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In this research, the numerical integral method procedure on uniform mesh is used to solve the singularly perturbed problem which has integral boundary value. This method also includes the trapezoid method, the finite difference method, and the Thomas algorithm. The problem is converted to finite difference problem by using finite difference approximations and trapezoid method. Finally, the convergence of the presented method is analyzed through sample application. Thus, the correctness and sufficiency of the method are shown. [ABSTRACT FROM AUTHOR]

Boundary conditions involving fractional derivatives of unknown functions are more general and can be used to generalize Dirichlet- or Neumann-type boundary conditions. In this article, we consider a class of nonlinear fractional differential equations having Atangana-Baleanu derivatives equipped with fractional boundary conditions. We employ some standard fixed-point theorems to establish the main results — Leray–Schauder alternative ensures the existence of solutions, whereas Banach contraction principle guarantees uniqueness. Furthermore, we present an implicit numerical scheme based on the trapezoidal method for obtaining a precise numerical estimation of the solution. Some examples are given to illustrate our analytical results and numerical findings. The main distinctive features of this work are as follows. (i) Some important properties of higher-order Atangana-Baleanu operators are introduced. (ii) Fractional boundary conditions are considered for the first time. (iii) A numerical approximation for the solution is expounded. [ABSTRACT FROM AUTHOR]

The Caputo fractional α -derivative, 0 < α < 1 , for non-smooth functions with 1 + α regularity is calculated by numerical computation. Let I be an interval and D α (I) be the set of all functions f (x) which satisfy f (x) = f (c) + f ′ (c) (x − a) + g c (x) (x − c) | (x − c) | α , where x , c ∈ I and g c (x) is a continuous function for each c. We first extend the trapezoidal method on the set D α (I) and rewrite the integrand of the Caputo fractional integral as a product of two differentiable functions. In this approach, the non-smooth function and the singular kernel could have the same impact. The trapezoidal method using the Riemann–Stieltjes integral (TRSI) depends on the regularity of the two functions in the integrand. Numerical simulations demonstrated that the order of accuracy cannot be increased as the number of zones increases using the uniform discretization. However, for a fixed coarsest grid discretization, a refinable mesh approach was employed; the corresponding results show that the order of accuracy is k α , where k is a refinable scale. Meanwhile, the application of the product of two differentiable functions can also be applied to some Riemann–Liouville fractional differential equations. Finally, the stable numerical scheme is shown. [ABSTRACT FROM AUTHOR]

In order to improve the imaging quality of magneto-acoustic concentration tomography for magnetic nanoparticles (MNPs) with magnetic induction (MACT-MI) and overcome the boundary singularity, this paper built a matrix model which shows the relationship between the partial derivative distribution of MNP concentration and the ultrasound signals, and focused on proposing a concentration reconstruction method based on the least squares QR factorization (LSQR) method–trapezoidal method. Firstly, simulation models with different shapes were established. Secondly, the magnetic and acoustic field simulation data was substituted into the inverse problem method based on LSQR–trapezoidal method for concentration reconstruction. Finally, the reconstructed images were analyzed and the effect of MNP cluster radius on the reconstruction was investigated. Considering the diffusely asymptotic concentration distribution of MNPs in actual biological tissue environment, an asymptotic concentration model was established and the reconstructed images were analyzed. The simulation results show that under the same conditions, compared with the reconstruction method based on the method of moments (MoM), LSQR–trapezoidal method has clearer image boundaries, more stable imaging results, and faster imaging speed. Compared with the uniform concentration model, LSQR–trapezoidal method is more applicable to the asymptotic concentration model. This study provides a basis for further reconstruction of the accuracy of experimental research. [ABSTRACT FROM AUTHOR]

The Caputo fractional α-derivative, 0<α<1, for non-smooth functions with 1+α regularity is calculated by numerical computation. Let I be an interval and Dα(I) be the set of all functions f(x) which satisfy f(x)=f(c)+f′(c)(x−a)+gc(x)(x−c)|(x−c)|α, where x,c∈I and gc(x) is a continuous function for each c. We first extend the trapezoidal method on the set Dα(I) and rewrite the integrand of the Caputo fractional integral as a product of two differentiable functions. In this approach, the non-smooth function and the singular kernel could have the same impact. The trapezoidal method using the Riemann–Stieltjes integral (TRSI) depends on the regularity of the two functions in the integrand. Numerical simulations demonstrated that the order of accuracy cannot be increased as the number of zones increases using the uniform discretization. However, for a fixed coarsest grid discretization, a refinable mesh approach was employed; the corresponding results show that the order of accuracy is kα, where k is a refinable scale. Meanwhile, the application of the product of two differentiable functions can also be applied to some Riemann–Liouville fractional differential equations. Finally, the stable numerical scheme is shown.

