SEMI-IMPLICIT SPECTRAL DEFERRED CORRECTION METHODS BASED ON SECOND-ORDER TIME INTEGRATION SCHEMES FOR NONLINEAR PDES

In[20],a semi-implicit spectral deferred correction(SDC)method was proposed,which is efficient for highly nonlinear partial differential equations(PDEs).The semi-implicit SDC method in[20]is based on first-order time integration methods,which are corrected iteratively,with the order of accuracy increased by one for each additional iteration.In this paper,we will develop a class of semi-implicit SDC methods,which are based on second-order time integration methods and the order of accuracy are increased by two for each additional iteration.For spatial discretization,we employ the local discontinuous Galerkin(LDG)method to arrive at fully-discrete schemes,which are high-order accurate in both space and time.Numerical experiments are presented to demonstrate the accuracy,efficiency and robustness of the proposed semi-implicit SDC methods for solving complex nonlinear PDEs.

A SIMPLE FAST METHOD IN FINDING PARTICULAR SOLUTIONS OF SOME NONLINEAR PDE

The 'trial function method' ( TFM for short) and a routine way in finding traveling,wave solutions to some nonlinear partial differential equations( PDE for short), wer explained. Two types of evolution equations are studied, one is a generalized Burgers or KdV equation, the other is the Fisher equation with special nonlinear forms of its reaction rate term. One can see that this method is simple, fast and allowing further extension.

Double-Parameter Solutions of Projective Riccati Equations and Their Applications

The purpose of the present paper is twofold. First, the projective Riccati equations （PREs for short） are resolved by means of a linearized theorem, which was known in the literature. Based on the signs and values of coeffcients of PREs, the solutions with two arbitrary parameters of PREs can be expressed by the hyperbolic functions, the trigonometric functions, and the rational functions respectively, at the same time the relation between the components of each solution to PREs is also implemented. Second, more new travelling wave solutions for some nonlinear PDEs, such as the Burgers equation, the mKdV equation, the NLS^＋ equation, new Hamilton amplitude equation, and so on, are obtained by using Sub-ODE method, in which PREs are taken as the Sub-ODEs. The key idea of this method is that the travelling wave solutions of nonlinear PDE can be expressed by a polynomial in two variables, which are the components of each solution to PREs, provided that the homogeneous balance between the higher order derivatives and nonlinear terms in the equation is considered.

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.

Medical X-Ray Image Enhancement Based on Kramer's PDE Model

The purpose of this study is to present an application of a novel enhancement technique for enhancing medical images generated from X-rays. The method presented in this study is based on a nonlinear partial differential equation （PDE） model, Kramer＇s PDE model. The usefulness of this method is investigated by experimental results. We apply this method to a medical X-ray image. For comparison, the X-ray image is also processed using classic Perona-Malik PDE model and Catte PDE model. Although the Perona-Malik model and Catte PDE model could also enhance the image, the quality of the enhanced images is considerably inferior compared with the enhanced image using Kramer＇s PDE model. The study suggests that the Kramer＇s PDE model is capable of enhancing medical X-ray images, which will make the X-ray images more reliable.

Maximal Dimension of Invariant Subspaces to Systems of Nonlinear Evolution Equations

In this paper, the dimension of invariant subspaces admitted by nonlinear sys- tems is estimated under certain conditions. It is shown that if the two-component nonlinear vector differential operator F = （F1, F2） with orders {k1, k2} （k1≥ k2） preserves the invariant subspace Wn1^1× Wn2^2 （n1 ≥ n2）, then n1 - n2 ≤ k2, n1 ≤2（k1 ＋ k2） ＋ 1, where Wnq^q is the space generated by solutions of a linear ordinary differential equation of order nq （q = 1, 2）. Several examples including the （1＋1）-dimensional diffusion system and Ito＇s type, Drinfel＇d-Sokolov-Wilson＇s type and Whitham-Broer-Kaup＇s type equations are presented to illustrate the result. Furthermore, the estimate of dimension for m-component nonlinear systems is also given.

Gradient estimates for nonlinear diffusion semigroups by coupling methods

In this paper, we obtain gradient estimates for certain nonlinear partial differential equations by coupling methods. First, we derive uniform gradient estimates for certain semi-linear PDEs based on the coupling method introduced by Wang in 2011 and the theory of backward SDEs. Then we generalize Wang's coupling to the G-expectation space and obtain gradient estimates for nonlinear diffusion semigroups, which correspond to the solutions of certain fully nonlinear PDEs.

