Class Of Partial Differential Equations Research Articles

ADAPTIVE REFINEMENT AND MULTISCALE MODELING IN 2D PERIDYNAMICS

The original peridynamics formulation uses a constant nonlocal region, the horizon, over the entire domain. We propose here adaptive refinement algorithms for the bond-based peridynamic model for solving statics problems in two dimensions that involve a variable horizon size. Adaptive refinement is an essential ingredient in concurrent multiscale modeling, and in peridynamics changing the horizon is directly related to multiscale modeling. We do not use any special conditions for the "coupling" of the large and small horizon regions, in contrast with other multiscale coupling methods like atomistic-to-continuum coupling, which require special conditions at the interface to eliminate ghost forces in equilibrium problems. We formulate, and implement in two dimensions, the peridynamic theory with a variable horizon size and we show convergence results (to the solutions of problems solved via the classical partial differential equations theories of solid mechanics in the limit of the horizon going to zero) for a number of test cases. Our refinement is triggered by the value of the nonlocal strain energy density. We apply the boundary conditions in a manner similar to the way these conditions are enforced in, for example, the finite-element method, only on the nodes on the boundary. This, in addition to the peridynamic material being effectively "softer" near the boundary (the so-called "skin effect") leads to strain energy concentration zones on the loaded boundaries. Because of this, refinement is also triggered around the loaded boundaries, in contrast to what happens in, for example, adaptive finite-element methods.

Using reproducing kernel for solving a class of singularly perturbed problems

This paper is concerned with a new algorithm for giving the analytical and approximate solutions of a class of partial differential equations with a singularly perturbed term in a reproducing kernel space. Two numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.

Continuous and discontinuous finite element methods for a peridynamics model of mechanics

In contrast to classical partial differential equation models, the recently developed peridynamic nonlocal continuum model for solid mechanics is an integro-differential equation that does not involve spatial derivatives of the displacement field. As a result, the peridynamic model admits solutions having jump discontinuities so that it has been successfully applied to fracture problems. The peridynamic model features a horizon which is a length scale that determines the extent of the nonlocal interactions. Based on a variational formulation, continuous and discontinuous Galerkin finite element methods are developed for the peridynamic model. Discontinuous discretizations are conforming for the model without the need to account for fluxes across element edges. Through a series of simple, one-dimensional computational experiments, we investigate the convergence behavior of the finite element approximations and compare the results with theoretical estimates. One issue addressed is the effect of the relative sizes of the horizon and the grid. For problems with smooth solutions, we find that continuous and discontinuous piecewise-linear approximations result in the same accuracy as that obtained by continuous piecewise-linear approximations for classical models. Piecewise-constant approximations are less robust and require the grid size to be small with respect to the horizon. We then study problems having solutions containing jump discontinuities for which we find that continuous approximations are not appropriate whereas discontinuous approximations can result in the same convergence behavior as that seen for smooth solutions. In case a grid point is placed at the locations of the jump discontinuities, such results are directly obtained. In the general case, we show that such results can be obtained through a simple, automated, abrupt, local refinement of elements containing the discontinuity. In order to reduce the number of degrees of freedom while preserving accuracy, we also briefly consider a hybrid discretization which combines continuous discretizations in regions where the solution is smooth with discontinuous discretizations in small regions surrounding the jump discontinuities.

A note on decay properties for the solutions of a class of partial differential equation with memory

The purpose of this article is to study decay properties for solutions of a class of PDEs with memory by Lyapunov functionals method. Moreover, we prove that when the kernels of the convolutions decay exponentially, the first and second order energy of the solutions decay exponentially. Also we show that when the kernels decay polynomially, these energies decay polynomially.

Viscous and inviscid regularizations in a class of evolutionary partial differential equations

We investigate solution properties of a class of evolutionary partial differential equations (PDEs) with viscous and inviscid regularization. An equation in this class of PDEs can be written as an evolution equation, involving only first-order spatial derivatives, coupled with the Helmholtz equation. A recently developed two-step iterative method (P.H. Chiu, L. Lee, T.W.H. Sheu, A dispersion-relation-preserving algorithm for a nonlinear shallow-water wave equation, J. Comput. Phys. 228 (2009) 8034–8052) is employed to study this class of PDEs. The method is in principle superior for PDE’s in this class as it preserves their physical dispersive features. In particular, we focus on a Leray-type regularization (H.S. Bhat, R.C. Fetecau, A Hamiltonian regularization of the Burgers equation, J. Nonlinear Sci. 16 (2006) 615–638) of the Hopf equation proposed in alternative to the classical Burgers viscous term. We show that the regularization effects induced by the alternative model can be vastly different from those induced by Burgers viscosity depending on the smoothness of initial data in the limit of zero regularization. We validate our numerical scheme by comparison with a particle method which admits closed form solutions. Further effects of the interplay between the dispersive terms comprising the Leray-regularization are illustrated by solutions of equations in this class resulting from regularized Burgers equation by selective elimination of dispersive terms.

