Difference Method In Order Research Articles

A computational study of the effects of graphene additions on electrical properties of polycrystalline copper

The addition of graphene has recently shown promise as a route for the significant improvement of the bulk electrical properties of metallic materials. We explore the effects these additions have on the net electrical conductivity of fabricated copper-graphene (Cu-Gr) nanocomposites as a function of grain structure and grain boundary properties. Synthetic 3D microstructures were generated to represent polycrystalline copper with different average grain diameters and twinned grain boundary fractions. Then, the Poisson equation of electrical transport was solved using a finite difference method in order to predict the net electrical conductivity of each microstructure. In this context, the potential effect of graphene on the conductivity of the composite was evaluated as a function of the number of affected grain boundaries. The results of these calculations indicate that 1.) as supported by literature, net electrical conductivity decreases with decreasing grain size, 2.) the presence of twinned grain boundaries results in smaller loss of conductivity than would otherwise be expected, and 3.) the presence of graphene on the grain boundaries can be expected to lead to improvements in net electrical conductivity. However, we also find that 4.) when the Cu grain structure becomes sufficiently refined, the addition of graphene could conceivably result in significant improvements in electrical conductivity over and above coarse-grained Cu. It is estimated from our calculations that, assuming microstructures with average grain sizes between 100 nm and 100 μm and graphene conductivity 1000 to 10,000 that of a typical Cu grain boundary, an improvement in electrical conductivity of approximately 17 % over that of bulk Cu may be attainable. Therefore, by performing this study we suggest a possible route for the improvement of Cu electrical properties through the addition of graphene.

A second order difference method combined with time two-grid algorithm for two-dimensional time-fractional Fisher equation

In this paper, a finite difference (FD) scheme based on time two-grid algorithm is proposed for solving the two-dimensional time-fractional Fisher equation (2D-TFFE). Firstly, the Caputo fractional derivative and the spatial derivative are discretized by the L 2 − 1 σ formula and the central difference formula, respectively. In order to improve the efficiency of computation, the time two-grid algorithm is then constructed. Secondly, based on the cut-off function technic, stability and convergence of the time two-grid FD scheme are obtained by the energy method, and the global convergence order is O ( τ F 2 + τ C 4 + h x 2 + h y 2 ) , where τ F and τ C represent time-step sizes on the fine and coarse grid, respectively, while h x and h y represent the space-step sizes. Finally, numerical experiments are presented to show the feasibility and efficiency of the algorithm.

A weighted combination of reproducing kernel particle shape functions with cardinal functions of scalable polyharmonic spline radial kernel utilized in Galerkin weak form of a mathematical model related to anti-angiogenic therapy

In this manuscript, a new localized meshfree (meshless) approximation, i.e., a combination of the reproducing kernel particle (RKP) shape functions with the cardinal functions of the scalable polyharmonic spline radial kernel with polynomial augmentation (PHS+poly) is introduced. It is called the RKP+PHS+poly approximation, and its convergence rate is of order O(hm+1), where m is the total degree of polynomials. We apply this method to construct the spaces of trial and test functions in a Galerkin scheme of an extended version of the biological mathematical model in two dimensions describing the interactions between endothelial cells, fibronectin, angiogenic growth factors, and fasentin concentrations. By considering the row-sum method, the eigenvalue stability is also numerically carried out for the discrete equations corresponding to the obtained weak formulation. Accordingly, a semi-implicit form of the backward difference method of order 1 (SBDF1) has been utilized to approximate the weak form in time. We complete our numerical algorithm by solving the obtained full-discretized problem overtime via the biconjugate gradient stabilized (BiCGSTAB) solver with a proper preconditioner. Some simulation results are investigated by estimating the maximum velocity parameter of an enzymatic reaction from the experimental dose–response curve of fasentin to demonstrate the effect of using fasentin drug.

Mathematical modeling of functionally graded porous geometrically nonlinear micro/nano cylindrical panels

Relevance. The study investigates the problem of stress-strain state and stability of porous functional-gradient size-dependent cylindrical panels. The composition and properties of alloys can differ and significantly affect the performance characteristics of products. Therefore, the research of material properties is relevant and contributes to the creation of new types of products demanded by the oil and gas industry. Aim. Development of a new model and creation of accurate methods for analyzing the stress-strain state of porous functional-gradient size-dependent micro/nano cylindrical panels taking into account geometrical nonlinearity. Methods. The method of variational iterations – the extended Kantorovich method is used to analyze the stress-strain state of cylindrical panels. The validity of the results is ensured by comparing the solutions obtained by the method of variational iterations in the first and second approximations with the solutions obtained by the authors, by the Bubnov–Galerkin method in higher approximations, by the finite difference method of the second order of accuracy, for which their convergence is investigated depending on a number of partitions of the integration area in the finite difference method and the number of series terms in the expansion of the basic functions in the Bubnov–Galerkin method. The results obtained by these methods are compared with the solutions obtained by other authors. It should be noted that the solutions obtained by the method of variational iterations for bending of functionally graded cylindrical panels under the action of transverse uniformly distributed load can be considered accurate. Results and conclusions. The authors have constructed the model of porous functional-gradient size-dependent cylindrical panels. Its use will allow studying the properties of alloys for producing drill pipes. The influence of material porosity type, porosity index, functional-gradient index, boundary conditions, size-dependent parameter, curvature parameters on the stress-strain state of cylindrical panels was analyzed using the developed method of variational iterations.