MIXED INTERIOR PENALTY DISCONTINUOUS GALERKIN METHODS FOR ONE-DIMENSIONAL FULLY NONLINEAR SECOND ORDER ELLIPTIC AND PARABOLIC EQUATIONS

This paper is concerned with developing accurate and efficient numerical methods for one-dimensional fully nonlinear second order elliptic and parabolic partial differential equations （PDEs）. In the paper we present a general framework for constructing high order interior penalty discontinuous Galerkin （IP-DG） methods for approximating viscosity solutions of these fully nonlinear PDEs. In order to capture discontinuities of the second order derivative uxx of the solution u, three independent functions p1,p2 and p3 are introduced to represent numerical derivatives using various one-sided limits. The proposed DG frame- work, which is based on a nonstandard mixed formulation of the underlying PDE, embeds a nonlinear problem into a mostly linear system of equations where the nonlinearity has been modified to include multiple values of the second order derivative uxz. The proposed framework extends a companion finite difference framework developed by the authors in [9] and allows for the approximation of fully nonlinear PDEs using high order polynomials and non-uniform meshes. In addition to the nonstandard mixed formulation setting, another main idea is to replace the fully nonlinear differential operator by a numerical operator which is consistent with the differential operator and satisfies certain monotonicity （called g-monotonicity） properties. To ensure such a g-monotonicity, the crux of the construction is to introduce the numerical moment, which plays a critical role in the proposed DG frame- work. The g-monotonicity gives the DG methods the ability to select the mathematically ＂correct＂ solution （i.e., the viscosity solution） among all possible solutions. Moreover, the g-monotonicity allows for the possible development of more efficient nonlinear solvers as the special nonlinearity of the algebraic systems can be explored to decouple the equations. This paper also presents and analyzes numerical results for several numerical test problems which are used to guage the accuracy and efficiency of the proposed DG methods.

The Improved Modified Decomposition Method for the Treatment of Systems of Singular Nonlinear Partial Differential Equations with Initial Conditions

New developed inverse differential operators incorporated into the semi- analytical treatment of the modified decomposition method （MDM） are used to solve the systems of first and second-order singular nonlinear partial differential equations （PDEs） with initial conditions arising in physics. The new proposed method is called the improved modified decomposition method （IMDM）, and is used to the treatment of a few case study initial-value problems. The results obtained by the IMDM are in full agreement with the existing exact analytical solutions.

An Efficient Numerical Solution of Nonlinear Hunter-Saxton Equation

In this paper, the nonlinear Hunter–Saxton equation, which is a famous partial differential equation,is solved by using a hybrid numerical method based on the quasilinearization method and the bivariate generalized fractional order of the Chebyshev functions(B-GFCF) collocation method. First, using the quasilinearization method,the equation is converted into a sequence of linear partial differential equations(LPD), and then these LPDs are solved using the B-GFCF collocation method. A very good approximation of solutions is obtained, and comparisons show that the obtained results are more accurate than the results of other researchers.

Nonlinear Stochastic Galerkin and Collocation Methods:Application to a Ferromagnetic Cylinder Rotating at High Speed

The stochastic Galerkin and stochastic collocation method are two state-ofthe-art methods for solving partial differential equations(PDE)containing random coefficients.While the latter method,which is based on sampling,can straightforwardly be applied to nonlinear stochastic PDEs,this is nontrivial for the stochastic Galerkin method and approximations are required.In this paper,both methods are used for constructing high-order solutions of a nonlinear stochastic PDE representing the magnetic vector potential in a ferromagnetic rotating cylinder.This model can be used for designing solid-rotor induction machines in various machining tools.A precise design requires to take ferromagnetic saturation effects into account and uncertainty on the nonlinear magnetic material properties.Implementation issues of the stochastic Galerkin method are addressed and a numerical comparison of the computational cost and accuracy of both methods is performed.The stochastic Galerkin method requires in general less stochastic unknowns than the stochastic collocation approach to reach a certain level of accuracy,however at a higher computational cost.

Reflected solutions of backward stochastic differential equations driven by G-Brownian motion

In this paper, we study the reflected solutions of one-dimensional backward stochastic differential equations driven by G-Brownian motion. The reflection keeps the solution above a given stochastic process. In order to derive the uniqueness of reflected G-BSDEs, we apply a "martingale condition" instead of the Skorohod condition. Similar to the classical case, we prove the existence by approximation via penalization. We then give some applications including a generalized Feynman-Kac formula of an obstacle problem for fully nonlinear partial differential equation and option pricing of American types under volatility uncertainty.

Numerical Solution of the Incompressible Navier-Stokes Equation by a Deep Branching Algorithm

We present an algorithm for the numerical solution of systems of fully nonlinear PDEs using stochastic coded branching trees.This approach covers functional nonlinearities involving gradient terms of arbitrary orders,and it requires only a boundary condition over space at a given terminal time T instead of Dirichlet or Neumann boundary conditions at all times as in standard solvers.Its implementation relies on Monte Carlo estimation,and uses neural networks that perform a meshfree functional estimation on a space-time domain.The algorithm is applied to the numerical solution of the Navier-Stokes equation and is benchmarked to other implementations in the cases of the Taylor-Green vortex and Arnold-Beltrami-Childress flow.

