Trapezoidal Rule Research Articles

Multi-Objective Optimization for Controlling the Dynamics of the Diabetic Population

To limit the adverse effects of diabetes, a personalized and long-term management strategy that includes appropriate medication, exercise and diet has become of paramount importance and necessity. Compartment-based mathematical control models for diabetes usually result in objective functions whose terms are conflicting, preventing the use of single-objective-based models for obtaining appropriate personalized strategies. Taking into account the conflicting aspects when controlling the diabetic population dynamics, this paper introduces a multi-objective approach consisting of four steps: (a) modeling the problem of controlling the diabetic population dynamics using a multi-objective mathematical model, (b) discretizing the model using the trapezoidal rule and the Euler–Cauchy method, (c) using swarm-intelligence-based optimizers to solve the model and (d) structuring the set of controls using soft clustering methods, known for their flexibility. In contrast to single-objective approaches, experimental results show that the multi-objective approach obtains appropriate personalized controls, where the control associated with the compartment of diabetics without complications is totally different from that associated with the compartment of diabetics with complications. Moreover, these controls enable a significant reduction in the number of diabetics with and without complications, and the multi-objective strategy saves up to 4% of the resources needed for the control of diabetes without complications and up to 18% of resources for the control of diabetes with complications.

Cooperative Lane-Change Motion Planning for Connected and Automated Vehicle Platoons in Multi-Lane Scenarios

Multi-vehicle motion planning (MVMP) has become an emerging paradigm in connected and automated vehicles (CAVs). The cooperative lane-change movements with the coexistence of platoons and CAVs are typical scenarios on the muti-lane roads. This paper proposes an optimal control framework with the advantages of completeness and universality for platoons and CAVs’ cooperative lane-change motion planning in different task scenarios. Two typical cooperative scenarios are designed for the subsequent study of optimal modeling. The platoons’ reconfiguration and original shape maintenance are considered to reflect the universality of moving objects and the diversity of cooperative tasks. Approximately geometric contour models, dynamic externally tangent rectangle and inflated rectangle, are utilized to describe the platoon’s profile. Analytical complete collision avoidance constraints among different motion objects are constructed effectively. Other necessary constraints and the weighted cost function that minimizes lane-change time and motion energy are comprehensively considered. The optimal control models are established for the desired scenarios. Moreover, a numerical solution method combined with the simultaneous direct collocation method based on the trapezoidal rule and the barrier function method is proposed to obtain the optimal schemes. Simulation and contrast experiments are conducted for two scenarios. The results indicate that the cost function’s weight coefficients and specific lane-change tasks influence the cooperative motion planning effects and verify that the proposed optimal control framework is of reasonability, effectiveness, and unification.

TrGANet: Transforming 3T to 7T dMRI using Trapezoidal Rule and Graph based Attention Modules.

Diffusion MRI (dMRI) is a non-invasive tool for assessing the white matter region of the brain by approximating the fiber streamlines, structural connectivity, and estimation of microstructure. This modality can yield useful information for diagnosing several mental diseases as well as for surgical planning. The higher angular resolution diffusion imaging (HARDI) technique is helpful in obtaining more robust fiber tracts by getting a good approximation of regions where fibers cross. Moreover, HARDI is more sensitive to tissue changes and can accurately represent anatomical details in the human brain at higher magnetic strengths. In other words, magnetic strengths affect the quality of the image, and hence high magnetic strength has good tissue contrast with better spatial resolution. However, a higher magnetic strength scanner (like 7T) is costly and unaffordable to most hospitals. Hence, in this work, we have proposed a novel CNN architecture for the transformation of 3T to 7T dMRI. Additionally, we have also reconstructed the multi-shell multi-tissue fiber orientation distribution function (MSMT fODF) at 7T from single-shell 3T. The proposed architecture consists of a CNN-based ODE solver utilizing the Trapezoidal rule and graph-based attention layer alongwith L1 and total variation loss. Finally, the model has been validated on the HCP data set quantitatively and qualitatively.

