The integrable 3rd-order Korteweg-de Vries (KdV) equation emerges uniquely at linear order in the asymptotic expansion for unidirectional shallow water waves. However, at quadratic order, this asymptotic expansion produces an entire family of shallow water wave equations that are asymptotically equivalent to each other, under a group of nonlinear, nonlocal, normal-form transformations introduce...

Here we present an approach to problems of diffraction that has its roots in a number of well-established theories such as the geometric theory of diffraction, the method of parabolic equations, the theory of Wiener functional integration, and the theory of stochastic processes. We start our analysis of the Helmholtz equation following closely the scheme of the ray method, but instead of approx...

In this paper, we consider the numerical solution of the Helmholtz equation, arising from the study of the wave equation in the frequency domain. The approach proposed here differs from those recently considered in the literature, in that it is based on a decomposition that is exact when considered analytically, so the only degradation in computational performance is due to discretization and r...

This paper presents a boundary-integral equation (BIE) method for the calculation of poloidal axisymmetric magnetic fields applicable in a wide range of ac frequencies. The method is based on the vector potential formulation and it uses the Green’s functions of Laplace and Helmholtz equations for the exterior and interior of conductors, respectively. The work is particularly focused on a calcul...

The paper contains a noniterative solver for the Helmholtz and the modified Helmholtz equations in a hexahedron. The solver is based on domain decomposition. The solution domain is divided into mostly parallelepiped subdomains. In each subdomain a particular solution of the nonhomogeneous Helmholtz equation is first computed by a fast spectral 3-D method which was developed in our earlier paper...

This paper primarily concerns the Whitham modulation equation of complex modified Korteweg–de Vries (cmKdV) with a step-like initial value. By utilizing Lax pair, we derive N-genus equations via averaging method. The can be integrated using hodograph transformation. We investigate Krichever’s algebro-geometric scheme to propose method for cmKdV integrable hierarchy and obtain velocities connect...

This paper first attempts to solve the transient heat conduction problem by combining recently proposed local knot method (LKM) with dual reciprocity (DRM). Firstly, temporal derivative is discretized a finite difference scheme, and thus governing equation of transfer transformed into non-homogeneous modified Helmholtz equation. Secondly, solution decomposed particular homogeneous solution. And...

Abstract. This paper presents new formulations of the boundary-domain integral equation (BDIE) and the boundary-domain integro-differential equation (BDIDE) methods for the numerical solution of the two-dimensional Helmholtz equation with variable coefficients. When the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equa...