most vertical-sided offshore structures when the velocity of the ice sheet falls within the range of the failure length divided by the natural period of the structure. This paper proposes that this relationship between ice failure length and the natural period of the structure can be used as one of the PLC occurrence conditions. The DDC failure mechanism provides a basis for another technical route to solve the PLC problem.

A novel discrete deep learning-based intelligent methodology for energy consumption classification

Energy consumption classification is one of the most widely-used approaches in the energy area that is applied in various applications such as household, commercial, urban, rural, industrial, etc. Energy consumption due to its substantial positive influence on the quality of made decisions, production and distribution management, and cost reduction, has considerable literature. In the classification literature, numerous statistical, intelligence, and hybrid methods have been developed and proposed in order to yield more accurate results. The common point of all these models is that whole of techniques follow a prevalent and repetitive procedure of learning process based on continuous distance-based cost function. While the objective function of classification is discrete. While the mismatch of the continuous cost function in classification problems with the discrete objective function causes them to be illogical or inefficient. Therefore, in this paper, a novel discrete learning process is offered to eliminate the inconsistency between the cost function and the classification objective function. The main difference between the proposed learning methodology rather than conventional versions is its cost function. In the proposed learning methodology, a mismatching function is considered as a cost function, which is dissimilar to previously developed ones, which are continuous functions based on distance, is a discrete function based on direction. In this way, in the proposed learning process, unknown parameters are discretely adjusted and at once jumped to the target. This is in contrast to conventional continuous learning algorithms in which the unknown parameters are continuously adjusted and step-by-step near the target. In this paper, deep multilayer neural networks (DMNs) have been exemplarily applied to implement the proposed learning algorithm. Although more consistency of goal and cost functions will logically has not a negative effect on the classification rate of classifiers. However, in this paper, to show the superiority of the proposed discrete deep multilayer neural network (DDMN) over conventional continuous-based classifiers, four energy consumption benchmark data sets have been employed. These data sets include the Tamilnadu Electricity Board Hourly Readings, Tetouan City Energy Consumption, Household Electric Power consumption, and steel industry Energy Consumption. Empirical results indicate that the proposed DDMN model, as pre-expected, can yield better performance in all cases than the DMN model. The DDMN model can achieve a 93.27% classification rate on average and upgrade its classic version approximately by 4.77%.

Limit theorems for discrete multitype branching processes counted with a characteristic

For a discrete time multitype supercritical Galton–Watson process (Zn)n∈N and corresponding genealogical tree T, we associate a new discrete time process (ZnΦ)n∈N such that, for each n∈N, the contribution of each individual u∈T to ZnΦ is determined by a (random) characteristic Φ evaluated at the age of u at time n. In other words, ZnΦ is obtained by summing over all u∈T the corresponding contributions Φu, where (Φu)u∈T are i.i.d. copies of Φ. Such processes are known in the literature under the name of Crump–Mode–Jagers (CMJ) processes counted with characteristicΦ. We derive a LLN and a CLT for the process (ZnΦ)n∈N in the discrete time setting, and in particular, we show a dichotomy in its limit behavior. By applying our main result, we also obtain a generalization of the results in Kesten and Stigum (1966).

A three-dimensional numerical pressure transient analysis model for fractured horizontal wells in shale gas reservoirs

Shale gas development has greatly contributed to the world's energy supply. Pressure transient analysis (PTA) has extremely helped the development of shale gas. In this paper, a three-dimensional numerical model for transient pressure analysis of multi-stage fractured horizontal wells (MFHWs) in shale gas reservoirs was proposed, which was based on the embedded discrete fracture model (EDFM). The adsorption effect of shale gas, stress sensitivity of shale reservoir, and high-velocity non-Darcy flow in fractures were considered in this model, and the model was solved by Matlab Reservoir Simulation Tool (MRST). The discrete process of governing equations is described in detail, the typical MFHW pressure transient response curve of a shale gas reservoir was obtained, different flow regimes were divided, and three-dimensional pressure distribution profile of different flow stages were drawn to clearly understand the flow states of shale gas in the reservoir. The effects of different flow mechanisms and fracture properties on the log–log curves of the MFHW pressure transient response were analyzed in detail. The numerical PTA model proposed in this paper not only considers a variety of complex flow mechanisms of shale gas, but also accurately describes the characteristics of complex fracture network of shale reservoirs in three-dimensional conditions through EDFM. In addition, the problems of the incomplete description of shale gas flow stages by conventional analytical methods and semi-analytical methods were avoided through the numerical solution. The pressure response behavior of fluid in shale gas reservoir can be better explained by using this model, which helps to reduce the uncertainty of shale reservoir development.

