Solution Of Problem Research Articles

Variable-step [formula omitted] method combined with time two-grid algorithm for multi-singularity problems arising from two-dimensional nonlinear delay fractional equations

In this paper, we focus on the numerical simulation for two-dimensional nonlinear fractional sub-diffusion equations in the presence of time delay. Firstly, we investigate the existence, uniqueness and regularity of the solution for such problems. The theoretical result implies that the solution at τ+ is smoother than that at 0+, where τ is a constant time delay, and this is an improvement for the work (Tan et al., 2022). Secondly, a high-order difference scheme based on L1 method is constructed. For the sake of repairing the convergence order in temporal direction and improving the computational efficiency, an efficient time two-grid algorithm based on nonuniform meshes is first developed. The convergence order of the two-grid scheme reaches O(NF−min{rα,2−α}+NC−min{2rα,4−2α}+h12+h22), where NF and NC represent the number of the fine and coarse grids respectively, while h1 and h2 are the space-step sizes. Furthermore, stability and convergence analysis of the proposed scheme are carefully verified by energy method. Finally, numerical experiments are carried out to show the validity of theoretical statements.

The Riemann Hilbert dressing method and wave breaking for two (2 + 1)-dimensional integrable equations

Abstract In this article, we present a method for generating (2 + 1)-dimensional integrable equations, resulting in the generalized Pavlov equation and dispersionless Kadomtsev–Petviashvili (dKP) equation, which can further be reduced to the standard Pavlov equation and dKP equation. Inspired by the inverse spectral transform presented in existing literature, we introduce the Riemann–Hilbert (RH) dressing method to construct the formal solutions of the Cauchy problems for the generalized Pavlov equation and dKP equation, providing a spectral representation of the solutions. Subsequently, we also extensively investigate the longtime behavior of solutions to these two equations in specific space regions. In particular, for the generalized dKP equation, we conduct a dedicated study on its implicit solutions expressed by arbitrary differential function through linearizing their RH problems. In the final section, we elaborate in detail on the analytic aspects of the wave breaking of a localized two-dimensional wave evolving according to the Hopf equation. With the assistance of a transformation, the longtime breaking of solutions to the generalized dKP equation can then be further characterized.

Wave propagation in the Kolmogorov–Petrovskii–Piskunov–Fischer equation with time delay and spatial deviation

For the logistic equation with diffusion two generalizations are considered: one with time delay and the other with deviation in the spatial variable. For these equations, the problem of density wave propagation is considered. These equations, known as Kolmogorov–Petrovsky–Piskunov–Fisher equations with delay or deviation (KPPF with delay or deviation), are investigated using both asymptotic and numerical methods.To investigate the qualitative behavior of solutions of such equations, the wave profile equation is considered and the conditions for the emergence of its oscillatory modes are identified.The local properties of solutions with periodic boundary conditions are examines, revealing that an increase in the period leads to the emergence of stable solutions with a more complex spatial structures.Numerical analysis of the wave propagation process indicates that at sufficiently small delay values, this equation has solutions that are close to those of the standard KPPF equation. Increasing the delay parameter leads first to the appearance of a damped oscillatory component in the spatial distribution of the solution. With further increases of this parameter, a complex spatial profile of the wave appears.In the final part of the chapter, which focuses on numerical calculation, the numerical analysis of the wave propagation process from two initial perturbations is performed. This analysis tracks the process of wave interaction. The complex spatially inhomogeneous structure that arises during wave propagation and interaction can be explained by the properties of the corresponding solutions of the periodic boundary value problem, especially as the interval of change in the spatial variable increases.

A neurodynamic approach for a class of pseudoconvex semivectorial bilevel optimization problems

The article proposes an exact approach to finding the global solution of a nonconvex semivectorial bilevel optimization problem, where the objective functions at each level are pseudoconvex, and the constraints are quasiconvex. Due to its non-convexity, this problem is challenging, but it attracts more and more interest because of its practical applications. The algorithm is developed based on monotonic optimization combined with a recent neurodynamic approach, where the solution set of the lower-level problem is inner approximated by copolyblocks in outcome space. From that, the upper-level problem is solved using the branch-and-bound method. Finding the bounds is converted to pseudoconvex programming problems, which are solved using the neurodynamic method. The algorithm's convergence is proved, and computational experiments are implemented to demonstrate the accuracy of the proposed approach.

