oscillatory properties for semilinear degenerate hyperbolic equations of second order we consider three kind of oscillatory properties of the solutions to semilinear degenerate hyperbolic equations several sufficient conditions for the oscillation or non oscillation are presented in particular they give us the positivity of the solutions for semilinear hyperbolic equations degenerating at initial point in one space dimension moreover we establish a few oscillatory conditions for the solutions of the mixed problem reduced to in one space dimension

critical exponents for semilinear equations of mixed elliptic hyperbolic and degenerate types for semilinear gellerstedt equations with tricomi goursat or dirichlet boundary conditions we prove poho aev type identities and derive nonexistence results that exploit an invariance of the linear part with respect to certain nonhomogeneous dilations a critical exponent phenomenon of power type in the nonlinearity is exhibited in these mixed elliptic hyperbolic or degenerate settings where the power is 1 less than the critical exponent in a relevant sobolev embedding c 2003 wiley periodicals inc

nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

conservation laws for equations of mixed elliptic hyperbolic and degenerate types for partial differential equations of mixed elliptic hyperbolic and degenerate types which are the euler lagrange equations for an associated lagrangian invariance with respect to changes in independent and dependent variables is investigated as are results in the classification of continuous one parameter symmetry groups for the variational and divergence symmetries conservation laws are derived via the method of multipliers the conservation laws resulting from anisotropic dilations are applied to prove uniqueness theorems for linear and nonlinear problems and the invariance under dilations of the linear part is used to derive critical exponent phenomena and to obtain localized energy estimates for supercritical problems

global existence for the one dimensional second order semilinear hyperbolic equations abstract the aim of the paper is to study necessary and sufficient conditions for the existence of the global solution of the one dimensional semilinear equation appearing in the boundary value problems of gas dynamics we investigate the cauchy problem for such equation in the domain where the operator is weakly hyperbolic we obtain the necessary condition for the existence of the self similar solutions for the semilinear gellerstedt type equation the approach used in the paper is based on the fundamental solution of the linear gellerstedt operator and the l p l q estimates

the self similar solutions of the one dimensional semilinear tricomi type equations abstract in this article we investigate the issue of existence of global in time solutions of semilinear tricomi type equations we give conditions that relate the nonlinearity the speed of propagation and the order of singularities of initial data these conditions guarantee existence of global in time solutions in particular we prove existence of solutions invariant under dilation by solving the cauchy problem with initial data which are homogeneous functions

on the long time integration of stochastic gradient systems this article addresses the weak convergence of numerical methods for brownian dynamics typical analyses of numerical methods for stochastic differential equations focus on properties such as the weak order which estimates the asymptotic stepsize h 0 convergence behaviour of the error of finite time averages recently it has been demonstrated by study of fokker planck operators that a non markovian numerical method generates approximations in the long time limit with higher accuracy order second order than would be expected from its weak convergence analysis finite time averages are first order accurate in this article we describe the transition from the transient to the steady state regime of this numerical method by estimating the time dependency of the coefficients in an asymptotic expansion for the weak error demonstrating that the convergence to second order is exponentially rapid in time moreover we provide numerical tests of the theory including comparisons of the efficiencies of the euler maruyama method the popular second order heun method and the non markovian method

decomposition formulas associated with the lauricella multivariable hypergeometric functions by making use of some techniques based upon certain inverse pairs of symbolic operators the authors investigate several decomposition formulas associated with lauricella s hypergeometric functions f b r f c r and f d r in r variables in the three variable case when r 3 some of these operational representations are constructed and applied in order to derive the corresponding decomposition formulas with the help of these inverse pairs of symbolic operators a total of 15 decomposition formulas are found which are expressed as products of hypergeometric functions of the gauss and appell types

spatial boundary problem with the dirichlet neumann condition for a singular elliptic equation abstract in the present work we find an explicit solution of a boundary problem with the dirichlet neumann condition for elliptic equation with singular coefficients in a quarter of ball for this aim the method of green s function has been used since found green s function contains a hypergeometric function of appell we had to deal with decomposition formulas formulas of differentiation and some adjacent relations for this hypergeometric function in order to get explicit solution of the formulated problem

