We consider the numerical evaluation of the Evans function, a Wronskian-like determinant that arises in the study of the stability of travelling waves. Constructing the Evans function involves matching the solutions of a linear ordinary differential equation depending on the spectral parameter. The problem becomes stiff as the spectral parameter grows. Consequently, the Gauss–Legendre method ha...

In this manuscript a new method is introduced for solving fractional differential equations. The fractional derivative is described in the Caputo sense. The main idea is to use fractional-order Legendre wavelets and operational matrix of fractional-order integration. First the fractional-order Legendre wavelets (FLWs) are presented. Then a family of piecewise functions is proposed, based on whi...

The most suitable and widely used numerical integration method for boundary integrals in the BEM method is Gauss-Legendre integration. But, this integration method is not appropriate for singular and nearly singular integrations in BEM. In this study, some criteria are introduced for recognizing nearly singular integrals in the integral form of the Laplace equation. At the rst stage, a criterio...

Abstract. We present a new tunably-accurate Laguerre Petrov-Galerkin spectral method for solving linear multi-term fractional initial value problems with derivative orders at most one and constant coe cients on the half line. Our method results in a matrix equation of special structure which can be solved in O(N logN) operations. We also take advantage of recurrence relations for the generalize...

In this paper, we report on very efficient algorithms for the spherical harmonic transform (SHT). Explicitly vectorized variations of the algorithm based on the Gauss-Legendre quadrature are discussed and implemented in the SHTns library which includes scalar and vector transforms. The main breakthrough is to achieve very efficient on-the-fly computations of the Legendre associated functions, e...

Measurement of transient pressure distribution on maritime structures is important for the assessment hydrodynamic loads applied. The commonly used sensors are mostly bulky, need to be bolted structure, and/or only provide point-wise measurements. In this paper, an elastic matrix layer with a network embedded piezoelectric proposed address these issues. For experimental validation, 400 × 5 mm e...

Abstract. Existing analysis shows that when the Gauss Runge-Kutta (GRK) (also called Legendre-Gauss collocation) formulation with s Gaussian nodes is applied to ordinary differential equation initial value problems, the discretization has order 2s (super-convergent) [8]. However, for time-dependent partial differential equations (PDEs) with boundary conditions, super-convergence is only observe...

This paper constructs highly accurate and efficient time integration methods for the solution of transient problems. The motion equations problems can be described by first-order ordinary differential equations, in which right-hand side is decomposed into two parts, a linear part nonlinear part. In proposed different orders, responses at previous step are transferred generalized Padé approximat...

In this paper it is proposed to compute the volume integral of certain functions whose antiderivates with respect to one of the variates (say either x or y or z) is available. Then by use of the well known Gauss Divergence theorem, it can be shown that the volume integral of such a function is expressible as sum of four integrals over the unit triangle. The present method can also evaluate the ...