Precision analysis of the integration rules used for transient simulation in electric networks

The adequate selection of the integration rule and time step is crucial to ensure accurate results in the simulation of an electric network. For this reason, this work has two main objectives: to analyze the distortion of the waves due to the integration rule and integration time step, and to provide guidelines for new circuit simulators’ users on how to properly choose an integration rule and a time step according to a desired precision. When discretizing a continuous system, the frequency response of the discrete system can be used to evaluate the precision of the mentioned rule, since a discrete system can be viewed as a filter. However, the frequency response provides the accuracy of the discrete system during the steady state; the transient-state accuracy can be analysed by evaluating the pole distortion due the discretization. This work focuses on the analysis of the steady-state accuracy and the transient-state accuracy of two discrete systems: an RL branch used as a fist order system and an RLC branch used as a second order system. For this analysis, two commonly used discretization rules are considered, backward Euler and trapezoidal rules.

Functional and Morphological Responses to Fluocinolone Acetonide 0.19 mg in Noninfectious Uveitic Macular Edema Evaluated as the Area-Under-the-Curve.

Purpose: This study investigated the impact of baseline clinical and optical coherence tomography (OCT) factors on the response to a 0.19-mg fluocinolone acetonide (FAc) implant in patients with noninfectious uveitic macular edema evaluated by the area under the curve over 24 months. Methods: A retrospective study was conducted of eyes of patients with noninfectious uveitic macular edema undergoing FAc treatment, with follow-up from baseline to 24 months. The area under the curve (AUC) of best-corrected visual acuity (BCVA) and the central macular thickness (CMT) were calculated using the trapezoidal rule. Clinical and OCT data at the time of FAc administration were collected, and associations with AUC of BCVA and CMT changes were investigated. Results: Twenty-three patients were enrolled. BCVA and CMT significantly improved after FAc implantation (P < 0.05). AUCBCVA and AUCCMT were 0.41 ± 0.33 logarithm of minimal angle of resolution/6 months and 320.15 ± 321.64 μm/6 months, respectively. Better baseline BCVA (coefficient [coef.] = 0.83, P < 0.001) and macular thickness reduction after FAc administration (coef. = -0.0001, P < 0.05) were associated with better BCVA after FAc treatment. In contrast, baseline OCT biomarkers such as ellipsoid zone reflectivity and choroidal vascularity index, sex, or disease duration before FAc injection showed no correlation with AUCBCVA and AUCCMT (P > 0.05). The younger the patient at the time of FAc injection, the greater the reduction in CMT (coef. = 1.76, P < 0.05). Conclusions: Among all clinical and morphological baseline factors, Baseline BCVA was the strongest predictor for AUCBCVA, while no association with baseline OCT features was observed. Overall, improvement of BCVA and CMT after FAc injection was maintained over 24 months. This study is registered in the German Clinical Trials Register under the DRKS-ID: DRKS00024399.

