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Present study reports the solution of generalized Rosenau–Kawahara-RLW equation. It includes motion of single solitary wave, interaction of two solitary waves along with the calculated invariants and error norms. Gaussian and undular bore initial conditions are studied to show evolution of solitons. Developed train of solitons and conservation of invariants are shown via figures and tables in the respective sections. Various case studies are presented to demonstrate the efficiency of the proposed numerical scheme. Solutions so produced may be helpful for explaining various nonlinear physical phenomena in nonlinear dynamical systems.

In this work, we developed a hybrid scheme for Gilson–Pickering equation by extracting advantageous features of collocation method and B-splines. Collocation method involves satisfying a differential equation to some tolerance at finite number of points and has low computational cost. The idea is to extract the superior accuracy and smoothness of B-spline functions with the low computational cost of collocation scheme. In the vision of a superficial periodic force, we obtain the quasiperiodic and chaotic behaviors of the forced Gilson–Pickering equation by perusing phase portrait technique, time series and Poincaré section. The sequel of frequency ($\omega$) of the superficial periodic force is very prime in case of transition from weakly chaotic feature to quasiperiodic feature of the forced Gilson–Pickering equation.

This paper studies Gardner equation, which represents long nonlinear internal waves. The collocation method based on B-splines is applied to the equation. The stability of the proposed numerical scheme is analyzed by using von Neumann theory. To observe some physical properties of long nonlinear internal waves, three test problems which contain the propagation of solitary waves, the interaction of solitary waves and evolution of solitons are considered. Also, the effect of nonlinearity on physical problems is investigated. In order to see this effect clearly, the same parameters are used during the computation for different degrees of nonlinearity in each problem.

We derive a Kronecker product approximation for the micromagnetic long-range interactions in a collocation framework by means of separable sinc quadrature. Evaluation of this operator for structured tensors (Canonical format, Tucker format, Tensor Trains) scales below linear in the volume size. Based on efficient usage of FFT for structured tensors, we are able to accelerate computations to quasi-linear complexity in the number of collocation points used in one dimension. Quadratic convergence of the underlying collocation scheme as well as exponential convergence in the separation rank of the approximations is proved. Numerical experiments on accuracy and complexity confirm the theoretical results.

Isogeometric analysis is applied to boundary integral equations corresponding to boundary-value problems governed by Laplace’s equation. It is shown that the smoothness of geometric parametrizations central to computer-aided design can be exploited for regularizing integral operators to obtain high-order collocation methods involving superior approximation and numerical integration schemes. The regularization is applicable to both singular and hyper-singular integral equations, and as a result one can formulate the governing integral equations so that the corresponding linear algebraic equations are well-conditioned. It is demonstrated that the proposed approach allows one to compute accurate approximate solutions which optimally converge to the exact ones.

Continuous interpolation of the sound pressure is favored in most applications of boundary element methods for acoustics. Few papers are known to the authors in which discontinuous elements are applied. Mostly they were used because they guarantee C¹ continuity of the geometry at element edges. The effect of superconvergence is known for boundary element collocation on discontinuous elements. This effect is observed if the collocation points are located at the zeroes of orthogonal functions, e.g. at the zeroes of the Legendre polynomials.

In this paper, we start with a review of related work. Then, formulation of discontinuous elements and position of nodes on the element are presented and discussed. One parameter controls the location of nodes on the element. The major part of the paper consists of the investigation of the computational example of a long duct. For that, the numeric solution is compared with the analytic solution of the corresponding one-dimensional problem. Error dependence in terms of frequency, element size and location of nodes on discontinuous elements is reported. It will be shown that the zeroes of the Legendre polynomials account for an optimal position of nodes. Similar results are observed for triangular elements. It can be seen that the error in the Euclidean norm changes by one or two orders of magnitude if the location of nodes is shifted over the element. It can be seen that the optimal location varies with the wave-number although remaining in the vicinity of the zeroes of orthogonal functions. The irregular mesh of a sedan cabin compartment accounts for the second example. Optimal choice of node position is confirmed for this example. One of the key results of this paper is that discontinuous boundary elements perform more efficiently than continuous ones, in particular for linear elements. This, however, further implies that nodes are located at the zeroes of orthogonal functions on the element.

In this short note alternative time domain boundary integral equations (TDBIE) for the scalar wave equation are formulated on a surface enclosing a volume. The technique used follows the traditional approach of subtracting and adding back relevant Taylor expansion terms of the field variable, but does not restrict this to the surface patches that contain the singularity only. From the divergence-free property of the added-back integrands, together with an application of Stokes' theorem, it follows that the added-back terms can be evaluated using line integrals defined on a cut between the surface and a sphere whose radius increases with time. Moreover, after a certain time, the line integrals may be evaluated directly. The results provide additional insight into the theoretical formulations, and might be used to improve numerical implementations in terms of stability and accuracy.

An implicit time integration meshfree collocation method for solving linear and nonlinear ordinary differential equations (ODEs) based on interpolating moving least squares technique, which uses singular weights for constructing ansatz functions, is presented. On an example of a system of equations for Foucault pendulum, the flexibility of the proposed approach is shown and the accuracy, convergence, and stability properties are investigated. In a nonlinear case, the method gives accurate results, which is demonstrated by the solution of Lorenz equations. The typical trajectory patterns, e.g. butterfly pattern, were observed and the properties of the method are compared to those of a higher-order time integration method.