Estimation of Soil Moisture Content Based on Fractional Differential and Optimal Spectral Index

Applying hyperspectral remote sensing technology to the prediction of soil moisture content (SMC) during the growth stage of soybean emerges as an effective approach, imperative for advancing the development of modern precision agriculture. This investigation focuses on SMC during the flowering stage under varying nitrogen application levels and film mulching treatments. The soybean canopy’s original hyperspectral data, acquired at the flowering stage, underwent 0–2-order differential transformation (with a step size of 0.5). Five spectral indices exhibiting the highest correlation with SMC were identified as optimal inputs. Three machine learning methods, namely support vector machine (SVM), random forest (RF), and back propagation neural network (BPNN), were employed to formulate the SMC prediction model. The results indicate the following: (1) The correlation between the optimal spectral index of each order, obtained after fractional differential transformation, and SMC significantly improved compared to the original hyperspectral reflectance data. The average correlation coefficient between each spectral index and SMC under the 1.5-order treatment was 0.380% higher than that of the original spectral index, with mNDI showing the highest correlation coefficient at 0.766. (2) In instances of utilizing the same modeling method with different input variables, the SMC prediction model’s accuracy follows the order: 1.5 order > 2.0 order > 1.0 order > 0.5 order > original order. Conversely, with consistent input variables and a change in the modeling method, the accuracy order becomes RF > SVM > BPNN. When comprehensively assessing model evaluation indicators, the 1.5-order differential method and RF method emerge as the preferred order differential method and model construction method, respectively. The R2 for the optimal SMC estimation model in the modeling set and validation set were 0.912 and 0.792, RMSEs were 0.005 and 0.004, and MREs were 2.390% and 2.380%, respectively. This study lays the groundwork for future applications of hyperspectral remote sensing technology in developing soil moisture content estimation models for various crop growth stages and sparks discussions on enhancing the accuracy of these different soil moisture content estimation models.

Time two-grid technique combined with temporal second order difference method for semilinear fractional diffusion-wave equations

Time two-grid technique combined with temporal second order difference method for semilinear fractional diffusion-wave equations

The Impact of Wind Power and Load Power Fluctuations on Energy Storage Sizing

The study presents a method of taking into account the impact of wind power and load power fluctuations on the energy storage sizing, comprised of batteries of identical capacity. To account the impact, two methods of calculating the difference between wind power generation and load consumption were presented over some time interval: 1st and 2nd order difference methods. Each of the methods can be parameterized and non-parameterized method with and without taking into account parameters respectively, where the parameters are: discharge current, required discharge duration, depth of discharge, battery capacity, Peukert’s constant, discharge time from 100% capacity, ambient temperature and wind power prediction error. Using the parameterized method compared allows to refine the value of the number of batteries. Using the 2nd order difference method compared to the 1st order difference method can significantly reduce the required number of batteries.

Improvement of streamflow simulation by combining physically hydrological model with deep learning methods in data-scarce glacial river basin

Robust streamflow simulation at glacial basins is essential for the improvement of water sustainability assessment, water security evaluation, and water resource management under the rapidly changing climate. Therefore, we proposed a hybrid modelling framework to link the SWAT+ model considering glacial hydrological processes (GSWAT+) with Gated Recurrent Unit (GRU) neural networks to improve the model simulations and to establish a framework for the robust simulation and forecast of high and low flows in glacial river basins, which could be further used for the explorations of extreme hydrological events under a warming climate. The performance of different models (GSWAT+, GRU, and GRU-GSWAT+, respectively) were thoroughly investigated based on numerical experiments for two data-scarce glacial watersheds in Northwest China. The results suggested that the hybrid model (GRU-GSWAT+) outperformed both the individual deep learning (DL) model (GRU) and the conventional hydrological model (GSWAT+) in terms of simulation and prediction accuracy. Notably, the proposed hybrid model considerably enhanced the simulations of low and high flows that the conventional GSWAT+ failed to capture. Furthermore, utilizing suitable data integration (DI) schemes on feature and target sequences can substantially help to strengthen model stability and representativeness for monthly and annual streamflow sequences. Specifically, introducing one order differential method and decomposition approach, such as ensemble empirical signal decomposition (EEMD) and complete EEMD with adaptive noise (CEEMDAN), into feature and target sequences enriched the learnable ancillary information, which consequently strengthened the predictive performance of the proposed model. Overall, the proposed hybrid model with the suitable DI scheme has the potential to significantly enhance the accuracy of streamflow simulation in data-scarce glacial river basins. This hybrid model not only upheld the fundamental physical principles from the GSWAT+ model, but also considerably mitigated the accumulated bias errors, which caused by the shortage of climate data and inadequate hydrological principles, by using DL based model and DI schemes.

