Iterative Decomposition Method Research Articles

Higher order Emden–Fowler type equations via uniform Haar Wavelet resolution technique

A uniform Haar wavelet collocation based method is proposed for finding the numerical results for a class of third order nonlinear (Emden–Fowler type) singular differential equations with initial and boundary conditions. At the point of singularity, the coefficient of the such equation blows up, that causes difficulties in capturing the numerical solutions near the point of singularity. Haar wavelet approach handles this peculiar situation efficiently. The proposed method is employed to reduce the IVPs/BVPs into the system of algebraic equations and the nonlinearity is taken care by Newton–Raphson method. It is demonstrated that, the method is appropriate for both initial as well as boundary conditions as these conditions are taken care automatically. Some numerical examples have been illustrated in order to demonstrate the ease of implementation and applicability of the method. The L2 norm and absolute errors further help to manifest the improvement of the findings with the increase in resolution J. We compare our results with other methods which exists in literature, e.g., variation iteration method (VIM), cubic B-spline method, Differential transformation method (DTM), Iterative decomposition method (IDM) and modified Adomian decomposition method (MADM). The second order convergence and error analysis of the proposed method is established to depict the accuracy and stability of the proposed method.

Application of numerical simulation in investigation of memristor structures based on oxides and chalcogenides

Models that describe bipolar resistive switching in planar microstructures based on oxide compounds (Bi2Sr2CaCu2O8+x, Nd2-xCexCuO4-y) and bismuth selenide are considered. Metal-isolator-metal planar-type meristor heterostructures were investigated, in which the micro-size is formed by an electrode whose diameter is much smaller than the total size of the structure (it can be both Chervinsky-type microjunctions and film electric electrodes). Another important feature of these heterostructures is the presence of a surface layer several tens of nanometers thick with specific conductivity significantly reduced relative to volume. The change in the resistive properties of such heterostructures is caused by the formation or destruction of the conductive channel through the above-mentioned layer. Numerical simulation has shown that the bipolar resistive switching is significantly influenced by the electrical field distribution topology. A “critical field” model is proposed to describe experimentally observed memristor effects in investigated heterostructures. In this model it is assumed that the change in specific conductivity occurs in those parts of the surface layer where the electric field strength exceeds some critical value. The model of the “critical field” is based on the numerical calculation of the distribution of electrical potential on the distribution of specific conductivity in the structure. In addition, the model allowing to analyze the influence of electrodiffusion of oxygen ions on resistive switching in heterostructures based on Bi2Sr2CaCu2O8+x is considered. At numerical realization of the models a combination of the integro-differential approximation of the differential equations, the multi-grid approach for localization of heterogeneities of physical characteristics, the iterative decomposition method and composite adaptive meshes was used. It allowed tracking the processes under investigation with necessary accuracy. The comparison of simulation results with experimental data is presented.

A Modified System of Nonlinear Fractional-Order Differential Equations in the Study of the Dynamics of Marital Relationships and their Behavioural Features

Urbanization and modernization have effects on marital relationships in Nigeria which led to high divorce rate among legitimate couples prompting unstable environment. This situation design and uses scientific means to study the dynamics of marital relationships and their behavioural features to check excesses in marriage and to promote stability. A modified system of nonlinear fractional-order differential equations was used to categorize people of different personalities and different Impact Factors of Memory, using different sets of parameters. The equations predict and interpret the features of the union of different individuals with external circumstance(s). Equations were adapted to a local environment where data collections were carried out to investigate factors affecting marriages. Data collected by the use of questionnaire validate the model. An Iterative Decomposition Method was adopted to solve the fractional system in which fractional derivatives were given in the Caputo sense; the obtained results were interpreted appropriately. The modified model shows the trajectory of the couple from the state of indifference and as the impact factor memory increases it affects their togetherness making the love between them to decay easily. Numerical simulation results were presented to show the effectiveness of the model and the accuracy of the statements established. Keywords: Differential model, dynamical system, impact factors of memory, iterative decomposition method, marital relationship

