On error estimation of finite element approximations to the elliptic equations in nonconvex polygonal domains

Numerical verification methods, so-called Nakao's methods, on existence or uniqueness of solutions to PDEs have been developed by Nakao and his group including the authors. They are based on the error estimation of approximate solutions which are mainly computed by FEM.     

PHYSICS INFORMED NEURAL NETWORKS (PINNs) FOR APPROXIMATING NONLINEAR DISPERSIVE PDEs

We propose a novel algorithm,based on physics-informed neural networks (PINNs) to efficiently approximate solutions of nonlinear dispersive PDEs such as the KdV-Kawahara,Camassa-Holm and Benjamin-Ono equations.The stability of solutions of these dispersive PDEs is leveraged to prove rigorous bounds on the resulting error.We present several numerical experiments to demonstrate that PINNs can approximate solutions of these dispersive PDEs very accurately.     

A note on the elliptic equation method

The elliptic equation method is improved for constructing exact travelling wave solutions of nonlinear partial differential equations (PDEs). The rational forms of Jacobi elliptic functions are presented. By using new Jacobi elliptic function solutions of the elliptic equation, new doubly periodic solutions are obtained for some important PDEs. This method can be applied to many other nonlinear PDEs.     

Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems

This paper presents two new approaches for finding the homogenized coefficients of multiscale elliptic PDEs. Standard approaches for computing the homogenized coefficients suffer from the so-called resonance error, originating from a mismatch between the true and the computational boundary conditions. Our new methods, based on solutions of parabolic and elliptic cell problems, result in an exponential decay of the resonance error.     

Asymptotically exact functional error estimators based on superconvergent gradient recovery

The use of dual/adjoint problems for approximating functionals of solutions of PDEs with great accuracy or to merely drive a goal-oriented adaptive refinement scheme has become well-accepted, and it continues to be an active area of research. The traditional approach involves dual residual weighting (DRW). In this work we present two new functional error estimators and give conditions under which we can expect them to be asymptotically exact. The first is of DRW type and is derived for meshes in which most triangles satisfy an -approximate parallelogram property. The second functional estimator involves dual error estimate weighting (DEW) using any superconvergent gradient recovery technique for the primal and dual solutions. Several experiments are done which demonstrate the asymptotic exactness of a DEW estimator which uses a gradient recovery scheme proposed by Bank and Xu, and the effectiveness of refinement done with respect to the corresponding local error indicators. Resubmitted to Numerische Mathematik, June 30, 2005, with changes suggested by referees.     

Optimal order a posteriori error estimates for a class of Runge–Kutta and Galerkin methods

We derive a posteriori error estimates, which exhibit optimal global order, for a class of time stepping methods of any order that include Runge–Kutta Collocation (RK-C) methods and the continuous Galerkin (cG) method for linear and nonlinear stiff ODEs and parabolic PDEs. The key ingredients in deriving these bounds are appropriate one-degree higher continuous reconstructions of the approximate solutions and pointwise error representations. The reconstructions are based on rather general orthogonality properties and lead to upper and lower bounds for the error regardless of the time-step; they do not hinge on asymptotics.     

Exact and analytical solutions for a nonlinear sigma model

We consider wave solutions to nonlinear sigma models in n dimensions. First, we reduce the system of governing PDEs into a system of ODEs through a traveling wave assumption. Under a new transform, we then reduce this system into a single nonlinear ODE. Making use of the method of homotopy analysis, we are able to construct approximate analytical solutions to this nonlinear ODE. We apply two distinct auxiliary linear operators and show that one of these permits solutions with lower residual error than the other. This demonstrates the effectiveness of properly selecting the auxiliary linear operator when performing homotopy analysis of a nonlinear problem. From here, we then obtain residual error‐minimizing values of the convergence control parameter. We find that properly selecting the convergence control parameter makes a drastic difference in the magnitude of the residual error. Together, appropriate selection of the auxiliary linear operator and of the convergence control parameter is shown to allow approximate solutions that quickly converge to the true solution, which means that few terms are needed in the construction of such solution. This, in turn, greatly improves computational efficiency. Copyright © 2013 John Wiley & Sons, Ltd.     

A numerical scheme for stochastic PDEs with Gevrey regularity

We consider strong approximations to parabolic stochastic PDEs.We assume the noise lies in a Gevrey space of analytic functions.This type of stochastic forcing includes the case of forcingin a finite number of Fourier modes. We show that with Gevreynoise our numerical scheme has solutions in a discrete equivalentof this space and prove a strong error estimate. Finally wepresent some numerical results for a stochastic PDE with a Ginzburg–Landaunonlinearity and compare this to the more standard implicitEuler–Maruyama scheme.     