An improved cubic approximation for Kepler’s equation

ABSTRACT A novel cubic Pade approximation of sin(e sin E) is used to solve Kepler’s equation and compute the eccentric anomaly with high accuracy without requiring iteration. It requires computation of sin, cos, atan, sqrt, and a cube root. A refinement of the higher order difference methods is described that is faster and gives improved numerical accuracy.

Effect of rotation on the suspension of phototactic bioconvection

In this article, we examine the effect of rotation on the suspension of the phototactic bioconvection model. Around a vertical axis, the suspension is rotated at a uniform angular velocity. During the study, two distinct combinations of the upper boundary conditions were considered. In order to solve the eigenvalue problem, the Newton–Raphson–Kantorovich finite difference method of order four is used. Linear analysis of the basic state is performed using neutral curves. We found that rotation has stabilizing effects on the system. According to observations, rigid boundaries should be preferred over other types of boundaries for preventing convection as they stabilize it more quickly. The results demonstrate a change in the most unstable mode from an overstable to a stationary state for particular parameters in response to a variation in the Taylor number. The rigid upper surface case often exhibits oscillatory instabilities at Taylor number increments. The impacts of the various other factors on the system's instability are discussed in detail for both upper boundaries.

Acceleration of Radiation Analysis Using an Arbitrary High Order Difference Method with Non-uniform Mesh

Acceleration of Radiation Analysis Using an Arbitrary High Order Difference Method with Non-uniform Mesh

Analysis of New Type of Second-order Fractional Linear Multi-step Method with Improved Stability

We present and investigate a new type of implicit fractional linear multi-step method of order two for fractional initial value problems. The method is obtained from the second-order superconvergence of the Grünwald-Letnikov approximation of the fractional derivative at a non-integer shift point. The method coincides with the backward difference method of order two for the classical initial value problem when the order of the derivative is one. The weight coefficients of the proposed method are obtained from the Grünwald weights and are hence computationally efficient compared with that of the fractional backward difference formula of order two. The stability properties are analyzed and it is shown that the stability region of the method is larger than that of the fractional Adams-Moulton method of order two and the fractional trapezoidal method. Numerical results and illustrations are presented to justify the analytical theories.

Pulsatile flow of thixotropic blood in artery under external body acceleration

This work presents a numerical technique for simulating non-Newtonian blood flow in human’s arteries driven by an oscillating pressure gradient. The blood is considered as a thixotropic fluid and its structural properties are considered to obey Moore’s thixotropic model as a constitutive equation. The equations of motion are simplified considering the flow laminar, axisymmetric and the fluid incompressible. A numerical solution is presented using finite difference method in order to compute the velocity field and wall shear stress distribution. The numerical results obtained have been validated with the analytical solution available in the literature. Furthermore, the effect of the structural properties, the average of the pressure gradient and the external acceleration on the velocity and wall shear stress distribution is investigated. These results reveal the influence of the different parameters studied on the pipe flow response of the thixotropic fluid.

Numerical prediction of vortex-induced vibrations of a long flexible riser with an axially varying tension based on a wake oscillator model

A numerical study based on the wake oscillator model has been conducted to determine the vortex-induced vibration (VIV) responses of a flexible riser with an axial time-varying tension. Three different types of flows, viz., linear shear, exponential shear, and real stepped flows, have been considered. The coupling equations of a structural oscillator and a wake oscillator have been solved using a standard central finite difference method of the second order. The VIV response characteristics including the structural displacement, structural frequency, displacement envelope, and displacement evolution for three different flow profiles have been systematically compared. Subsequently, for each flow profile, the effect of the different tension models on the VIV characteristics has been investigated. The numerical results indicated that, both the VIV displacement and VIV frequency in real stepped flow had diverse characteristics from those in linear shear flow. Compared with the corresponding VIV characteristics in real stepped flow, the VIV displacement in exponential shear flow was similar, however, the VIV frequency in exponential shear flow was disparate. The VIV displacement with the constant tension model had larger values than that with the real tension model.