Computation of Asymptotic Stability for a Class of Partial Differential Equations with Delay

In this paper the question of asymptotic stability for retarded functional reaction diffusion equations is faced. Due to the infinite dimension of the problem a numerical approach is necessary. Here we propose a technique based on a pseudospectral discretization in time and on a spectral discretization in space of the infinitesimal generator associated to the semigroup of solution operators. Some numerical experiments on a Hutchinson-type equation modeling heat conduction in a rod with spatially variable gain delayed feedback are performed to show the efficiency of the scheme. This work represents a nontrivial extension of previous work of the authors on the computation of asymptotic stability for delay differential equations.

A collocation and Cartesian grid methods using new radial basis function to solve class of partial differential equations

In the present paper, a class of partial differential equation represented by Poisson's type problems are solved using a proposed Cartesian grid method and a collocation technique using a new radial basis function. The advantage of using this new radial basis function represented by overcoming singularity from the diagonal elements when thin plate radial basis function is used. The new function is a combination of both multiquadric and thin plate radial basis functions. The new radial basis function contains a control parameter ϵ, that takes one when evaluating the singular elements and equals zero elsewhere. Collocation of the approximate solution of the potential over the governing and boundary condition equations leads to a double linear system of equations. A proposed algebraic procedure is then developed to solve the double system. Examples of Poisson and Helmholtz equations are solved and the present results are compared with the their analytical solutions. A good agreement with analytical results is achieved.

Numerical solution of KKT systems in PDE-constrained optimization problems via the affine scaling trust-region approach†

A recently proposed trust-region approach for bound-constrained nonlinear equations is applied to the Karush-Kuhn-Tucker (KKT) system arising from the discretization of a class of partial differential equation (PDE)-constrained optimization problems. Two different implementations are developed that take into account the large dimension and the special structure of the problems. The linear algebra phase is analysed considering the possibility of solving the arising linear systems by either direct methods or short-recurrence iterative linear solvers. Viability of the approach is proved through several numerical experiments on large KKT systems arising from the discretization of control problems.

Modeling a Diving Board

The beam equation is a classic fourth-order partial differential equation used to de scribe the transverse displacement of a vibrating beam. One may encounter it in an introductory course on boundary value problems or mathematical physics. One par ticular type of beam modeled with this equation is the cantilever, which is a long, thin, rectangular bar with one end clamped and the other end free to move up and down [3]. Cantilever beams appear in a variety of applications, including bridge con struction [13], aircraft wing design [1], architecture [9,11], microwave imaging [15], temperature measurement [19], and virus detection [7]. This paper came about as the result of an undergraduate Honors Thesis project at Ball State University [16]. The goal of the thesis was to address the question: Since we can think of a diving board as a cantilever, how well does the beam equation model a diving board? To answer this question, we created a simple experimental set-up con sisting of a two-meter stick with one end clamped to a table and the free end set in motion. Vertical displacements at tape-marked positions along the beam were cap tured with a digital video camera, analyzed with the freely available World-in-Motion physics demonstration software [4], and compared to a model based on the standard beam equation. As is often the case with models of real-world phenomena, our initial model failed to match the measured data. Using the wave equation as a model [2], we first introduced a damping term to improve results and then a forcing term to arrive at a reasonable model.

A compendium of partial differential equation models: method of lines analysis with Matlab

A Compendium of Partial Differential Equation Models presents numerical methods and associated computer codes in Matlab for the solution of a spectrum of models expressed as partial differential equations (PDEs), one of the mostly widely used forms of mathematics in science and engineering. The authors focus on the method of lines (MOL), a well-established numerical procedure for all major classes of PDEs in which the boundary value partial derivatives are approximated algebraically by finite differences. This reduces the PDEs to ordinary differential equations (ODEs) and thus makes the computer code easy to understand, implement, and modify. Also, the ODEs (via MOL) can be combined with any other ODEs that are part of the model (so that MOL naturally accommodates ODE/PDE models). This book uniquely includes a detailed line-by-line discussion of computer code as related to the associated equations of the PDE model.