A Cooperative Co-Evolutionary Approach to Large-Scale Multisource Water Distribution Network Optimization

Potable water distribution networks (WDNs) are important infrastructures of modern cities. A good design of the network can not only reduce the construction expenditure but also provide reliable service. Nowadays, the scale of the WDN of a city grows dramatically along with the city expansion, which brings heavy pressure to its optimal design. In order to solve the large-scale WDN optimization problem, a cooperative co-evolutionary algorithm is proposed in this paper. First, an iterative trace-based decomposition method is specially designed by utilizing the information of water tracing to divide a large-scale network into small subnetworks. Since little domain knowledge is required, the decomposition method has great adaptability to multiform networks. Meanwhile, during optimization, the proposed algorithm can gradually refine the decomposition to make it more accurate. Second, a new fitness function is devised to handle the pressure constraint of the problem. The function transforms the constraint into a part of the objective to punish the infeasible solutions. Finally, a new suite of benchmark networks are created with both balanced and imbalanced cases. Experimental results on a widely used real network and the benchmark networks show that the proposed algorithm is promising.

An Approximation Technique for Fractional Order Delay Differential Equations

In this research article, an Iterative Decomposition Method is applied to approximate linear and non-linear fractional delay differential equation. The method was used to express the solution of a Fractional delay differential equation in the form of a convergent series of infinite terms which can be effortlessly computable.The method requires neither discretization nor linearization. Solutions obtained for some test problems using the proposed method were compared with those obtained from some methods and the exact solutions. The outcomes showed the proposed approach is more efficient and correct.

Multi-stage contingency-constrained co-planning for electricity-gas systems interconnected with gas-fired units and power-to-gas plants using iterative Benders decomposition

The increasing share of variable renewable energy sources and the improving requirements on system security and reliability are calling for important changes in the integrated energy systems. Hence this paper proposes a multi-stage contingency-constrained co-planning model for electricity-gas systems interconnected with gas-fired units and power-to-gas plants considering the uncertainties of load demand and wind power. The proposed model considers the long-term co-planning for electricity-gas systems with the short-term operation constraints and N-1 reliability criterion, which is formulated as a mixed-integer linear programming problem. First, a scenario reduction method based on affinity propagation clustering is utilized to handle the uncertainties. Afterwards, the N-1 criterion is tackled by a contingency screening method based on line outage distribution factors. Finally, an iterative Benders decomposition method is developed to divide the programming problem into one master investment problem and three subproblems. The iterative process will continue between master problem and each subproblem until an economic, reliable and fuel-supply feasible solution is obtained. In order to verify the feasibility of the proposed model and the effectiveness of the proposed method, case studies are conducted on two different scales of electricity-gas systems.

Quantitative Reconstruction of Dual K-Edge Contrast Agent Distribution for Photon- Counting Computed Tomography

Contrast-enhanced Computed Tomography (CT) imaging is very helpful for the detection of tumor metastasis and cancer cells. It is one of the most effective means of clinical imaging examination. The objective of this paper is to realize the quantitative reconstruction of two contrast agents at the same time and obtain the non-contrast-enhanced images in four energy bins as an additional product. This paper presents an iterative material decomposition method based on volume conservation and K-edge characteristics. Compared with other K-edge based material decomposition algorithms, the proposed method can simultaneously quantify the two contrast agents and make better use of the energy information provided by the photon-counting detector. The proposed algorithm was compared with the image-domain K-edge subtraction, Angular Rejection and filtered back projection (FBP) reconstruction. Numerical simulation and Monte Carlo simulation were used to verify the effectiveness of the proposed algorithm. We found that the proposed algorithm has higher efficiency and is meaningful for the quantification of contrast agent concentration. The proposed algorithm can achieve the quantitative separation of dual K-edge contrast agent by a single scan, which is significant for reducing the radiation dose to patients and improving the efficiency of material decomposition.