Numerical solution of the boundary value problem of elliptic equation by Levi function scheme

AbstractFor boundary value problem of an elliptic equation with variable coefficients describing the physical field distribution in inhomogeneous media, the parametrix can represent the solution in terms of volume and surface potentials, with the drawback that the volume potential involving in the solution expression requires heavy computational costs as well as the solvability of the integral equations with respect to the density pair. We introduce an modified integral expression for the solution to an elliptic equation in divergence form under the parametrix framework. The well‐posedness of the linear integral system with respect to the density functions to be determined is rigorously proved. Based on the singularity decomposition for the parametrix, we propose two schemes to deal with the volume integrals so that the density functions can be solved efficiently. One method is an adaptive discretization scheme for computing the integrals with continuous integrands, leading to the uniform accuracy of the integrals in the whole domain, and consequently the efficient computations for the density functions. The other method is the dual reciprocity method which is a meshless approach converting the volume integrals into boundary integrals equivalently by expressing the volume density as the combination of the radial basis functions determined by the interior grids. The proposed schemes are justified numerically to be of satisfactory computation costs. Numerical examples in 2‐dimensional and 3‐dimensional cases are presented to show the validity of the proposed schemes.

Emission Rate Estimation of Industrial Air Pollutant Emissions Based on Mobile Observation

Mobile observation has been widely used in the monitoring of air pollution. However, studies on pollution sources and emission characteristics based on mobile navigational observation are rarely reported in the literature. A method for quantitative source analysis for industrial air pollutant emissions based on mobile observations is introduced in this paper. NOx pollution identified in mobile observations is used as an example of the development of the method. A dispersion modeling scheme that fine-tuned the meteorological parameters according to the actual meteorological conditions was adopted to minimize the impact of uncertainties in meteorological conditions on the accuracy of small-scale dispersion modeling. The matching degree between simulated and observed concentrations was effectively improved through this optimization search. In response to the efficiency requirements of source resolution for multiple sources, a random search algorithm was first used to generate candidate solution samples, and then the solution samples were evaluated and optimized. Meanwhile, the new index Smatch was established to evaluate the quality of candidate samples, considering both numerical error and spatial distribution error of concentration, in order to address the non-uniqueness of the solution in the multi-source problem. Then, the necessity of considering the spatial distribution error of concentration is analyzed with the case study. The average values of NOx emission rates for the two study cases were calculated as 69.8 g/s and 70.8 g/s. The Smatch scores were 0.92–0.97 and 0.92–0.99. The results were close to the online monitoring data, and this kind of pollutant emission monitoring based on the mobile observation experiment was initially considered feasible. Additional analysis and clarifications were provided in the discussion section on the impact of uncertainties in meteorological conditions, the establishment of a priori emission inventories, and the interpretation of inverse calculation results.

A class of one-parameter filled functions with the same local minima as the objective function for global optimization problems

As an effective global optimization method, the filled function algorithm obtains the optimal solution of optimization problems by alternately minimizing the objective function and filled function. One development direction of the filled function method is to construct the filled function with good properties, which directly affects the efficiency of the algorithm and has attracted the attention of scholars. This paper presents a class of one-parameter filled function that is easy to adjust. It is proved theoretically that the filled function is continuously differentiable and has the same local minimizers as the objective function, and these minimizers are all better than the current local minimizer of the objective function. Meanwhile, some natures of the constructed filled function are conducted as followed by a new algorithm is presented. The new filled function algorithm optimizes the iterative framework of the conventional filled function method, improving the efficiency and decreasing the computational cost. The numerical experiments of the new algorithm on several optimization issues are reported with satisfactory computational results, verifying the feasibility and efficiency of the algorithm.