higher order pathwise numerical approximations of spdes with additive noise in this article the pathwise numerical approximation of semilinear parabolic stochastic partial differential equations spdes driven by additive noise is considered a new numerical scheme for the time and space discretization of such spdes is proposed and this scheme is shown to converge for such spdes faster than standard numerical schemes such as the linear implicit euler scheme or the linear implicit crank nicholson scheme the suggested scheme takes advantage of the smoothing effect of the dominant linear operator and of two linear functionals of the noise process of the spde the abstract setting under which the scheme is analyzed includes spdes driven by fractional brownian motions too

weighted strichartz estimates and global existence for semilinear wave equations in this paper we prove a sharp global existence theorem in all dimensions for nonlinear wave equations with power type nonlinearities the proof is based on a weighted strichartz estimate involving powers of the lorentz distance

fundamental solutions of the wave equation in robertson walker spaces abstract in this article we construct the fundamental solutions for the wave equation in the robertson walker spaces arising in the de sitter model of the universe we then use these fundamental solutions to represent solutions of the cauchy problem for the equation with and without a source term

stability analysis of numerical schemes for stochastic differential equations stochastic differential equations sdes represent physical phenomena dominated by stochastic processes as for deterministic ordinary differential equations odes various numerical schemes are proposed for sdes in this paper we study the stability of numerical schemes for scalar sdes with respect to the mean square norm which we call ms stability we will show some figures of the ms stability domain or regions for some numerical schemes and present numerical results which confirm it this notion is an extension of absolute stability in numerical methods for odes

existence of solution for the n dimension second order semilinear hyperbolic equations abstract in the paper we establish the local and global existence of solution for the n dimensional second order semilinear hyperbolic equation with a strongly singular coefficient which appears in the boundary value problems of fluid dynamics based on the analysis about the loss of regularity on the line t 0 for the solution of the corresponding linear equation and the decay at infinity which caused by the singular coefficient we obtain the existence of a small solution for the semilinear equation by use of fixed point theorem

a shift in the strauss exponent for semilinear wave equations with a not effective damping abstract in this note we study the global existence of small data solutions to the cauchy problem for the semilinear wave equation with a not effective scale invariant damping term namely v t t v 2 1 t v t v p v 0 x v 0 x v t 0 x v 1 x where p 1 n 2 we prove blow up in finite time in the subcritical range p 1 p 2 n and existence theorems for p p 2 n n 2 3 in this way we find the critical exponent for small data solutions to this problem our results lead to the conjecture p 2 n p 0 n 2 for n 2 where p 0 n is the strauss exponent for the classical semilinear wave equation with power nonlinearity

price s law on nonstationary space times abstract in this article we study the pointwise decay properties of solutions to the wave equation on a class of nonstationary asymptotically flat backgrounds in three space dimensions under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a t 3 local uniform decay rate price s law price 1972 54 for linear waves as a corollary we also prove price s law for certain small perturbations of the kerr metric this result was previously established by the second author in tataru 65 on stationary backgrounds the present work was motivated by the problem of nonlinear stability of the kerr schwarzschild solutions for the vacuum einstein equations which seems to require a more robust approach to proving linear decay estimates

weak approximation of stochastic partial differential equations the nonlinear case we study the error of the euler scheme applied to a stochastic partial dierential equation we prove that as it is often the case the weak order of convergence is twice the strong order a key ingredient in our proof is malliavin calculus which enables us to get rid of the irregular terms of the error we apply our method to the case a semilinear stochastic heat equation driven by a space time white noise

on the existence of low regularity solutions to semilinear generalized tricomi equations in mixed type domains abstract in 20 21 we have established the existence and singularity structures of low regularity solutions to the semilinear generalized tricomi equations in the degenerate hyperbolic regions and to the higher order degenerate hyperbolic equations respectively in the present paper we shall be concerned with the low regularity solution problem for the semilinear mixed type equation t 2 u t 2 l 1 u f t x u with an initial data u 0 x x h s r n 0 s n 2 where t x r r n n 2 l n f t x u is c 1 smooth in its arguments and has compact support with respect to the variable x under the assumption of the subcritical growth of f t x u on u we will show the existence and regularity of the considered solution in the mixed type domain t 0 t 0 r n for some fixed constant t 0 0