In this paper, we formulate an effective method to find an optimal solution of trapezoidal intuitionistic fuzzy fractional transportation problem[TIFFTP] of type-2. The proposed method achieve its goal successively when compared to the existing methods (Gupta and Anupum). Trapezoidal ranking method is used, which is based on the area of both membership and non membership parts of the numbers. An illustrative example is provided to demonstrate the feasibility of this method.

The parallel propagating waves (Langmuir, R and L waves) are studies in different plasma environments by choosing the different values of $\eta =({mc^{2}}/{k_{B}T})$ and $({\omega ^{2}_{\mathrm{ pe}}}/{\omega ^{2}_{\mathrm{ ce}}})$. The dispersion relations of these waves are derived under a weak magnetic field limit for a fully relativistic plasma environment. The analytical solution of the integrals in the dispersion relations is not possible, so we use a numerical quadrature approach for their solution. We observe that in both relativistic and weakly relativistic plasmas, these modes shifted to lower values of frequency. The cutoff points of R- and L-waves shifted to further lower values in high-density plasma compared to low density. [ABSTRACT FROM AUTHOR]

In this paper, we formulate an effective method to find an optimal solution of trapezoidal intuitionistic fuzzy fractional transportation problem[TIFFTP] of type-2. The proposed method achieve its goal successively when compared to the existing methods (Gupta and Anupum). Trapezoidal ranking method is used, which is based on the area of both membership and non membership parts of the numbers. An illustrative example is provided to demonstrate the feasibility of this method. [ABSTRACT FROM AUTHOR]

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An accurate estimation of evapotranspiration (ET) from crops is crucial in irrigation management, crop yield assessment, and optimal allocation of water resources, particularly in arid regions. This study explores the estimation of seasonal evapotranspiration for crops using multisource remote sensing images. The proposed estimation framework starts with estimating daily evapotranspiration (ETd) values, which are then used to calculate ET estimates during the crop growing season (ETs). We incorporated Landsat images into the surface energy balance algorithm over land (SEBAL) model, and we used the trapezoidal and sinusoidal methods to estimate the seasonal ET. The trapezoidal method used multitemporal ETd images, while the sinusoidal method employs time-series Moderate Resolution Imaging Spectroradiometer (MODIS) images and multitemporal ETd images. Experiments were implemented in the agricultural lands of the Kai-Kong River Basin, Xinjiang, China. The experimental results show that the obtained ETd estimates using the SEBAL model are comparable with those from the Penman–Monteith method. The ETs obtained using the trapezoidal and sinusoidal methods both have a relatively high spatial resolution of 30 m. The sinusoidal method performs better than the trapezoidal method when using low temporal resolution Landsat images. We observed that the omission of Landsat images during the middle stage of crop growth has the greatest impact on the estimation results of ETs using the sinusoidal method. Based on the results of the study, we conclude that the proposed sinusoidal method, with integrated multisource remote sensing images, offers a useful tool in estimating seasonal evapotranspiration for crops in arid regions.

This study proposed the nodal dynamic equation to solve the electromagnetic transient of linear switching circuit. The nodal dynamic equation is derived from the principle of energy conservation. By using trapezoidal method and time step to discrete the dynamic equation, its time discrete format is equivalent to the EMTP equation. Comparing with the EMTP method, the proposed method can change its numerical integration method and time step flexibly in the simulation. It is more suitable to solve the switching circuit. [ABSTRACT FROM AUTHOR]

In this paper, American options on a discount bond are priced under the Cox-Ingrosll-Ross (CIR) model. The linear complementarity problem of the option value is solved numerically by a penalty method. The problem is transformed into a nonlinear partial differential equation (PDE) by adding a power penalty term. The solution of the penalized problem converges to the one of the original problem. To numerically solve this nonlinear PDE, we use the horizontal method of lines to discretize the temporal variable and the spatial variable by means of trapezoidal method and a cubic spline collocation method, respectively. We show that this full discretization scheme is second order convergent, and hence the convergence of the numerical solution to the viscosity solution of the continuous problem is guaranteed. Numerical results are presented and compared with other collocation methods given in the literature.

In this paper, a singularly perturbed system of reaction–diffusion Boundary Value Problems (BVPs) is examined. To solve such a type of problems, a Modified Initial Value Technique (MIVT) is proposed on an appropriate piecewise uniform Shishkin mesh. The MIVT is shown to be of second order convergent (up to a logarithmic factor). Numerical results are presented which are in agreement with the theoretical results.