A practical path planning method for optimal repair paths between multiple small-size defects

Purpose This study aims to solve the problem of repair path planning between multiple small-size defects in the field of additive manufacturing (AM) repair by using Python-based ant colony algorithm (ACO). The optimal parameter combination scheme is obtained by discussing the influencing factors of parameters in the ACO. Design/methodology/approach The effects of the information heuristic factor α, the expected heuristic factor ß and the pheromone volatile factor ρ on the simulation results were investigated by designing a three-factor and three-level orthogonal experiment. The fast convergence of ACO in finding the optimal solution of multiple small-size defect repair path problem is proved by comparing the simulation results with those of genetic algorithm (GA) on the same data set. Findings The ACO can effectively solve the repair path planning problem between multiple small-size defects by optimizing the parameters. In the case of 50 defect locations, the simulation results of the ACO with optimized parameters are 159.8 iterations and 3,688 average path lengths, while the GA has 4,027.2 average path lengths under the same data set and the same number of iterations, and by comparison, it is proved that the ACO can find the optimal solution quickly in the small-size defects repair path planning problem, which greatly improves the efficiency of defect repair. Originality/value The parameter-optimized ACO can be quickly applied to the planning problem of repair paths between multiple small-size defects in the field of AM repair, which can better improve the defect repair efficiency and reduce the waste of resources.

Stress-deformation analysis of the cracked elastic body

A new analytical plane Elastic Fracture Mechanics (LEFM) model of the effective elastic moduli exhibited by an isotropic elastic cracked solid in tensile and compressive stress fields is presented. The effective Young’s modulus and Poisson’s ratio E¯,v¯, respectively, relate the stresses applied to the cracked solid with its macroscopic deformation. In the case of a compressive stress field the effective elastic moduli depend solely on properly oriented cracks that are either open or closed. This model is then used for the stress and displacement estimations around a circular axisymmetric hole under internal pressure and external hydrostatic stress. The creation of the hole alters the stresses in its neighborhood, and induces a deterioration of the elastic moduli. In this manner, the effective elastic moduli depend on the radial distance from the hole’s boundary. The validation of the algorithm was done against Kirsch’s analytical solution of the axisymmetric hole problem for nearly zero density of cracks as well as for large values of the friction angle. The model predicts a lower tangential stress concentration at the wall of the hole compared to that predicted by the constant linear elasticity solution and a maximum stress concentration that is not necessarily located at the wall of the hole. This apparent enhancement of rock strength adjacent to the wall is more pronounced for larger concentrations of open cracks.

Numerical study of the influence of the parameters of statistical distribution of the structural elements’ ultimate strength on deformable bodies’ fracture processes

This work is dedicated to the numerical study of the influence of the parameters of statistical distribution of the structural elements’ ultimate strength on fracture processes of deformable bodies with stress concentrators. The loading diagrams, the macro level behavior and the damaging process kinetics have been studied using various probability distribution laws and their standard deviations. The main types of the damage accumulation process have been identified. The approach basing on the analysis of the solutions of the boundary value problems within the elasticity theory has been proposed in order to predict the fracture process kinetics, the corresponding parameters have been introduced. Advantages, limitations and potential practical use of the proposed methodology have been discussed. It has been concluded that the statistical distribution of the structural elements’ strength properties should be taken into account in the numerical modeling of the fracture processes to ensure the reliability and safety of critical structures.

A Systematic Literature Review on the Use of Federated Learning and Bioinspired Computing

Federated learning (FL) and bioinspired computing (BIC), two distinct, yet complementary fields, have gained significant attention in the machine learning community due to their unique characteristics. FL enables decentralized machine learning by allowing models to be trained on data residing across multiple devices or servers without exchanging raw data, thus enhancing privacy and reducing communication overhead. Conversely, BIC draws inspiration from nature to develop robust and adaptive computational solutions for complex problems. This paper explores the state of the art in the integration of FL and BIC, introducing BIC techniques and discussing the motivations for their integration with FL. The convergence of these fields can lead to improved model accuracy, enhanced privacy, energy efficiency, and reduced communication overhead. This synergy addresses inherent challenges in FL, such as data heterogeneity and limited computational resources, and opens up new avenues for developing more efficient and autonomous learning systems. The integration of FL and BIC holds promise for various application domains, including healthcare, finance, and smart cities, where privacy-preserving and efficient computation is paramount. This survey provides a systematic review of the current research landscape, identifies key challenges and opportunities, and suggests future directions for the successful integration of FL and BIC.