on the analyticity of solutions to semilinear differential equations degenerated on a submanifold abstract we investigate the analyticity of solutions to semilinear elliptic equations degenerated on a submanifold we introduce a new weighted sobolev space which is appropriate for studying such equations the technique for linear equations using cut off functions cannot be applied and we need to use a representation formula which requires a fundamental solution

similarity solutions of f 3 x f xx f y d y d with applications to transonic aerodynamics of den se gases similarity solutions of the nonlinear small disturbance equation phi x 3 phi xx phi tilde y tilde y which describes a two dimensional near sonic potential flow of dense gases are studied the analysis is based on a coordinate transformation from the cartesian physical plane to the hodograph velocity plane at which the governing equation becomes linear the relations between the similarity parameters in these two planes are constructed the first order similarity solution of the velocity potential phi in the physical plane is described analytically by a parametric representation in terms of the hodograph similarity variable the new solutions are applied to the problem of a near sonic small disturbance flow of dense gases in the surroundings of two dimensional slender semi infinite bodies with x n generators where 2 7 le n 1 and x is the distance along the body axis in this problem the similarity term describes a symmetric flow about the body axis in the case when

mean square and asymptotic stability of the stochastic theta method stability analysis of numerical methods for ordinary differential equations odes is motivated by the question for what choices of stepsize does the numerical method reproduce the characteristics of the test equation we study a linear test equation with a multiplicative noise term and consider mean square and asymptotic stability of a stochastic version of the theta method we extend some mean square stability results in saito and mitsui siam j numer anal 33 1996 pp 2254 2267 in particular we show that an extension of the deterministic a stability property holds we also plot mean square stability regions for the case where the test equation has real parameters for asymptotic stability we show that the issue reduces to finding the expected value of a parametrized random variable we combine analytical and numerical techniques to get insights into the stability properties for a variant of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region this allows us to prove a number of results the technique introduced is widely applicable and we use it to show that a fully implicit method suggested by kloeden and platen numerical solution of stochastic differential equations springer verlag 1992 has an asymptotic stability extension of the deterministic a stability property we also use the approach to explain some numerical results reported in milstein platen and schurz siam j numer anal 35 1998 pp 1010 1019

nonlinear liouville theorems for grushin and tricomi operators the aim of this paper is to study necessary conditions for existence of weak solutions of the inequality l x y dx dy u u q x 1 y 2 x rd y rk where l is a quasi homogeneous differential operator q 1 1 2 r and k d 1 as special cases our results can be applied when l is tricomi or grushin type operators

a note on the fundamental solution for the tricomi type equation in the hyperbolic domain abstract barros neto and gelfand duke math j 98 3 1999 465 duke math j 117 2 2003 561 constructed for the tricomi operator y x 2 y 2 on the plane the fundamental solutions with the supports in the regions related to the geometry of the characteristics of the tricomi operator we give for the tricomi type operator t 2 t m x a fundamental solution relative to an arbitrary point of r n 1 with the support in the region t 0 where the operator is hyperbolic our key observation is that the fundamental solution for the tricomi type operator can be written like an integral of the distributions generated by the fundamental solution of the cauchy problem for the wave equation the application of that fundamental solution to the l p l q estimate for the forced tricomi type equation is given as well

fundamental solutions for the tricomi operator in this paper we explicitly calculate fundamental solutions for the tricomi operator relative to an arbitrary point in the plane and show that all such fundamental solutions originate from the hypergeometric function f 1 6 1 6 1 that is obtained when we look for homogeneous solutions to the reduced hyperbolic tricomi equation

analyticity of solutions of semi linear equations with double characteristics we prove the analyticity and gevrey regularity of solutions of elliptic degenerate semi linear differential equations principle part of which is a linear operator with double characteristics considered first by gilioli and treves a new elementary proof for hypoellipticity in the weak sense is given

weak order for the discretization of the stochastic heat equation in this paper we study the approximation of the distribution of x t hilbert valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as dx t ax t dt q 1 2 d w t quad x 0 x in h quad t in 0 t driven by a gaussian space time noise whose covariance operator q is given we assume that a alpha is a finite trace operator for some alpha 0 and that q is bounded from h into d a beta for some beta geq 0 it is not required to be nuclear or to commute with a the discretization is achieved thanks to finite element methods in space parameter h 0 and implicit euler schemes in time parameter delta t t n we define a discrete solution x n h and for suitable functions varphi defined on h we show that e varphi x n h e varphi x t o h 2 gamma delta t gamma noindent where gamma 1 alpha beta let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations

postprocessed integrators for the high order integration of ergodic sdes the concept of effective order is a popular methodology in the deterministic literature for the construction of efficient and accurate integrators for differential equations over long times the idea is to enhance the accuracy of a numerical method by using an appropriate change of variables called the processor we show that this technique can be extended to the stochastic context for the construction of new high order integrators for the sampling of the invariant measure of ergodic systems the approach is illustrated with modifications of the stochastic theta method applied to brownian dynamics where postprocessors achieving order two are introduced numerical experiments including stiff ergodic systems illustrate the efficiency and versatility of the approach

the dirichlet problem for a 3d elliptic equation with two singular coefficients abstract in the present work we investigate the dirichlet problem for a three dimensional 3d elliptic equation with two singular coefficients we find four fundamental solutions of the equation containing hypergeometric functions of appell then using an a b c method the uniqueness for the solution of the dirichlet problem is proved applying a method of green s function we are able to find the solution of the problem in an explicit form moreover decomposition formulas formulas of differentiation and some adjacent relations for appell s hypergeometric functions were used in order to find the explicit solution for the formulated problem

long time accuracy of lie trotter splitting methods for langevin dynamics a new characterization of sufficient conditions for the lie trotter splitting to capture the numerical invariant measure of nonlinear ergodic langevin dynamics up to an arbitrary order is discussed our characterization relies on backward error analysis and needs weaker assumptions than assumed so far in the literature in particular neither high weak order of the splitting scheme nor symplecticity are necessary to achieve high order approximation of the invariant measure of the langevin dynamics numerical experiments confirm our theoretical findings

postprocessing for stochastic parabolic partial differential equations we investigate the strong approximation of stochastic parabolic partial differential equations with additive noise we introduce postprocessing in the context of a standard galerkin approximation although other spatial discretizations are possible in time we follow g j lord and j rougemont ima j numer anal 24 2004 pp 587 604 and use an exponential integrator we prove strong error estimates and discuss the best number of postprocessing terms to take numerically we evaluate the efficiency of the methods and observe rates of convergence some experiments with the implicit euler maruyama method are described

nonlinear waves on 3d hyperbolic space in this article global in time dispersive estimates and strichartz estimates are explored for the wave equation on three dimensional hyperbolic space due to the negative curvature extra dispersion is noted as compared to the euclidean case and a wider range of strichartz estimates is proved using these small data global existence to semilinear wave equations is shown for a range of powers that is broader than that known for euclidean space

bessel integrals and fundamental solutions for a generalized tricomi operator abstract the method of partial fourier transform is used to find explicit formulas for two remarkable fundamental solutions for a generalized tricomi operator these fundamental solutions reflect clearly the mixed type of the tricomi operator in proving these results we establish explicit formulas for fourier transforms of some functions involving bessel functions

long run accuracy of variational integrators in the stochastic context this paper presents a lie trotter splitting for inertial langevin equations geometric langevin algorithm and analyzes its long time statistical properties the splitting is defined as a composition of a variational integrator with an ornstein uhlenbeck flow assuming that the exact solution and the splitting are geometrically ergodic the paper proves the discrete invariant measure of the splitting approximates the invariant measure of inertial langevin equations to within the accuracy of the variational integrator in representing the hamiltonian in particular if the variational integrator admits no energy error then the method samples the invariant measure of inertial langevin equations without error numerical validation is provided using explicit variational integrators with first second and fourth order accuracy

on semilinear tricomi equations with critical exponents or in two space dimensions abstract this paper is a complement of our recent works on the semilinear tricomi equations in 9 and 10 for the semilinear tricomi equation t 2 u t u u p with initial data u 0 t u 0 u 0 u 1 where t 0 x r n n 3 p 1 and u i c 0 r n i 0 1 we have shown in 9 and 10 that there exists a critical exponent p crit n 1 such that the solution u in general blows up in finite time when 1 p p crit n and there is a global small solution for p p crit n in the present paper firstly we prove that the solution of t 2 u t u u p will generally blow up for the critical exponent p p crit n and n 2 secondly we establish the global existence of small data solution to t 2 u t u u p for p p c r i t n and n 2 thus we have given a systematic study on the blowup or global existence of small data solution u to the equation t 2 u t u u p for n 2