35-OR: Incretin Production Is Impaired in Atypical Low BMI Diabetes Mellitus

Introduction: Atypical Low BMI Diabetes Mellitus (ALBDM) constitutes an unclassified form of diabetes that is likely to be prevalent in low- and middle-income countries. Recently, we reported in Diabetes Care that such individuals demonstrate striking defects in insulin secretion, despite absence of visible pancreatic pathology (PMID: 35522035). This study aims to determine whether defects in the incretin axis might contribute to impaired insulin secretion in ALBDM. Methodology: We assessed the glucagon and incretin (GLP-1, GIP, and Oxyntomodulin) responses to an oral glucose tolerance test (OGTT) in patients with ALBDM. Ten male subjects demonstrating features of ALBDM (n=10, age 33±8.3 years, BMI 18.5±1.1 kg/m2, HbA1c 7.5±0.7%) were compared with 7 healthy BMI-matched normoglycemic (HbA1c: 5.0±0.3%) controls. All subjects underwent an OGTT, with blood sampling for C-peptide and incretin levels at nine specific time points. Insulin secretion rate (ISR) was calculated by C-peptide deconvolution using ISEC software, and total secretion rate for incretins were calculated as area under the curve (AUC) using the Trapezoid method with linear interpolation. Results: As we previously reported, ALBDM subjects displayed impaired insulin secretion (50% reduction in post-OGTT insulin AUC, p=0.007). There was a profound reduction of OGTT-stimulated rise in the incretins GLP-1 (80% reduction; p=0.055), GIP (79% reduction; p=0.02) and oxyntomodulin (75% reduction; p=0.007) in ALBDM subjects compared to controls. In ALBDM subjects, mild hyperglucagonemia was observed in the early postprandial phase at 15 min of the OGTT followed by slow postprandial suppression. Conclusions: Significant reductions in levels of GLP-1, GIP and Oxyntomodulin and an accompanying early phase of hyperglucagonemia suggest potential defects in development and/or function of enteroendocrine cells, which may contribute to defective insulin secretion and hyperglycemia in patients with Atypical Low BMI Diabetes Mellitus. Disclosure F.Jebasingh: None. N.Jiang: None. M.Hawkins: None. N.Thomas: None. S.Anoop: None. M.E.Kurian: None. V.N.Sandeep: None. F.Christina: None. S.Jennifer: None. A.Jose: None. P.Christudoss: None. G.Rebekah: None. Funding Endocrine Society of India

The explicit β1/β2-Bathe time integration method

In this paper we introduce the explicit β1/β2-Bathe method for solving dynamic problems, in particular wave propagations. Like for the implicit β1/β2-Bathe method, the proposed explicit scheme uses two sub-steps per time step and can be used directly as a first-order and a second-order method with the capability to suppress high spurious frequency response. In both sub-steps, standard Taylor series are employed resulting in an explicit solution scheme. The novelty is that we calculate the final displacements and velocities in each sub-step by applying correction terms using the generalized trapezoidal rule with control parameters β1 and β2. This approach makes the method a quite simple scheme. We consider the stability, accuracy and numerical dispersion and give recommendations on the parameter values β1 and β2 to be used in practice. We give the solutions of four problems, three of which are wave propagation problems, and compare the results with those obtained using other methods. While more experience in the use of the procedure is needed to understand its full solution capabilities, we can already conclude that the proposed method is effective in some wave propagation analyses.

#2668 SUPERIOR BIOCOMPATIBILITY OF REGIONAL CITRATE ANTICOAGULATION COMPARED TO HEPARIN DURING EXPANDED HEMODIALYSIS WITH THERANOVA® MEMBRANE

