spectral methods for weakly singular volterra integral equations with smooth solutions we propose and analyze a spectral jacobi collocation approximation for the linear volterra integral equations vies of the second kind with weakly singular kernels in this work we consider the case when the underlying solutions of the vies are sufficiently smooth in this case we provide a rigorous error analysis for the proposed method which shows that the numerical errors decay exponentially in the infinity norm and weighted sobolev space norms numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence

mach reflection for the two dimensional burgers equation abstract we study shock reflection for the two 2d burgers equation this model equation is an asymptotic limit of the euler equations and retains many of the features of the full equations a von neumann type analysis shows that the 2d burgers equation has detachment sonic and crocco points in complete analogy with gas dynamics numerical solutions support the detachment sonic criterion for transition from regular to mach reflection there is also strong numerical evidence that the reflected shock in the 2d burgers mach reflection forms a smooth wave near the mach stem as proposed by colella and henderson in their study of the euler equations

diffraction of a weak shock by a wedge we match formal asymptotic expansions with differently scaled variables to obtain a uniform approximation to the similarity solution of the shock wedge diffraction problem

mixed boundary value problems for elliptic and parabolic differential equations of second order on considere les problemes aux valeurs limites mixtes suivants lu f 1 dans mu f 2 sur u f 3 sur avec l operateur elliptique et pu f 1 dans 0 t mu f 2 sur 0 t u f 3 sur 0 t 0 pour des operateurs p paraboliques pour des operateurs obliques generaux m et pour des domaines ou et se rencontrent a des angles arbitraires on montre l existence de solutions classiques quand les angles sont non nuls et f 3 est continue de holder

stability of a mach configuration we prove the stability of a mach configuration which occurs in shock reflection off an obstacle or shock interaction in compressible flow the compressible flow is described by a full steady euler system of gas dynamics the unperturbed mach configuration is composed of three straight shock lines and a slip line carrying contact discontinuity among four regions divided by these four lines in the neighborhood of the intersection two are supersonic regions and other two are subsonic regions we prove that if the constant states in the supersonic regions are slightly perturbed then the structure of the whole configuration holds while the other two shock fronts and the slip line as well as the flow field in the subsonic regions are also slightly perturbed such a conclusion asserts the existence and stability of the general mach configuration in shock theory in order to prove the result we reduce the problem to a free boundary value problem where two unknown shock fronts are free boundaries while the slip line is transformed to a fixed line by a lagrange transformation in the region where the solution is to be determined we have to deal with an elliptic hyperbolic composed system by decoupling this system and combining the technique for both hyperbolic equations and elliptic equations we establish the required estimates which are crucial in the proof of the existence of a solution to the free boundary value problem 2005 wiley periodicals inc

a linear approximation for the regular reflection of a weak shock at a wedge abstract the problem of shock reflection by a wedge in the flow dominated by the unsteady potential flow equation is an important problem in weak regular reflection the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream the reflected shock is transonic its position is a free boundary for the unsteady potential equation which is degenerate at the sonic line in self similar coordinates applying the special partial hodograph transformation used in zhouping xin huicheng yin transonic shock in a nozzle i 2 d case comm pure appl math lvii 2004 1 51 zhouping xin huicheng yin transonic shock in a nozzle ii 3 d case ims preprint 2003 we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain when the angle between incident shock and wedge is small we can see the weak regular reflection as the disturbance of normal reflection as in chen shuxing linear approximation of shock reflection at a wedge with large angle comm partial differential equations 21 78 1996 1103 1118 by linearizing the resulted nonlinear equation and boundary conditions with the above viewpoint in chen shuxing linear approximation of shock reflection at a wedge with large angle comm partial differential equations 21 78 1996 1103 1118 we obtain a linear degenerate elliptic equation with mixed boundary conditions in a curved quadrilateral domain by means of elliptic regularization techniques a delicate a priori estimate and compact arguments we show that the solution of the linearized problem is smooth in the interior and lipschitz continuous up to the degenerate boundary

