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This study aims to apply a numerical meshless method, namely, the boundary knot method (BKM) combined with the meshless analog equation method (MAEM) in space and use a semi-implicit scheme in time for finding a new numerical solution of the advection–reaction–diffusion and reaction–diffusion systems in two-dimensional spaces, which arise in biology.

First, the BKM is applied to approximate the spatial variables of the studied mathematical models. Then, this study derives fully discrete scheme of the studied models using a semi-implicit scheme based on Crank–Nicolson idea, which gives a linear system of algebraic equations with a non-square matrix per time step that is solved by the singular value decomposition. The proposed approach approximates the solution of a given partial differential equation using particular and homogeneous solutions and without considering the fundamental solutions of the proposed equations.

This study reports some numerical simulations for showing the ability of the presented technique in solving the studied mathematical models arising in biology. The obtained results by the developed numerical scheme are in good agreement with the results reported in the literature. Besides, a simulation of the proposed model is done on buttery shape domain in two-dimensional space.

This study develops the BKM combined with MAEM for solving the coupled systems of (advection) reaction–diffusion equations in two-dimensional spaces. Besides, it does not need the fundamental solution of the mathematical models studied here, which omits any difficulties.

This study aims to propose a new numerical method for solving non-linear partial differential equations on irregular domains.

The main aim of the current paper is to propose a local meshless collocation method to solve the two-dimensional Klein-Kramers equation with a fractional derivative in the Riemann-Liouville sense, in the time term. This equation describes the sub-diffusion in the presence of an external force field in phase space.

First, the authors use two finite difference schemes to discrete temporal variables and then the radial basis function-differential quadrature method has been used to estimate the spatial direction. To discrete the time-variable, the authors use two different strategies with convergence orders O(τ1+γ) and O(τ2−γ) for 0 < γ < 1. Finally, some numerical examples have been presented to show the high accuracy and acceptable results of the proposed technique.

The proposed numerical technique is flexible for different computational domains.

The purpose of this study is to use the method of lines to solve the two-dimensional nonlinear advection–diffusion–reaction equation with variable coefficients.

The strictly positive definite radial basis functions collocation method together with the decomposition of the interpolation matrix is used to turn the problem into a system of nonlinear first-order differential equations. Then a numerical solution of this system is computed by changing in the classical fourth-order Runge–Kutta method as well.

Several test problems are provided to confirm the validity and efficiently of the proposed method.

For the first time, some famous examples are solved by using the proposed high-order technique.

To optimize the shape of a cascade where velocity (or pressure) distribution is optimally given.

The semi‐inverse method suggested by Ji‐Huan He is applied to establishment of a variational theory for the discussed inverse problem. The boundary conditions on unknown shape are converted into the natural boundary conditions of the obtained variational functional.

Ji‐Huan He's semi‐inverse method is a powerful tool to the search for the variational formulation for the discussed problem. The derivation procedure is very simple and convenient; the finite element method based on the variational theory with moving boundary provides a very effective and robust numerical approach to the inverse problem.

The design method is limited to frictionless flow.

The numerical method based on variational principle with moving boundary can be readily extended to other cases with moving surfaces or free boundaries.

The suggested numerical method can satisfy the demand of various cascade designs, where the velocity (or pressure) distribution can be optimally given from different aspects of engineering requirement: aerodynamics, strength, manufacture, etc.

The purpose of this paper is to develop a transform method and a deep learning model to identify the inner surface shape based on the measurement temperature at the outer boundary of the pipe.

The training process is assisted by the finite element method (FEM) simulation which solves the direct problem for the data preparation. To avoid re-meshing the domain when the inner surface shape varies, a new transform method is proposed to transform the shape identification problem into the effective thermal conductivity identification problem. The deep learning model is established to set up the relationship between the measurement temperature and the effective thermal conductivity. Then the unknown geometry shape is acquired by the mapping between the inner shape and the effective thermal conductivity through the inverse transform method.

The new method is successfully applied to identify the internal boundary of a pipe with eccentric circle, ellipse and nephroid inner geometries. The results show that as the measurement points increased and the measurement error decreased, the results became more accurate. The position of the measurement point and mesh density of the FEM model have less effect on the results.

The deep learning model and the transform method are developed to identify the pipe inner surface shape. There is no need to re-mesh the domain during the computation progress. The results show that the proposed method is a fast and an accurate tool for identifying the pipe inner surface.

The purpose of this paper is to describe a novel universal error estimator and the adaptive time-stepping process in the generalized single-step single-solve (GS4-1) computational framework, applied for the fluid dynamics with illustrations to incompressible Navier–Stokes equations.

The proposed error estimator is universal and versatile that it works for the entire subsets of the GS4-1 framework, encompassing the nondissipative Crank–Nicolson method, the most dissipative backward differential formula and anything in between. It is new and novel that the cumbersome design work of error estimation for specific time integration algorithms can be avoided. Regarding the numerical implementation, the local error estimation has a compact representation that it is determined by the time derivative variables at four successive time levels and only involves vector operations, which is simple for numerical implementation. Additionally, the adaptive time-stepping is further illustrated by the proposed error estimator and is used to solve the benchmark problems of lid-driven cavity and flow past a cylinder.