multidomain spectral method for the helically reduced wave equation we consider the 2 1 and 3 1 scalar wave equations reduced via a helical killing field respectively referred to as the 2 dimensional and 3 dimensional helically reduced wave equation hrwe the hrwe serves as the fundamental model for the mixed type pde arising in the periodic standing wave psw approximation to binary inspiral we present a method for solving the equation based on domain decomposition and spectral approximation beyond describing such a numerical method for solving strictly linear hrwe we also present results for a nonlinear scalar model of binary inspiral the psw approximation has already been theoretically and numerically studied in the context of the post minkowskian gravitational field with numerical simulations carried out via the eigenspectral method despite its name the eigenspectral technique does feature a finite difference component and is lower order accurate we intend to apply the numerical method described here to the theoretically well developed post minkowski psw formalism with the twin goals of spectral accuracy and the coordinate flexibility afforded by global spectral interpolation

a competition between fujita and strauss type exponents for blow up of semi linear wave equations with scale invariant damping and mass abstract we obtain a blow up result for solutions to a semi linear wave equation with scale invariant dissipation and mass and power non linearity in the case in which the model has a wave like behavior we perform a change of variables that transforms our starting equation in a strictly hyperbolic semi linear wave equation with time dependent speed of propagation applying kato s lemma we prove a blow up result for solutions to the transformed equation under some assumptions on the initial data the limit case that is when the exponent p is exactly equal to the upper bound of the range of admissible values of p yielding blow up needs special considerations in this critical case an explicit integral representation formula for solutions of the corresponding linear cauchy problem in 1d is derived finally carrying out the inverse change of variables we get a non existence result for global in time solutions to the original model

integral transform approach to generalized tricomi equations abstract in this article we introduce the integral transform that can be used to obtain solutions of the generalized tricomi equation via solutions of a simpler equation we employed a particular version of this transform in 13 14 41 42 43 44 45 46 to investigate in a unified way several equations such as the linear and semilinear tricomi equations gellerstedt equation the wave equation in the einstein de sitter spacetime the wave and the klein gordon equations in the de sitter and anti de sitter spacetimes

high order integrator for sampling the invariant distribution of a class of parabolic stochastic pdes with additive space time noise we introduce a time integrator to sample with high order of accuracy the invariant distribution for a class of semilinear spdes driven by an additive space time noise combined with a postprocessor the new method is a modification with negligible overhead of the standard linearized implicit euler maruyama method we first provide an analysis of the integrator when applied for sdes finite dimension where we prove that the method has order 2 for the approximation of the invariant distribution instead of 1 we then perform a stability analysis of the integrator in the semilinear spde context and we prove in a linear case that a higher order of convergence is achieved numerical experiments including the semilinear heat equation driven by space time white noise confirm the theoretical findings and illustrate the efficiency of the approach

weak convergence analysis of the linear implicit euler method for semilinear stochastic partial differential equations with additive noise abstract in this paper we analyze the weak error of a semi discretization in time by the linear implicit euler method for semilinear stochastic partial differential equations spdes with additive noise the main result reveals how the weak order depends on the regularity of noise and that the order of weak convergence is twice that of strong convergence in particular the linear implicit euler method for spdes driven by trace class noise achieves an almost optimal order 1 for arbitrarily small 0

on fundamental solutions for multidimensional helmholtz equation with three singular coefficients abstract the main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with three singular coefficients which could be expressed in terms of a confluent hypergeometric function of four variables in addition the order of the singularity is determined and the properties of the found fundamental solutions that are necessary for solving boundary value problems for degenerate elliptic equations of second order are found

high order numerical approximation of the invariant measure of ergodic sdes we introduce new sufficient conditions for a numerical method to approximate with high order of accuracy the invariant measure of an ergodic system of stochastic differential equations independently of the weak order of accuracy of the method we then present a systematic procedure based on the framework of modified differential equations for the construction of stochastic integrators that capture the invariant measure of a wide class of ergodic sdes brownian and langevin dynamics with an accuracy independent of the weak order of the underlying method numerical experiments confirm our theoretical findings

