The Bratu's equation is a strongly nonlinear second-order ordinary differential equation that arises in electrospinning process and models temperature distribution within a flame in combustion theory. Bratu-type equations are used to simulate the ignition of flammable gases and flame propagation. In this paper, two methods are proposed to obtain highly accurate and reliable approximate solutions of Bratu-type boundary value problems. The first technique is a power series method which is based on the generalised Cauchy product that simplifies the difficulty associated with the nonlinear terms. Subsequently, explicit recurrence relations for the expansion coefficients of the series solutions are obtained. The second approach uses a twelfth-order second derivative backward differentiation formula that is implemented as a boundary value method. This numerical method is referred to as second derivative backward differentiation boundary value method. Three examples are given to illustrate the effectiveness, reliability, and accuracy of the proposed methods. The results obtained from both methods are in excellent agreement with the known exact solution. Comparison of the approximate and exact solutions shows that the proposed methods are reliable and accurate in solving a class of strongly nonlinear boundary value problems of Bratu-type.

Invariant-preserving schemes with arbitrarily high-order accuracy for the two-component Camassa–Holm system via Hamiltonian boundary value methods

This paper presents the development and analysis of a Toeplitz five-band monotone convergence boundary value method for the numerical solution of fourth-order boundary value problems (BVPs). The proposed approach is constructed by assembling linear multistep schemes into a block framework, derived systematically through the interpolation and collocation techniques. The resulting block method produces a Toeplitz coefficient structure with a five-band form, ensuring computational efficiency and numerical stability. The convergence properties of the method are rigorously examined, with proofs showing that the scheme is both consistent and stable, thereby guaranteeing monotone convergence. Special attention is given to the ability of the method to handle the complexity inherent in fourth-order BVPs, which frequently arise in engineering and applied sciences, particularly in beam theory, elasticity, and fluid mechanics. To demonstrate the efficiency and reliability of the method, several benchmark fourth-order boundary value problems are solved and compared against existing methods in the literature. Numerical results show that the proposed Toeplitz five-band block method achieves higher accuracy with fewer computational steps, while maintaining stability across a wide range of test problems. The performance improvements highlight its superiority in terms of accuracy, convergence rate, and computational cost. Overall, the Toeplitz five-band monotone convergence boundary value method provides a robust and efficient tool for solving fourth-order BVPs, contributing a significant advancement in numerical methods for higher-order differential equations.

Hamiltonian boundary value methods applied to KdV-KdV systems

Abstract The present investigation examined how stamping affects dimpled beam fundamental frequencies. Modal analysis on a finite element (FE) dimpled beam with single and multiple dimples was effectively created. This three-dimensional model properly depicts transverse, lateral, and torsional vibration modes. Thus, it furnishes the experimental dimpled beam analysis more accurately. Comparison studies of the initial five natural frequencies determined by the Finite Element Analysis (FEA), Boundary Value Method (BVM), and experimental modal analysis are carried out. This study employed free-free beams with one dimple, two dimples in the same direction, and two dimples in opposite directions. First, FEA results were compared to empirical measurements. The variance from experimentation was observed to be less than 2.5%. Thus, the methodology provided in this work may be utilized to accurately determine the natural frequency range of dimpled beams with unconstrained boundary conditions at both ends. FEA and BVM were compared for dimpled beams and boundary conditions. FEA estimated natural frequencies better than BVM in all cases.

This study aims to develop an indicator system for assessing the humanistic care competencies of nurses in infectious disease hospitals and provide a scientific measurement tool to understand the current humanistic care competencies level of infectious disease nurses. A mixed-methods design integrating qualitative interviews and a modified Delphi study. Initially, we derived a list of potential indicators of humanistic care for nurses in infectious disease hospitals from literature reviews and interviews with a nominal group technique (n=41). Following this, 26 experts from across China participated in two Delphi rounds from May to July 2023. Then the indicators were screened, revised and supplemented using the boundary value method and expert opinions. Next, the hierarchical analysis method was utilised to determine the weights of the indicators. The average effective response rate across the two Delphi rounds was 94%. The authority coefficients for the first and second rounds were 0.85 and 0.90, respectively, suggesting the experts were highly authoritative. There was a consistent rating among experts with a coordination coefficient for each indicator (p < 0.001). Ultimately, this study identified 4 primary indicators, 8 secondary indicators, and 35 tertiary indicators. The four primary indicators and their weights are basic care competency (0.158), therapeutic care competency (0.544), spiritual care competency (0.158) and safety care competency (0.140). This research provides a scientifically rigorous and comprehensive framework to evaluate the humanistic care competencies of nurses in infectious disease hospitals in China. This system will serve as an effective tool for evaluating the humanistic care competencies of nurses in specialized infectious disease hospitals in China and other overseas regions. This study provides a new tool to assess the humanistic care competencies of nurses in infectious disease hospitals. Form an effective humanistic care competencies index system that can be used to build and develop the need for nurses to possess different aspects of humanistic care competencies tailored to infectious disease patients in hospitals. No patients or public contribution.

