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"On a System of Linear Partial Differential Equations of the Hyperbolic Type" is an article from The Annals of Mathematics, Volume 20 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1967120

"On a System of Linear Partial Differential Equations of the Hyperbolic Type" is an article from The Annals of Mathematics, Volume 20 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1967120

"On a System of Linear Partial Differential Equations of the Hyperbolic Type" is an article from The Annals of Mathematics, Volume 20 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1967120

"On a System of Linear Partial Differential Equations of the Hyperbolic Type" is an article from The Annals of Mathematics, Volume 20 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1967120

"On a System of Linear Partial Differential Equations of the Hyperbolic Type" is an article from The Annals of Mathematics, Volume 20 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1967120

"A New Canonical Form for Systems of Partial Differential Equations" is an article from American Journal of Mathematics, Volume 39 . View more articles from American Journal of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2370446

522 pages : 24 cm

522 pages : 24 cm

522 pages : 24 cm

522 pages : 24 cm

522 pages : 24 cm

Numerical solutions of partial differential equations (PDEs) on manifolds continues to generate a lot of interest among scientists in the natural and applied sciences. On the other hand, recent developments of 3D scanning and computer vision technologies have produced a large number of 3D surface models represented as point clouds. Herein, we develop a simple and efficient method for solving PDEs on closed surfaces represented as point clouds. By projecting the radial vector of standard radial basis function(RBF) kernels onto the local tangent plane, we are able to produce a representation of functions that permits the replacement of surface differential operators with their Cartesian equivalent. We demonstrate, numerically, the efficiency of the method in discretizing the Laplace Beltrami operator.

It is proved that the members of the Riccati hierarchy, the so-called Riccati chain equations, can be considered as particular cases of projective Riccati equations, which greatly simplifies the study of the Riccati hierarchy. This also allows us to characterize Riccati chain equations geometrically in terms of the projective vector fields of a flat Riemannian metric and to easily derive their associated superposition rules. Next, we establish necessary and sufficient conditions under which it is possible to map second-order Riccati chain equations into conformal Riccati equations through a local diffeomorphism. This fact can be used to determine superposition rules for particular higher-order Riccati chain equations which depend on fewer particular solutions than in the general case. Therefore, we analyze the properties of Euclidean, hyperbolic and projective vector fields on the plane in detail. Finally, the use of contact transformations enables us to apply the derived results to the study of certain integrable partial differential equations, such as the Kaup-Kupershmidt and Sawada-Kotera equations.

It is proved that the members of the Riccati hierarchy, the so-called Riccati chain equations, can be considered as particular cases of projective Riccati equations, which greatly simplifies the study of the Riccati hierarchy. This also allows us to characterize Riccati chain equations geometrically in terms of the projective vector fields of a flat Riemannian metric and to easily derive their associated superposition rules. Next, we establish necessary and sufficient conditions under which it is possible to map second-order Riccati chain equations into conformal Riccati equations through a local diffeomorphism. This fact can be used to determine superposition rules for particular higher-order Riccati chain equations which depend on fewer particular solutions than in the general case. Therefore, we analyze the properties of Euclidean, hyperbolic and projective vector fields on the plane in detail. Finally, the use of contact transformations enables us to apply the derived results to the study of certain integrable partial differential equations, such as the Kaup-Kupershmidt and Sawada-Kotera equations.

We extend the invariant manifold method for analyzing the asymptotics of dissipative partial differential equations on unbounded spatial domains to treat equations in which the linear part has order greater than two. One important example of this type of equation which we analyze in some detail is the Cahn-Hilliard equation. We analyze the marginally stable solutions of this equation in some detail. A second context in which such equations arise is in the Ginzburg-Landau equation, or other pattern forming equations, near a codimension-two bifurcation.