Optimized Adaptive Local Iterative Filtering Algorithm Based on Permutation Entropy for Rolling Bearing Fault Diagnosis.

The characteristics of the early fault signal of the rolling bearing are weak and this leads to difficulties in feature extraction. In order to diagnose and identify the fault feature from the bearing vibration signal, an adaptive local iterative filter decomposition method based on permutation entropy is proposed in this paper. As a new time-frequency analysis method, the adaptive local iterative filtering overcomes two main problems of mode decomposition, comparing traditional methods: modal aliasing and the number of components is uncertain. However, there are still some problems in adaptive local iterative filtering, mainly the selection of threshold parameters and the number of components. In this paper, an improved adaptive local iterative filtering algorithm based on particle swarm optimization and permutation entropy is proposed. Firstly, particle swarm optimization is applied to select threshold parameters and the number of components in ALIF. Then, permutation entropy is used to evaluate the mode components we desire. In order to verify the effectiveness of the proposed method, the numerical simulation and experimental data of bearing failure are analyzed.

A comparative analysis for the solution of nonlinear Burgers' equation.

In this work, a comparative study of seven well-known mathematical techniques for the coupled Burgers' equations is reported. The techniques involve in this comparison are as follows: Laplace transform Adomian decomposition method, Laplace transform homotopy perturbation method, Variational iteration method, Variational iteration decomposition method, Variational iteration homotopy perturbation method, the optimal homotopy asymptotic method, and OHAM with Daftardar-Jafari polynomial. Here we considered a practical example which consists of coupled Burgers' equations with the kinematic viscosity ε=1. Convergence and stability analysis is a major part of this analysis. After a careful observation, it is found that the variational iteration method has faster convergence than all the remaining methods. Adomian decomposition method and Homotopy perturbation method show weaker stability in comparison with other involved techniques.

Image decomposition and texture analysis via combined bi‐dimensional Bedrosian's principles

Image decomposition is an important issue in image processing. The existing approaches including the bi-dimensional empirical mode decomposition (BEMD) still fail to separate monocomponents in multicomponents in many cases. To solve this problem, this study proposes a new image decomposition method based on the new derived combined bi-dimensional Bedrosian's principle that has not been reported anywhere else for image processing. First, this study investigates a few bi-dimensional Bedrosian's principles according to the bi-dimensional Hilbert transforms. Second, based on the derived bi-dimensional Bedrosian's principles and the original multicomponents, the authors provide the combined bi-dimensional Bedrosian's principle and the assisted components obtained through projections via optimisation so that these monocomponents in multicomponents can be separated in the case that the existing methods fail. Third, an iterative image decomposition method is proposed via the above principles to decompose the multicomponent image into true monocomponents. The proposed method can solve the problems caused by the cross-angle and amplitude ratio and frequency ratio between these components that BEMD fails to solve. Also, the phase and amplitude are estimated for texture analysis after the decomposition is demonstrated. Experiments are shown to support the proposed methods.

Application of New Iterative Method and Adomian Decomposition Method to Hamel’s Flow Problem

Application of New Iterative Method and Adomian Decomposition Method to Hamel’s Flow Problem

Low-grazing angle propagation and scattering by an object above a highly conducting rough sea surface in a ducting environment from an accelerated MoM

In a previous paper, by combining three techniques, i.e. Subdomain Decomposition Iterative Method (SDIM), Adaptive Cross Approximation (ACA), and Forward–Backward Spectral Acceleration (FBSA), from the Method of Moments (MoM), a high-efficiency calculation of the propagation and scattering in ducting maritime environments has been proposed. In this paper, this algorithm is updated by adding a perfectly conducting object above the sea surface, assumed to be highly conducting, which makes the environment very complex. Then, to quantify the effect of the object on the total scattered field, the coherent and incoherent powers, with and without object, are simulated by considering a surface of 800,000 unknowns (length of 6 km and a frequency of 5 GHz).

