An innovative subdivision collocation algorithm for heat conduction equation with non-uniform thermal diffusivity.

This paper presents a subdivision collocation algorithm for numerically solving the heat conduction equation with non-uniform thermal diffusivity, considering both initial and boundary conditions. The algorithm involves transforming the differential form of the heat conduction equation into a system of equations and discretizing the time variable using the finite difference formula. The numerical solution of the system of heat conduction equations is then obtained. The feasibility of the algorithm is verified through theoretical and numerical analyses. Additionally, numerical and graphical representations of the obtained numerical solutions are provided, along with a comparison to existing methods. The results demonstrate that our proposed method outperforms the existing methods in terms of accuracy.

Computational investigation of stochastic Zika virus optimal control model using Legendre spectral method.

This study presents a computational investigation of a stochastic Zika virus along with optimal control model using the Legendre spectral collocation method (LSCM). By accumulation of stochasticity into the model through the proposed stochastic differential equations, we appropriating the random fluctuations essential in the progression and disease transmission. The stability, convergence and accuracy properties of the LSCM are conscientiously analyzed and also demonstrating its strength for solving the complex epidemiological models. Moreover, the study evaluates the various control strategies, such as treatment, prevention and treatment pesticide control, and identifies optimal combinations that the intervention costs and also minimize the proposed infection rates. The basic properties of the given model, such as the reproduction number, were determined with and without the presence of the control strategies. For R 0 < 0 , the model satisfies the disease-free equilibrium, in this case the disease die out after some time, while for R 0 > 1 , then endemic equilibrium is satisfied, in this case the disease spread in the population at higher scale. The fundamental findings acknowledge the significant impact of stochastic phonemes on the robustness and effectiveness of control strategies that accelerating the need for cost-effective and multi-faceted approaches. In last the results provide the valuable insights for public health department to enabling more impressive mitigation of Zika virus outbreaks and management in real-world scenarios.

A novel collocation method with a coronavirus optimization algorithm for the optimal control of COVID-19: A case study of Wuhan, China.

In this study, we develop a numerical optimization approach to address the challenge of optimal control in the spread of COVID-19. We evaluate the impact of various control strategies aimed at reducing the number of exposed and infectious individuals. Our novel approach employs Legendre wavelets, their derivative operational matrix, and a collocation method to transform the COVID-19 transmission optimal control model into a nonlinear programming (NLP) problem. To solve this problem, we employ a coronavirus optimization algorithm (COVIDOA) to determine the optimal control, state variables, and objective value. We investigate three control plans for this highly contagious disease, focusing on individual protection, rapid detection and treatment, detection with delay in treatment, and environmental viral dispersion as time-based control functions. These strategies are applied within an SEIR-type control model specific to COVID-19 in China, designed to mitigate disease spread. Lastly, we analyze the effects of various parameters within the COVID-19 spread model. Our numerical results highlight the significant impact of strategies that minimize the number of exposed and infectious individuals, particularly those related to rapid detection, detection delay, and environmental viral dispersion, in controlling and preventing the transmission of the COVID-19 virus.

Bernstein collocation technique for a class of Sturm-Liouville problems.

Sturm-Liouville problems have yielded the biggest achievement in the spectral theory of ordinary differential operators. Sturm-Liouville boundary value issues appear in many key applications in natural sciences. All the eigenvalues for the standard Sturm-Liouville problem are guaranteed to be real and simple, and the related eigenfunctions form a basis in a suitable Hilbert space. This article uses the weighted residual collocation technique to numerically compute the eigenpairs of both regular and singular Strum Liouville problems. Bernstein polynomials over [0,1] has been used to develop a weighted residual collocation approach to achieve an improved accuracy. The properties of Bernstein polynomials and the differentiation formula based on the Bernstein operational matrix are used to simplify the given singular boundary value problems into a matrix-based linear algebraic system. Keeping this fact in mind such a polynomial with space defined collocation scheme has been studied for Strum Liouville problems. The main reasons to use the collocation technique are its affordability, ease of use, well-conditioned matrices, and flexibility. The weighted residual collocation method is found to be more appealing because Bernstein polynomials vanish at the two interval ends, providing better versatility. A multitude of test problems are offered along with computation errors to demonstrate how the suggested method behaves. The numerical algorithm and its applicability to particular situations are described in detail, along with the convergence behavior and precision of the current technique.