In this paper, we study the possible second order Lax operators for all the possible (1+1)-dimensional models with Schwarz variants and some special types of high dimensional models. It is shown that for every (1+1)-dimensional model and some special types of high dimensional models which possess Schwarz variants may have a second order Lax pair. The exiplicit Lax pairs for (1+1)-dimensional Korteweg de Vries equation, Harry Dym equation, Boussinesq equation, Caudry-Dodd-Gibbon-Sawada-Kortera equation, Kaup-Kupershmidt equation, Riccati equation, (2+1)-dimensional breaking soliton equation and a generalized (2+1)-dimensional fifth order equation are given.

Following I. S. Krasilshchik and A. M. Vinogradov, we regard PDEs as infinite-dimensional manifolds with involutive distributions and consider their special morphisms called differential coverings, which include constructions like Lax pairs and Backlund transformations. We show that, similarly to usual coverings in topology, at least for some PDEs differential coverings are determined by actions of a sort of fundamental group. This is not a group, but a certain system of Lie algebras, which generalize Wahlquist-Estabrook algebras. From this we deduce an algebraic necessary condition for two PDEs to be connected by a Backlund transformation. We compute these infinite-dimensional Lie algebras for several KdV type equations and prove non-existence of Backlund transformations. As a by-product, for some class of Lie algebras g we prove that any subalgebra of g of finite codimension contains an ideal of g of finite codimension.

v p., 1 \U+fffd\., 108 p. 22 cm

In this paper parabolic random partial differential equations and parabolic stochastic partial differential equations driven by a Wiener process are considered. A deterministic, tensorized evolution equation for the second moment and the covariance of the solutions of the parabolic stochastic partial differential equations is derived. Well-posedness of a space-time weak variational formulation of this tensorized equation is established.

A new iterative technique is presented for solving of initial value problem for certain classes of multidimensional linear and nonlinear partial differential equations. Proposed iterative scheme does not require any discretization, linearization or small perturbations and therefore significantly reduces numerical computations. Rigorous convergence analysis of presented technique and an error estimate are included as well. Several numerical examples for high dimensional initial value problem for heat and wave type partial differential equations are presented to demonstrate reliability and performance of proposed iterative scheme.

We apply the Tensor Train (TT) decomposition to construct the tensor product Polynomial Chaos Expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization, and to compute some quantities of interest (mean, variance, exceedance probabilities). We assume that the random diffusion coefficient is given as a smooth transformation of a Gaussian random field. In this case, the PCE is delivered by a complicated formula, which lacks an analytic TT representation. To construct its TT approximation numerically, we develop the new block TT cross algorithm, a method that computes the whole TT decomposition from a few evaluations of the PCE formula. The new method is conceptually similar to the adaptive cross approximation in the TT format, but is more efficient when several tensors must be stored in the same TT representation, which is the case for the PCE. Besides, we demonstrate how to assemble the stochastic Galerkin matrix and to compute the solution of the elliptic equation and its post-processing, staying in the TT format. We compare our technique with the traditional sparse polynomial chaos and the Monte Carlo approaches. In the tensor product polynomial chaos, the polynomial degree is bounded for each random variable independently. This provides higher accuracy than the sparse polynomial set or the Monte Carlo method, but the cardinality of the tensor product set grows exponentially with the number of random variables. However, when the PCE coefficients are implicitly approximated in the TT format, the computations with the full tensor product polynomial set become possible. In the numerical experiments, we confirm that the new methodology is competitive in a wide range of parameters, especially where high accuracy and high polynomial degrees are required.

In this work we present a Lyapunov inequality for linear and quasilinear elliptic differential operators in $N-$dimensional domains $\Omega$. We also consider singular and degenerate elliptic problems with $A_p$ coefficients involving the $p-$Laplace operator with zero Dirichlet boundary condition. As an application of the inequalities obtained, we derive lower bounds for the first eigenvalue of the $p-$Laplacian, and compare them with the usual ones in the literature.