A linearly approximated iterative Gaussian decomposition method for waveform LiDAR processing

Full-waveform LiDAR (FWL) decomposition results often act as the basis for key LiDAR-derived products, for example canopy height, biomass and carbon pool estimation, leaf area index calculation and under canopy detection. To date, the prevailing method for FWL product creation is the Gaussian Decomposition (GD) based on a non-linear Levenberg-Marquardt (LM) optimization for Gaussian node parameter estimation. GD follows a “greedy” approach that may leave weak nodes undetected, merge multiple nodes into one or separate a noisy single node into multiple ones. In this manuscript, we propose an alternative decomposition method called Linearly Approximated Iterative Gaussian Decomposition (LAIGD method). The novelty of the LAIGD method is that it follows a multi-step “slow-and-steady” iterative structure, where new Gaussian nodes are quickly discovered and adjusted using a linear fitting technique before they are forwarded for a non-linear optimization. Two experiments were conducted, one using real full-waveform data from NASA’s land, vegetation, and ice sensor (LVIS) and another using synthetic data containing different number of nodes and degrees of overlap to assess performance in variable signal complexity. LVIS data revealed considerable improvements in RMSE (44.8% lower), RSE (56.3% lower) and rRMSE (74.3% lower) values compared to the benchmark GD method. These results were further confirmed with the synthetic data. Furthermore, the proposed multi-step method reduces execution times in half, an important consideration as there are plans for global coverage with the upcoming Global Ecosystem Dynamics Investigation LiDAR sensor on the International Space Station.

Pyroshock Acceleration Field Reconstruction in Temporal and Spectral Domains Based on Laser Shock Scanning and Iterative Decomposition and Synthesis Considering Stop Band Effects

Pyrotechnic devices are used to separate substructures from main structures. Pyroshock can cause failure in electronic components that are sensitive to high frequency shock. Most of the existing methods to analyze pyroshock have limitations for high frequency simulations and are only available for simulation of point explosive-induced pyroshock. To solve the problem of existing methods, we developed a laser shock-based pyroshock reconstruction algorithm covering high frequency range that can predict linear explosive-induced pyroshock, as well as point explosive-induced ones. The developed algorithm reconstructs pyroshock from laser shock test in both temporal and spectral domains using an iterative signal decomposition and synthesis method. In the signal decomposition and synthesis process, unremoved signals in the stopbands occurred and were compensated by iteration to improve the results. At the end of this paper, various types of pyroshock were processed through the proposed method. Pyroshock wave propagation images and shock response spectrum images were presented as a result. To verify the algorithm, we compared the obtained result with a real pyroshock. The time domain signal was reconstructed with an averaged peak to peak acceleration difference of 20.21%, and the shock response spectrum was reconstructed with an average mean acceleration difference of 25.86%.

Propagation and Scattering in Ducting Maritime Environments From an Accelerated Boundary Integral Equation

For a 2-D problem and at microwave radar frequencies, in a previous paper, the field scattered by a highly conducting large rough sea surface in the presence of a duct having a linear-square refractive index profile has been computed. The forward–backward (FB) method has been applied to solve the linear system obtained by discretizing the boundary integral equation from the method of moments. To reduce again the complexity in order to solve huge problems, the subdomain decomposition iterative method is applied and combined with the FB spectral-acceleration for the calculation of the surface currents on each subsurface of the sea. The adaptive cross approximation algorithm is also applied to accelerate the coupling steps between the subsurfaces. Finally, this method is tested on the medium having a refractive index of parabolic profile, which is more realistic than a linear profile.