The Rates of Convergence for Functional Limit Theorems with Stable Subordinators and for CTRW Approximations to Fractional Evolutions

From the initial development of probability theory to the present days, the convergence of various discrete processes to simpler continuous distributions remains at the heart of stochastic analysis. Many efforts have been devoted to functional central limit theorems (also referred to as the invariance principle), dealing with the convergence of random walks to Brownian motion. Though quite a lot of work has been conducted on the rates of convergence of the weighted sums of independent and identically distributed random variables to stable laws, the present paper is the first to supply the rates of convergence in the functional limit theorem for stable subordinators. On the other hand, there is a lot of activity on the convergence of CTRWs (continuous time random walks) to processes with memory (subordinated Markov process) described by fractional PDEs. Our second main result is the first one yielding rates of convergence in such a setting. Since CTRW approximations may be used for numeric solutions of fractional equations, we obtain, as a direct consequence of our results, the estimates for error terms in such numeric schemes.

Amino-acid classification based on terahertz absorption spectroscopy with Gaussian process and maximum likelihood

The rotational and vibrational energy levels of numerous biomolecules lie in the terahertz (THz) band, which makes THz spectroscopy a viable option for biosensing and monitoring the chemical processes such as the formation of hydrogen bonds. However, classifying the fingerprint THz spectra of various molecules has constantly challenged real-world applications. Here, a machine learning model based on a discrete Gaussian process and maximum likelihood estimation (DGPML) for classifying an ensemble of 20 amino acids is proposed. The model demonstrates a prediction accuracy of 99.5% in the presence of scattering noise and various experimental conditions and is able to distinguish spectra without significant absorption peaks. Compared to the conventional Gaussian process classifier, which generally exhibits less time and space complexity than neural-network models, DGPML requires 99% less training time and achieves 1530 times faster processing speed with around 0.29 million parameters. Regarding the DGPML, a database that encloses the THz absorption spectrum and its covariance matrix, which describes the deviation of the absorption spectrum under various experimental conditions, is designed. Due to the compact size and high efficiency of the proposed model, it has the potential to power the THz chemical sensor in the future.

THEORETICAL FOUNDATIONS OF FORECASTING THE RESOURCE OF HYDRAULIC STRUCTURES

It is established that in terms of the discrete Markov process, the problem is reduced to the search for unconditional probabilities of the system S at an arbitrary step k in state Si, that is, obtaining a transition probabilities matrix. In this formulation, the model is used for assessing the technical condition of the element; assessing the level of safety of operation of structural elements; ranking elements according to the need for repairs, reconstruction or replacement; in strategic planning of repair or reconstruction costs in conditions of limited funding and forecasting the remaining resource of elements. It is established that the theoretical basis of the study, which aims to predict the resource of hydraulic structures in operation, is the Markov theory of random processes. For a mathematical description of the process of element degradation, the most successful is the mathematical apparatus of the Markov random processes. Determination of the failure intensity parameter is the dominant feature of the Markov phenomenological model of damage accumulation to hydraulic structures' elements. The only parameter of lifecycle management is the failure rate . In the model under consideration, the parameter is determined based on the initial conditions for an individual element obtained from the survey results. Because the parameter λ is determined for an individual element and must be specified each time after the next survey, the accuracy of the model will increase. The proposed model is integral. It does not contain an explicit theoretical apparatus for a material-sensitive element, its static scheme, construction technology, environmental conditions, etc. On the other hand, all these factors and many other secondary ones are taken into account in the model at the moment the state of the element is determined using classification tables containing physical and mechanical signs of degradation. In the theory of structures, the statistical approach to formulating the transition matrix is widespread and is based on historical data from the structure operation system. It is believed that the transition matrix based on the data of the operating system is a more realistic basis for predicting the processes of structures degradation. A large number of foreign studies are devoted to the practical application of the transition matrix based on statistical data, which consider the features of transition matrices related to the bridge operation system in different countries. In this formulation, each element of the transition probability matrix P is the probability that the system in the state will move to state j in one step (i.e., in one year). At the same time, it is considered that there are no operational interventions, so the sub-diagonal elements are zero. As before, the sum of elements of the same line is 1 and the element pjj = 1 because state j is absorbing. For the implementation algorithm of the Markov chain model for forecasting the technical condition of hydraulic structures in general, the initial data are: statistical data of the distribution of structures by the state at the time of the forecast, the rating assessment of the structure is calculated by an expert according to the scale and the forecast time in years. It is established that the degradation properties of structural designs are described by two parameters: the degradation criterion and the failure rate. Any factor of the stress-strain state can be taken as a degradation criterion: reliability, internal forces, or deformations. The degradation criterion can be an arbitrary rating assessment. In our case, the reliability of the element is taken as the degradation criterion, as the most general factor of the stress-strain state.