Iterated Stochastic Processes: Simulation and Relationship with High Order Partial Differential Equations

In this paper, we consider the composition of two independent processes: one process corresponds to position and the other one to time. Such processes will be called iterated processes. We first propose an algorithm based on the Euler scheme to simulate the trajectories of the corresponding iterated processes on a fixed time interval. This algorithm is natural and can be implemented easily. We show that it converges almost surely, uniformly in time, with a rate of convergence of order 1/4 and propose an estimation of the error. We then extend the well known Feynman-Kac formula which gives a probabilistic representation of partial differential equations (PDEs), to its higher order version using iterated processes. In particular we consider general position processes which are not necessarily Markovian or are indexed by the real line but real valued. We also weaken some assumptions from previous works. We show that intertwining diffusions are related to transformations of high order PDEs. Combining our numerical scheme with the Feynman-Kac formula, we simulate functionals of the trajectories and solutions to fourth order PDEs that are naturally associated to a general class of iterated processes.     

Asymptotic expansions for degenerate parabolic equations

We prove asymptotic convergence results for some analytical expansions of solutions to degenerate PDEs with applications to financial mathematics. In particular, we combine short-time and global-in-space error estimates, previously obtained in the uniformly parabolic case, with some a priori bounds on “short cylinders”, and we achieve short-time asymptotic convergence of the approximate solution in the degenerate parabolic case.     

From Inviscid to Viscous Aerodynamical Global Optimal Shape's Design

The refinement of the global aerodynamical optimal design (OD) of the shape of a flying configuration (FC) can be performed by improving of the start solutions for the optimization and/or of the optimization strategy itself. The proposed strategy is the own developed iterative optimum–optimorum theory. The study is here focused on the further improvement of her new, original, reinforced, zonal, spectral solutions for the partial–differential equations (PDEs) of the three–dimensional compressible Navier–Stokes layer (NSL), which govern the flow over the FCs, in subsonic and supersonic flow. These NSL's solutions, which are good suited for the computation and, especially, for the global optimal design, use the analytical potential solutions of the flow over the same FC twice: firstly as outer flow, at the NSL's edge (instead of the parallel flow used by Prandtl in his boundary layer theory) and, secondly, the velocity's components are products between the corresponding potential velocities and polynomial expansions with arbitrary coefficients, which are used to satisfy the NSL's PDEs. The use of analytical elliptical potential solutions leads to subsonic and the use of hyperbolical potential solutions leads to supersonic stabilized and rapid convergent NSL's solutions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-classical laws with conserved quantities from the arbitrary functions included in general solution to elasticity and application

The general solution to static and/or dynamic linear elasticity is a transformation between the displacements and new arbitrary functions, whose conservativeness depends on some independent partial differential equations (PDEs) satisfied by the new arbitrary functions. Zhang's general solutions are mathematically appropriate since the displacements are expressed in terms of two new arbitrary functions, and the sum of the highest order derivative added together from the independent PDEs satisfied by the two new arbitrary functions is the same as that of Navier–Cauchy equations. Therefore, the following points should be emphasized: (i) the independent PDEs come from the Laplace and D'Alembert operators acting on the two new arbitrary functions in static and dynamic general solutions, respectively, and it is found that the two new arbitrary functions are related to the rotations, first strain invariant and distortion; (ii) especially, conservation laws constructed from the equations satisfied by the spatial integrals of functions hold true, although some arbitrary functions of the spatial integrals have been canceled. Based on these facts, since Noether's identity not only can be applied to a Lagrangian but also can be used to construct a functional for widespread PDEs, the functionals relating to the rotations, first strain invariant and distortion are constructed with arbitrary integer order spatial derivative or integral, and the conservation laws follow. This kind of non-classical conservation laws does not come from the Lagrangian density of an elastic body and belongs to the deep-level natures of symmetries of elastic field derived by standard techniques. Availability is shown by two examples, from which the field intensity of a vertical load applied to the surface of an elastic half-space and the path-independent integrals in a coordinate system moving with Galilean transformation are presented for comparison.     

Estimates of blow‐up times of a system of semilinear SPDEs

In this paper, blow‐up property to a system of nonlinear stochastic PDEs driven by two‐dimensional Brownian motions is investigated. The lower and upper bounds for blow‐up times are obtained. When the system parameters satisfy certain conditions, the explicit solutions of a related system of random PDEs are deduced, which allows us to use Yor's formula to obtain the distribution functions of several blow‐up times. Particularly, the impact of noises on the life span of solutions is studied as the system parameters satisfy different conditions. Copyright © 2016 John Wiley & Sons, Ltd.     