L-stable and A-stable numerical method of order two for stiff differential equation

This article presents a finite difference method of order two for stiff differential equation that will overcome the effects of stiffness since it is A-stable. And, reflect the asymptotic behavior of the solution of the stiff problem since it is L-stable. The classical explicit Runge–Kutta methods of order two are not suitable for stiff problems since the numerical solution will not overcome the effects of stiffness and will not reflect the asymptotic behavior of the solution of the stiff problem (for example, Dahlquist problem). And, they are not both A-stable and L-stable. On applying Euler’s implicit method, the numerical solution will overcome the effects of stiffness and will reflect the asymptotic behavior of the solution of the stiff problem. And hence, it is both A-stable and L-stable. But it is of order one method. A new numerical method which is both A-stable and L-stable and of order two for the numerical solution of a stiff differential equation is presented in this article. It will overcome the effects of stiffness and reflects the asymptotic behavior of the solution. The new method works well for large values of step size and works well in large domain. The new method is a modified form of the mid-point rule for the stiff problems. The rate of order of convergence is proved to be two both theoretically and numerically. Experimental results show the performance of the method based on the metrics such as stability function, stability region, order star fingers, the rate of order of theoretical and numerical convergence, absolute and relative errors, percentage of numerical solution accuracy and local and global truncation errors both numerically and graphically.

Numerical solution of singularly perturbed Fredholm integro-differential equations by homogeneous second order difference method

This work presents a computational approximate to solve singularly perturbed Fredholm integro-differential equation with the reduced second type Fredholm equation. This problem is discretized by a finite difference approximate, which generates second-order uniformly convergent numerical approximations to the solution. Parameter-uniform approximations are generated using Shishkin type meshes. The performance of the numerical scheme is tested which supports the effectiveness of the technique.

Time two-grid technique combined with temporal second order difference method for two-dimensional semilinear fractional sub-diffusion equations

In this paper, we construct a high order difference scheme for two-dimensional semilinear fractional sub-diffusion equations at first. To reduce the computation time, an efficient time two-grid algorithm is then proposed. The global convergence order of the two-grid scheme reaches O(τF2+τC4+h12+h22), where τF and τC represent the time-step sizes on the fine and coarse grids respectively, while h1 and h2 are the space-step sizes. Furthermore, stability and convergence of the scheme are carefully studied by some skills. At last, the theoretical statements are verified by numerical experiments.

Mathematical modeling of transdermal drug delivery using microneedle

This work presents a closed-form analytical and numerical technique that is developed to investigate drug diffusion from sustained release systems (transdermal drug delivery using microneedle). Diffusion is known as predominant transport phenomenon in drug delivery systems. Therefore, parameter optimization in relation to diffusional mass transport, such as diffusion coefficient, initial concentration and device length were the essential keys to achieve perfect drug control. A closed form analytical solution was derived from Fick's second law and a numerical solution is presented using finite difference method in order to compute the concentration distribution. Furthermore, the effect of the diffusion coefficient, initial concentration and device length on the concentration diffusion is investigated. Results reveal the influence of the different parameters studied on diffusion and concentration over time and space.

Convergence analysis of the homogeneous second order difference method for a singularly perturbed Volterra delay-integro-differential equation

• A new approach for designing the computational method has been considered. • This new approach results in a local truncation error containing much lower derivatives of exact solution compared to classical methods. • The method allows to weaken or get rid of the smoothness of the data functions, a determining factor for convergence analysis. • The previous works related to Volterra integro-differential equation were only concerned with regular cases. • Problems involving boundary layers have a solution with bad behaviours in applications encountered in different fields of engineering. A linear Volterra delay-integro-differential equation with a singular perturbation parameter ε is considered. The problem is discretized using exponentially fitted schemes on the Shishkin type meshes. It is proved that the numerical approximations generated by this method are O ( N − 2 ln N ) convergent in the discrete maximum norm, where N is the mesh parameter. Numerical results show a good agreement with the theoretical analysis.

Three-dimensional vortex-induced vibrations of a circular cylinder predicted using a wake oscillator model

This paper reports a numerical study based on a wake oscillator model, to determine the three-dimensional vortex-induced vibration (VIV) responses on a flexible cylinder with pinned-pinned boundary conditions subjected to a uniform flow. Four different aspect ratios have been selected for the study. The coupling equations of the structural oscillator models and wake oscillator models in both the cross-flow (CF) and in-line (IL) directions have been solved using a standard central difference method of the second order. The structural displacement, structural frequency, response wave pattern, response trajectory, and lift force coefficient, for four aspect ratios, have been compared. The numerical results establish that, for a small aspect ratio, the CF displacements have absolute standing wave behaviors without travelling wave behaviors, and the IL displacements have dominant standing wave behaviors with slight travelling wave behaviors. Further, the VIV trajectory is repeatable and displays figure-eight shapes. However, for a large aspect ratio, the CF displacements display identical characteristics with the IL displacements for a small aspect ratio, and the IL displacements for a large aspect ratio are simultaneously dominated by standing and travelling wave behaviors. Moreover, the VIV trajectory is apparently aperiodic and shows chaotic shapes.