numerical analysis on ergodic limit of approximations for stochastic nls equation via multi symplectic scheme we consider a finite dimensional approximation of the stochastic nonlinear schrodinger equation driven by multiplicative noise which is derived by applying a symplectic method to the original equation in spatial direction both the unique ergodicity and the charge conservation law for this finite dimensional approximation are obtained on the unit sphere to simulate the ergodic limit over long time for the finite dimensional approximation we discretize it further in the temporal direction to obtain a fully discrete scheme which inherits not only the stochastic multi symplecticity and charge conservation law of the original equation but also the unique ergodicity of the finite dimensional approximation the temporal average of the fully discrete numerical solution is proved to converge to the ergodic limit with order 1 with respect to the time step for a fixed spatial step numerical experiments verify our theoretical results on charge conservation ergodicity and weak convergence

the wave equation on hyperbolic spaces abstract we study the dispersive properties of the wave equation associated with the shifted laplace beltrami operator on real hyperbolic spaces and deduce new strichartz estimates for a large family of admissible pairs as an application we obtain local well posedness results for the nonlinear wave equation

semilinear hypoelliptic differential operators with multiple characteristics in this paper we consider the regularity of solutions of semilinear differential equations principal parts of which consist of linear polynomial operators constructed from real vector fields based on the study of fine properties of the principal linear parts we then obtain the regularity of solutions of the nonlinear equations some results obtained in this article are also new in the frame of linear theory

a generalization of the weighted strichartz estimates for wave equations and an application to self similar solutions weighted strichartz estimates with homogeneous weights with critical exponents are proved for the wave equation without a support restriction on the forcing term the method of proof is based on expansion by spherical harmonics and on the sobolev space over the unit sphere by which the required estimates are reduced to the radial case as an application of the weighted strichartz estimates the existence and uniqueness of self similar solutions to nonlinear wave equations are proved on up to five space dimensions 2006 wiley periodicals inc

global solutions and self similar solutions of semilinear wave equation we prove existence uniqueness and regularity results for the global solutions of the semilinear wave equations in particular we show existence of regular self similar solutions also we build some finite energy asymptotically self similar solutions

computing ergodic limits for langevin equations abstract to evaluate averages with respect to the invariant law for langevin type equations one has to integrate a system over long time intervals especially when dissipation is small this is a challenging problem from the computational point of view since coefficients of the langevin equations are typically not globally lipschitz the difficulties are redoubled in the nonglobal lipschitz case we observe an exploding behavior of some approximate trajectories due to many usual numerical methods in g n milstein m v tretyakov numerical integration of stochastic differential equations with nonglobally lipschitz coefficients siam j numer anal 43 2005 1139 1154 we propose and justify a concept which in principle allows us to apply any numerical method to stochastic differential equations with nonglobally lipschitz coefficients here quasi symplectic integrators from g n milstein m v tretyakov quasi symplectic methods for langevin type equations ima j numer anal 23 2003 593 626 they are the most appropriate methods for solving langevin type equations especially when damping is small and the concept of g n milstein m v tretyakov numerical integration of stochastic differential equations with nonglobally lipschitz coefficients siam j numer anal 43 2005 1139 1154 are applied for calculation of ergodic limits special attention is paid to the case when the invariant measure is known e g when this measure is gibbsian some computer experiments with three model problems van der pol s equation with additive noise one dimensional arrays of oscillators in thermal equilibrium and a physical pendulum with linear friction and additive noise are presented

an l p a priori estimate for the tricomi equation in the upper half space we establish an lp a priori estimate for the tricomi equation our main tool is mihlin s multiplier theorem combined with well known estimates of the newtonian potential 0 introduction the purpose of this paper is to establish an lp a priori estimate for the tricomi equation in the upper half space the tricomi equation arises in transonic gas dynamics and is a typical model equation of changing type it has been extensively investigated from the various viewpoints the tricomi equation can be interpreted as an elliptic equation which degenerates on the boundary which is our viewpoint in this paper we can formulate the dirichlet boundary value problem in the upper half space as follows 02 yacuz f in r 0 2 u 0 x x on mt where r y x y 0 x e rn and x is the laplacian in the x variable an l2 a priori estimate can be obtained very easily following the presentation in 8 we suppose that u e co rn and multiply the equation 0 1 by 92u then we integrate over r r to obtain 0 3 ir dy2x 2jj v dydx rn 1 o 9y2 rn 1 o a y 2irn1l vq dx j 1 f 3 dydx where vx stands for the gradient in x e r n this yields 0 4 2u f 1 yx 4y21 l2 rn dy l2 rn l2 rn m f 1 l2 rn 11vx01il2 rn 1 7 received by the editors december 30 1996 and in revised form february 10 1998 1991 mathematics subject classification primary 35j70 35b45