Graded mesh B-spline collocation method for two parameters singularly perturbed boundary value problems.

The solutions of two parameters singularly perturbed boundary value problems typically exhibit two boundary layers. Because of the presence of these layers standard numerical methods fail to give accurate approximations. This paper introduces a numerical treatment of a class of two parameters singularly perturbed boundary value problems whose solution exhibits boundary layer phenomena. A graded mesh is considered to resolve the boundary layers and collocation method with cubic B-splines on the graded mesh is proposed and analyzed. The proposed method leads to a tri-diagonal linear system of equations. The stability and parameters uniform convergence of the present method are examined. To verify the theoretical estimates and efficiency of the method several known test problems in the literature are considered. Comparisons to some existing results are made to show the better efficiency of the proposed method. Summing up:•The present method is found to be stable and parameters uniform convergent and the numerical results support the theoretical findings.•Experimental results show that the present method approximates the solution very well and has a rate of convergence of order two in the maximum norm.•Experimental results show that cubic B-spline collocation method on graded mesh is more efficient than cubic B-spline collocation method on Shishkin mesh and some other existing methods in the literature.

Double-time-scale non-probabilistic reliability-based controller optimization for manipulator considering motion error and wear growth.

Motion error and wear growth are two crucial factors that determine the performance of the manipulator system. The two factors are distinct from the sensitivity to time and respectively indicate its control performance and lifespan level. In this paper, a double-time-scale non-probabilistic reliability (DTSNPR)-based optimization method, which considers time-sensitive factor motion error as time-dependent reliability (TDR) and time-insensitive factor clearance wear growth as time-independent reliability (TIR), is proposed to comprehensively evaluate and optimize the controller of the manipulator system in consideration of these multi-scale factors. Meanwhile, for a highly nonlinear response problem of a manipulator system, the adaptive subinterval collocation method (ASICM) which transfers the highly nonlinear uncertainty propagation problem into a sequence of small subintervals, is adopted to obtain the response range more precisely. The proposed DTSNPR-based optimization method is applied to three numerical manipulator systems and ensures each of them under a predefined level of reliability upon both motion error and wear growth. The results also indicate that the proposed ASICM owns an advantage over computational cost and precision, with only 0.4% error and 1% computational cost compared with the Monte-Carlo methods.

Linear barycentric rational collocation method for solving generalized Poisson equations.

We consider the Poisson equation by collocation method with linear barycentric rational function. The discrete form of the Poisson equation was changed to matrix form. For the basis of barycentric rational function, we present the convergence rate of the linear barycentric rational collocation method for the Poisson equation. Domain decomposition method of the barycentric rational collocation method (BRCM) is also presented. Several numerical examples are provided to validate the algorithm.

Analyses of entropy generation in micropolar liquid using overlapping multi-domain spectral quasilinearization method.

In this article, entropy generation on Micropolar fluid through a perpendicular hot sheet is studied. The governing equations of the mathematical model of this article will transfer to non-dimensional system of ordinary differential equations by similarity transformations. This system of non-dimensional ODEs will be solved by a newly developed spectral collocation method. The dimensionless mathematical calculations are handled through utilizing the spectral quasilinearization method (SQLM) along with the concept of overlapping grids. Condition numbers, residual error and solution error norms estimations are provided to appraise accuracy, convergence and the stability of this numerical method. Using the overlapping multi-domain (M-D) approach has been found to produce most accurate, stable and convergent solutions when contrasted to the single-domain (S-D) technique. The outcomes are displayed and depicted graphically and through tabular forms. The effects of different non-dimensional parameters on flow, temperatures, concentration and entropy generation are studied. The accuracy of the numerical method has been checked through comparison with previously published articles and error analyses test.

Full-Waveform Inversion for Breast Ultrasound Tomography Using Line-Shape Modeled Elements.