A least-squares meshfree collocation method is presented. The method is based on the first-order differential equations in order to result in a better conditioned linear algebraic equations, and to obtain the primary variables (displacements) and the dual variables (stresses) simultaneously with the same accuracy. The moving least-squares approximation is employed to construct the shape functions. The sum of squared residuals of both differential equations and boundary conditions at nodal points is minimized. The present method does not require any background mesh and additional evaluation points, and thus is a truly meshfree method. Unlike other collocation methods, the present method does not involve derivative boundary conditions, therefore no stabilization terms are needed, and the resulting stiffness matrix is symmetric positive definite. Numerical examples show that the proposed method possesses an optimal rate of convergence for both primary and dual variables, if the nodes are uniformly distributed. However, the present method is sensitive to the choice of the influence length. Numerical examples include one-dimensional diffusion and convection-diffusion problems, two-dimensional Poisson equation and linear elasticity problems.

We consider the solutions of a space–time fractional diffusion equation on the interval [-1, 1]. The equation is obtained from the standard diffusion equation by replacing the second-order space derivative by a Riemann–Liouville fractional derivative of order between one and two, and the first-order time derivative by a Caputo fractional derivative of order between zero and one. As the fundamental solution of this fractional equation is unknown (if exists), an eigenfunction approach is applied to obtain approximate fundamental solutions which are then used to solve the space–time fractional diffusion equation with initial and boundary values. Numerical results are presented to demonstrate the effectiveness of the proposed method in long time simulations.

In this paper, an alternative meshfree approach is presented for transient heat conduction analysis of nonlinear functionally graded materials (FGMs). The main idea behind the introduced approach is to use collocation in local domains containing of sets of regular or scattered nodes and approximating the solution by B-spline basis functions. It combines the favorable properties of B-spline basis functions in having arbitrary degree for better resolution of solution, partition of unity and the Kronecker delta properties with low computational effort of collocation. The method is called as local B-spline collocation method. It is mathematically simple, efficient to program and truly meshless. The method is applied for analyzing transient heat conduction in a wide range of FGMs with various material gradation models, in both 2D and 3D domains. The results obtained agree well with those computed by analytical solution and other well-known methods, confirming the suitability and efficacy of the presented scheme.

This paper summarizes the main principles of the solution structure method and presents it in combination with atomic basis functions and a collocation technique. The solution of a boundary value problem is expressed in the form of formulae called solution structures, which depend on three components: the first component describes the geometry of the domain exactly in the analytical form, the second describes all boundary conditions exactly, and the third component, that contains information about the differential equation, is the unknown component represented by a linear combination of atomic basis functions. The proposed method is applied to solve the torsion problem.

This work proposes a hybrid block numerical method of tenth order for the direct solution of fifth-order initial value problems. The formulas that constitute the block method are derived from a continuous approximation obtained through interpolation and collocation techniques. In order to obtain better accuracy, sixth-order derivatives are incorporated to develop the formulas. The main characteristics of the method are analyzed, namely, the order, local truncation errors, zero-stability, consistency and convergence. The proposed strategy performs well, as shown by some numerical examples and the corresponding efficiency curves. Compared to existing numerical methods in the literature, the proposed method is competitive and the numerical approximations it provides are significantly close to the precise solutions.

The aim of this paper is to use higher-order energy balance method as a novel solution procedure for investigation the nonlinear oscillation of various systems. The method is associated with collocation method and employed for a group of nonlinear problems including pure cubic nonlinearity, fractional elastic force and duffing harmonic terms. Obtained results are evaluated by comparing with the classical balance method and also the exact solutions computed numerically. It is shown our solution procedure achieves more accurate results versus the classical solution and higher agreement are observed between the newer solutions and the exact ones for the systems. Effect of initial conditions in the nonlinear natural frequencies are carried out for a range of small and large values and it is proved the proposed method is not only simple but also more reliable for analysis of such systems.

In this study, our goal was to find out how the verbs 幫忙 bāngmáng and 幫助 bāngzhù differ in usage. In particular, we aimed to determine the characteristics of the co-occurrences for the two verbs to investigate whether the verb 幫忙 bāngmáng typically takes an event as an object with positive or neutral collocates and whether the verb 幫助 bāngzhù typically takes a person/an organization as an object with negative collocates. Data for the two synonymous verbs were collected from the Chinese GigaWord Corpus using Chinese Word Sketch. As we predicted, our corpus data showed that 幫忙 bāngmáng typically took an event as an object and 幫助 bāngzhù typically took a person/an organization as an object, and that the verb 幫助 bāngzhù was more often linked with lexical items that had negative meanings. Based on the results presented, this study has practical implications for second language acquisition, translation, and dictionary compiling.

An implementation of the standard collocation method based on polynomial interpolation is presented in a matrix framework in this paper. The underlying differentiation matrix can be partitioned to yield a superconvergent implicit multistep-like method to solve the initial value problem numerically. The first- and second-order versions of this method are L-stable.

In this paper, a new local RBF collocation approach based on point interpolation method, which can assure that its coefficient matrices are of bandwidth, has been presented. The main feature of this approach is to use a Hermite-type interpolation scheme, namely making use of the normal gradient at Neumann boundary so that this computational accuracy has greatly improved. The accuracy and simplicity of this presented approach will be shown numerically by well-known 2-D linear elastic Cook's membrane benchmark.