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The purpose of this paper is to analyse a novel technique for an efficient numerical approximation of systems of highly oscillatory ordinary differential equations (ODEs) that arise in electronic systems subject to modulated signals.

The paper combines a Filon‐type method with waveform relaxation techniques for nonlinear systems of ODEs.

The analysis includes numerical examples to compare with traditional methods such as the trapezoidal rule and Runge‐Kutta methods. This comparison shows that the proposed approach can be very effective when dealing with systems of highly oscillatory differential equations.

The present paper constitutes a preliminary study of Filon‐type methods applied to highly oscillatory ODEs in the context of electronic systems, and it is a starting point for future research that will address more general cases.

The proposed method makes use of novel and recent techniques in the area of highly oscillatory problems, and it proves to be particularly useful in cases where standard methods become expensive to implement.

This paper aims to explore a new approach for time-domain modelling of interconnects with highly oscillatory modulated sources.

The paper uses an asymptotic method in conjunction with the Green’s function of the telegrapher’s equations. The Green’s function is expressed as a series of rational functions in the Laplace domain and are converted to pole-residue form, thereby enabling time-domain implementation.

The results indicate that the method is accurate for modelling interconnects when wide-varying frequencies are present in the sources.

The technique is important in circuit design for assessing signal integrity and in electromagnetic compatibility testing.

The purpose of this paper is to apply a novel technique for the simulation of nonlinear systems subject to modulated chirp signals.

The simulation technique is first described and its salient features are presented. Two examples are given to confirm the merits of the method.

The results indicate that the method is appropriate for simulating nonlinear systems subject to modulated chirp signals. In particular, the efficiency and accuracy of the method is seen to improve as the chirp frequency increases. In addition, error bounds are given for the method.

Chirp signals are employed in several important applications such as representing biological signals and in spread spectrum communications. Analysis of systems involving such signals requires accurate, appropriate and effective simulation techniques.

This chapter considers the lag structures of dynamic models in economics, arguing that the standard approach is too simple to capture the complexity of actual lag structures arising, for example, from production and investment decisions. It is argued that recent (1990s) developments in the the theory of functional differential equations provide a means to analyze models with generalized lag structures. The stability and asymptotic stability of two growth models with generalized lag structures are analyzed. The chapter's penultimate section includes a speculative discussion of time-varying parameters.

The paper proposes an efficient and insightful approach for solving neutral delay differential equations (NDDE) with high-frequency inputs. This paper aims to overcome the need to use a very small time step when high frequencies are present. High-frequency signals abound in communication circuits when modulated signals are involved.

The method involves an asymptotic expansion of the solution and each term in the expansion can be determined either from NDDE without oscillatory inputs or recursive equations. Such an approach leads to an efficient algorithm with a performance that improves as the input frequency increases.

An example shall indicate the salient features of the method. Its improved performance shall be shown when the input frequency increases. The example is chosen as it is similar to that in literature concerned with partial element equivalent circuit (PEEC) circuits (Bellen et al., 1999). Its structure shall also be shown to enable insights into the behaviour of the system governed by the differential equation.

The method is novel in its application to NDDE as arises in engineering applications such as those involving PEEC circuits. In addition, the focus of the method is on a technique suitable for high-frequency signals.

The purpose of this study is an application of the multi-wavelets Galerkin method to delay differential equations with vanishing delay known as Pantograph equation.

The method consists of expanding the required approximate solution at the elements of the Alpert multi-wavelets. Using the operational matrices of integration and wavelet transform matrix, the authors reduce the problem to a set of algebraic equations.

Because of the large size of the system, thresholding is used to obtain a new sparse system, and then this new system is solved to reduce the computational effort and related computer run time. The authors demonstrate that the solutions may be efficiently represented in a multi-wavelets basis because of flexible vanishing moments property of this type of multi-wavelets.

The L₂ convergence of the scheme for the proposed equation has been investigated. A series of numerical tests is provided to demonstrate the validity and applicability of the technique.

This study aims to simulate the dendritic growth in Stokes flow by iteratively coupling a domain and boundary type meshless method.