Abstract Background and Aims Regional citrate anticoagulation (RCA) is known to improve the biocompatibility of hemodialysis (HD) by reducing complement, leukocyte and platelet activation. As continuous exposure to bio-incompatible HD may accelerate cardiovascular disease and increase mortality, improved biocompatibility may be important for clinical outcomes, including long-term survival of chronic HD patients. We found no reports of RCA during expanded hemodialysis (HDx), except our recent pilot study [1]. The aim of our trial was to compare RCA and full dose unfractionated heparin (UFH) during HDx with respect to biocompatibility parameters. Methods We performed a randomized, cross-over trial in 32 patients on chronic HD. Each had 2 HDx procedures with Theranova® membrane: one with UFH and one with RCA. RCA was achieved with 8% trisodium citrate at a flow rate of 150 mL/min. Anticoagulation with UFH was achieved by a bolus of 30 I.U. per kg of body weight, followed by an infusion of 18 I.U. per kg of body weight per hour. Anticoagulation was monitored by post-filter ionized calcium (RCA) or activated partial thromboplastin time (UFH). The primary outcomes were serum concentrations of C3a (marker of complement activation) and myeloperoxidase (MPO; marker of leukocyte activation) and plasma concentrations of platelet factor 4 (PF4; marker of platelet activation). Blood was drawn prior to, after 15 minutes and just before the end of HDx. The area under the biocompatibility marker concentration-time curve (AUC) was calculated for an individual biocompatibility marker from all three concentrations using the linear trapezoidal rule. As a secondary outcome, adhesion of blood cells and activation of clotting on inner surfaces of dialyzer fibers were visualized using scanning electron microscopy (SEM) in a subset of patients (n = 5). Biocompatibility parameters were compared between the groups using paired t-test or Wilcoxon signed-rank test. Results The trial included 22 males and 10 females. Patients were of mean age 65 ± 12 years with median dialysis vintage of 31 (IQR 17-89) months. Mean duration of HDx was 4.4 ± 0.4 hours (always the same for an individual patient). All patients had arteriovenous fistulas. Analysis of PF4 included only 8 patients, while the rest had concentrations of PF4 above the measuring range, probably due to unspecific platelet degranulation because of preanalytical factors. The concentrations of C3a, MPO and PF4 rose significantly after 15 minutes of HDx with UFH, while we observed no significant rises during HDx with RCA (Table 1). The AUC for biocompatibility markers were significantly smaller during HDx with RCA in comparison to UFH: 1163.0 ± 306.8 vs. 1444.1 ± 385.9 µgh/L for C3a (p&amp;lt;0.001); 78.2 (IQR 39.0-131.2) vs. 275.1 (IQR 136.9-492.9) µgh/L for MPO (p&amp;lt;0.001) and 100.0 (IQR 51.8-196.6) vs. 379.4 (IQR 295.0-553.0) IUh/mL for PF4 (p = 0.04). SEM showed almost no deposition on the membrane with comparable clotting score: 2 (IQR 1-3) for RCA vs. 2 (IQR 1-2.25) for UFH (p = 0.40). Conclusion Given the higher activation of complement, leukocytes and platelets during HDx with UFH, we conclude that RCA offers superior biocompatibility compared to UFH during HDx. Whether improved biocompatibility has clinical relevance remains to be explored.

Pemetaan Morfologi Ambang Galala Poka (APG)

The Galala Poka (APG) sill is in the form of an underwater hill located in the narrow waters of the TAD and TAL transition zones. The morphology of APG is important to map because it influences the dynamics of water masses during the tidal period in the TAL and TAD Transitional areas, and can even explain the phenomena of cohesive sediment transport and deposition as well as tidal mixing phenomena that often occur around APG. This research aims to map the topographical profile of the APG seabed and calculate its spatial dimensions.The GIS-based mapping method is used to create an APG seabed topographic model using data compiled from sounding results, digital Navionic Sonar Chart data and digital shipping channel bathymetry data. The trapezoidal rule model is used to calculate the APG dimensions. APG seabed 2D and 3D spatial models mapped with ArcGIS 10.8 and Surfer 23.The results of the analysis found the APG boundaries at coordinates 128.19316 – 128.19829E and 3.66027 – 3.56659S and UTM coordinates 410399.1390 ​​– 410969.1292 mT and 9594764.3000 – 9595384.3000 mU. The topography of APG is generally convex with a flat peak at the center. There are variations in the slope of the seabed, especially on the outer side of the APG. APG volume with a reference depth of 0m (LWS) 1,304,828.05 m3, surface area 151,954.65 m2. At a reference depth of 8m, the APG volume is 20,071.95 m3, the area is 94,887.23 m2. At a reference depth of 10m, the APG volume is 351,296.95 m3, and the surface area is 73,012.06 m2.

