Boundary-value Problems For Partial Differential Equations Research Articles

New benchmark solutions for free vibration of clamped rectangular thick plates and their variants

It is of significance to explore benchmark analytic free vibration solutions of rectangular thick plates without two parallel simply supported edges, because the classic analytic methods are usually invalid for the problems of this category. The main challenge is to find the solutions meeting both the governing higher order partial differential equations (PDEs) and boundary conditions of the plates, i.e., to analytically solve associated complex boundary value problems of PDEs. In this letter, we extend a novel symplectic superposition method to the free vibration problems of clamped rectangular thick plates, with the analytic frequency solutions obtained by a brief set of equations. It is found that the analytic solutions of clamped plates can simply reduce to their variants with any combinations of clamped and simply supported edges via an easy relaxation of boundary conditions. The new results yielded in this letter are not only useful for rapid design of thick plate structures but also provide reliable benchmarks for checking the validity of other new solution methods.

A segmented and weighted Adomian decomposition algorithm for boundary value problem of nonlinear groundwater equation

AbstractBased on Adomian decomposition method, a new algorithm for solving boundary value problem (BVP) of nonlinear partial differential equations on the rectangular area is proposed. The solutions obtained by the method precisely satisfy all boundary conditions, except the small pieces near the four corners of the rectangular area. A theorem on the boundary error is given. Hence, the Adomian decomposition method is more efficiently applied to BVPs for partial differential equations. Segmented and weighted analytical solutions with a high accuracy for the BVP of nonlinear groundwater equations on a rectangular area are obtained by the present algorithm. Copyright © 2013 John Wiley & Sons, Ltd.

A Substructuring Domain Decomposition Scheme for Unsteady Problems

AbstractDomain decomposition methods are used for the approximate solution of boundary-value problems for partial differential equations on parallel computing systems. Specific features of unsteady problems are fully taken into account in iteration-free domain decomposition schemes. Regionally-additive schemes are based on various classes of splitting schemes. In this paper we highlight a class of domain decomposition schemes which are based on the partition of the initial domain into subdomains with common boundary nodes. Using a partition of unity we construct and analyze unconditionally stable schemes for domain decomposition based on a two-component splitting: the problem within each subdomain and the problem at their boundaries. As an example we consider a Cauchy problem of first or second order in time with a non-negative self-adjoint second order operator in space. The theoretical discussion is supplemented with the numerical solution of a model problem for a two-dimensional parabolic equation.

Comparative study of Fourier transform and an approach of fast Fourier transform in solving the boundary value problem of PDE

The article compares Fast Fourier Transform (FFT) against the Fourier Transform in solving the boundary value problems (BVP) of partial differential equations, in particular, on wave equation. The approach of FFT used here is based on Bill Davis and Jerry Uhl’s work with FFT, known as Fast Fourier Fit (FFF) in differential equation. In our work MATHEMATICA and FORTRANcodeswere used for both Fourier Transformand Fast Fourier Fit to implement the solution procedure of BVP of partial differential equations. Comparison of the two approaches shows similar result but FFF does the job more efficiently than the Fourier Transform

A collection of computational techniques for solving singular boundary-value problems

The objective of this paper is a wide survey and classification of various computational techniques which are used for solving singular boundary-value problems. Here, we have considered various research papers within last 10 years to get a better the know-how of present scenario. This paper contains summary of linear, non-linear second-order two-point singular boundary-value problems (BVPs), two-point singularly perturbed BVPs in ordinary differential equations and singularly elliptic BVPs in partial differential equations.

Probabilistic representation of solutions of hydrodynamic equations

It is well known that the relationships between the theory of partial differential equations (PDE) and probability theory are very deep and allow one to derive new nontrivial results concerning the behavior of random processes and properties of the solutions of boundary-value problems of PDE. It is natural to make an attempt to construct a probabilistic representation of the solution of a boundary problem for the Navier-Stokes system and other hydrodynamic equations. We construct such a representation in terms of a diffusion process. Bibliography: 16 titles.

A numerical verification method for the existence of weak solutions for nonlinear boundary value problems

In this paper, we consider a numerical technique which enables us to verify the existence of solutions for nonlinear two point boundary value problems (BVP). We formulate the problem as a fixed point of a Newton-like operator and present a verification algorithm by computer based on Sadovskii's fixed point theorem. In order to compute the Newton type iterative sequences in infinite dimensional space, two concepts, the rounding and the rounding error, are introduced. Also some numerical examples are illustrated. We emphasize that, since the present method uses the weak formulation of BVP and the finite element approximations instead of the existing validation techniques, it is aslo applicable, in principle, to the BVP for partial differential equations as it stands.