dispersion for the wave equation inside strictly convex domains i the friedlander model case we consider a model case for a strictly convex domain r d of dimension d 2 with smooth boundary 6 and we describe dispersion for the wave equation with dirichlet boundary conditions more specically we obtain the optimal xed time decay rate for the smoothed out green function a t 1 4 loss occurs with respect to the boundary less case due to repeated occurrences of swallowtail type singularities in the wave front set

numerical methods for strong solutions of stochastic differential equations an overview this paper gives a review of recent progress in the design of numerical methods for computing the trajectories sample paths of solutions to stochastic differential equations we give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important with a particular focus on computational biology applications and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes we present the stochastic taylor series expansion as the fundamental mechanism for constructing effective numerical methods give general results that relate local and global order of convergence and mention the magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem we also present various classes of explicit and implicit methods for strong solutions based on the underlying structure of the problem finally we discuss implementation issues relating to maintaining the brownian path efficient simulation of stochastic integrals and variable step size implementations based on various types of control

towards a systematic linear stability analysis of numerical methods for systems of stochastic differential equations we develop two classes of test equations for the linear stability analysis of numerical methods applied to systems of stochastic ordinary differential equations of ito type sodes motivated by the theory of stochastic stabilization and destabilization these test equations capture certain fundamental effects of stochastic perturbation in systems of sodes while remaining amenable to analysis before and after discretization we then carry out a linear stability analysis of the theta maruyama method applied to these test equations investigating mean square and almost sure asymptotic stability of the test equilibria we discuss the implications of our work for the notion of a stability of the theta maruyama method and use numerical simulation to suggest extensions of our results to test systems with nonnormal drift coefficients

sonic supersonic solutions for the two dimensional steady full euler equations this paper focuses on the structure of classical sonic supersonic solutions near sonic curves for the two dimensional full euler equations in gas dynamics in order to deal with the parabolic degeneracy near the sonic curve a novel set of dependent and independent variables are introduced to transform the euler equations into a new system of governing equations which displays a clear regularity singularity structure with the help of technical characteristic decompositions the existence of a local smooth solution for the new system is first established in a weighted metric space by using the iteration method and then expressed in terms of the original physical variables this is the first construction of a classical sonic supersonic solution near a sonic curve for the full euler equations

on the summation of series involving bessel or struve functions abstract the subject called summation of series can be viewed in two different ways from one point of view it means numerical summation which involves acceleration of convergence and from the other it represents a variety of summation formulas which are considered important but which are often not useful the aim of this paper is to produce a compromise between the two opposite approaches when series of bessel or struve functions are considered the proposed method leads to series of riemann zeta and related functions which converge much faster than the originals in the most significant cases closed form formulas were obtained

spectral bounds for tricomi problems and application to semilinear existence and existence with uniqueness results abstract for the linear tricomi problem it is shown that real eigenvalues corresponding to generalized eigenfunctions must be positive and that the energy integral methods used to prove solvability results can give lower bounds on the spectrum exploiting the linear solvability theory and spectral information standard nonlinear analysis tools are employed to yield results on existence and uniqueness for semilinear problems in particular using the leray schauder principle existence of generalized solutions with sublinear nonlinearities is established for sublinear or asymptotically linear nonlinearities that satisfy a lipschitz condition the contraction mapping principle is employed to give results on existence with uniqueness the lipschitz constant depends on lower bounds for the spectrum of the linear problem for certain superlinear problems maximum principles for the linear problem are used via the method of upper and lower solutions to give results on existence