Chaotic dynamics of the fractional order Schnakenberg model and its control

The Schnakenberg model is thought to be the Caputo fractional derivative. In order to create caputo fractional differential equations for the Schnakenberg model, a discretization process is first used. The fixed points in the model are categorized topologically. Then, we show analytically that, under certain parametric conditions, a Neimark-Sacker (NS) bifurcation and a Flip-bifurcation are supported by a fractional order Schnakenberg model. Using central manifold and bifurcation theory, we demonstrate the presence and direction of NS and Flip bifurcations. The parameter values and the initial conditions have been found to have a profound impact on the dynamical behavior of the fractional order Schnakenberg model. Numerical simulations are shown to demonstrate chaotic behaviors like bifurcations, phase portraits, period 2, 4, 7, 8, 10, 16, 20 and 40 orbits, invariant closed cycles, and attractive chaotic sets in addition to validating analytical conclusions. In order to support the system’s chaotic characteristics, we also compute the maximal Lyapunov exponents and fractal dimensions quantitatively. Finally, the chaotic trajectory of the system is stopped using the OGY approach, hybrid control method, and state feedback method.

Adams-based hierarchical features fusion network for image dehazing

Recent developments in Convolutional Neural Networks (CNNs) have made them one of the most powerful image dehazing methods. In particular, the Residual Networks (ResNets), which can avoid the vanishing gradient problem effectively, are widely deployed. To understand the success of ResNets, recent mathematical analysis of ResNets reveals that a ResNet has a similar formulation as the Euler method in solving the Ordinary Differential Equations (ODE’s). Hence, image dehazing which can be formulated as an optimal control problem in dynamical systems can be solved by a single-step optimal control method, such as the Euler method. This optimal control viewpoint provides a new perspective to address the problem of image restoration. Motivated by the advantages of multi-step optimal control solvers in ODE’s, which include better stability and efficiency than single-step solvers, e.g. Euler, we propose the Adams-based Hierarchical Feature Fusion Network (AHFFN) for image dehazing with modules inspired by a multi-step optimal control method named the Adams–Bashforth method. Firstly, we extend a multi-step Adams–Bashforth method to the corresponding Adams block, which achieves a higher accuracy than that of single-step solvers because of its more effective use of intermediate results. Then, we stack multiple Adams blocks to mimic the discrete approximation process of an optimal control in a dynamical system. To improve the results, the hierarchical features from stacked Adams blocks are fully used by combining Hierarchical Feature Fusion (HFF) and Lightweight Spatial Attention (LSA) with Adams blocks to form a new Adams module. Finally, we not only use HFF and LSA to fuse features, but also highlight important spatial information in each Adams module for estimating the clear image. The experimental results using synthetic and real images demonstrate that the proposed AHFFN obtains better accuracy and visual results than that of state-of-the-art methods.

Pre-birth acquisition of personhood: Incremental accrual of attributes as the framework for individualization by serial and concurrently acting developmental factors.