existence and regularity of solution of mixed boundary value problem of a busemann equation with nonlinear absorb term abstract the busemann equation is a classical equation coming from fluid dynamics the well posed problem and regularity of solution of busemann equation with nonlinear term are interesting and important the busemann equation is elliptic in y 0 and is degenerate at the line y 0 in r 2 with a special nonlinear absorb term we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain by means of elliptic regularization technique a delicate prior estimate and compact argument we show that the solution of mixed boundary value problem of busemann equation is smooth in the interior and lipschitz continuous up to the degenerate boundary on some conditions the result is better than the result of classical boundary degenerate elliptic equation

decision making generalized by a cumulative probability weighting function abstract typical examples of intertemporal decision making involve situations in which individuals must choose between a smaller reward but more immediate and a larger one delivered later analogously probabilistic decision making involves choices between options whose consequences differ in relation to their probability of receiving in economics the expected utility theory eut and the discounted utility theory dut are traditionally accepted normative models for describing respectively probabilistic and intertemporal decision making a large number of experiments confirmed that the linearity assumed by the eut does not explain some observed behaviors as nonlinear preference risk seeking and loss aversion that observation led to the development of new theoretical models called non expected utility theories neut which include a nonlinear transformation of the probability scale an essential feature of the so called preference function of these theories is that the probabilities are transformed by decision weights by means of a cumulative probability weighting function w p we obtain in this article a generalized function for the probabilistic discount process this function has as particular cases mathematical forms already consecrated in the literature including discount models that consider effects of psychophysical perception we also propose a new generalized function for the functional form of w the limiting cases of this function encompass some parametric forms already proposed in the literature far beyond a mere generalization our function allows the interpretation of probabilistic decision making theories based on the assumption that individuals behave similarly in the face of probabilities and delays and is supported by phenomenological models

existence and regularity of the solution of a mixed boundary value problem for the keldysh equation with a nonlinear absorption term abstract the keldysh equation is a more general form of the classic tricomi equation from fluid dynamics its well posedness and the regularity of its solution are interesting and important the keldysh equation is elliptic in y 0 and is degenerate at the line y 0 in r 2 adding a special nonlinear absorption term we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain similar to the potential fluid shock reflection problem by means of an elliptic regularization technique a delicate a priori estimate and compact argument we show that the solution of a mixed boundary value problem of the keldysh equation is smooth in the interior and lipschitz continuous up to the degenerate boundary under some conditions we believe that this kind of regularity result for the solution will be rather useful

psychophysics of time perception and intertemporal choice models intertemporal choice and psychophysics of time perception have been attracting attention in econophysics and neuroeconomics several models have been proposed for intertemporal choice exponential discounting general hyperbolic discounting exponential discounting with logarithmic time perception of the weber fechner law a q exponential discount model based on tsallis s statistics simple hyperbolic discounting and stevens power law exponential discounting exponential discounting with stevens power time perception in order to examine the fitness of the models for behavioral data we estimated the parameters and aicc akaike information criterion with small sample correction of the intertemporal choice models by assessing the points of subjective equality indifference points at seven delays our results have shown that the orders of the goodness of fit for both group and individual data were weber fechner discounting general hyperbola stevens power law discounting simple hyperbolic discounting exponential discounting indicating that human time perception in intertemporal choice may follow the weber fechner law indications of the results for neuropsychopharmacological treatments of addiction and biophysical processing underlying temporal discounting and time perception are discussed

