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We propose a tailored finite point method (TFPM) for solving time fractional convection dominated diffusion equations in this paper. The main idea of TFPM is to firstly approximate the diffusion, convection coefficient near each grid by a constant, and then determine the weights of the finite difference scheme by using the exact solution of the convection diffusion equation with constant coefficients. This adaptation perfectly captures the rapid transition of the solutions which contain sharp boundary layers even with coarse meshes. The accuracy and stability of the scheme are rigorously analyzed. Numerical examples are shown to verify the accuracy and reliability of the proposed scheme. [ABSTRACT FROM AUTHOR]

Anomalous diffusion in the presence or absence of an external force field is often modelled in terms of the fractional evolution equations, which can involve the hyper-singular source term. For this case, conventional time stepping methods may exhibit a severe order reduction. Although a second-order numerical algorithm is provided for the subdiffusion model with a simple hyper-singular source term t μ , - 2 < μ < - 1 in [arXiv:2207.08447], the convergence analysis remain to be proved. To fill in these gaps, we present a simple and robust smoothing method for the hyper-singular source term, where the Hadamard finite-part integral is introduced. This method is based on the smoothing/IDm-BDFk method proposed by Shi and Chen (SIAM J Numer Anal 61:2559–2579, 2023) for the subdiffusion equation with a weakly singular source term. We prove that the kth-order convergence rate can be restored for the diffusion-wave case γ ∈ (1 , 2) and sketch the proof for the subdiffusion case γ ∈ (0 , 1) , even if the source term is hyper-singular and the initial data is not compatible. Numerical experiments are provided to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

The diffusive viscous (DV) model is a useful tool for interpreting low-frequency seismic attenuation and the influence of fluid saturation on frequency-dependent reflections. Among present methods for the numerical solution of the corresponding DV wave equation, the finite-difference frequency-domain (FDFD) method with complex-valued adaptive coefficients (CVAC) has the advantage of efficiently suppressing both numerical dispersion and numerical attenuation. In this research, the FDFD method with CVAC is first generalized to a 3D DV equation. In addition, the current calculation of CVAC involves the numerical integration of propagation angles, conjugate gradient (CG) iterative optimization, and the sequential selection of initial values, which is difficult and inefficient for implementation. An improved method is developed for calculating CVAC, in which a complex-valued least-squares problem is constructed by substituting the 3D complex-valued plane-wave solutions into the FDFD scheme. The QR-decomposition method is used to efficiently solve the least-squares problem. Numerical dispersion and attenuation analyses reveal that the FDFD method with CVAC requires ∼2.5 spatial points in a wavelength within a dispersion deviation of 1% and an attenuation deviation of 10% for the 3D DV equation. An analytic solution for 3D DV wave equation in homogeneous media is proposed to verify the effectiveness of the proposed method. Numerical examples also demonstrate that the FDFD method with CVAC can obtain accurate wavefield modelling results for 3D DV models with a limited number of spatial points in a wavelength, and the FDFD method with QR-based CVAC requires less computational time than the FDFD method with CG-based CVAC. [ABSTRACT FROM AUTHOR]

The time-fractional Cattaneo (TFC) equation is a practical tool for simulating anomalous dynamics in physical diffusive processes. The existing numerical solutions to the TFC equation generally deal with the Dirichlet boundary conditions. In this paper, we incorporate the absorbing boundary condition as a complex-frequency-shifted (CFS) perfectly matched layer (PML) into the TFC equation. Then, we develop an adaptive-coefficient (AC) finite-difference frequency-domain (FDFD) method for solving the TFC with CFS PML. The corresponding analytical solution for homogeneous TFC equation with a point source is proposed for validation. The effectiveness of the developed AC FDFD method is verified by the numerical examples of four typical TFC models, including the different orders of time-fractional derivatives for both the homogeneous model and the layered model. The numerical examples show that the developed AC FDFD method is more accurate than the traditional second-order FDFD method for solving the TFC equation with the CFS PML absorbing boundary condition, while requiring similar computational costs. [ABSTRACT FROM AUTHOR]

