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In this study, a second-order standard wave equation extended to a two-dimensional viscous wave equation with timely differentiated advection-diffusion terms has been solved by differential quadrature methods (DQM) using a modification of cubic B-spline functions. Two numerical schemes are proposed and compared to achieve numerical approximations for the solutions of nonlinear viscous wave equations.

Two schemes are adopted to reduce the given system into two systems of nonlinear first-order partial differential equations (PDE). For each scheme, modified cubic B-spline (MCB)-DQM is used for calculating the spatial variables and their derivatives that reduces the system of PDEs into a system of nonlinear ODEs. The solutions of these systems of ODEs are determined by SSP-RK43 scheme. The CPU time is also calculated and compared. Matrix stability analysis has been performed for each scheme and both are found to be unconditionally stable. The results of both schemes have been extensively discussed and compared. The accuracy and reliability of the methods have been successfully tested on several examples.

A comparative study has been carried out for two different schemes. Results from both schemes are also compared with analytical solutions and the results available in literature. Experiments show that MCB-DQM with Scheme II yield more accurate and reliable results in solving viscous wave equations. But Scheme I is comparatively less expensive in terms of CPU time. For MCB-DQM, less depository requirements lead to less aggregation of approximation errors which in turn enhances the correctness and readiness of the numerical techniques. Approximate solutions to the two-dimensional nonlinear viscous wave equation have been found without linearizing the equation. Ease of implementation and low computation cost are the strengths of the method.

For the first time, a comparative study has been carried out for the solution of nonlinear viscous wave equation. Comparisons are done in terms of accuracy and CPU time. It is concluded that Scheme II is more suitable.

The purpose of this paper is to present a general formulation of the quadrature element method (QEM). The method is then used to investigate the free vibration of functionally graded (FG) beams with general boundary conditions and different variations of material properties.

The quadrature elements with arbitrary number of nodes and nodal distributions are established on the basis of two types of FG Timoshenko beam theories. One called TBT-1 takes the cross-sectional rotation as the unknown function and the other called TBT-2 uses the transverse shear strain as the unknown function. Explicit formulas are provided via the help of the differential quadrature (DQ) rule and thus the elements can be implemented adaptively with ease.

The suitability and computational efficiency of the proposed quadrature elements for the vibration analysis of FG beams are demonstrated. The convergence rate of the proposed method is high. The elements are shear-locking free and can yield accurate solutions with a small number of nodes for both thin and moderately thick beams. The performance of the element based on TBT-1 is better than the one based on TBT-2.

The present QEM is different from the existing one which exclusively uses Gauss–Lobatto–Legendre (GLL) nodes and GLL quadrature and thus is more general. The element nodes can be either the same or different from the integration points, making the selection of element nodes more flexible. Presented data are accurate and may be a reference for other researchers to develop new numerical methods. The QEM may be also useful in multi-scale modeling and in the analysis of civil infrastructures.

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.

The purpose of this paper is to highlight the implementation of a recently developed weak form quadrature element method for nonlinear free vibrations of Timoshenko beams subjected to three different boundary conditions.

The design of the paper is based on considering the geometrically nonlinear effects of axial strain, bending curvature, and shear strain. Then the quadrature element formulation of the beam is introduced.

The efficiency of the method is demonstrated by a convergence study. Ratios of the nonlinear fundamental frequencies to the corresponding linear frequencies are extracted. Their variations with the ratio of amplitude to radius of gyration and the slenderness ratio are examined. The effects of the nonlinearity on higher order frequencies and mode shapes are also investigated.

The computed results show fast convergence and compare well with available literature results.

The purpose of this paper is to propose a semi‐analytical method for studying the interaction between reservoir and concrete gravity dams.

The reservoir is assumed to be unbounded at the far end and the solution is sought for incompressible and in‐viscid fluid. A concrete gravity dam is assumed to behave as a cantilever beam of variable section, and the inclination of the neutral axis is ignored.

It is shown that use of the differential quadrature method (DQM), with a few grid points in conjunction with the finite difference method (FDM), yields an acceptable convergence of results. Comparing the results of the proposed method with those of the literature shows the competency of the method.

DQM for space derivatives and FDM for time derivatives are used to discretize the partial differential equation of motion.

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 the method is to develop a numerical method for the solution of nonlinear partial differential equations.

A new numerical approach based on Barycentric Rational interpolation has been used to solve partial differential equations.

A numerical technique based on barycentric rational interpolation has been developed to investigate numerical simulation of the Burgers’ and Fisher’s equations. Barycentric interpolation is basically a variant of well-known Lagrange polynomial interpolation which is very fast and stable. Using semi-discretization for unknown variable and its derivatives in spatial direction by barycentric rational interpolation, we get a system of ordinary differential equations. This system of ordinary differential equation’s has been solved by applying SSP-RK43 method. To check the efficiency of the method, computed numerical results have been compared with those obtained by existing methods. Barycentric method is able to capture solution behavior at small values of kinematic viscosity for Burgers’ equation.

To the best of the authors’ knowledge, the method is developed for the first time and validity is checked by stability and error analysis.

The purpose of this paper is to propose a new approach to determine the aeroelastic instability of truncated conical shells. In the proposed approach the governing equation of flutter for a truncated conical shell is established using Love's thin shell theory and the quasi-steady first-order piston theory.

The derivatives in both the governing equations and the boundary conditions are discretized with the differential quadrature method, and the critical flutter chamber pressure is obtained by eigenvalue analysis.

The influence of the shell geometry parameters, such as semi-cone angle, radius-thickness ratio and length-radius ratio, on the critical flutter chamber pressure is studied. Results are also presented to indicate the stabilizing effects of aerodynamic damping and the destabilizing effects of the curvature correction term of piston theory on flutter of truncated conical shell.

The present approach is an efficient method for the free vibration and flutter analysis of truncated conical shells due to its high order of accuracy and less requirement of virtual storage and computational effort.

This paper aims to capture the effective behaviour of nonlinear coupled advection-diffusion-reaction systems and develop a new computational scheme based on differential quadrature method.

The developed scheme converts the coupled system into a system of ordinary differential equations. Finally, the obtained system is solved by a four-stage RK4 scheme.

The developed scheme helped to capture the different types of patterns of nonlinear time-dependent coupled advection-diffusion-reaction systems such as Brusselator model, Chemo-taxis model and linear model which are similar to the existing patterns of the models.

The originality lies in the fact that the developed scheme is new for coupled advection-diffusion-reaction systems such as Brusselator model, Chemo-taxis model and linear models. Second, the captured pattern is similar to the existing patterns of the models.

The multi‐domain generalized differential quadrature method is applied in this paper to simulate the flows in Czochralski crystal growth. The effect of interface treatment on the numerical solution is studied through four types of interface approximations. The performance of those four interface approximations is validated by a benchmark problem suggested by Wheeler. It is demonstrated in this study that the multi‐domain GDQ approach is an efficient method which can obtain accurate numerical solutions by using very few grid points, and the overlapped interface approximation provides the most accurate numerical results.