An analysis of the conservation laws for certain third-grade fluids

The nature of the flow of a third-grade fluid with regard to its underlying conservation laws is studied. In particular, the fluid occupying the space over a wall is considered. At the surface of the wall, suction or blowing velocity is applied. By introducing a velocity field, the governing equations are reduced to a class of partial differential equations (pdes). A complete class of conservation laws for the resulting equations is constructed and analyzed using invariance properties of the corresponding multipliers/characteristics.

On the positivstellensatz in Weyl’s algebra

We use harmonic analysis tools on the Heisenberg group to prove decay and regularity properties for a certain class of partial differential equations. These properties allow us to relax a technical condition for a sum of squares decomposition for the Weyl algebra.

Sequential Monte Carlo methods for diffusion processes

In this paper, we show how to use sequential Monte Carlo methods to compute expectations of functionals of diffusions at a given time and the gradients of these quantities w.r.t. the initial condition of the process. In some cases, via the exact simulation of the diffusion, there is no time discretization error, otherwise the methods use Euler discretization. We illustrate our approach on both high- and low-dimensional problems from optimal control and establish that our approach substantially outperforms standard Monte Carlo methods typically adopted in the literature. The methods developed here are appropriate for solving a certain class of partial differential equations as well as for option pricing and hedging.

FROBENIUS INTEGRABLE DECOMPOSITIONS FOR TWO CLASSES OF NONLINEAR EVOLUTION EQUATIONS WITH VARIABLE COEFFICIENTS

Frobenius integrable decompositions are introduced for partial differential equations with variable coefficients. Two classes of partial differential equations with variable coefficients are transformed into Frobenius integrable ordinary differential equations. The resulting solutions are illustrated to describe the solution phenomena shared with the KdV and potential KdV equations, the Boussinesq equation and the Camassa–Holm equation with variable coefficients.

Analysis and application of finite volume element methods to a class of partial differential equations

The finite volume element (FVE) methods for a class of partial differential equations are discussed and analyzed in this paper. The new initial values are introduced in the finite volume element schemes, and we obtain optimal error estimates in Lp and W1,p (2⩽p⩽∞) as well as some superconvergence estimates in W1,p (2⩽p⩽∞). The main results in this paper perfect the theory of the finite volume element methods.

Breakdown of weak-turbulence and nonlinear wave condensation

The formation of a large-scale coherent structure (a condensate) as a result of the long time evolution of the initial value problem of a classical partial differential nonlinear wave equation is considered. We consider the nonintegrable and unforced defocusing NonLinear Schrödinger (NLS) equation as a representative model. In spite of the formal reversibility of the NLS equation, the nonlinear wave exhibits an irreversible evolution towards a thermodynamic equilibrium state. The equilibrium state is characterized by a homogeneous solution (condensate), with small-scale fluctuations superposed (uncondensed particles), which store the information necessary for “time reversal”. We analyze the evolution of the cumulants of the random wave as originally formulated by D.J. Benney and P.G. Saffman [D.J. Benney, P.G. Saffman, Proc. Roy. Soc. London A 289 (1966) 301] and A.C. Newell [A.C. Newell, Rev. Geophys. 6 (1968) 1]. This allows us to provide a self-consistent weak-turbulence theory of the condensation process, in which the nonequilibrium formation of the condensate is a natural consequence of the spontaneous regeneration of a non-vanishing first-order cumulant in the hierarchy of the cumulants’ equations. More precisely, we show that in the presence of a small condensate amplitude, all relevant statistical information is contained in the off-diagonal second order cumulant, as described by the usual weak-turbulence theory. Conversely, in the presence of a high-amplitude condensate, the diagonal second-order cumulants no longer vanish in the long time limit, which signals a breakdown of the weak-turbulence theory. However, we show that an asymptotic closure of the hierarchy of the cumulants’ equations is still possible provided one considers the Bogoliubov’s basis rather than the standard Fourier’s (free particle) basis. The nonequilibrium dynamics turns out to be governed by the Bogoliubov’s off-diagonal second order cumulant, while the corresponding diagonal cumulants, as well as the higher order cumulants, are shown to vanish asymptotically. The numerical discretization of the NLS equation implicitly introduces an ultraviolet frequency cut-off. The simulations are in quantitative agreement with the weak turbulence theory without adjustable parameters, despite the fact that the theory is expected to breakdown nearby the transition to condensation. The fraction of condensed particles vs energy is characterized by two distinct regimes: For small energies ( H ≪ H c ) the Bogoliubov’s regime is established, whereas for H ≲ H c the small-amplitude condensate regime is described by the weak-turbulence theory. In both regimes we derive coupled kinetic equations that describe the coupled evolution of the condensate amplitude and the incoherent field component. The influence of finite size effects and of the dimensionality of the system are also considered. It is shown that, beyond the thermodynamic limit, wave condensation is reestablished in two spatial dimensions, in complete analogy with uniform and ideal 2D Bose gases.

