The method of particular solutions for solving axisymmetric polyharmonic and poly-Helmholtz equations

In this paper we derive analytical particular solutions for the axisymmetric polyharmonic and poly-Helmholtz partial differential operators using Chebyshev polynomials as basis functions. We further extend the proposed approach to the particular solutions of the product of Helmholtz-type operators. By using this formulation, we can approximate the particular solution when the forcing term of the differential equation is approximated by a truncated series of Chebyshev polynomials. These formulas were further implemented to solve inhomogeneous partial differential equations (PDEs) in which the homogeneous solutions were obtained by the method of fundamental solutions (MFS). Several numerical experiments were carried out to validate our newly derived particular solutions. Due to the exponential convergence of Chebyshev interpolation and the MFS, our numerical results are extremely accurate.     

Particular solutions of splines and monomials for polyharmonic and products of Helmholtz operators

This paper presents the particular solutions for the polyharmonic and the products of Helmholtz partial differential operators with polyharmonic splines and monomials right-hand side. By the application of the Hörmander linear partial differential operator theory, many of the systems can be reduced to a single equation involving the polyharmonic or the product of Helmholtz differential operators. If the inhomogeneous right-hand side of these operators can be removed by the method of particular solutions, then boundary-type numerical methods, such as the boundary element method, the method of fundamental solutions, and the Trefftz method, can be applied to solve these differential equations.     

Finite element formulations that include particular solutions of governing equations

The correspondence between the homogeneous and particular solutions of differential equations and the related solutions of the discrete equations is brought to light in this study. Guided by this correspondence, the trial functions for finite elements are written in two parts—one the homogeneous and the other a prescribed particular solution. It is shown that this approach allows one to simplify the computations of element matrices and also yields better results, throughout the element, than those obtained by the conventional approach. Two simple examples are provided for illustration.     

Comparing RBF‐FD approximations based on stabilized Gaussians and on polyharmonic splines with polynomials

In this work, we are concerned with radial basis function–generated finite difference (RBF‐FD) approximations. Numerical error estimates are presented for stabilized flat Gaussians (RBF(SGA)‐FD) and polyharmonic splines with supplementary polynomials (RBF(PHS)‐FD) using some analytical solutions of the Poisson equation in a square domain. Both structured and unstructured point clouds are employed for evaluating the influence of cloud refinement, size of local supports, and maximal permissible degree of the polynomials in RBF(PHS)‐FD. High order of accuracy was attained with both RBF(SGA)‐FD and RBF(PHS)‐FD especially for unstructured clouds. Absolute errors in the first and second derivatives were also estimated at all points of the domain using one of the analytical solutions. For RBF(SGA)‐FD, this test showed the occurrence of improprieties of some decentered supports localized on boundary neighborhoods. This phenomenon was not observed with RBF(PHS)‐FD.     

The annihilator method for computing particular solutions to partial differential equations

We generalize the well-known annihilator method, used to find particular solutions for ordinary differential equations, to partial differential equations. This method is then used to find particular solutions of Helmholtz-type equations when the right hand side is a linear combination of thin plate and higher order splines. These particular solutions are useful in numerical algorithms for solving boundary value problems for a variety of elliptic and parabolic partial differential equations.     

A class of particular solutions of one-dimensional nonlinear equations of heat conduction

A class of solutions is obtained for the heat-conduction equations in the case of a power relation between the coefficient of thermal conductivity and the temperature.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 32, No. 3, pp. 508–511, March, 1977.     

Multiple positive solutions of integral BVPs for high-order nonlinear fractional differential equations with impulses and distributed delays

In this article, we study the existence of multiple solutions of the integral boundary value problems for high-order nonlinear fractional differential equations with impulses and distributed delays. Some sufficient criteria will be established by the fixed point index theorem in cones. As application, one example is given to demonstrate the validity of our main results.     

Nodal DG-FEM solution of high-order Boussinesq-type equations

A discontinuous Galerkin finite-element method (DG-FEM) solution to a set of high-order Boussinesq-type equations for modelling highly nonlinear and dispersive water waves in one horizontal dimension is presented. The continuous equations are discretized using nodal polynomial basis functions of arbitrary order in space on each element of an unstructured computational domain. A fourth-order explicit Runge-Kutta scheme is used to advance the solution in time. Methods for introducing artificial damping to control mild nonlinear instabilities are also discussed. The accuracy and convergence of the model with both h (grid size) and p (order) refinement are confirmed for the linearized equations, and calculations are provided for two nonlinear test cases in one horizontal dimension: harmonic generation over a submerged bar, and reflection of a steep solitary wave from a vertical wall. Test cases for two horizontal dimensions will be considered in future work.     

