Exact solutions of Euler equations of ideal gasdynamics via Lie group analysis

In this paper, we explicitly characterize a class of solutions to the first order quasilinear system of partial differential equations (PDEs), governing one dimensional unsteady planar and radially symmetric flows of an adiabatic gas involving shock waves. For this, Lie group analysis is used to identify a finite number of generators that leave the given system of PDEs invariant. Out of these generators, two commuting generators are constructed involving some arbitrary constants. With the help of canonical variables associated with these two generators, the assigned system of PDEs is reduced to an autonomous system, whose simple solutions provide non trivial solutions of the original system. It is interesting to remark that one of the special solutions obtained here, using this approach, is precisely the blast wave solution known in the literature.      

Lie group analysis and propagation of weak discontinuity in one-dimensional ideal isentropic magnetogasdynamics

The aim of this paper is to carry out symmetry group analysis to obtain important classes of exact solutions from the given system of nonlinear partial differential equations (PDEs). Lie group analysis is employed to derive some exact solutions of one dimensional unsteady flow of an ideal isentropic, inviscid and perfectly conducting compressible fluid, subject to a transverse magnetic field for the magnetogasdynamics system. By using Lie group theory, the full one-parameter infinitesimal transformations group leaving the equations of motion invariant is derived. The symmetry generators are used for constructing similarity variables which leads the system of PDEs to a reduced system of ordinary differential equations; in some cases, it is possible to solve these equations exactly. Further, using the exact solution, we discuss the evolutionary behavior of weak discontinuity.     

Similarity analysis of modified shallow water equations and evolution of weak waves

In this paper, we obtain exact solutions to the nonlinear system of partial differential equations (PDEs), describing the one dimensional modified shallow water equations, using invariance group properties of the governing system. Lie group of point symmetries with commuting infinitesimal operators, are presented. The symmetry generators are used for constructing similarity variables which lead the governing system of PDEs to system of ordinary differential equations (ODEs); in some cases, it is possible to solve these equations exactly. A particular solution to the governing system, which exhibits space-time dependence, is used to study the evolutionary behavior of weak discontinuities.     

Symmetry analysis and exact solutions of magnetogasdynamic equations

Using the invariance group properties of the governing systemof partial differential equations (PDEs), admitting Lie groupof point transformations with commuting infinitesimal generators,we obtain exact solutions to the system of PDEs describing one-dimensionalunsteady planar and cylindrically symmetric motions in magnetogasdynamicsinvolving shock waves. Some appropriate canonical variablesare characterised that transform the equations at hand to anequivalent autonomous form, the constant solutions of whichcorrespond to non-constant solutions of the original system.The governing system of PDEs includes as a special case theEuler's equations of non-isentropic gasdynamics. It is interestingto remark that in the absence of magnetic field, one of theexact solutions obtained here is precisely the blast wave solutionobtained earlier using a different method of approach. A particularsolution to the governing system, which exhibits space–timedependence, is used to study the wave pattern that finally developswhen a magnetoacoustic wave impacts with a shock. The influenceof magnetic field strength on the evolutionary behaviour ofincident and reflected waves and the jump in shock acceleration,after collision, are studied.     

Isovector fields and similarity solutions of Einstein vacuum equations for rotating fields

The isovector fields (infinitesimal generators of Lie groups) of Einstein vacuum equations for stationary axially symmetric rotating fields, in conventional form, that is a coupled system of nonlinear partial differential equations (PDEs) of second order are derived using the geometric prolongation technique. Some symmetry transformations and similarity (exact) solutions of Einstein vacuum equations are obtained.     

Symmetry group analysis and exact solutions of isentropic magnetogasdynamics

In this paper, we obtain exact solutions to the nonlinear system of partial differential equations (PDEs), describing the one dimensional unsteady simple flow of an isentropic, inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field. Lie group of point transformations are used for constructing similarity variables which lead the governing system of PDEs to system of ordinary differential equations (ODEs); in some cases, it is possible to solve these equations exactly. A particular solution to the governing system, which exhibits space-time dependence, is used to study the evolutionary behavior of weak discontinuities.     

On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case

We prove a result of existence and uniqueness of solutions to forward–backward stochastic differential equations, with non-degeneracy of the diffusion matrix and boundedness of the coefficients as functions of x as main assumptions.This result is proved in two steps. The first part studies the problem of existence and uniqueness over a small enough time duration, whereas the second one explains, by using the connection with quasi-linear parabolic system of PDEs, how we can deduce, from this local result, the existence and uniqueness of a solution over an arbitrarily prescribed time duration. Improving this method, we obtain a result of existence and uniqueness of classical solutions to non-degenerate quasi-linear parabolic systems of PDEs.This approach relaxes the regularity assumptions required on the coefficients by the Four-Step scheme.     

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.     