Ament Model for Multiple Rebounds From Rough Sea Surfaces in a Stratified Medium and Validation From the MoM-Based Multilevel SDIM

In this paper, the well-known Ament model is generalized to multiple rebounds (reflections) from rough sea surfaces in a stratified medium. To solve this propagation-scattering problem in the radar microwave domain, the physical optics (PO) approximation is applied to determine the currents on the rough sea surface. Next, to solve the propagation problem, i.e., to calculate the scattered field in the duct, an appropriate Green function of the duct (which is a stratified medium with linear-square refraction index profile) is used to radiate the currents from Huygens’ principle. Due to refraction phenomenon, this field induces new currents on the surface (second rebound), which are again evaluated by PO. The process is iterated. Then, for each rebound, the coherent components of the surface currents and the scattered field are calculated and compared with those of a flat surface in order to derive the generalized Ament model. In addition, this model is compared with the results computed by the rigorous method of moments and accelerated by the multilevel subdomain decomposition iterative method combined with the adaptive cross approximation.

Legendre Polynomial and Iterative Decomposition Methods for Linear Fractional Integro-Differential Equations

Legendre Polynomial and Iterative Decomposition Methods for Linear Fractional Integro-Differential Equations

Numerical Solutions of Three-Dimensional Coupled Burgers’ Equations by Using Some Numerical Methods

In this paper, we found the numerical solution of three-dimensional coupled Burgers’ Equations by using more efficient methods: Laplace Adomian decomposition method, Laplace transform homotopy perturbation method, variational iteration method, variational iteration decomposition method and variational iteration homotopy perturbation method. Example is examined to validate the efficiency and accuracy of these methods and they reduce the size of computation without the restrictive assumption to handle nonlinear terms and it gives the solutions rapidly.

Neuroanatomic localization of priming effects for famous faces with latency-corrected event-related potentials

The late components of event-related brain potentials (ERPs) pose a difficult problem in source localization. One of the reasons is the smearing of these components in conventional averaging because of trial-to-trial latency-variability. The smearing problem may be addressed by reconstructing the ERPs after latency synchronization with the Residue Iteration Decomposition (RIDE) method. Here we assessed whether the benefits of RIDE at the surface level also improve source localization of RIDE-reconstructed ERPs (RERPs) measured in a face priming paradigm. Separate source models for conventionally averaged ERPs and RERPs were derived and sources were localized for both early and late components. Jackknife averaging on the data was used to reduce the residual variance during source localization compared to conventional source model fitting on individual subject data. Distances between corresponding sources of both ERP and RERP models were measured to check consistency in both source models. Sources for activity around P100, N170, early repetition effect (ERE/N250r) and late repetition effect (LRE/N400) were reported and priming effects in these sources were evaluated for six time windows. Significant improvement in priming effect of the late sources was found from the RERP source model, especially in the Medio-Temporal Lobe, Prefrontal Cortex, and Anterior Temporal Lobe. Consistent with previous studies, we found early priming effects in the right hemisphere and late priming effects in the left hemisphere. Also, the priming effects in right hemisphere outnumbered the left hemisphere, signifying dominance of right hemisphere in face recognition. In conclusion, RIDE reconstructed ERPs promise a comprehensive understanding of the time-resolved dynamics the late sources play during face recognition.

A Decomposition Method for a Unilateral Contact Problem with Tresca Friction Arising in Electro-elastostatics

This article is concerned with the numerical modeling of unilateral contact problems in an electro-elastic material with Tresca friction law and electrical conductivity condition. First, we prove the existence and uniqueness of the weak solution of the model. Rather than deriving a solution method for the full coupled problem, we present and study a successive iterative (decomposition) method. The idea is to solve successively a displacement subproblem and an electric potential subproblem in block Gauss-Seidel fashion. The displacement subproblem leads to a constraint non-differentiable (convex) minimization problem for which we propose an augmented Lagrangian algorithm. The electric potential unknown is computed explicitly using the Riesz's representation theorem. The convergence of the iterative decomposition method is proved. Some numerical experiments are carried out to illustrate the performances of the proposed algorithm.