Comparisons of fundamental solutions and particular solutions for Trefftz methods

In the Trefftz method (TM), the admissible functions satisfying the governing equation are chosen, then only the boundary conditions are dealt with. Both fundamental solutions (FS) and particular solutions (PS) satisfy the equation. The TM using FS leads to the method of fundamental solutions (MFS), and the TM using PS to the method of particular solutions (MPS). Since the MFS is one of TM, we may follow our recent book [20], [21] to provide the algorithms and analysis. Since the MFS and the MPS are meshless, they have attracted a great attention of researchers. In this paper numerical experiments are provided to support the error analysis of MFS in Li [15] for Laplace's equation in annular shaped domains. More importantly, comparisons are made in analysis and computation for MFS and MPS. From accuracy and stability, the MPS is superior to the MFS, the same conclusion as given in Schaback [24]. The uniform FS is simpler and the algorithms of MFS are easier to carry out, so that the computational efforts using MFS are much saved. Since today, the manpower saving is the most important criterion for choosing numerical methods, the MFS is also beneficial to engineering applications. Hence, both MFS and MPS may serve as modern numerical methods for PDE.     

A one-parameter solution of the laminar boundary-layer equations in a gas with arbitrary outer-flow velocity and arbitrary temperature difference

The author presents the methods and results of numerical integration of universal laminar boundary-layer equations (in the one-parameter approximation) for a gas flow with large velocities, a Prandtl number equal to unity, and a linear relation between the dynamic viscosity and temperature.     

The application of shape-preserving splines for the solution of differential equations

This paper investigates the use of shape-preserving interpolants based on Bernstein polynomials for the numerical solution of differential equations. A simple and accurate algorithm is presented for the integration of the initial value ordinary differential equation, as well as for the partial differential equation of the Burger type.     

Particular solutions of Helmholtz-type operators using higher order polyhrmonic splines

We obtain explicit analytical particular solutions for Helmholtz-type operators, using higher order splines. These results generalize those in Golberg, Chen and Rashed (1998) and Chen and Rashed (1998) for thin plate splines. This enables one to substantially improve the accuracy of algorithms for solving boundary value problems for Helmholtz-type equations.     

General features of spatiotemporal instability induced by arbitrary high-order nonlinear dispersions in metamaterials

Spatiotemporal instability (STI) in metamaterials (MMs) with a Kerr-type nonlinear polarization is investigated. A general expression for instability gain, taking account of the effects of arbitrary high-order linear and nonlinear dispersions, is derived. Special attention is paid to the general features of instability induced by nonlinear dispersion effects originating from the combination of dispersive magnetic permeability and nonlinear polarization. It is shown that, just like their linear counterparts, all even-order nonlinear dispersions not only deform the original instability regions, but also may lead to the appearance of new instability regions. However, all odd-order nonlinear dispersions always suppress STI irrespective of their signs, quite different from their linear counterparts which exert no influence on instability. Moreover, we find that, unlike the linear dispersions, the nonlinear dispersions lead to the dependence of gain peak on the spatial modulation frequency. The role of the first-order nonlinear dispersion, namely self-steepening (SS), and second-order nonlinear dispersion (SND) in STI is particularly analyzed to illustratively demonstrate the general results.     

Cauchy problems of Laplace's equation by the methods of fundamental solutions and particular solutions

The Cauchy problems of Laplace's equation are ill-posed with severe instability. In this paper, numerical solutions are solicited by the method of fundamental solutions (MFS) and the method of particular solutions (MPS). We focus on the analysis of the MFS, and derive the bounds of errors and condition numbers. The analysis for the MPS can also be obtained similarly. Numerical experiments and comparisons are reported for the Cauchy and Dirichlet problems by the MPS and the MFS. The Cauchy noise data and the regularization are also adopted in numerical experiments. Both the MFS and the MPS are effective to Cauchy problems. The MPS is superior in accuracy and stability; but the MFS owns simplicity of algorithms, and earns flexibility for a wide range of applications, such as Cauchy problems. These conclusions also coincide with [37]. The basic analysis of error and stability is explored in this paper, and applied to the Cauchy data. There are many reports on numerical Cauchy problems, see the survey paper in [12]; most of them are of computational aspects. The strict analysis of this paper may, to a certain degree, fill up the existing gap between theory and computation of Cauchy problems by the MFS and the MPS. Moreover, comprehensive analysis and compatible computation are two major characteristics of this paper, which may enhance the study of numerical Cauchy problems forward to a higher and advanced level.     

Spatial discretization of hyperbolic equations with periodic solutions

We investigate the Cauchy problem for hyperbolic equations for which the frequencies of the main Fourier components in the solution are located in a given frequency interval. Difference formulae for the spatial derivatives are constructed that are tuned to the given intervals of frequencies. Numerical examples illustrating these special discretizations are given both for linear and non-linear problems.     

Low-complexity computation of plate eigenmodes with Vekua approximations and the method of particular solutions

This paper extends the method of particular solutions (MPS) to the computation of eigenfrequencies and eigenmodes of thin plates, in the framework of the Kirchhoff-Love plate theory. Specific approximation schemes are developed, with plane waves (MPS-PW) or Fourier-Bessel functions (MPS-FB). This framework also requires a suitable formulation of the boundary conditions. Numerical tests, on two plates with various boundary conditions, demonstrate that the proposed approach provides competitive results with standard numerical schemes such as the finite element method, at reduced complexity, and with large flexibility in the implementation choices.     

Particular solutions of Laplacian,Helmholtz-type,and polyharmonic operators involving higher order radial basis functions

Particular solutions of higher order radial basis functions of conical and spline types are obtained for the Laplacian, Helmholtz-type, and polyharmonic operators. These particular solutions are needed in the implementation of the Dual Reciprocity Boundary Element Method.