OBJECTIVE: The objective of the work described here was to incorporate the spatial shapes of the transducer elements into the framework of the full-waveform inversion. METHODS: An element is treated as its cross-section in the 2-D imaging plane, that is, a line segment. The elements are not simply modeled as a set of point sources on their surface to avoid staircasing artifacts. By use of the Fourier collocation method, an element is spatially represented as the discrete convolution between its spatial distribution and a band-limited delta function. The excitation pulses on the emitters and recorded signals on the receivers are then weighted based on the discrete convolution results. Digital and physical experiments are implemented to validate the method. DISCUSSION: It is meaningful to model the shapes of the elements if their spatial sizes are similar to or larger than the acoustic wavelengths. It should, however, be noted that because this article focuses on 2-D imaging, the inter-plane effects are not considered. CONCLUSION: The approach helps reduce the root mean square errors and increase the structural similarity of the reconstructed images. It also helps to improve the stability of convergence and to accelerate the convergence speed.

A new spline technique for the time fractional diffusion-wave equation.

The current research article proposes an approximate solution of the fractional diffusion wave equation (FDWE) by using a new collocation method based on the cubic B-splines. The fractional derivative in the time direction is considered in Caputo form. The theoretical research of the proposed algorithm is discussed with L ∞ and H 1 norms. The method presented in this article is found to be of order (∆t 3- α + h 4). The highlights of the current technique proposed in this article are as under:•The method is high-order collocation and uses a compact stencil. The error analysis is discussed to authenticate the order of convergence of the proposed numerical approximation.•The comparisons of errors with the already existing methods are done and observed that our method produces more accurate results than the methods presented in the literature.

A two-stage dimension-reduced dynamic reliability evaluation (TD-DRE) method for vibration control structures based on interval collocation and narrow bounds theories.

Due to the uncertainties from modeling, manufacturing, and working environments, many vibration active control systems usually show dynamic uncertain properties. Hence structural reliability estimation benchmarking to full-cycle vibratory responses is vitally important. In this study, a novel two-stage dimension-reduced dynamic reliability evaluation (TD-DRE) method for linear quadratic regulator (LQR) controlled structures is developed. This method combines interval uncertainties and the time-variant reliability (TVR) concept. In the first stage, the Taylor series expansion is employed to analyze several typical limit states for definition of the time-discretized dynamic reliability. Then the interval collocation method tackles the solution. In the second stage, the TVR problem is indeed transformed to a time-invariant reliability (TIR) problem. Furthermore, the narrow bounds theorem deduces the presented TD-DRE index. Eventually, two application examples are utilized to verify the effectiveness and accuracy of the proposed method. The proposed TD-DRE is more accurate than the traditional first-order Taylor expansion and more effective than the first-passage reliability evaluation method. This method can provide a reference and an initial value for further design, and improve the efficiency of LQR controller design in practical engineering.

Significance of Convection and Internal Heat Generation on the Thermal Distribution of a Porous Dovetail Fin with Radiative Heat Transfer by Spectral Collocation Method.

A variety of methodologies have been used to explore heat transport enhancement, and the fin approach to inspect heat transfer characteristics is one such effective method. In a broad range of industrial applications, including heat exchangers and microchannel heat sinks, fins are often employed to improve heat transfer. Encouraged by this feature, the present research is concerned with the temperature distribution caused by convective and radiative mechanisms in an internal heat-generating porous longitudinal dovetail fin (DF). The Darcy formulation is considered for analyzing the velocity of the fluid passing through the fin, and the Rosseland approximation determines the radiation heat flux. The heat transfer problem of an inverted trapezoidal (dovetail) fin is governed by a second-order ordinary differential equation (ODE), and to simplify it to a dimensionless form, nondimensional terms are utilized. The generated ODE is numerically solved using the spectral collocation method (SCM) via a local linearization approach. The effect of different physical attributes on the dimensionless thermal field and heat flux is graphically illustrated. As a result, the temperature in the dovetail fin transmits in a decreasing manner for growing values of the porosity parameter. For elevated values of heat generation and the radiation-conduction parameter, the thermal profile of the fin displays increasing behavior, whereas an increment in the convection-conduction parameter downsizes the thermal dispersal. It is found that the SCM technique is very effective and more conveniently handles the nonlinear heat transfer equation. Furthermore, the temperature field results from the SCM-based solution are in very close accordance with the outcomes published in the literature.