A preconditioned phase-field model for dendritic solidification of a pure supercooled melt is solved by the strong-form space-time adaptive approach based on dynamic quadtree domain decomposition. The domain-type space discretisation relies on monomial augmented polyharmonic splines interpolation. The forward Euler scheme is used for time evolution. The boundary-type meshless method solves the Stokes flow around the dendrite based on the collocation of the moving and fixed flow boundaries with the regularised Stokes flow fundamental solution. Both approaches are iteratively coupled at the moving solid–liquid interface. The solution procedure ensures computationally efficient and accurate calculations. The novel approach is numerically implemented for a 2D case.

The solution procedure reflects the advantages of both meshless methods. Domain one is not sensitive to the dendrite orientation and boundary one reduces the dimensionality of the flow field solution. The procedure results agree well with the reference results obtained by the classical numerical methods. Directions for selecting the appropriate free parameters which yield the highest accuracy and computational efficiency are presented.

A combination of boundary- and domain-type meshless methods is used to simulate dendritic solidification with the influence of fluid flow efficiently.

The purpose of this paper is to present a novel gradient‐based iterative algorithm for the joint diagonalization of a set of real symmetric matrices. The approximate joint diagonalization of a set of matrices is an important tool for solving stochastic linear equations. As an application, reliability analysis of structures by using the stochastic finite element analysis based on the joint diagonalization approach is also introduced in this paper, and it provides useful references to practical engineers.

By starting with a least squares (LS) criterion, the authors obtain a classical nonlinear cost‐function and transfer the joint diagonalization problem into a least squares like minimization problem. A gradient method for minimizing such a cost function is derived and tested against other techniques in engineering applications.

A novel approach is presented for joint diagonalization for a set of real symmetric matrices. The new algorithm works on the numerical gradient base, and solves the problem with iterations. Demonstrated by examples, the new algorithm shows the merits of simplicity, effectiveness, and computational efficiency.

A novel algorithm for joint diagonalization of real symmetric matrices is presented in this paper. The new algorithm is based on the least squares criterion, and it iteratively searches for the optimal transformation matrix based on the gradient of the cost function, which can be computed in a closed form. Numerical examples show that the new algorithm is efficient and robust. The new algorithm is applied in conjunction with stochastic finite element methods, and very promising results are observed which match very well with the Monte Carlo method, but with higher computational efficiency. The new method is also tested in the context of structural reliability analysis. The reliability index obtained with the joint diagonalization approach is compared with the conventional Hasofer Lind algorithm, and again good agreement is achieved.

The purpose of this paper is to upgrade our previous developments of Local Radial Basis Function Collocation Method (LRBFCM) for heat transfer, fluid flow and electromagnetic problems to thermoelastic problems and to study its numerical performance with the aim to build a multiphysics meshless computing environment based on LRBFCM.

Linear thermoelastic problems for homogenous isotropic body in two dimensions are considered. The stationary stress equilibrium equation is written in terms of deformation field. The domain and boundary can be discretized with arbitrary positioned nodes where the solution is sought. Each of the nodes has its influence domain, encompassing at least six neighboring nodes. The unknown displacement field is collocated on local influence domain nodes with shape functions that consist of a linear combination of multiquadric radial basis functions and monomials. The boundary conditions are analytically satisfied on the influence domains which contain boundary points. The action of the stationary stress equilibrium equation on the constructed interpolation results in a sparse system of linear equations for solution of the displacement field.

The performance of the method is demonstrated on three numerical examples: bending of a square, thermal expansion of a square and thermal expansion of a thick cylinder. Error is observed to be composed of two contributions, one proportional to a power of internodal spacing and the other to a power of the shape parameter. The latter term is the reason for the observed accuracy saturation, while the former term describes the order of convergence. The explanation of the observed error is given for the smallest number of collocation points (six) used in local domain of influence. The observed error behavior is explained by considering the Taylor series expansion of the interpolant. The method can achieve high accuracy and performs well for the examples considered.

The method can at the present cope with linear thermoelasticity. Other, more complicated material behavior (visco-plasticity for example), will be tackled in one of our future publications.

LRBFCM has been developed for thermoelasticity and its error behavior studied. A robust way of controlling the error was devised from consideration of the condition number. The performance of the method has been demonstrated for a large number of the nodes and on uniform and non-uniform node arrangements.