The proposed computational procedure is capable of eliminating the nonphysical oscillations in GS4-1(1,1)/Crank–Nicolson method; being CPU-efficient in both dissipative and nondissipative schemes with better solution accuracy; and detecting the complex physics and hence selecting a suitable time step according to the user-defined error threshold.

To the best of the authors’ knowledge, for the first time, this study applies the general purpose GS4-1 family of time integration algorithms for transient simulations of incompressible Navier–Stokes equations in fluid dynamics with constant and adaptive time steps via a novel and universal error estimator. The proposed computational framework is simple for numerical implementation and the time step selection based on the proposed error estimation is efficient, benefiting to the computational expense for transient simulations.

This paper aims to develop a new isogeometric boundary element formulation based on non-uniform rational basis splines (NURBS) curves for solving Reissner’s shear-deformable plates.

The generalized displacements and tractions along the problem boundary are approximated as NURBS curves having the same rational B-spline basis functions used to describe the geometrical boundary of the problem. The source points positions are determined over the problem boundary by the well-known Greville abscissae definition. The singular integrals are accurately evaluated using the singularity subtraction technique.

Numerical examples are solved to demonstrate the validity and the accuracy of the developed formulation.

This formulation is considered to preserve the exact geometry of the problem and to reduce or cancel mesh generation time by using NURBS curves employed in computer aided designs as a tool for isogeometric analysis. The present formulation extends such curves to be implemented as a stress analysis tool.

This paper proposes a novel approach to impose the Neumann boundary condition for isogeometric analysis (IGA) of Euler–Bernoulli beam with 1-D formulation. The proposed method is for only IGA in which it is difficult to handle the Neumann boundary conditions. The control points of B-spline are equivalent to nodes in finite element method. With 1-D formulation, it is not possible to accommodate multiple degrees of freedom in IGA. This case arises in the analysis of beams. The paper aims to propose a way to work around this issue in a simple way.

Neumann boundary conditions, which are even-order derivatives (example: double derivative) of the primary variable, are inherently satisfied in the weak form. Boundary conditions with an odd number of derivatives (example: slope) are imposed with the introduction of a new penalty matrix.

The proposed method can impose a slope boundary condition for IGA of a beam using 1-D formulation.

From the literature, it can be observed that the beam is formulated in 1-D by considering it as either a rotation-free element or a 2-D formulation by considering shear strain along with the normal strain. The work represents 1-D formulation of a beam while considering the slope boundary condition, which is easy and effective to formulate, compared with the slope boundary conditions reported in previous works.

Drawing from findings of a case study of inter‐organisational collaboration, this paper aims to employ organisational theory to examine the potential learning that opens between educational organisations. The focus is discursive practices. Two questions guide the analysis. What (unique) practices are implicated in the “knotworking” of inter‐organisational collaboration? What knowledge and capacities are learned in these discursive practices?

A case study was conducted of a collaboration between a university unit, school district, elementary school and parent executive board to govern a laboratory school. Documents were examined and 17 interviews conducted and analysed inductively. Document analysis and second stage transcript analysis employed methods of discourse analysis.

The case analysis suggests that collaborations open unique sites for organizational learning. Actors (teachers, administrators, parents) engage with various discourses in the “knots” of inter‐organisational networks. Those who thrive in the “knot” of collaboration learn how to be flexibly attuned to shifting elements that emerge in negotiations. Further, these actors appear to develop capacities of mapping, translating, rearticulating, and spanning boundaries among the diverse positions of organisations.

The case study is limited in scope in order to allow in‐depth discourse analysis of the data.

The combination of theories employed here – a practice‐based organizational learning theory called “knotworking” and critical organisational discourse analysis – is unique in educational administration research. It is argued that together, these theories provide a useful analytic approach for administrators wanting to understand and work through the cultural and political complexities of inter‐organisational collaborations.

The purpose of this paper is to present some modifications in the spline‐based differential quadrature method (DQM), in order to accelerate the convergence of the method. The improvements are explained and examined by the examples of the free vibration of conical shells. The composite laminated shell, as well as isotropic one, are taken under consideration.

To determine weighting coefficients for the DQM, the spline interpolation with non‐standard definitions of the end conditions is used. One of these definitions combines natural and not‐a‐knot end conditions, while the other one uses the boundary conditions for considered problem as the end conditions. The weighting coefficients can be determined by solving set of equations arising from spline interpolation.

It is shown that the proposed modifications significantly improve the convergence of the method, especially when the boundary conditions are introduced at the stage of the computation of the weighting coefficients. Unfortunately, the use of this approach is limited to some types of boundary conditions.

The paper describes development of the modified spline interpolation dedicated to DQM and examines the possibility of building boundary conditions into the weighting coefficients at the stage of the computation of these coefficients.