Differential characteristic set algorithm for the complete symmetry classification of partial differential equations

In this paper, we present a differential polynomial characteristic set algorithm for the complete symmetry classification of partial differential equations (PDEs) with some parameters. It can make the solution to the complete symmetry classification problem for PDEs become direct and systematic. As an illustrative example, the complete potential symmetry classifications of nonlinear and linear wave equations with an arbitrary function parameter are presented. This is a new application of the differential form characteristic set algorithm, i.e., Wu’s method, in differential equations.

The generating analytic element approach with application to the modified Helmholtz equation

In this paper a new method for obtaining functions with a given singular behavior that satisfy a class of partial differential equations is presented. Differential equations of this class contain operators of the form $${\nabla^{2n}}$$ , where n is a positive integer. The method uses Wirtinger calculus which enables one to invert the Laplacian in combination with the decomposition method introduced by Adomian at the end of the twentieth century. The procedure uses a singular holomorphic function as its basis, and constructs the solution term by term as an infinite series of functions; the process consists of an infinite number of steps of integration. This method is applied to construct a number of singular solutions to the modified Helmholtz equation in the context of groundwater flow. These functions are discharge potentials, which are two-dimensional functions by definition. The gradient of the discharge potential is the vertically integrated flow over the thickness of an aquifer, or water-bearing layer. The discharge potentials of interest here are those used in the analytic element method. This method, as originally conceived, relies on the superposition of suitably chosen holomorphic functions, and is a form of a method known as the Trefftz method, not to be confused with the Trefftz method applied to finite element techniques. The main analytic elements used are singular line elements, characterized by either a jump along the element in the tangential or the normal component of the discharge vector. The analytic line elements for the case of divergence-free irrotational flow are well established and many of these are forms of singular Cauchy integrals. Application of the analytic element method to more general cases of flow, governed for example by the modified Helmholtz equation (flow in systems of aquifers separated by leaky layers) and the heat equation (transient flow) is possible using the method presented in this paper. The latter application is beyond the scope of this paper, but it is worth noting that for that case the constant that occurs in the modified Helmholtz is replaced by a general function of time and application of Laplace transforms can be avoided. A method for constructing such functions is presented; the procedure for constructing these functions is referred to as the generating analytic element approach. Application of this approach requires the existence of the holomorphic singular line element. The approach is discussed and an example for the case of a line-sink for a system of two aquifers separated by a leaky layer and bounded above by in impermeable boundary is presented.

Existence of solutions for a class of neutral partial differential equations with nonlocal conditions in the [formula omitted]-norm

This paper is concerned with the existence of mild and strong solutions for some neutral partial differential equations with nonlocal conditions in the α -norm. The linear part of the equations is assumed to generate a compact analytic semigroup of uniformly bounded linear operators, and the nonlinear part satisfies some Lipschitz conditions with respect to the α -norm. By using a fixed point theorem of Sadovskii, we establish some existence results, which generalize the recent conclusions on this issue. In the end, the obtained results will be applied to a class of partial differential equations with nonlocal conditions.

Fuzzy Solutions to Partial Differential Equations: Adaptive Approach

A new technique using an adaptive fuzzy algorithm to obtain the solutions to a class of partial differential equations (PDEs) is presented. The design objective is to find a fuzzy solution to satisfy precisely the PDEs with boundary conditions. According to the adaptive scheme of fuzzy logic systems, a fuzzy solution with adjustable parameters for the PDE is first described. Then, a set of adaptive laws for tuning the free parameters in the consequent part is derived from minimizing an appropriate error function. In addition, an elegant approximated error bound between the exact solution and the proposed fuzzy solution with respect to the number of fuzzy rules and solution errors has also been derived. Furthermore, the convergence of error equations in mesh points is also discussed from the energy perspective. In this paper, we show that the proposed method can solve a variety of PDEs encountered in engineering. Comparisons are also made with solutions obtained by the finite-element method.