global uniqueness of transonic shocks in two dimensional steady compressible euler flows we prove that for the two dimensional steady complete compressible euler system with given uniform upcoming supersonic flows the following three fundamental flow patterns special solutions in gas dynamics involving transonic shocks are all unique in the class of piecewise c1 smooth functions under appropriate conditions on the downstream subsonic flows i the normal transonic shocks in a straight duct with finite or infinite length after fixing a point the shock front passing through ii the oblique transonic shocks attached to an infinite wedge iii a flat mach configuration containing one supersonic shock two transonic shocks and a contact discontinuity after fixing a point where the four discontinuities intersect these special solutions are constructed traditionally under the assumption that they are piecewise constant and they have played important roles in the studies of mathematical gas dynamics our results show that the assumption of a piecewise constant can be replaced by some weaker assumptions on the downstream subsonic flows which are sufficient to uniquely determine these special solutions mathematically these are uniqueness results on solutions of free boundary problems of a quasi linear system of elliptic hyperbolic composite mixed type in bounded or unbounded planar domains without any assumptions on smallness the proof relies on an elliptic system of pressure p and the tangent of the flow angle w v u obtained by decomposition of the euler system in lagrangian coordinates and a newly developed method for the l estimate that is independent of the free boundaries by combining the maximum principles of elliptic equations and careful analysis of the shock polar applied on the maybe curved shock fronts

probability and time trade off probability and time are integral dimensions of virtually any decision to treat them together we consider the prospect of receiving outcome x with a probability p at time t we define risk and time distance and show that if these two distances are traded off linearly then preferences are characterized by three functions a value function a probability discount rate function and a psychological distance function the concavity of the psychological distance function explains the common ratio and common difference effects a decreasing probability discount rate accounts for the magnitude effect the discount rate and the risk premium depend on the shape of these three functions this paper was accepted by peter wakker decision analysis

global solutions of shock reflection by large angle wedges for potential flow when a plane shock hits a wedge head on it experiences a reflectiondiffraction process and then a self similar reflected shock moves outward as the original shock moves forward in time experimental computational and asymptotic analysis has shown that various patterns of shock reflection may occur including regular and mach reflection however most of the fundamental issues for shock reflection have not been understood including the global structure stability and transition of the different patterns of shock reflection therefore it is essential to establish the global existence and structural stability of solutions of shock reflection in order to understand fully the phenomena of shock reflection on the other hand there has been no rigorous mathematical result on the global existence and structural stability of shock reflection including the case of potential flow which is widely used in aerodynamics such problems involve several challenging difficulties in the analysis of nonlinear partial differential equations such as mixed equations of elliptic hyperbolic type free boundary problems and corner singularity where an elliptic degenerate curve meets a free boundary in this paper we develop a rigorous mathematical approach to overcome these difficulties i nvolved and establish a global theory of existence and stability for shock reflection by large angle wedges for potential flow the techniques and ideas developed here will be useful for other nonlinear problems involving similar difficulties

existence and uniqueness analysis of a detached shock problem for the potential flow abstract we study a problem for two dimensional steady potential and isentropic euler equations in a bounded domain where an artificial detached shock interacts with a wedge using the stream function we obtain a free boundary problem for the subsonic state and the detached artificial shock curve and we prove that such configuration admits a unique solution in certain weighted holder spaces the proof is based on various holder and schauder estimates for second order elliptic equations and fixed point theorems moreover we pose an energy principle and remark that the physical attached shock is the minimizer of the energy functional

the amount based deformation of the q exponential discount function a joint analysis of delay and magnitude effects abstract the magnitude effect is one of the main anomalies shown by people in the context of intertemporal choice thus the main objective of this paper is to provide a general procedure to obtain a general discount function able to explain this paradox by this methodology we will to mathematically deform time in a discount function more specifically to multiply the variable time by a certain power of the reward amount however in most cases the existing literature presents some ad hoc discounting models to describe a unique anomaly in order to solve this limitation we will apply the former methodology to the q exponential discount function introduced by cajueiro 2006 in this way we will obtain a mathematical setting able to jointly explain the delay and the magnitude effects depending on the values of parameters q and involved in the model moreover another advantage of this amount based q exponential function is its convergence to the time deformed exponential hyperbolic and linear discounting depending on the value of q in the joint interval 1 1