Discrete events and processes influence development of individual humans. Attribution of personhood to any individual human being cannot be disconnected from the underlying biological events and processes of early human development. Nonetheless, the philosophical, sociological and legal components that are integral to the meaning of the term as commonly used cannot be deduced from biology alone. The challenge for biomedical scientists to inform discussion in this arena then rests on profiling the key biological events and processes that must be assessed when considering how one might objectively reason about the task of superimposing the concept of personhood onto the developing biological entity of a potential human being. Endogenous genetic and epigenetic events and exogenous developmental milieu processes diversify developmental trajectories of potential individual humans prior to livebirth. First, fertilization and epigenetic resetting of each individual's organismic clock to time zero (t = 0) at the gastrulation/primitive streak stage (day 15 of embryogenesis), are two discrete unseen biological events that impact a potential individual human's attributes. Second, those two discrete unseen biological events are immersed in the continuous developmental process spanning pre-fertilization and gestation, further driving individualization of diverse attributes of each future human before the third discrete and blatant biological event of parturition and livebirth. Exposures of the gravida to multiple diverse exogenous exposures means that morphogenesis and physiogenesis of every embryo/fetus has individualized attributes for its future human lifespan. Our proposed framework based on the biological discrete events and processes spanning pre-fertilization and prenatal development, implies that personhood should be incrementally attributed, and societal protections should be graduated and applied progressively across the pre-birth timespan.

Smart factory mapping and design: methodological approaches

Several studies show a demand for further transfer of Industry 4.0 concepts and technologies into practice. This paper describes how companies can be supported methodologically to identify individual opportunities for optimization within their future smart factories. Existing methods are screened, recombined, and enhanced to create a sufficient demand-oriented approach, based on the lean value stream mapping and design. The final approach covers a mapping phase of the current state and a design phase of a future state. It follows the general principle of process optimization prior to digitalization. Being a highly visual method, it can be used to explain the needs and the motivation for the identified improvement initiatives to all involved parties, including non-IT-specialists. This paper also presents an exemplary result and several learnings from tests with three companies with discrete manufacturing processes.

Advances in Zebrafish for Diabetes Mellitus with Wound Model.

Diabetic foot ulcers cause great suffering and are costly for the healthcare system. Normal wound healing involves hemostasis, inflammation, proliferation, and remodeling. However, the negative factors associated with diabetes, such as bacterial biofilms, persistent inflammation, impaired angiogenesis, inhibited cell proliferation, and pathological scarring, greatly interfere with the smooth progress of the entire healing process. It is this impaired wound healing that leads to diabetic foot ulcers and even amputations. Therefore, drug screening is challenging due to the complexity of damaged healing mechanisms. The establishment of a scientific and reasonable animal experimental model contributes significantly to the in-depth research of diabetic wound pathology, prevention, diagnosis, and treatment. In addition to the low cost and transparency of the embryo (for imaging transgene applications), zebrafish have a discrete wound healing process for the separate study of each stage, resulting in their potential as the ideal model animal for diabetic wound healing in the future. In this review, we examine the reasons behind the delayed healing of diabetic wounds, systematically review various studies using zebrafish as a diabetic wound model by different induction methods, as well as summarize the challenges and improvement strategies which provide references for establishing a more reasonable diabetic wound zebrafish model.

Preconditioned successive over relaxation iterative method via semi-approximate approach for Burgers’ equation

&lt;span lang="EN-US"&gt;This paper proposes the combination of a preconditioner applied with successive over relaxation (SOR) iterative method for solving a sparse and huge scale linear system (LS) in which its coefficient matrix is a tridiagonal matrix. The purpose for applying the preconditioner is to enhance the convergence rate of SOR iterative method. Hence, in order to examine the feasibility of the proposed iterative method which is preconditioner SOR (PSOR) iterative method, first we need to derive the approximation equation of one-dimensional (1D) Burgers’ equation through the discretization process in which the second-order implicit finite difference (SIFD) scheme together with semi-approximate (SA) approach have been applied to the proposed problem. Then, the generated LS is modified into preconditioned linear system (PLS) to construct the formulation of PSOR iterative method. Furthemore, to analyze the feasibility of PSOR iterative method compared with other point iterative methods, three examples of 1D Burgers’ equation are considered. In conclusion, the PSOR iterative method is superior than PGS iterative method. The simulation results showed that our proposed iterative method has low iteration numbers and execution time.&lt;/span&gt;