We consider a generalization of the projecting operators method for the case of Cauchy problem for systems of 1D evolution differential equations of first order with variable coefficients. It is supposed that the coefficients dependence on the only variable x is weak, that is described by a small parameter introduction. Such problem corresponds, for example, to the case of wave propagation in a weakly inhomogeneous medium. As an example, we specify the problem to adiabatic acoustics. For the Cauchy problem, to fix unidirectional modes, the projection operators are constructed. The method of successive approximations (perturbation theory) is developed and based on pseudodifferential operators theory. The application of these projection operators allows to obtain approximate evolution equations corresponding to the separated directed waves.

In this paper, we introduce a class of stochastic partial differential equations (SPDEs) with fractional time-derivatives, and study the $L_2$-theory of the equations. This class of SPDEs can be used to describe random effects on transport of particles in medium with thermal memory or particles subject to sticking and trapping.

In this paper we study fractal solutions of linear and nonlinear dispersive PDE on the torus. In the first part we answer some open questions on the fractal solutions of linear Schr\"odinger equation and equations with higher order dispersion. We also discuss applications to their nonlinear counterparts like the cubic Schr\"odinger equation (NLS) and the Korteweg-de Vries equation (KdV). In the second part, we study fractal solutions of the vortex filament equation and the associated Schr\"odinger map equation (SM). In particular, we construct global strong solutions of the SM in $H^s$ for $s>\frac32$ for which the evolution of the curvature is given by a periodic nonlinear Schr\"odinger evolution. We also construct unique weak solutions in the energy level. Our analysis follows the frame construction of Chang {\em et al.} \cite{csu} and Nahmod {\em et al.} \cite{nsvz}.

We investigate parabolic Muckenhoupt weights and functions of bounded mean oscillation (BMO) related to nonlinear parabolic partial differential equations. The main result gives a full characterization of weak and strong type weighted norm inequalities for parabolic forward in time maximal operators. In addition, we give a Jones type factorization result for the parabolic Muckenhoupt weights and a Coifman-Rochberg type characterization of the parabolic BMO from Moser's seminal paper through parabolic Muckenhoupt weights and maximal functions.

The Hodge-de Rham Theorem is introduced and discussed. This result has implications for the general study of several partial differential equations. Some propositions which have applications to the proof of this theorem are used to study some related results concerning a class of partial differential equation in a novel way.

The solution of a parabolic stochastic partial differential equation (SPDE) driven by an infinite-dimensional Brownian motion is in general not a semi-martingale anymore and does in general not satisfy an It\^{o} formula like the solution of a finite-dimensional stochastic ordinary differential equation (SODE). In particular, it is not possible to derive stochastic Taylor expansions as for the solution of a SODE using an iterated application of the It\^{o} formula. Consequently, until recently, only low order numerical approximation results for such a SPDE have been available. Here, the fact that the solution of a SPDE driven by additive noise can be interpreted in the mild sense with integrals involving the exponential of the dominant linear operator in the SPDE provides an alternative approach for deriving stochastic Taylor expansions for the solution of such a SPDE. Essentially, the exponential factor has a mollifying effect and ensures that all integrals take values in the Hilbert space under consideration. The iteration of such integrals allows us to derive stochastic Taylor expansions of arbitrarily high order, which are robust in the sense that they also hold for other types of driving noise processes such as fractional Brownian motion. Combinatorial concepts of trees and woods provide a compact formulation of the Taylor expansions.

Nonlinear equations come to us in tremendous variety, each with its own questions and its own difficulties. At one extreme are the completely integrable equations, with constants of the motion and a rich algebraic structure. At the other extreme  is  chaos, with turbulent solutions and statistical averages. Between these two possibilities, algebraic and ergodic, lies the full range of nonlinear phenomena. There are smooth solutions which develop shocks, or bifurcate, or maintain slow and nearly periodic variations that imitate the linear theory. Each of  these questions requires a separate treatment, and the subject would be simpler if we know for every equation which behavior to expect.  Nevertheless these equations, the nonlinear partial differential equations  which arise in applications, share one crucial property. They are all vulnerable when the right pattern in found. It is a slow process, to uncover and reveal their structure, but  it  is moving forward.