potential theory for regular and mach reflection of a shock at a wedge if a plane shock hits a wedge a self similar pattern of reflected shocks travels outward as the shock moves forward in time the nature of the pattern is explored for weak incident shocks strength b and small wedge angles 2 w through potential theory a number of different scalings some study of mixed equations and matching asymptotics for the different scalings the self similar equations are of mixed type a linearization gives a linear mixed flow valid away from a sonic curve near the sonic curve a shock solution is constructed in another scaling except near the zone of interaction between the incident shock and the wall where a special scaling is used the parameter c1 2w 1 b ranges from 0 to here is the polytropic constant and c1 is the sound speed behind the incident shock for 2 regular reflection weak or strong can occur and the whole pattern is reconstructed to lowest order in shock strength for 1 2 mach reflection occurs and the flow behind the reflection is subsonic and can be constructed in principle with an open elliptic problem and matched the case 0 can be solved for 1 2 2 or even larger the flow behind a mach reflection may be transonic and further investigation must be made to determine what happens the basic pattern of reflection is an almost semi circular shock issuing for regular reflection from the reflection point on the wedge and for mach reflection matched with a local interaction flow assuming their nature choosing the least entropy generation the weak regular reflection will occur for sufficiently large von neumann paradox an accumulation point of vorticity occurs on the wedge above the leading point 1994 john wiley sons inc

high performance parallel numerical methods for volterra equations with weakly singular kernels non stationary discrete time waveform relaxation methods for abel systems of volterra integral equations using fractional linear multistep formulae are introduced fully parallel discrete waveform relaxation methods having an optimal convergence rate are constructed a significant expression of the error is proved which allows us to estimate the number of iterations needed to satisfy a prescribed tolerance and allows us to identify the problems where the optimal methods offer the best performance the numerical experiments confirm the theoretical expectations

fast numerical solution of weakly singular volterra integral equations abstract we give a fortran program for the numerical solution of abel volterra convolution equations of the second and first kind we use a fractional linear multistep method bdf4 1 2 and use fast fourier transform techniques to exploit the convolution structure a detailed description of the computation of the quadrature weights using fft is given numerical examples are included

the transonic shock in a nozzle 2 d and 3 d complete euler systems abstract in this paper we study a transonic shock problem for the euler flows through a class of 2 d or 3 d nozzles the nozzle is assumed to be symmetric in the diverging or converging part if the supersonic incoming flow is symmetric near the divergent or convergent part of the nozzle then as indicated in section 147 of r courant k o friedrichs supersonic flow and shock waves interscience publ new york 1948 there exist two constant pressures p 1 and p 2 with p 1 p 2 such that for given constant exit pressure p e p 1 p 2 a symmetric transonic shock exists uniquely in the nozzle and the position and the strength of the shock are completely determined by p e moreover it is shown in this paper that such a transonic shock solution is unique under the restriction that the shock goes through the fixed point at the wall in the multidimensional setting furthermore we establish the global existence stability and the long time asymptotic behavior of an unsteady symmetric transonic shock under the exit pressure p e when the initial unsteady shock lies in the symmetric diverging part of the 2 d or 3 d nozzle on the other hand it is shown that an unsteady symmetric transonic shock is structurally unstable in a global in time sense if it lies in the symmetric converging part of the nozzle

stability of transonic shocks in steady supersonic flow past multidimensional wedges abstract we are concerned with the stability of multidimensional m d transonic shocks in steady supersonic flow past multidimensional wedges one of our motivations is that the global stability issue for the m d case is much more sensitive than that for the 2 d case which requires more careful rigorous mathematical analysis in this paper we develop a nonlinear approach and employ it to establish the stability of weak shock solutions containing a transonic shock front for potential flow with respect to the m d perturbation of the wedge boundary in appropriate function spaces to achieve this we first formulate the stability problem as a free boundary problem for nonlinear elliptic equations then we introduce the partial hodograph transformation to reduce the free boundary problem into a fixed boundary value problem near a background solution with fully nonlinear boundary conditions for second order nonlinear elliptic equations in an unbounded domain to solve this reduced problem we linearize the nonlinear problem on the background shock solution and then after solving this linearized elliptic problem develop a nonlinear iteration scheme that is proved to be contractive