Suboptimality of Gauss–Hermite Quadrature and Optimality of the Trapezoidal Rule for Functions with Finite Smoothness

Suboptimality of Gauss–Hermite Quadrature and Optimality of the Trapezoidal Rule for Functions with Finite Smoothness

Two families of second-order fractional numerical formulas and applications to fractional differential equations

Two families of second-order difference formulas called fractional BT- $$\theta $$ and fractional BN- $$\theta $$ for the fractional calculus are proposed by introducing a free parameter $$\theta $$ aiming to generalize the classic fractional BDF2, the fractional trapezoidal rule (FTR) and the second-order generalized Newton-Gregory formula (GNGF2). The error bounds for these two families of novel formulas are analysed indicating that fractional BT- $$\frac{1}{2}$$ (or FTR) is superior to other formulas for integral cases. At least two advantages can be observed for the introduction of $$\theta $$ : (i) The fractional BN- $$\theta $$ exhibits superconvergence for an appropriate $$\theta $$ which permits us to develop more robust third-order schemes for problems such as the diffusion-wave equation, compared with the fractional BDF3. (ii) The fractional BT- $$\theta $$ allows us to take the advantage of FTR that a sub-optimal error bound can be obtained (by taking $$\theta $$ close to $$\frac{1}{2}$$ ) when directly discretizing fractional derivatives. Further, some correction techniques are discussed to overcome the solution initial singularity in numerical tests. The numerical results confirm the correctness of the theoretical analysis and the efficiency of our scheme.

NUMERICAL SOLUTIONS OF THIRD ORDER FREDHOLM INTEGRO DIFFERENTIAL EQUATION VIA LINEAR MULTISTEP-QUADRATURE FORMULAE

In this study, we present a 6-step linear multistep mehod in union with some newton-cote quadrature family for solving third order Linear Fredholm Integro-Differential Equation(LIDE). The schemes were derived using Vieta-Pell-Lucas Polynomial as the approximating function. The linear multi-step component is for the non integral part while the quadrature family is for the Integral part. The quaudrature methods Boole , Simpson 3/8 and Trapezoidal rule were separately combined with the linear multistep method. The qualititaive analysis of the scheme revealed that the method is consistence, stable and convergent. In order to further attest to the behavioural attribute of the methods, numerical experiments were carried out on some selected Initial Value Problems.The results from the tested problems and their absolute errors of deviation revealed that the new method is very suitbale for solution to the tested problems. The scheme when combined with Boole and Simpson 3/8 merhod, performed better with Tracendental function than when combined with Trapezoidal rule and vice –versa. The results further showed that the proposed method perform creditably well with lesser computional steps when compared with some existing methods when applied to the selected examples.

Reconstruction of Timewise Dependent Coefficient and Free Boundary in Nonlocal Diffusion Equation with Stefan and Heat Flux as Overdetermination Conditions

The problem of reconstruction of a timewise dependent coefficient and free boundary at once in a nonlocal diffusion equation under Stefan and heat Flux as nonlocal overdetermination conditions have been considered. A Crank–Nicolson finite difference method (FDM) combined with the trapezoidal rule quadrature is used for the direct problem. While the inverse problem is reformulated as a nonlinear regularized least-square optimization problem with simple bound and solved efficiently by MATLAB subroutine lsqnonlin from the optimization toolbox. Since the problem under investigation is generally ill-posed, a small error in the input data leads to a huge error in the output, then Tikhonov’s regularization technique is applied to obtain regularized stable results.

Nonsymmetric interior penalty Galerkin method for nonlinear time-fractional integro-partial differential equations

This article discusses an efficient numerical method for solving nonlinear time-fractional integro-partial differential initial–boundary-value problems. To tackle the non-linearity of the problem, we first use the Newton linearization process. Then we apply the non-symmetric interior penalty Galerkin method for the spatial variable, and obtain a semi-discrete problem in the time variable. Finally, to obtain the fully-discrete scheme, we use both L1-scheme and L2-scheme for time-fractional derivative, and trapezoidal rule for integral term in the semi-discrete problem. We derive L2-norm stability and error estimates. The validity of the error analysis is supported by numerical experiments.