In this paper, we propose a novel variable-separation (NVS) method for generic multivariate functions. The idea of NVS is extended to to obtain the solution in tensor product structure for stochastic partial differential equations (SPDEs). Compared with many widely used variation-separation methods, NVS shares their merits but has less computation complexity and better efficiency. NVS can be used to get the separated representation of the solution for SPDE in a systematic enrichment manner. No iteration is performed at each enrichment step. This is a significant improvement compared with proper generalized decomposition. Because the stochastic functions of the separated representations obtained by NVS depend on the previous terms, this impacts on the computation efficiency and brings great challenge for numerical simulation for the problems in high stochastic dimensional spaces. In order to overcome the difficulty, we propose an improved least angle regression algorithm (ILARS) and a hierarchical sparse low rank tensor approximation (HSLRTA) method based on sparse regularization. For ILARS, we explicitly give the selection of the optimal regularization parameters at each step based on least angle regression algorithm (LARS) for lasso problems such that ILARS is much more efficient. HSLRTA hierarchically decomposes a high dimensional problem into some low dimensional problems and brings an accurate approximation for the solution to SPDEs in high dimensional stochastic spaces using limited computer resource. A few numerical examples are presented to illustrate the efficacy of the proposed methods.

In this paper, we propose a novel variable-separation (NVS) method for generic multivariate functions. The idea of NVS is extended to to obtain the solution in tensor product structure for stochastic partial differential equations (SPDEs). Compared with many widely used variation-separation methods, NVS shares their merits but has less computation complexity and better efficiency. NVS can be used to get the separated representation of the solution for SPDE in a systematic enrichment manner. No iteration is performed at each enrichment step. This is a significant improvement compared with proper generalized decomposition. Because the stochastic functions of the separated representations obtained by NVS depend on the previous terms, this impacts on the computation efficiency and brings great challenge for numerical simulation for the problems in high stochastic dimensional spaces. In order to overcome the difficulty, we propose an improved least angle regression algorithm (ILARS) and a hierarchical sparse low rank tensor approximation (HSLRTA) method based on sparse regularization. For ILARS, we explicitly give the selection of the optimal regularization parameters at each step based on least angle regression algorithm (LARS) for lasso problems such that ILARS is much more efficient. HSLRTA hierarchically decomposes a high dimensional problem into some low dimensional problems and brings an accurate approximation for the solution to SPDEs in high dimensional stochastic spaces using limited computer resource. A few numerical examples are presented to illustrate the efficacy of the proposed methods.

With human social behaviors influence, some boyciana-fish reaction-diffusion system coupled with elliptic human distribution equation is considered. Firstly, under homogeneous Neumann boundary conditions and ratio-dependent functional response the system can be described as a nonlinear partial differential algebraic equations (PDAEs) and the corresponding linearized system is discussed with singular system theorem. In what follows we discuss the elliptic subsystem and show that the three kinds of nonnegative are corresponded to three different human interference conditions: human free, overdevelopment and regular human activity. Next we examine the system persistence properties: absorbtion region and the stability of positive steady states of three systems. And the diffusion-driven unstable property is also discussed. Moreover, we propose some energy estimation discussion to reveal the dynamic property among the boyciana-fish-human interaction systems.Finally, using the realistic data collected in the past fourteen years, by PDAEs model parameter optimization, we carry out some predicted results about wetland boyciana population. The applicability of the proposed approaches are confirmed analytically and are evaluated in numerical simulations.

It is proved that a differentiable with respect to each variable function $f:\mathbb R^2\to\mathbb R$ is a solution of the equation $ \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y}=0$ if and only if there exists a function $\varphi:\mathbb R\to\mathbb R$ such that $f(x,y)=\varphi(x-y)$. This gives a positive answer to a question of R.~Baire. Besides, we use this result to solving analogous partial differential equations in abstract spaces and partial differential equations of higher-order.