In this work, the problem of finite generalized and viscoelastic deformation of thin membranes with different geometries, made of incompressible hyperelastic materials, is formulated. The multiplicative decomposition of the deformation gradient tensor into elastic and viscous parts, and making use of dissipation inequality, nonlinear evolution equations for the internal variables of the models are obtained. The mechanical behavior of the dampers is assumed to be linearly viscous. Therefore, the Cauchy-like stress in the dampers is similar to that in Newtonian fluids and includes terms for the hydrostatic pressure and viscosity. The implicit and second-order accurate trapezoidal method is employed for the time integration of the evolution equations. Due to the highly nonlinear governing differential equations including the effects of geometric nonlinearity and viscoelasticity, a nonlinear finite element formulation based on isoparametric elements is developed. The accuracy and performance of the developed formulation and time-dependent solutions are verified by studying several numerical examples. The obtained results are compared with theoretical and experimental data available in the literature. The proposed formulation can appropriately predict the experimental results of viscoelastic membranes for both in-plane and out-of-plane deformations. [ABSTRACT FROM AUTHOR]

The ground truth for cascading failure in power system can only be obtained through a detailed dynamic model involving nonlinear differential and algebraic equations whose solution process is computationally expensive. This has prohibited adoption of such models for cascading failure simulation. To solve this, we propose a fast cascading failure simulation approach based on implicit Backward Euler method (BEM) with stiff decay property. Unfortunately, BEM suffers from hyperstability issue in case of oscillatory instability and converges to the unstable equilibrium. We propose a predictor–corrector approach to fully address the hyperstability issue in BEM. The predictor identifies oscillatory instability based on eigendecomposition of the system matrix at the post-disturbance unstable equilibrium obtained as a byproduct of BEM. The corrector uses right eigenvectors to identify the group of machines participating in the unstable mode. This helps in applying appropriate protection schemes as in ground truth. We use Trapezoidal method (TM)-based simulation as the benchmark to validate the results of the proposed approach on the IEEE 118-bus network, 2383-bus Polish grid, and IEEE 68-bus system. The proposed approach is able to track the cascade path and replicate the end results of TM-based simulation with very high accuracy while reducing the average simulation time by ≈ 10 − 35 fold. The proposed approach was also compared with the partitioned method, which led to similar conclusions. • Fast dynamic cascading failure simulation approach using Backward Euler method (BEM). • Hyperstability of BEM addressed through a new predictor–corrector (PC) approach. • Adaptive Center of Inertia (COI) frame for faster convergence of Newton iterations. • Comprehensive comparison of proposed BEM-PC approach against state-of-art. • BEM-PC replicates cascade path and end-results with ≈ 10 − 35 -fold average speedup. [ABSTRACT FROM AUTHOR]

Integral and differential equations have a fundamental importance in the functional analysis and the practice problems, and many domains of scientific research. However, the resolution of differential equations with constant coefficients is easy, but the resolution of these equations with variable coefficients is practically difficult or impossible in more part of the cases. This work present a analytical method which it transform a differential equations with initial conditions to a Volterra equations of second kind, efficient methods for approximate numerical solution of these equations, the analysis of the existence of their solutions, and the convergence of the error. [ABSTRACT FROM AUTHOR]

In the past 10 years, the 'parareal' (parallel-in-time) algorithm has attracted lots of attention thanks to its excellent performance in scientific computing. The parareal algorithm is iterative and is characterized by two propagators G and F which are associated with a coarse step size ΔT and a fine step size Δt, respectively, where ΔT =JΔt and J ≥ 2 is an integer. When we apply this algorithm to large-scale systems of ordinary differential equations obtained by semidiscretizing partial differential equations, two questions arise naturally. (I) Is the error between the iterate and the target solution contractive at each iteration for any choice of the discretization parameters ΔT, J and Δx? (II) How small can the contraction factor be and can such a contraction factor be independent of the discretization parameters? For linear problems u' =Au + g with symmetric negative-definite matrix A, when the implicit Euler method is used as both the G- and F-propagators, positive answers to these two questions were given by Mathew et al. (2010, SIAM J. Sci. Comput., 32, 1180-1200) and the contraction factor can be bounded by 0.298 for any choice of the discretization parameters. In this paper, for the case that the implicit Euler method is used as the G-propagator, we provide a positive answer to (I) for three second-order Fpropagators: the trapezoidal method, the TR/BDF2 method and the two-stage diagonally implicit Runge- Kutta (2s-DIRK) method. For (II), we prove that the contraction factors can be bounded by 0.316 and 13 for the 2s-DIRK method and the TR/BDF2 method (provided the parameter γ involved in TR/BDF2 satisfies γ ∈ [0.043, 0.977]), respectively, and both bounds are independent of the discretization parameters. For the trapezoidal method, we show that a uniform bound (less than 1) of the contraction factor does not exist. Numerical results are presented to validate the theoretical prediction. [ABSTRACT FROM AUTHOR]