Computational Approach for Differential Equations with Local and Nonlocal Fractional-Order Differential Operators

Laplace transform has been used for solving differential equations of fractional order either PDEs or ODEs. However, using the Laplace transform sometimes leads to solutions in Laplace space that are not readily invertible to the real domain by analytical techniques. Therefore, numerical inversion techniques are then used to convert the obtained solution from Laplace domain into time domain. Various famous methods for numerical inversion of Laplace transform are based on quadrature approximation of Bromwich integral. The key features are the contour deformation and the choice of the quadrature rule. In this work, the Gauss–Hermite quadrature method and the contour integration method based on the trapezoidal and midpoint rule are tested and evaluated according to the criteria of applicability to actual inversion problems, applicability to different types of fractional differential equations, numerical accuracy, computational efficiency, and ease of programming and implementation. The performance and efficiency of the methods are demonstrated with the help of figures and tables. It is observed that the proposed methods converge rapidly with optimal accuracy without any time instability.

Three optimal families of three‐sub‐step dissipative implicit integration algorithms with either second, third, or fourth‐order accuracy for second‐order nonlinear dynamics

SummaryThis paper reviews the published composite three‐sub‐step implicit algorithms all of which adopt the trapezoidal rule in the first sub‐step. Three optimal families of three‐sub‐step implicit algorithms are developed to achieve second, third, and fourth‐order accuracy. The present second‐ and third‐order methods achieve identical effective stiffness matrices, thus embedding optimal spectral characteristics. Besides, both of them impose two dissipative parameters ( and ) to control numerical dissipation imposed in the second and third sub‐steps. The parameter adjusts overall numerical dissipation in the whole integration schemes, while the firstly used parameter can change numerical dissipation in the second sub‐step. The numerical example has shown the superiority of controlling middle sub‐step dissipation via . The present fourth‐order method is moderately dissipative due to achieving higher‐order accuracy, but it presents a more reasonable sub‐step division than the published fourth‐order three‐sub‐step trapezoidal rule. Linear and nonlinear examples are simulated to show the numerical performance and superiority of the three novel methods. This paper recommends using the proposed second‐ and third‐order three‐sub‐step methods to solve various dynamic problems.

Quantitative LC-MS/MS. 1. Impact of Points across a Peak on the Accuracy and Precision of Peak Area Measurements.

The number of points across a chromatographic peak has long been recognized as a key determinant of the accuracy and precision of the measured peak area. In LC-MS-based quantitation experiments in drug discovery and development, the "rule-of-thumb" has been to use 15 or more points. This "rule" is based on the literature describing chromatographic methods where the goal was to achieve the lowest possible imprecision in the measurements, especially when unknown analytes are being detected. Restricting methods to the requirement of at least 15 points across a peak can be detrimental to the development methods that fully optimize the signal-to-noise ratio for the assay using longer dwell times and/or transition summing. This study aims to show that 7 points across the peak for peaks that are 9 s or less wide provide more than sufficient accuracy and precision for drug quantitation studies. Data from simulated Gaussian curves using a sampling interval of 7 points across the peak gave peak area calculations within 1% of the expected total peak area using the Trapezoidal and Riemann rules and 0.6% for the Simpson rule. Low and high concentration samples (n = 5) were assayed using three different LC methods on three different days on two different instruments (API5000 and API5500). The difference in peak area (%ΔPA) and relative standard deviation of the peak areas (%RSD) was less than ∼5%. No significant difference was observed from the data that were obtained from different sampling intervals, different peak widths, different days, different peak sizes, and different instruments. Three core analytical runs were performed on three different days. In each core run, the lower limit of quantitation (LLOQ, n = 5), low quality control (LQC, n = 5), middle quality control (MQC, n = 5), and high-quality control samples (HQC, n = 5) were processed and run simultaneously with a standard curve. The range of the intra- and interday accuracy and precision for 3 core runs was 98.0-105% and 0.9-3.0% for 7 data points and 97.5-105% and 0.8-4.3% for 17 data points, respectively. No significant difference was observed for the different sampling intervals. The results show a sampling interval of 7 points for peaks up to 9 s wide is sufficient to define a peak accurately and precisely for drug quantitation studies in drug discovery and development.