Task-Importance-Oriented Task Selection and Allocation Scheme for Mobile Crowdsensing

In Mobile Crowdsensing (MCS), sensing tasks have different impacts and contributions to the whole system or specific targets, so the importance of the tasks is different. Since resources for performing tasks are usually limited, prioritizing the allocation of resources to more important tasks can ensure that key data or information can be collected promptly and accurately, thus improving overall efficiency and performance. Therefore, it is very important to consider the importance of tasks in the task selection and allocation of MCS. In this paper, a task queue is established, the importance of tasks, the ability of participants to perform tasks, and the stability of the task queue are considered, and a novel task selection and allocation scheme (TSAS) in the MCS system is designed. This scheme introduces the Lyapunov optimization method, which can be used to dynamically keep the task queue stable, balance the execution ability of participants and the system load, and perform more important tasks in different system states, even when the participants are limited. In addition, the Double Deep Q-Network (DDQN) method is introduced to improve on the traditional solution of the Lyapunov optimization problem, so this scheme has a certain predictive ability and foresight on the impact of future system states. This paper also proposes action-masking and iterative training methods for the MCS system, which can accelerate the training process of the neural network in the DDQN and improve the training effect. Experiments show that the TSAS based on the Lyapunov optimization method and DDQN performs better than other algorithms, considering the long-term stability of the queue, the number and importance of tasks to be executed, and the congestion degree of tasks.

Approximation Solution for Initial Value Problem of Singularly Perturbed Integro-Differential Equation

Adomian decomposition method (ADM) is used to approximately solve the initial value problem of the singularly perturbed Volterra and Fredholm differential equation. With this method, the desired accurate results are obtained in only a few terms. The approach is simple and effective. An example application is made to demonstrate the effectiveness of ADM. The result obtained is compared with the exact solution. Convergence analysis of the method was performed.

Deep learning solutions for inverse problems in advanced biomedical image analysis on disease detection

Inverse problems in biomedical image analysis represent a significant frontier in disease detection, leveraging computational methodologies and mathematical modelling to unravel complex data embedded within medical images. These problems include deducing the unknown properties of biological structures or tissues from the observed imaging data, presenting a unique challenge in decoding intricate biological phenomena. Regarding disease detection, this technique has played a critical role in optimizing diagnostic efficiency by extracting meaningful insights from different imaging modalities like molecular imaging, MRI, and CT scans. Inverse problems contribute to uncovering subtle abnormalities by employing iterative optimization techniques and sophisticated algorithms, enabling precise and early disease detection. Deep learning (DL) solutions have emerged as robust mechanisms for addressing inverse problems in biomedical image analysis, especially in disease recognition. Inverse problems involve reconstructing unknown structures or parameters from observed data, and the DL model excels in learning complex representations and mappings. This study develops a DL Solution for Inverse Problems in the Advanced Biomedical Image Analysis on Disease Detection (DLSIP-ABIADD) technique. The DLSIP-ABIADD technique exploits the DL approach to solve inverse problems and detect the presence of diseases on biomedical images. To solve the inverse problem, the DLSIP-ABIADD technique uses a direct mapping approach. Bilateral filtering (BF) is used for image preprocessing. Besides, the MobileNetv2 model derives feature vectors from the input images. Moreover, the Henry gas solubility optimization (HGSO) method is applied for optimal hyperparameter selection of the MobileNetv2 model. Furthermore, a bidirectional long short-term memory (BiLSTM) model is deployed to identify diseases in medical images. Extensive simulations have been involved to illustrate the better performance of the DLSIP-ABIADD technique. The experimentation outcomes stated that the DLSIP-ABIADD technique performs better than other models.

Ultra-Low-Power Hybrid Synaptic Memristor via Simultaneous-Controlled Mechanism of Cations and Anions for Hyper-Scale Synaptic Cores