We consider linear and nonlinear hyperbolic SPDEs with mixed derivatives with additive space-time Gaussian white noise of the form $Y_{xt}=F(Y) + \sigma W_{xt}.$ Such equations, which transform to linear and nonlinear wave equations, including Klein-Gordon, Liouville's and the sine-Gordon equation, are related to what Zimmerman (1972) called a diffusion equation. An explicit numerical scheme is employed in both deterministic and stochastic examples. The scheme is checked for accuracy against known exact analytical solutions for deterministic equations. In the stochastic case with $F=0$, solutions yield sample paths for the Brownian sheet whose statistics match well exact values. Generally the boundary conditions are chosen to be initial values $Y(x,0)$ and boundary values $Y(0,t)$ on the quarter-plane or subsets thereof, which have been shown to lead to existence and uniqueness of solutions. For the linear case solutions are compared at various grid sizes and wave-like solutions were found, with and without noise, for non-zero initial and boundary conditions. Surprisingly, wave-like structures seemed to emerge with zero initial and boundary conditions and purely noise source terms with no signal. Equations considered with nonlinear $F$ included quadratic and cubic together with the sine-Gordon equation. For the latter, wave-like structures were apparent with $\sigma \le 0.25$ but they tended to be shattered at larger values of $\sigma$. Previous work on stochastic sine-Gordon equations is briefly reviewed.

In this paper we develop an adaptive procedure for the numerical solution of general, semilinear elliptic problems with possible singular perturbations. Our approach combines both a prediction-type adaptive Newton method and an adaptive finite element discretization (based on a robust a posteriori error analysis), thereby leading to a fully adaptive Newton-Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for different examples.

"Part II. Partial Differential Equations of the Second Order" is an article from The Annals of Mathematics, Volume 4 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1967327

Multivariate median filters have been proposed as generalisations of the well-established median filter for grey-value images to multi-channel images. As multivariate median, most of the recent approaches use the $L^1$ median, i.e.\ the minimiser of an objective function that is the sum of distances to all input points. Many properties of univariate median filters generalise to such a filter. However, the famous result by Guichard and Morel about approximation of the mean curvature motion PDE by median filtering does not have a comparably simple counterpart for $L^1$ multivariate median filtering. We discuss the affine equivariant Oja median and the affine equivariant transformation--retransformation $L^1$ median as alternatives to $L^1$ median filtering. We analyse multivariate median filters in a space-continuous setting, including the formulation of a space-continuous version of the transformation--retransformation $L^1$ median, and derive PDEs approximated by these filters in the cases of bivariate planar images, three-channel volume images and three-channel planar images. The PDEs for the affine equivariant filters can be interpreted geometrically as combinations of a diffusion and a principal-component-wise curvature motion contribution with a cross-effect term based on torsions of principal components. Numerical experiments are presented that demonstrate the validity of the approximation results.

We apply the method of simplest equation for obtaining exact solitary traveling-wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives. We consider first the general case of presence of monomials of the both (odd and even) grades and then turn to the two particular cases of nonlinear equations that contain only monomials of odd grade or only monomials of even grade. The methodology is illustrated by numerous examples.

We consider time fractional stochastic heat type equation $$\partial^\beta_tu(t,x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[\sigma(u)\stackrel{\cdot}{W}(t,x)]$$ in $(d+1)$ dimensions, where $\nu>0$, $\beta\in (0,1)$, $\alpha\in (0,2]$, $d

We derive computable error estimates for finite element approximations of linear elliptic partial differential equations (PDE) with rough stochastic coefficients. In this setting, the exact solutions contain high frequency content that standard a posteriori error estimates fail to capture. We propose goal-oriented estimates, based on local error indicators, for the pathwise Galerkin and expected quadrature errors committed in standard, continuous, piecewise linear finite element approximations. Derived using easily validated assumptions, these novel estimates can be computed at a relatively low cost and have applications to subsurface flow problems in geophysics where the conductivities are assumed to have lognormal distributions with low regularity. Our theory is supported by numerical experiments on test problems in one and two dimensions.