Objectives: This study aimed to compare the incidence, clinicopathological characteristics and survival results of pancreatic signet ring cell carcinoma (PSRCC) and pancreatic adenocarcinomas (PDAC), as well as to analyze the clinical characteristics related to the overall survival (OS) of PSRCC, and to establish an effective prognostic nomogram to predict the risks associated with patient outcomes. Methods: A total of 85,288 eligible patients including 425 PSRCC and 84,863 PDAC cases were retrieved from the Surveillance, Epidemiology, and End Results database. The survival curve was calculated using the Kaplan–Meier method and differences in them were measured by Log-rank tests. The Cox proportional hazards regression model was used to identify independent predictors of OS in patients with PSRCC. A nomogram was constructed to predict 1-, 3-, and 5-year OS. The performance of the nomogram was measured by C-index, receiver operating characteristic (ROC) curve, decision curve analysis (DCA). Results: The incidence of PSRCC is much lower than that of PDAC (10.798 V.S. 0.349 per millions). PSRCC is an independent predictor of pancreatic cancer with a poorer histological grade, a higher rate of lymph node and distant metastasis, and a poorer prognosis. We identified four independent prognostic factors including grade, American Joint Committee on Cancer Tumor-Node-Metastasis (TNM) stage, surgery and chemotherapy based on the Cox regression model. The C-index and DCA curves showed better performance of the nomogram than TNM stage. ROC curve analysis also showed that the nomogram had good discrimination, with an area under the curve of 0.840, 0.896, and 0.923 for 1-, 3-, and 5-year survival. The calibration curves showed good agreement between the prediction by the nomogram and actual observations. Conclusion: PSRCC is a rare but fatal subtype of pancreatic cancer. The constructed nomogram in this study accurately predicted the prognosis of PSRCC, performed better than the TNM stage. [ABSTRACT FROM AUTHOR]

The subdiffusion equations with a Caputo fractional derivative of order α ∈ (0 , 1) arise in a wide variety of practical problems, which describe the transport processes, in the force-free limit, slower than Brownian diffusion. In this work, we derive the correction schemes of the Lagrange interpolation with degree k ( k ≤ 6 ) convolution quadrature, called L k approximation, for the subdiffusion. The key step of designing correction algorithm is to calculate the explicit form of the coefficients of L k approximation by the polylogarithm function or Bose-Einstein integral. To construct a τ 8 approximation of Bose-Einstein integral, the desired (k + 1 - α) th-order convergence rate can be proved for the correction L k scheme with nonsmooth data, which is higher than kth-order BDFk method in [Jin, Li, and Zhou, SIAM J. Sci. Comput., 39 (2017), A3129–A3152; Shi and Chen, J. Sci. Comput., (2020) 85:28]. The numerical experiments with spectral method are given to illustrate theoretical results. [ABSTRACT FROM AUTHOR]

The time-fractional Cattaneo equation is an equation where the fractional order α ∈ (1 , 2) has the capacity to model the anomalous dynamics of physical diffusion processes. In this paper, we consider an efficient scheme for solving such an equation in two space dimensions. First, we obtain the space's semi-discrete numerical scheme by using the compact difference operator in the spatial direction. Then, the semi-discrete scheme is converted to a low-order system by means of order reduction, and the fully discrete compact difference scheme is presented by applying the L2-1 σ formula. To improve the computational efficiency, we adopt the fast discrete Sine transform and sum-of-exponentials techniques for the compact difference operator and L2-1 σ difference operator, respectively, and derive the improved scheme with fast computations in both time and space. That aside, we also consider the graded meshes in the time direction to efficiently handle the weak singularity of the solution at the initial time. The stability and convergence of the numerical scheme under the uniform meshes are rigorously proven, and it is shown that the scheme has second-order and fourth-order accuracy in time and in space, respectively. Finally, numerical examples with high-dimensional problems are demonstrated to verify the accuracy and computational efficiency of the derived scheme. [ABSTRACT FROM AUTHOR]

This paper aims to numerically study the time-fractional Allen-Cahn equation, where the time-fractional derivative is in the sense of Caputo with order α ∈ (0 , 1) . Considering the weak singularity of the solution u (x , t) at the starting time, i.e., its first and/or second derivatives with respect to time blowing-up as t → 0 + albeit the function itself being right continuous at t = 0 , two well-known difference formulas, including the nonuniform L1 formula and the nonuniform L2- 1 σ formula, which are used to approximate the Caputo time-fractional derivative, respectively, and the local discontinuous Galerkin (LDG) method is applied to discretize the spatial derivative. With the help of discrete fractional Gronwall-type inequalities, the stability and optimal error estimates of the fully discrete numerical schemes are demonstrated. Numerical experiments are presented to validate the theoretical results. [ABSTRACT FROM AUTHOR]