Mathematical modeling of atmospheric internal waves phenomenon and its solution by Elzaki Adomian decomposition method

This article involves the study of atmospheric internal waves phenomenon,also referred to as gravity waves.This phenomenon occurs inside the fluid,not on the surface.The model is based on a shallow fluid hypothesis represented by a system of nonlinear partial differential equations.The basic assumption of the shallow flow model is that the horizontal size is much larger than the vertical size.Atmospheric internal waves can be perfectly represented by this model as the waves are spread over a large horizontal area.Here we used the Elzaki Adomian Decomposition Method(EADM)to obtain the solution for the considered model along with its convergence analysis.The Adomian decomposition method together with the Elzaki transform gives the solution in a convergent series without any linearization or perturbation.Comparisons are built between the results obtained by EADM and HAM to examine the accuracy of the proposed method.

An Efficient Operator-Splitting Method for Noise Removal in Images

In this work,noise removal in digital images is investigated.The importance of this problem lies in the fact that removal of noise is a necessary pre-processing step for other image processing tasks such as edge detection,image segmentation,image compression,classification problems,image registration etc.A number of different approaches have been proposed in the literature.In this work,a non-linear PDE-based algorithm is developed based on the ideas proposed by Lysaker,Osher and Tai[IEEE Trans.Image Process.,13(2004),1345-1357].This algorithm consists of two steps:flow field smoothing of the normal vectors,followed by image reconstruction.We propose a finite-difference based additive operator-splitting method that allows for much larger time-steps.This results in an efficient method for noise-removal that is shown to have good visual results.The energy is studied as an objective measure of the algorithm performance.

Affine processes under parameter uncertainty

We develop a one-dimensional notion of affine processes under parameter uncertainty,which we call nonlinear affine processes.This is done as follows:given a setof parameters for the process,we construct a corresponding nonlinear expectation on the path space of continuous processes.By a general dynamic programming principle,we link this nonlinear expectation to a variational form of the Kolmogorov equation,where the generator of a single affine process is replaced by the supremum over all corresponding generators of affine processes with parameters in.This nonlinear affine process yields a tractable model for Knightian uncertainty,especially for modelling interest rates under ambiguity.We then develop an appropriate Ito formula,the respective term-structure equations,and study the nonlinear versions of the Vasiˇcek and the Cox–Ingersoll–Ross(CIR)model.Thereafter,we introduce the nonlinear Vasicek–CIR model.This model is particularly suitable for modelling interest rates when one does not want to restrict the state space a priori and hence this approach solves the modelling issue arising with negative interest rates.

Neural Network-Based Variational Methods for Solving Quadratic Porous Medium Equations in High Dimensions

In this paper,we propose and study neural network-based methods for solutions of high-dimensional quadratic porous medium equation(QPME).Three variational formulations of this nonlinear PDE are presented:a strong formulation and two weak formulations.For the strong formulation,the solution is directly parameterized with a neural network and optimized by minimizing the PDEresidual.It can be proved that the convergence of the optimization problem guarantees the convergence of the approximate solution in the L^(1)sense.Theweak formulations are derived following(Brenier in Examples of hidden convexity in nonlinear PDEs,2020)which characterizes the very weak solutions of QPME.Specifically speaking,the solutions are represented with intermediate functions who are parameterized with neural networks and are trained to optimize the weak formulations.Extensive numerical tests are further carried out to investigate the pros and cons of each formulation in low and high dimensions.This is an initial exploration made along the line of solving high-dimensional nonlinear PDEs with neural network-based methods,which we hope can provide some useful experience for future investigations.

NUMERICAL COMPUTATION OF BOUNDED SOLUTIONS FORA SEMILINEAR ELLIPTIC EQUATION ON AN INFINITE STRIP

In this paper, we consider the computation of bounded solutions of a semilinear elliptic equation on an infinite strip. The dynamical system approach and reduction on center manifold are used to overcome the difficulties in numerical procedure.

Continuity of Almost Harmonic Maps with the Perturbation Term in a Critical Space

The authors study the continuity estimate of the solutions of almost harmonic maps with the perturbation term f in a critical integrability class(Zygmund class)L^(n/2)log^(q) L,n is the dimension with n≥3.They prove that when q>n/2 the solution must be continuous and they can get continuity modulus estimates.As a byproduct of their method,they also study boundary continuity for the almost harmonic maps in high dimension.