I provide partial differential equations (PDEs) - in finite cases - and functional differential equations (FDEs) - in field-theoretic cases - which determine observables or beables in the senses of Kuchar and of Dirac. I consider such for a wide range of relational particle mechanics as well as for Electromagnetism, Yang-Mills Theory and General Relativity. I give an underlying reason why pure-configuration Kuchar observables or beables are already well-known: notions of shape, E-fields, B-fields, loops and 3-geometries. I additionally pose the PDEs or FDEs for pure-momentum observables or beables, and for observables or beables which have a mixture of configuration and momentum functional dependence.

We explore variational Poisson-Nijenhuis structures on nonlinear PDEs and establish relations between Schouten and Nijenhuis brackets on the initial equation with the Lie bracket of symmetries on its natural extensions (coverings). This approach allows to construct a framework for the theory of nonlocal structures.

It is shown that existence of a global solution to a particular nonlinear system of second order partial differential equations on a complete connected Riemannian manifold has topological and geometric implications and that in the domain of positivity of such solution its reciprocal is the radial function of only one of the following rotationally symmetric hypersurfaces in $\Rn$: paraboloid, ellipsoid, one sheet of a two-sheeted hyperboloid, and a hyperplane.

A generalized KdV equation is formulated as an exterior differential system, which is used to determine the prolongation structure of the equation. The prolongation structure is obtained for several cases of the variable powers, and nontrivial algebras are determined. The analysis is extended to a differential system which gives the Camassa-Holm equation as a particular case. The subject of conservation laws is briefly discussed for each of the equations. A Backlund transformation is determined using one of the prolongations.

The paper deals with a formally self-adjoint first order linear differential operator acting on m-columns of complex-valued half-densities over an n-manifold without boundary. We study the distribution of eigenvalues in the elliptic setting and the propagator in the hyperbolic setting, deriving two-term asymptotic formulae for both. We then turn our attention to the special case of a two by two operator in dimension four. We show that the geometric concepts of Lorentzian metric, Pauli matrices, spinor field, connection coefficients for spinor fields, electromagnetic covector potential, Dirac equation and Dirac action arise naturally in the process of our analysis.

An algorithm for solving a large class of two- and three-dimensional nonseparable elliptic partial differential equations (PDE's) is developed and tested. It uses a modified D'Yakanov-Gunn iterative procedure in which the relaxation factor is grid-point dependent. It is easy to implement and applicable to a variety of boundary conditions. It is also computationally efficient, as indicated by the results of numerical comparisons with other established methods. Furthermore, the current algorithm has the advantage of possessing two important properties which the traditional iterative methods lack; that is: (1) the convergence rate is relatively insensitive to grid-cell size and aspect ratio, and (2) the convergence rate can be easily estimated by using the coefficient of the PDE being solved.

In this paper we study symmetry reductions of a class of nonlinear fourth order partial differential equations \be u_{tt} = \left(\kappa u + \gamma u^2\right)_{xx} + u u_{xxxx} +\mu u_{xxtt}+\alpha u_x u_{xxx} + \beta u_{xx}^2, \ee where $\alpha$, $\beta$, $\gamma$, $\kappa$ and $\mu$ are constants. This equation may be thought of as a fourth order analogue of a generalization of the Camassa-Holm equation, about which there has been considerable recent interest. Further equation (1) is a ``Boussinesq-type'' equation which arises as a model of vibrations of an anharmonic mass-spring chain and admits both ``compacton'' and conventional solitons. A catalogue of symmetry reductions for equation (1) is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole. In particular we obtain several reductions using the nonclassical method which are no} obtainable through the classical method.