Background: Metastatic pancreatic cancer (mPC) is a highly lethal malignancy with poorer survival. However, chemotherapy alone was unable to maintain long‐term survival. This study aimed to evaluate the individualized survival benefits of pancreatectomy plus chemotherapy (PCT) for mPC. Methods: A total of 4546 patients with mPC from 2004 to 2015 were retrieved from the Surveillance, Epidemiology, and End Results database. The survival curve was calculated using the Kaplan-Meier method and differences in survival curves were tested using log-rank tests. Cox proportional hazards regression analyses were performed to evaluate the prognostic value of involved variables. A new nomogram was constructed to predict overall survival based on independent prognosis factors. The performance of the nomogram was measured by concordance index, calibration plot, and area under the receiver operating characteristic curve. Results: Compared to pancreatectomy or chemotherapy alone, PCT can significantly improve the prognosis of patients with mPC. In addition, patients with well/moderately differentiated tumors, age ≤66 years, tumor size ≤42 mm, or female patients were more likely to benefit from PCT. Multivariate analysis showed that age at diagnosis, sex, marital status, grade, tumor size, and treatment were independent prognostic factors. The established nomogram has a good ability to distinguish and calibrating. Conclusion: PCT can prolong survival in some patients with mPC. Our nomogram can individualize predict OS of pancreatectomy combined with chemotherapy in patients with concurrent mPC. [ABSTRACT FROM AUTHOR]

The research of finding hidden attractors in nonlinear dynamical systems has attracted much consideration because of its practical and theoretical importance. A new fractional order four-dimensional system, which can exhibit some hidden hyperchaotic attractors, is proposed in this paper. The predictor–corrector method of the Adams–Bashforth–Moulton algorithm and the parameter switching algorithm are used to numerically study this system. It is interesting that three different kinds of hidden hyperchaotic attractors with two positive Lyapunov exponents are found, and the fractional order system can have a line of equilibria, no equilibrium point, or only one stable equilibrium point. Moreover, a self-excited attractor is also recognized with the change of its parameters. Finally, the synchronization behavior is studied by using a linear feedback control method. [ABSTRACT FROM AUTHOR]

The barrier Lyapunov function (BLF) and its variants such as the asymmetric barrier Lyapunov function (ABLF) combined with backstepping technique have recently been employed to handle constraints for a class of non‐linear systems. However, the repeated differentiation in backstepping will result in the requirement of high‐order differentiability and the complexity of controllers in the multiple‐state high‐order systems. This study introduces dynamic surface control (DSC) to deal with these problems. The authors propose a backstepping DSC scheme based on the ABLF to address time‐varying output constraints for a class of non‐linear systems. The proposed control scheme can avoid the proliferation and singularity of repeated differentiation. As a consequence, it can relax the requirements of high‐order differentiability for stabilising functions and high power of output tracking error transformation involved in ABLF synthesis. The new controller can be proved to guarantee that all the closed‐loop signals remain bounded, and to ensure output constraints never violated. Comparison studies with previous work validate that DSC incorporated with ABLF can achieve favourable performance bounded within constraints, and it can also ensure fast and stable tracking convergence in the presence of disturbance. Experimental results illustrate the effectiveness of the presented method. [ABSTRACT FROM AUTHOR]

In this paper, we derive a high-order compact finite difference scheme for solving the reaction–subdiffusion equation with Neumann boundary value condition. The L1 method is used to approximate the temporal Caputo derivative, and the compact difference operator is applied for spatial discretization. We prove that the compact finite difference method is unconditionally stable and convergent with orderO(τ2−α+h4) inL2norm, where τ, α, andhare the temporal step size, the order of time fractional derivative and the spatial step size, respectively. Finally, some numerical experiments are carried out to show the effectiveness of the proposed difference scheme. [ABSTRACT FROM AUTHOR]

We report two posttraumatic fungal endophthalmitis cases with anterior and entire segment involvement caused, respectively, by Candida albicans and Aspergillus fumigatus. The anterior endophthalmitis induced by C. albicans presented multifocal lesions with hyphael structures in anterior chamber, but without involvement of posterior vitreous. The pan-endophthalmitis caused by A. fumigatus displayed significant hypopyon and vitritis. Treatment strategies are anterior chamber cleaning for anterior endophthalmitis and vitrectomy for pan-endophthalmitis, excepting systemic and topical antifungal drug applications including intravitreal injection for both types of endophthalmitis. There is a much better outcome in the case with anterior segment involvement than that with entire segment involvement. This case report provides evidence that the different types of posttraumatic fungal endophthalmitis related to different fungal infections indeed existed and should be managed with different strategies. [ABSTRACT FROM AUTHOR]

In this paper, we derive two novel finite difference schemes for two types of time-space fractional diffusion equations by adopting weighted and shifted Grünwald operator, which is used to approximate the Riemann-Liouville fractional derivative to the second order accuracy. The stability and convergence of the schemes are analyzed via mathematical induction. Moreover, the illustrative numerical examples are carried out to verify the accuracy and effectiveness of the schemes. [ABSTRACT FROM AUTHOR]

In this work, the classical (1+1)-dimensional Klein-Gordon-Schrödinger (KGS) system is studied. The ansatz and the homogeneous balance principle are employed in searching for particular soliton solutions, such as bright and dark solitons. Regrettably, dark solitons cannot be captured. However, this procedure leads to a series of new singular solitons and explicit periodic wave solutions. [ABSTRACT FROM AUTHOR]