Objective: To present an evaluation indicator system for access to cancer screening services. Methods: The evaluation indicator pool was constructed through a scoping review. The theoretical framework was constructed based on the multi-source indicators, and the qualitative expert consultation method was employed to form the initial version of the three-level evaluation indicator system. Delphi expert consultation method was conducted in two rounds to evaluate the relevance, importance, and availability of the proposed evaluation indicator system. The expert positive coefficient, authority coefficient, coordination degree of expert opinions, and concentration of expert opinions were subjected to analysis. Subsequently, the three-level evaluation indicator system for access to cancer screening services was adjusted and determined based on the boundary value method and the open opinions of experts. Finally, the combination weight method was employed to determine the weight. Results: The initial version of the indicator system comprised 3 primary (first-level) indicators, 11 secondary (second-level) indicators, and 46 tertiary (third-level) indicators. Delphi expert consultation was conducted for the initial version, and 17 experts ultimately completed it, exhibiting a positive coefficient of 100% and an authority coefficient of 0.87. In comparison to the initial round of consultation, Kendall's W coefficient ranges (0.15-0.43, all P<0.05) of relevance, importance, and availability scores for each tertiary indicator in the second round exhibited an improvement. The analysis of the importance dimension indicates that expert opinions are also more concentrated, as evidenced by an increase of 8.5% and 7.0% in the proportion of the tertiary indicators with an arithmetic mean above 8 and a full mark ratio above 0.5, respectively. The final evaluation indicator system comprises three primary indicators, with the weights of structure evaluation, process evaluation, and outcome evaluation being 0.338, 0.378, and 0.285, respectively. It also comprises 11 secondary indicators and 45 tertiary indicators. Conclusions: The evaluation indicator system developed in this article can be an effective evaluation tool for quantitative comparison of access to cancer screening services across different populations, cancer types, and before and after intervention. Furthermore, it is recommended that the system undergo continuous optimization concerning its application.

Hamiltonian Boundary Value Methods (HBVMs) for functional differential equations with piecewise continuous arguments

Scientific research activity in hospitals is important for promoting the development of clinical medicine, and the scientific literacy of medical staff plays an important role in improving the quality and competitiveness of hospital research. To date, no index system applicable to the scientific literacy of medical staff in China has been developed that can effectively evaluate and guide scientific literacy. This study aimed to establish an index system for the scientific literacy of medical staff in China and provide a reference for improving the evaluation of this system. In this study, a preliminary indicator pool for the scientific literacy of medical staff was constructed through the nominal group technique (n = 16) with medical staff. Then, two rounds of Delphi expert consultation surveys (n = 20) were conducted with clinicians, and the indicators were screened, revised and supplemented using the boundary value method and expert opinions. Next, the hierarchical analysis method was utilized to determine the weights of the indicators and ultimately establish a scientific literacy indicator system for medical staff. Following expert opinion, the index system for the scientific literacy of medical staff featuring 2 first-level indicators, 9second-level indicators, and 38 third-level indicators was ultimately established, and the weights of the indicators were calculated. The two first-level indicators were research literacy and research ability, and the second-level indicators were research attitude (0.375), ability to identify problems (0.2038), basic literacy (0.1250), ability to implement projects (0.0843), research output capacity (0.0747), professional capacity (0.0735), data-processing capacity (0.0239), thesis-writing skills (0.0217), and ability to use literature (0.0181). This study constructed a comprehensive scientific literacy index system that can assess medical staff's scientific literacy and serve as a reference for evaluating and improving their scientific literacy.

Solve Riemann–Liouville boundary value problems using collocation boundary value methods with the graded mesh

Potentiality of the HOFiD_bvp code in solving different kind of second order boundary value problems

Numerical solutions of fractional differential equation with multiple delays via block boundary value method

Block boundary value methods (BBVMs) are extended in this paper to obtain the numerical solutions of nonlinear delay-differential-algebraic equations with singular perturbation (DDAESP). It is proved that the extended BBVMs in some suitable conditions are globally stable and can obtain a unique exact solution of the DDAESP. Besides, whenever the classic Lipschitz conditions are satisfied, the extended BBVMs are preconsistent and $p$th order consistent. Moreover, through some numerical examples, the correctness of the theoretical results and computational validity of the extended BBVMs is further confirmed.