Unravelled multilevel transformation networks for predicting sparsely observed spatio-temporal dynamics.

In this paper, we address the problem of predicting complex, nonlinear spatio-temporal dynamics when available data are recorded at irregularly spaced sparse spatial locations. Most of the existing deep learning models for modelling spatio-temporal dynamics are either designed for data in a regular grid or struggle to uncover the spatial relations from sparse and irregularly spaced data sites. We propose a deep learning model that learns to predict unknown spatio-temporal dynamics using data from sparsely-distributed data sites. We base our approach on the radial basis function (RBF) collocation method which is often used for meshfree solution of partial differential equations. The RBF framework allows us to unravel the observed spatio-temporal function and learn the spatial interactions among data sites on the RBF-space. The learned spatial features are then used to compose multilevel transformations of the raw observations and predict its evolution in future time steps. We demonstrate the advantage of our approach using both synthetic and real-world climate data. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.

Effect of vaccination on the transmission dynamics of COVID-19 in Ethiopia.

Governments and health officials are eager to gain a thorough understanding of the dynamics of COVID-19 transmission in order to devise strategies to mitigate the pandemic's negative effects. As a result, we created a new fractional order mathematical model to investigate the dynamics of Covid-19 vaccine transmission in Ethiopia. The nonlinear system of differential equations for the model is represented using Atangana-Baleanu fractional derivative in Caputo sense and the Jacobi spectral collocation method is used to convert this system into an algebraic system of equations, which is then solved using inexact Newton's method. The fundamental reproduction number, R 0 for the proposed model is determined using the next generation matrix approach.

Transmission dynamic of stochastic hepatitis C model by spectral collocation method.

In this research work, we present an exciting mathematical analysis of a stochastic model, using a standard incidence function, for infectious disease hepatitis C transmission dynamics. In this model, we divided the infected population into three different classes with two different infection stages known as chronic class and acute class while the third is an isolation class. We also presents briefly the Legendre spectral method for the numerical solution to the proposed model. It is observed that the disease-free equilibrium is asymptotically stable, when basic reproduction number R0<1. It is also shown that the proposed model has a stable endemic equilibrium when the reproduction number R0>1. Also, sensitivity analysis is carried out to study and identify the effect of parameters on R0. Moreover, we have performed numerical simulations to study the influence of disease free equilibrium and endemic equilibrium. Legendre polynomial and Legendre weight function are used to solve the proposed stochastic system numerically. Numerical results are compared against the basic reproduction number.

Radial basis collocation method with parameters optimized for estimating pollutant release history.

The identification of pollutant source release history in rivers is important for emergency response of pollution accidents and formulating remediation strategies. Space-time radial basis collocation method (RBCM), as a meshless method with strong applicability, can directly estimate the release history from the concentration data measured at downstream observation sites. However, the uncertainty of specific parameters in space-time RBCM is the main factor affecting the accuracy of estimation. Therefore, a way to solve the parameters efficiently and accurately is essential. For this purpose, a new model which combines space-time RBCM and differential evolution algorithm (DEA) is established to identify the source release history. First of all, efficient parameter optimizer DEA is introduced to search the parameters that affect the estimation accuracy of space-time RBCM. Then, a new loss function considering the imbalance configuration of RBCM nodes is designed to ensure the rationality of the parameters obtained by DEA. The results of numerical cases and real field case show that the proposed method can accurately estimate the real release history with low time consumption. It is also demonstrated that DEA is more efficient than k-fold cross-validation in searching the optimal parameters for space-time RBCM, and the parameters obtained from the new loss function can make the estimated release history more precise.

The triple-deck stage of marginal separation.