Symbolic computation of exact solutions expressible in rational formal hyperbolic and elliptic functions for nonlinear partial differential equations

With the aid of symbolic computation, some algorithms are presented for the rational expansion methods, which lead to closed-form solutions of nonlinear partial differential equations (PDEs). The new algorithms are given to find exact rational formal polynomial solutions of PDEs in terms of Jacobi elliptic functions, solutions of the Riccati equation and solutions of the generalized Riccati equation. They can be implemented in symbolic computation system Maple. As applications of the methods, we choose some nonlinear PDEs to illustrate the methods. As a result, we not only can successfully obtain the solutions found by most existing Jacobi elliptic function methods and Tanh-methods, but also find other new and more general solutions at the same time.     

一种求解跳系数椭圆标量方程的渐近展开方法的误差分析

本文研究了具有跳系数(或多尺度性质)的椭圆型PDEs问题.利用渐近展开方法,将具有多尺度性质的PDEs模型问题解耦为若干光滑系数(或单尺度性)的PDEs问题进行求解,获得了和经典线性有限元方法在.0意义下同样的误差阶,表明了该方法的有效性.     

Existence and Uniqueness of Bounded Weak Solutions of a Semilinear Parabolic PDE

This paper has two parts. In part I, the existence and uniqueness are established for Sobolev solutions of a class of semilinear parabolic partial differential equations. Moreover, a probabilistic interpretation of the solutions in terms of backward stochastic differential equations is obtained. In part II, the existence for viscosity solutions of PDEs with obstacle and Neumann boundary condition is proved.     

Exact Traveling-Wave Solutions for Model Geophysical Systems

Exact traveling-wave solutions of time-dependent nonlinear inhomogeneous PDEs, describing several model systems in geophysical fluid dynamics, are found. The reduced nonlinear ODEs are treated as systems of linear algebraic equations in the derivatives. A variety of solutions are found, depending on the rank of the algebraic systems. The geophysical systems include acoustic gravity waves, inertial waves, and Rossby waves. The solutions describe waves which are, in general, either periodic or monoclinic. The present approach is compared with the earlier one due to Grundland (1974) for finding exact solutions of inhomogeneous systems of nonlinear PDEs.     

Finite difference method and its convergent error analyses for thermistor problem

The thermistor problem is an initial-boundary value problem of coupled nonlinear differential equations. The nonlinear PDEs consist of a heat equation with the Joule heating as asource and a current conservation equation with temperature-dopendent electrical conductivity.This problem has important opplicatioJls in industry. In this paper, A new finite differencescheme is proposed on nonuniform rectangular partition for the thermistor problem. In the theo-retical analyses,the second-order error estimates are obtained for electrical potential in discrete L^2 and H^1 norms,and for the temperature in L^2 norm. In order to get these second-order errorestimates,the Joule heating source is used in a changed equivalent form.     

广义射影Riccati方程方法与(2+1)维色散长波方程新的精确行波解

助于符号计算软件Maple,通过一种构造非线性偏微分方程更一般形式行波解的直接方 法,即改进的广义射影Ricccati方程方法,求解(2 1)维色散长波方程,得到该方程的新的 更一般形式的行波解,包括扭状孤波解,钟状解,孤子解和周期解．并对部分新形式孤波解画 图示意．     

The particular solutions for thin plates resting on Pasternak foundations under arbitrary loadings

Analytical particular solutions of splines and monomials are obtained for problems of thin plate resting on Pasternak foundation under arbitrary loadings, which are governed by a fourth‐order partial differential equation (PDEs). These analytical particular solutions are valuable when the arbitrary loadings are approximated by augmented polyharmonic splines (APS) constructed by splines and monomials. In our derivations, the real coefficient operator in the governing equation is decomposed into two complex coefficient operators whose particular solutions are known in literature. Then, we use the difference trick to recover the analytical particular solutions of the original operator. In addition, we show that the derived particular solution of spline with its first few directional derivatives are bounded as r → 0. This solution procedure may have the potential in obtaining analytical particular solutions of higher order PDEs constructed by products of Helmholtz‐type operators. Furthermore, we demonstrate the usages of these analytical particular solutions by few numerical cases in which the homogeneous solutions are complementarily solved by the method of fundamental solutions (MFS). © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010