self similar solutions of semilinear wave equation with variable speed of propagation abstract we investigate the issue of existence of the self similar solutions of the generalized tricomi equation in the half space where the equation is hyperbolic we look for the self similar solutions via the cauchy problem an integral transformation suggested in k yagdjian a note on the fundamental solution for the tricomi type equation in the hyperbolic domain j differential equations 206 2004 227 252 is used to represent solutions of the cauchy problem for the linear tricomi type equation in terms of fundamental solutions of the classical wave equation this representation allows us to prove decay estimates for the linear tricomi type equation with a source term obtained in k yagdjian the self similar solutions of the tricomi type equations z angew math phys in press doi 10 1007 s00033 006 5099 2 estimates for the self similar solutions of the linear tricomi type equation are the key tools to prove existence of the self similar solutions

overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space time noise we consider the numerical approximation of parabolic stochastic partial differential equations driven by additive space time white noise we introduce a new numerical scheme for the time discretization of the finite dimensional galerkin stochastic differential equations which we call the exponential euler scheme and show that it converges in the strong sense faster than the classical numerical schemes such as the linear implicit euler scheme or the crank nicholson scheme for this equation with the general noise in particular we prove that our scheme applied to a semilinear stochastic heat equation converges with an overall computational order 1 3 which exceeds the barrier order 1 6 for numerical schemes using only basic increments of the noise process reported previously by contrast our scheme takes advantage of the smoothening effect of the laplace operator and of a linear functional of the noise and therefore overcomes this order barrier

on closed boundary value problems for equations of mixed elliptic hyperbolic type for partial differential equations of mixed elliptic hyperbolic type we prove results on existence and existence with uniqueness of weak solutions for closed boundary value problems of dirichlet and mixed dirichlet conormal types such problems are of interest for applications to transonicflow and are overdetermined for solutions with classical regularity the method employed consists in variants of the a b c integral method of friedrichs in sobolev spaces with suitable weights particular attention is paid to the problem of attaining results with a minimum of restrictions on the boundary geometry and the form of the type change function in addition interior regularity results are also given in the important special case of the tricomi equation c

interior regularity of the dirichlet problem for the tricomi equation for the tricomi equation with dirichlet boundary conditions we study the relationship between singularites at the boundary and singularities in the interior of a bounded planar region with smooth non characteristic boundary necessary and sufficient conditions for interior smoothness are stated in terms of microlocal regularity at the boundary and are proven via known microlocal propagation of singularities results along the generalized bicharacteristic flow in particular a trapped gliding ray phenomenon at parabolic boundary points is demonstrated under a sharp geometric hypothesis which provides a microlocal explanation for the possibility of having only isolated singularities at the boundary which is a question left open in the work of morawetz

novel similarity solutions of the sonic small disturbance equation with applications to airfoil transonic aerodynamics similarity solutions of the small disturbance equation for a two dimensional near sonic potential flow are studied the analysis uses basic similarity solutions of the problem in the hodograph plane the first order similarity solution of the velocity potential in the physical plane is described analytically by a parametric representation in terms of the hodograph similarity variable and function the second order similarity term is governed by a linear equation and is described analytically as a product of two functions one is the power of the basic hodograph similarity function and the other is a solution of the basic hodograph hypergeometric equation with a different constant the application of the present solutions to the far field approximation of a sonic flow about a thin airfoil results in a relation between frankl s special similarity parameter and the hodograph similarity variable the new solutions are also applied to the problem of a near sonic small disturbance flow in the surroundings of a

transonic flow around optimum critical airfoils optimum critical nonlifting airfoils that have the highest free stream mach number m infty for a given thickness ratio delta have recently been presented by schwendeman kropinski and cole z angew math phys 44 1993 pp 556 571 the asymptotic shape of these airfoils as m infty to 1 and delta to 0 consists of a sonic arc with a x 2 5 nose in the present paper the transonic potential flow around these asymptotic shapes is analyzed asymptotic expansions of the velocity potential function are constructed in terms of delta in an outer region around the airfoil and in an inner region near the nose the outer expansion consists of the transonic small disturbance theory where a leading edge singularity appears the inner expansion accounts for the flow around the nose where a stagnation point exists the matching between the expansions results in a boundary value problem in the inner region for the solution of a uniform sonic flow about a two dimensional x 2 5

the weighted w2 p estimate on the solution of the gellerstedt equation in the upper half space abstract in this note we are concerned with the weighted w 2 p estimate for the gellerstedt equation which is degenerate on the boundary r 2 based on the estimates of the generalized airy functions and mihlin s multiplier theorem we establish a weighted w 2 p a priori estimate