The method of matched asymptotic expansions is applied to the investigation of transitional separation bubbles. The problem-specific Reynolds number is assumed to be large and acts as the primary perturbation parameter. Four subsequent stages can be identified as playing key roles in the characterization of the incipient laminar-turbulent transition process: due to the action of an adverse pressure gradient, a classical laminar boundary layer is forced to separate marginally (I). Taking into account viscous-inviscid interaction then enables the description of localized, predominantly steady, reverse flow regions (II). However, certain conditions (e.g. imposed perturbations) may lead to a finite-time breakdown of the underlying reduced set of equations. The ensuing consideration of even shorter spatio-temporal scales results in the flow being governed by another triple-deck interaction. This model is capable of both resolving the finite-time singularity and reproducing the spike formation (III) that, as known from experimental observations and direct numerical simulations, sets in prior to vortex shedding at the rear of the bubble. Usually, the triple-deck stage again terminates in the form of a finite-time blow-up. The study of this event gives rise to a noninteracting Euler-Prandtl stage (IV) associated with unsteady separation, where the vortex wind-up and shedding process takes place. The focus of the present paper lies on the triple-deck stage III and is twofold: firstly, a comprehensive numerical investigation based on a Chebyshev collocation method is presented. Secondly, a composite asymptotic model for the regularization of the ill-posed Cauchy problem is developed. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1007/s10665-021-10125-3.

Elastic Properties Measurement Using Guided Acoustic Waves.

Nondestructive evaluation of elastic properties plays a critical role in condition monitoring of thin structures such as sheets, plates or tubes. Recent research has shown that elastic properties of such structures can be determined with remarkable accuracy by utilizing the dispersive nature of guided acoustic waves propagating in them. However, existing techniques largely require complicated and expensive equipment or involve accurate measurement of an additional quantity, rendering them impractical for industrial use. In this work, we present a new approach that requires only a pair of piezoelectric transducers used to measure the group velocities ratio of fundamental guided wave modes. A numerical model based on the spectral collocation method is used to fit the measured data by solving a bound-constrained nonlinear least squares optimization problem. We verify our approach on both simulated and experimental data and achieve accuracies similar to those reported by other authors. The high accuracy and simple measurement setup of our approach makes it eminently suitable for use in industrial environments.

Numerical computing approach for solving Hunter-Saxton equation arising in liquid crystal model through sinc collocation method.

In this study, numerical treatment of liquid crystal model described through Hunter-Saxton equation (HSE) has been presented by sinc collocation technique through theta weighted scheme due to its enormous applications including, defects, phase diagrams, self-assembly, rheology, phase transitions, interfaces, and integrated biological applications in mesophase materials and processes. Sinc functions provide the procedure for function approximation over all types of domains containing singularities, semi-infinite or infinite domains. Sinc functions have been used to reduce HSE into an algebraic system of equations that makes the solution quite superficial. These algebraic equations have been interpreted as matrices. This projected that sinc collocation technique is considerably efficacious on computational ground for higher accuracy and convergence of numerical solutions. Stability analysis of the proposed technique has ensured the accuracy and reliability of the method, moreover, as the stability parameter satisfied the condition the proposed solution of the problem converges. The solution of the HSE is presented through graphical figures and tables for different cases that are constructed on various values of θ and collocation points. The accuracy and efficiency of the proposed technique is analyzed on the basis of absolute errors.

Kansa Method for Unsteady Heat Flow in Nonhomogenous Material with a New Proposal of Finding the Good Value of RBF's Shape Parameter.

New engineering materials exhibit a complex internal structure that determines their properties. For thermal metamaterials, it is essential to shape their thermophysical parameters' spatial variability to ensure unique properties of heat flux control. Modeling heterogeneous materials such as thermal metamaterials is a current research problem, and meshless methods are currently quite popular for simulation. The main problem when using new modeling methods is the selection of their optimal parameters. The Kansa method is currently a well-established method of solving problems described by partial differential equations. However, one unsolved problem associated with this method that hinders its popularization is choosing the optimal shape parameter value of the radial basis functions. The algorithm proposed by Fasshauer and Zhang is, as of today, one of the most popular and the best-established algorithms for finding a good shape parameter value for the Kansa method. However, it turns out that it is not suitable for all classes of computational problems, e.g., for modeling the 1D heat conduction in non-homogeneous materials, as in the present paper. The work proposes two new algorithms for finding a good shape parameter value, one based on the analysis of the condition number of the matrix obtained by performing specific operations on interpolation matrix and the other being a modification of the Fasshauer algorithm. According to the error measures used in work, the proposed algorithms for the considered class of problem provide shape parameter values that lead to better results than the classic Fasshauer algorithm.