Hölder stability and uniqueness for the mean field games system via Carleman estimates

We are concerned with the mathematical study of the mean field games system (MFGS). In the conventional setup, the MFGS is a system of two coupled nonlinear parabolic partial differential equation (PDE)s of the second order in a backward–forward manner, namely, one terminal and one initial condition are prescribed, respectively, for the value function and the population density . In this paper, we show that uniqueness of solutions to the MFGS can be guaranteed if, among all four possible terminal and initial conditions, either only two terminals or only two initial conditions are given. In both cases, Hölder stability estimates are proven. This means that the accuracies of the solutions are estimated in terms of the given data. Moreover, these estimates readily imply uniqueness of corresponding problems for the MFGS. The main mathematical apparatus to establish those results is two new Carleman estimates, which may find application in other contexts associated with coupled parabolic PDEs.     

On the evolution equation for magnetic geodesics

Orbits of charged particles under the effect of a magnetic field are mathematically described by magnetic geodesics. They appear as solutions to a system of (nonlinear) ordinary differential equations of second order. But we are only interested in periodic solutions. To this end, we study the corresponding system of (nonlinear) parabolic equations for closed magnetic geodesics and, as a main result, eventually prove the existence of long time solutions. As generalization one can consider a system of elliptic nonlinear partial differential equations (PDEs) whose solutions describe the orbits of closed p-branes under the effect of a “generalized physical force”. For the corresponding evolution equation, which is a system of parabolic nonlinear PDEs associated to the elliptic PDE, we can establish existence of short time solutions.     

The use of factors to discover potential systems or linearizations

Factors of a given system of PDEs are solutions of an adjoint system of PDEs related to the system's Fréchet derivative. In this paper, we introduce the notion of potential conservation laws, arising from specific types of factors, which lead to useful potential systems. Point symmetries of a potential system could yield nonlocal symmetries of the given system and its linearization by a noninvertible mapping.We also introduce the notion of linearizing factors to determine necessary conditions for the existence of a linearization of a given system of PDEs.     

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.     

The Nullstellensatz for Systems of PDE

Given an ideal I in , the polynomial ring in n-indeterminates, the affine variety of I is the set of common zeros in ⁿ of all the polynomials that belong to I, and the Hilbert Nullstellensatz states that there is a bijective correspondence between these affine varieties and radical ideals of . If, on the other hand, one thinks of a polynomial as a (constant coefficient) partial differential operator, then instead of its zeros in ⁿ, one can consider its zeros, i.e., its homogeneous solutions, in various function and distribution spaces. An advantage of this point of view is that one can then consider not only the zeros of ideals of , but also the zeros of submodules of free modules over (i.e., of systems of PDEs). The question then arises as to what is the analogue here of the Hilbert Nullstellensatz. The answer clearly depends on the function–distribution space in which solutions of PDEs are being located, and this paper considers the case of the classical spaces. This question is related to the more general question of embedding a partial differential system in a (two-sided) complex with minimal homology. This paper also explains how these questions are related to some questions in control theory.     

On the summability of divergent power series satisfying singular PDEs

The aim of this note is to apply the Borel–Laplace summation method studied by H. Chen, Z. Luo and C. Zhang (Summability of formal solutions of singular PDEs by means of two-dimensional Borel–Laplace method, preprint) to the divergent power series solutions to two families of nonlinear PDEs. The first one contains particularly a two-dimensional version of the so-called Euler equation (ODE), while the second is called totally characteristic type PDE by H. Chen and H. Tahara (On the holomorphic solution of non-linear totally characteristic equations, Math. Nachr. 219 (2000) 85–96).     

Analytic approximate solutions to Burgers,Fisher, Huxley equations and two combined forms of these equations

In this paper, we are giving analytic approximate solutions to a class of nonlinear PDEs using the homotopy analysis method (HAM). The Burgers, Fisher, Huxley, Burgers–Fisher and Burgers–Huxley equations are considered. We aim two goals: one is to highlight the efficiency of HAM in solving this class of PDEs and the other is that, although the considered equations have different combinations of nonlinear terms, when applying HAM, we use the same initial guess, the same auxiliary linear operator and the same auxiliary function for all of them.     

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.     

半圆域内的二维线性椭圆偏微分方程

本文研究了半圆域内的二维线性椭圆偏微分方程.利用Fokas提出的求解凸多边形区域内的线性椭圆偏微分方程的变换方法,我们改进了这个方法来研究半圆域内Laplace方程,修改Helmholtz方程和Helmholtz方程的解,并且导出了这些方程解的积分表达式,讨论了Helmholtz方程的广义Dirichlet到Neumann映射.     

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.     

Well-posedness for a class of fourth order diffusions for image processing

A number of image denoising models based on higher order parabolic partial differential equations (PDEs) have been proposed in an effort to overcome some of the problems attendant to second order methods such as the famous Perona–Malik model. However, there is little analysis of these equations to be found in the literature. In this paper, methods of maximal regularity are used to prove the existence of unique local solutions to a class of fourth order PDEs for noise removal. The proof is laid out explicitly for two newly proposed fourth order models, and an outline is given for how to apply the techniques to other proposed models.     

A low-dimensional conjugacy for elliptic equations and symmetry breaking on rotated domains

A topological conjugacy is established between certain elliptic PDEs with one unbounded time direction and a simple second-order differential equation, admitting the dynamics of such PDEs to be examined on a two-dimensional submanifold. By this means, periodic solutions can be obtained to elliptic equations as perturbations of those that are independent of time.