Membrane proteins typically deform the surrounding lipid bilayer membrane, which can play an important role in the function, regulation, and organization of membrane proteins. Membrane elasticity theory provides a beautiful description of protein-induced lipid bilayer deformations, in which all physical parameters can be directly determined from experiments. While analytic solutions of protein-induced elastic bilayer deformations are most easily developed for proteins with approximately circular cross sections, structural biology has shown that membrane proteins come in a variety of distinct shapes, with often considerable deviations from a circular cross section. We develop here a boundary value method (BVM) that permits the construction of analytic solutions of protein-induced elastic bilayer deformations for protein shapes with arbitrarily large deviations from a circular cross section, for constant as well as variable boundary conditions along the bilayer-protein interface. We apply this BVM to protein-induced lipid bilayer thickness deformations. Our BVM reproduces available analytic solutions for proteins with circular cross sectionand yields, for proteins with noncircular cross section, excellent agreement with numerical, finite element solutions. On this basis, we formulate a simple analytic approximation of the bilayer thickness deformation energy associated with general protein shapes and show that, for modest deviations from rotational symmetry, this analytic approximation is in good agreement with BVM solutions. Using the BVM, we survey the dependence of protein-induced elastic bilayer thickness deformations on protein shape, and thus explore how the coupling of protein shape and bilayer thickness deformations affects protein oligomerization and transitions in protein conformational state.

An efficient new thermal-wave inverse-problem approach based on an integral-equation boundary-value method coupled with an imperialist competitive algorithm was developed. The methodology was successfully applied to simultaneously reconstruct density and thermal conductivity depth profiles in a sintered powder metallurgy sample from an industrial automotive manufacturer with a surface layer of higher density than the bulk. The density and thermal conductivity depth profiles were validated independently using the manufacturer's data and in-house temperature and porosity measurements. The present non-destructive inverse problem approach represents a generalized formalism to thermal-wave reconstruction of dual depth profiles using frequency scan data measured from the interrogated surface. From a fundamental viewpoint, the method adds significant insights into the relationship between thermal conductivity and density distributions in inhomogeneous solids.

The large storage requirement is a critical issue in cross-correlation imaging-condition based reverse time migration (RTM), because it requires the operation of the source and receiver wavefields at the same time. The boundary value method (BVM), based on the finite difference method (FDM), can be used to reconstruct the source wavefield in the reverse time propagation in the same way as the receiver wavefield, which can reduce the storage burden of the RTM data. Considering that the FDM cannot well handle models with discontinuous material properties and rough interfaces, we develop a source wavefield reconstruction strategy based on the finite element method (FEM), using proper orthogonal decomposition (POD) to enhance computational efficiency. In this method, we divide the whole time period into several segments, and construct the POD basis functions to get a reduced order model (ROM) for the source wavefield reconstruction in each segment. We show the corresponding quantitative analysis of the storage requirement of the POD-FEM. Numerical tests on the homogeneous model show the effectiveness of the proposed method, while the layered model and part of the Marmousi model tests indicate that the POD-FEM can keep an excellent balance between computational efficiency and memory usage compared with the full-stored method (FSM) and the BVM, and can be effectively applied in imaging.

Recently, the efficient numerical solution of Hamiltonian problems has been tackled by defining the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Their derivation relies on the expansion of the vector field along the Legendre orthonormal basis. Interestingly, this approach can be extended to cope with other orthonormal bases and, in particular, we here consider the case of the Chebyshev polynomial basis. The corresponding Runge-Kutta methods were previously obtained by Costabile and Napoli [33]. In this paper, the use of a different framework allows us to carry out a novel analysis of the methods also when they are used as spectral formulae in time, along with some generalizations of the methods.

We present a class of Boundary Value Methods (BVMs) that can be applied to semi-stable and unstable Partial Differential Equation (PDE) models. Step-by-step (initial value) methods, such as Runge–Kutta and linear multistep methods have numerical stability regions which do not intersect with certain regions of the complex plane that are significant to the time-integration of unstable PDEs. BVMs, which need extra numerical conditions at the final time, are global methods and are, in some sense, free of such barriers. We discuss BVMs based on generalized midpoint methods, combined with appropriate numerical initial and final conditions. The stability regions of these methods intersect with a significant part of the complex plane. In several numerical experiments we will illustrate the usefulness of such methods when applied to PDE models, such as a dispersive wave equation, a space-fractional PDE and the backward heat equation.

A class of linearization-based collocation methods for initial value and boundary value engineering problems

Source wavefield reconstruction based on an implicit staggered-grid finite-difference operator for seismic imaging