Recently, memristive devices such as phase change random access memory (PCRAM), resistive random access memory (ReRAM), and ferroelectric random access memory (FeRAM) have been actively researched to implement hyper-scale synaptic cores of various ANNs, i.e., deep neural networks (DNNs), spiking neural networks (SNNs), convolutional neural networks (CNNs), and binarized neural networks (BNNs). In particular, ReRAM is currently being highlighted as an artificial synaptic device because of its multilevel capability (> 11 bits)1, high switching speed (< 100 ps)2, high endurance (> 1012 cycles)3, high scalability, and complementary metal-oxide-semiconductor (CMOS) compatibility4. ReRAM is generally divided into two types: valence change memory (VCM) cells having oxygen vacancy filaments and electrochemical metallization (ECM) cells having conductive metal filaments. VCM cells have great retention but exhibit high switching current (~ 100 μA). In contrast, ECM cells exhibit relatively lower switching current (~10 uA) than VCM cells but have poor retention due to diffusion phenomena at high external temperatures5. Due to their high switching current and poor reliability, VCM and ECM cells have limitations in their application as synaptic devices in next-generation neuromorphic systems for energy consumption similar to that of the human brain (~20 W). In this study, for the first time, we developed a Ru-based ultra-low-power (< 1μA) hybrid synaptic memristor having a simultaneous-controlled mechanism of Ru cations and oxygen anions in a resistive switching layer. To elucidate the simultaneous-controlled mechanism of Ru cations and oxygen anions, we fabricated the VCM, ECM, and Ru-based hybrid memristor and performed x-ray photoelectron spectroscopy (XPS) and time-of-flight secondary ion mass spectrometry (ToF-SIMS) analysis depending on the resistance states (i.e., pristine, set, and reset). In addition, the mobility of mobile species in VCM, ECM, and Ru-based hybrid memristors was calculated indirectly via electrical properties (switching speed, operating voltage) and ToF-SIMS depth profiles, revealing the mechanism by which Ru-based hybrid memristor has low switching currents (< 1μA). Finally, the power consumption of active synaptic cores in the training and inference process of the designed DNN was evaluated for VCM, ECM, and Ru-based hybrid memristor. The simultaneous-controlled mechanism of Ru cations and oxygen anions of the Ru-based hybrid memristor and its application to DNN will be presented in detail. Acknowledgement This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. RS-2023-00260527) and Institute of Information & communications Technology Planning & Evaluation (IITP) under the artificial intelligence semiconductor support program to nurture the best talents (IITP-(2023)-RS-2023-00253914) grant funded by the Korea government(MSIT)". References Rao, M. et al. Thousands of conductance levels in memristors integrated on CMOS. Nature 615, 823–829 (2023).Yu, S., Wu, Y., Jeyasingh, R., Kuzum, D. & Wong, H. S. P. An electronic synapse device based on metal oxide resistive switching memory for neuromorphic computation. IEEE Trans. Electron Devices 58, 2729–2737 (2011).Lee, H. Y. et al. Evidence and solution of over-RESET problem for HfOX based resistive memory with sub-ns switching speed and high endurance. Tech. Dig. - Int. Electron Devices Meet. IEDM 7–10 (2010) doi:10.1109/IEDM.2010.5703395.Wong, H. S. P. et al. Metal-oxide RRAM. Proc. IEEE 100, 1951–1970 (2012).Yoon, J. H. et al. A Low-Current and Analog Memristor with Ru as Mobile Species. Adv. Mater. 32, 1–9 (2020). Figure 1

(Invited) 3D Printing of 2D Materials for Optimized Electrochemical Performance

Electrochemical energy storage (EES) and conversion devices (e.g. batteries, supercapacitors, and reactors) are emerging as primary methods for global efforts to shift energy dependence from limited fossil fuels towards sustainable and renewable resources. These devices, while showing great potential for meeting some key metrics set by conventional technologies, still face significant limitations. For example, an EES device tends to exhibit large energy density (e.g. lithium-ion battery) or power density (e.g. supercapacitor), but not both. This inability of a single device to simultaneously achieve both metrics represents a major obstacle to widespread adoption of EES devices. Similarly, many catalytic processes depend on expensive platinum group metals (PGM) because non-PGM materials have poor performance or durability. Though the integration of 2D materials (e.g. graphene, dichalcogenides, MXene, etc.) into electrochemical devices has yielded some exciting results towards tackling these issues, significant improvements are still needed. One approach to optimizing the performance of these devices is to focus on one of the fundamental processes that occur in these systems: mass (or charge) transport. The efficient transport of ions within EES and conversion devices is critical to realizing better performance and durability. The pore structure of the electrode is a key factor in determining this transport phenomena, but in many cases, engineering the pore structure in a highly deterministic fashion can be challenging. In this work, we explore a number of additive manufacturing methods (e.g. direct ink write, projection microstereolithography, etc.) to engineer the pore structure of device electrodes. We also determine effective electrode geometries using both simple theory and topology optimization techniques. The topology optimization couples the solution of the forward electrochemical problem over the full electrode domain with gradient-based optimization. The output of our code is a three-dimensional CAD representation which optimizes over specific performance metrics and which can be used to print functional electrodes. This work provides a systematic path toward automated design and fabrication of engineered electrodes with precise control over the fluid and species distribution.This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