To obtain the pore structure eigenvalue, the following two methods are usually applied: the graphical method and the trapezoidal method. Both methods have intrinsic disadvantages. For the graphical method, the pore size distribution curve must satisfy the normal distribution and some of the data from the cumulative pore size distribution curve are selected and averaged; this makes the method inflexible and causes inevitable computation errors. For the trapezoidal method, the saturation range is restricted, which causes the obtained results to deviate from the true value. To overcome these disadvantages, we propose a new method that is based on the trapezoidal method. The software 1stOpt is applied to get the capillary pressure function by fitting with the measured data. By substituting the pore radius function derived by the capillary pressure function into the computed formula of the new method, we obtain the pore structure eigenvalues. In comparison to the conventional methods, the results obtained by the new method are closer to the true value because this method takes into account the entire saturation range for integrating. In addition, it can be applied in all kinds of pore size distributions. [ABSTRACT FROM AUTHOR]

Electromagnetic transient (EMT) simulations are now often used for system studies including power electronics converters which control voltages and currents using switching devices. The trapezoidal method, used in Electro-Magnetic Transients Program (EMTP) for numerical integration, produces "fictitious" sustained numerical oscillation, when an inductor current or a capacitor voltage is suddenly changed especially by switching. To cope with this problem, the research group of the authors has proposed applying the two-stage diagonally implicit Runge--Kutta (2S-DIRK) method which is an inherently "oscillation-free" numerical integration method. In order to bring out full accuracy of the 2S-DIRK method, supplementary numerical techniques are required for the representation of voltage sources, current sources and switches. This paper illustrates these techniques with numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we propose a hybrid initial-value technique for singularly perturbed boundary value problems. First, we develop a hybrid scheme to solve the singularly perturbed initial-value problems, and then, the hybrid scheme is used to solve the singularly perturbed boundary value problems. The scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter. Necessary error estimates are derived for the scheme. To verify computational efficiency and accuracy, some numerical examples are provided. [ABSTRACT FROM AUTHOR]

In this paper, the trapezoidal method is proposed to solve the "fuzzy initial value problem". The method is obtained by combining the Euler method and a trapezoidal rule calculating integration. Convergence of the proposed methods are also presented in detail. These methods are illustrated by solving some examples. [ABSTRACT FROM AUTHOR]

Abstract: Numerical methods for Volterra integral equations with discontinuous kernel need to be tuned to their peculiar form. Here we propose a version of the trapezoidal direct quadrature method adapted to such a type of equations. In order to delineate its stability properties, we first investigate about the behavior of the solution of a suitable (basic) test equation and then we find out under which hypotheses the trapezoidal direct quadrature method provides numerical solutions which inherit the properties of the continuous problem. [Copyright &y& Elsevier]

Abstract: A mathematical model has been developed that could predict kinetic parameters for the adsorption of divalent cations (lead, copper and zinc) onto low-grade rock phosphate using experimental data. The experiments were conducted with the initial concentrations of metal ions ranging from 10 to 100 mg/L. The mathematical model is based on application of Freundlich isotherm to mass transfer across the film surrounding the adsorbent. A code in C programming is used to numerically integrate the model equation, and to obtain the best simulated values of Freundlich constants K, N, order of reaction n, and film transfer coefficient, α. It is observed that the adsorption of metal ions on rock phosphate is more sensitive to , and α in comparison to K, and lead is adsorbed more favorably than copper and zinc. [Copyright &y& Elsevier]

Potassium spectra with low counting statistics were measured with a NaI detector from a water phantom, simulating a brain, and were analyzed for error propagation in determination of K employing either the Trapezoidal Method or the Library Least-Squares method. We demonstrate, using measured and synthetic spectra, that a smaller error is obtained in the analysis of potassium when using the Library Least-Squares method. [Copyright &y& Elsevier]

Convolution and deconvolution are the classical in-vitro-in-vivo correlation tools to describe the relationship between input and weighting/response in a linear system, where input represents the drug release in vitro, weighting/response any body response in vivo. While functional treatment, e.g. in terms of polyexponential or Weibull distribution, is more appropriate for general survey or prediction, numerical algorithms are useful for treating actual experimental data. Deconvolution is not considered an algorithm by its own, but the inversion of a corresponding convolution. MS Excel is shown to be a useful tool for all these applications. [Copyright &y& Elsevier]