Experimental investigation of holdup fraction using the trapezoidal rule, Simpson's rule and the average offset formula in perforated horizontal wellbore

This study aimed to investigate the holdup fraction in a perforated horizontal well. This article examined the surface of flow patterns (bubble, dispersed bubble, elongated dispersed bubble/slug, plug, stratified wave, half-slug, stratified wave flow, and stratified transition wave/semi-stratified flow). Numerical methods include the trapezoidal rule, Simpson's rule, and the mean displacement formula used to calculate the liquid surface area under a curve for irregular shapes. The average ratio of the flow pattern surface areas by the trapezoidal rule, the Simpsons rule and the average displacement formula were 81.3%, 79.8% and 81.14%, respectively. In addition, the surface area of the liquid phase (holdup fraction) which increases with bubble flow is 108.6 cm 2 and decreases with semi-stratified flow was 51.9 cm 2. However, the ratio of the liquid surface area that increased with Case A (bubble flow) was 32%, and it decreased by 15% with Case L (semi-stratified flow) due to the increased air-phase flow (void fraction). Water phase flow (holdup fraction) was kept constant. The percent error minimum of the liquid surface was 9.42%, 11%, and 9.50% by the Trapezoidal Rule, Simpsons Rule, and Average Displacement Formula occurring in Case A (bubble flow), respectively. In contrast, the maximum values of the surface defect percentage were 56.58%, 57.25%, and 56.75%, respectively, occurring in Case L (semi-stratified flow).

On the integral solution of elliptic Kepler’s equation

In a recent paper, Philcox, Goodman and Slepian obtain an explicit solution of the elliptic Kepler’s equation (KE) as a quotient of two contour integrals along a Jordan curve mathcal {C} = mathcal {C}(M,e) that contains the unique real solution of KE but not includes other complex zeros of KE in its interior. The aim of this paper is to study the main issues that arise in the practical implementation of this integral solution. Thus, after a study of the complex zeros of KE, several families of Jordan contours mathcal {C} = mathcal {C}(M,e) that are suitable for this integral solution are proposed. Since contours with minimal length turn out to be the more accurate for numerical purposes, several families that minimize their length are constructed. Secondly, the approximation of the contour integrals by the composite trapezoidal rule is considered. Recall that this rule is employed in the fast Fourier transform and, in spite of its lower order, displays a spectral convergence as a function of the number of nodes, which implies a very fast convergence. Finally, the results of some numerical experiments are presented to show that such a combination of appropriate contours with the composite trapezoidal rule leads to a powerful numerical method to solve KE with any desired accuracy for all values of eccentricity.

Optimal error analysis of space–time second-order difference scheme for semi-linear non-local Sobolev-type equations with weakly singular kernel

In this paper, we construct and analyze a Crank–Nicolson difference scheme for solving the semilinear Sobolev-type equation with the Riemann–Liouville fractional integral of order α∈(0,1). The proposed scheme consists of two main stages. First, to compensate for the singular behavior of the exact solution at the initial time t=0, a Crank–Nicolson scheme and the product trapezoidal integration rule are proposed on graded meshes for time discretization. Second, a classical finite difference operator on uniform meshes is used for space discretization. Under suitable assumptions of the regularity condition, the stability and convergence analysis of this scheme in a new norm are given by the energy argument. It is shown that how the mesh grading exponent γ affects the temporal convergence rate of the presented scheme in the final convergence result. This scheme can attain a second-order accuracy in both time and space by choosing an optimal grading parameter γ. Numerical results confirm that the error analysis is sharp, and comparisons with results on uniform temporal meshes are included to indicate the effectiveness of our method.