A comprehensive isogeometric analysis of frictional Hertz contact problem

The development of computational methods has advanced the solving of complex contact problems. Despite this, reproducing the analytical solution of a classical Hertz contact problem using computational methods remains challenging. Finite element method (FEM) has gained popularity, with many studies solving frictionless Hertz contact problems. However, the more general Hertz contact problem with finite friction lacks an analytical solution and is more complex. FEM was first used for this in the 1970s. Recently, isogeometric analysis (IGA) has shown enhanced efficiency than FEM. However, comprehensive studies on the frictional Hertz contact problem using IGA are lacking. The present study aims to fill this gap by examining a benchmark example and exploring contact simulation parameters using the developed in-house MATLAB® code.

A novel methodology for the diffusion coefficient determination in porous media by X-ray high-resolution radiographies

The first step in diagnosing a patient with cancer is the biopsy and the study of the tissue sample taken, suitably processed, is essential for a correct diagnosis. Tissue processors are machines that automatically perform the entire processing protocol, often relying on empirical timing related to chemical diffusions in the biological samples. The determination of the diffusion coefficient in porous materials, such as histological tissue, is still under study with multiple methods often resulting in inaccurate predictions and with few amounts of literature data for biological samples. The authors present an experimental technique, based on a multitude of X-ray radiographs acquired over time, where the intensity of the attenuation to the ionising beam is proportional to the concentration of the solvent that diffuses in the histological preparation. The paper so describes a proof-of-concept of a novel method to estimate the diffusion coefficient in a histological sample, based on the acquisition of time-dependent X-ray radiographs and the solution of an inverse diffusion problem.

The Existence and Uniqueness Results of Solutions for a Fractional Hybrid Integro-differential System

This paper discuss two important results for a fractional hybrid boundary value problem of Riemann-Liouville integro-differential systems, the researches and the advance in this field and also the importance of this subject in the modeling of nonlinear real phenomena corresponding to a great variety of events gives the motivation to study this boundary value problem. The results are as follow, the first result consider the existence and uniqueness results of solutions for a fractional hybrid boundary value problem of Riemann-Liouville integro-differential system this result based on Krasnoslskii fixed point theorem for a sum of two operators, the second result is the uniqueness of solution for fractional hybrid boundary value problem of Riemann-Liouville integro-differential systems, the main result is based on Banach fixed point theorem, both results comes after transforming the system into Volterra integral system then transform again into operator system, then using fixed point theory to prove the results, this articule was ended buy an example to well illustrat the results and ideas of proof.

Finite Element Approach for Rheological Behavior in Colloidal Electrolytes in Lithium-Ion Battery Performance.

The main implication of articulating electrolyte performance is studying the energy density, charging aspects, formation of precipitates, thermal fluctuations during charging-discharging, and safety of batteries against fire or spark. One of the most significant aspects is the ability to design colloidal electrolytes that can enhance the overall performance of batteries along with dealing with all internal problems within a battery system. Through this optimization progression, the general performance and efficiency of Li-ion batteries can be improved. This work is presented in the study of the boundary value problem for rheological properties of colloidal electrolytes as a fourth grade fluid for lithium ion (Li-ion) batteries down a vertical cylinder. They have exceptional characteristics, such as low volatility and high thermal stability. The practical usage of the exact flow is restricted, as it involves very complicated integrals. The nonlinear problem that arises is solved by Galerkin's finite element approach based on the weighted-residual formulation, which is used to find the approximate solutions of the fourth-grade problem. This approach utilizes a piecewise linear approximation using linear Lagrange polynomials. Convergence of the solutions, which briefly describes the flow characteristics, includes the effects of the emerging parameters. The results obtained after implementation are not restrictive to small values of the flow parameters. Numerical studies have shown the superior accuracy and lesser computational cost of this scheme in comparison to collocation, the homotopy analysis method, and the homotopy perturbation method. The impact of the relevant parameters is examined through graphical results after implementation of a number of iterations.