stability of transonic shocks in supersonic flow past a wedge abstract in this paper we study the stability of transonic shocks in steady supersonic flow past a wedge we take the potential flow equation as the mathematical model to describe the compressible flow it is known that in generic case such a problem admits two possible location of shock connecting the flow ahead it and behind it they can be distinguished as supersonic supersonic shock and supersonic subsonic shock or transonic shock both these possible shocks satisfy the rankine hugoniot conditions and entropy condition in this paper we prove that the transonic shock is also stable under perturbation of the coming flow provided the pressure at infinity is well controlled

regularity of solutions to regular shock reflection for potential flow the shock reflection problem is one of the most important problems in mathematical fluid dynamics since this problem not only arises in many important physical situations but also is fundamental for the mathematical theory of multidimensional conservation laws that is still largely incomplete however most of the fundamental issues for shock reflection have not been understood including the regularity and transition of different patterns of shock reflection configurations therefore it is important to establish the regularity of solutions to shock reflection in order to understand fully the phenomena of shock reflection on the other hand for a regular reflection configuration the potential flow governs the exact behavior of the solution in c 1 1 across the pseudo sonic circle even starting from the full euler flow that is both of the nonlinear systems are actually the same in a physically significant region near the pseudo sonic circle thus it becomes essential to understand the optimal regularity of solutions for the potential flow across the pseudo sonic circle the transonic boundary from the elliptic to hyperbolic region and at the point where the pseudo sonic circle the degenerate elliptic curve meets the reflected shock a free boundary connecting the elliptic to hyperbolic region in this paper we study the regularity of solutions to regular shock reflection for potential flow in particular we prove that the c 1 1 regularity is optimal for the solution across the pseudo sonic circle and at the point where the pseudo sonic circle meets the reflected shock we also obtain the c 2 regularity of the solution up to the pseudo sonic circle in the pseudo subsonic region the problem involves two types of transonic flow one is a continuous transition through the pseudo sonic circle from the pseudo supersonic region to the pseudo subsonic region the other a jump transition through the transonic shock as a free boundary from another pseudo supersonic region to the pseudo subsonic region the techniques and ideas developed in this paper will be useful to other regularity problems for nonlinear degenerate equations involving similar difficulties

the existence and stability of conic shock waves we study the existence of the nonsymmetrical conic shock wave produced by a supersonic flow past a distorted conic projectile for the weak conic shock wave we establish the existence and its linear stability using the mathematical model of an isentropic irrotational flow

a parallel algorithm for large systems of volterra integral equations of abel type a significative number of recent applications require numerical solution of large systems of abel volterra integral equations here we propose a parallel algorithm to numerically solve a class of these systems designed for a distributed memory mimd architecture in order to achieve a good efficiency we employ a fully parallel and fast convergent waveform relaxation wr method and evaluate the lag term by using fft techniques to accelerate the convergence of the wr method and to best exploit the parallel architecture we develop special strategies the performances of the resulting code nswr4 are illustrated on some examples

discrete time waveform relaxation volterra runge kutta methods convergence analysis the discrete time relaxation methods based on volterra runge kutta methods for solving large system of second kind volterra integral equations are proposed convergence of the discrete time iteration process with particular attention to parallel methods is investigated

the uniqueness of transonic shocks in supersonic flow past a 2 d wedge abstract we have proved the uniqueness of transonic shocks in steady supersonic flows past a slightly perturbed two dimensional infinite wedge under appropriate conditions on the downstream subsonic flows we formulate it to a mathematical problem of the uniqueness of solutions of nonlinear partial differential equations of hyperbolic elliptic mixed type with a free boundary by working on several elliptic equations of physical quantities separately we obtain a priori estimates of them and then prove the uniqueness without assumptions on high regularity moreover uniform estimates on the ellipticity and the positive lower bound of the speed are achieved under a geometrical condition on the wedge the mathematical ideas and techniques developed here will also be useful for other related problems involving similar analytical difficulties

self similar solutions for weak shock reflection we present numerical solutions of a two dimensional riemann problem for the unsteady transonic small disturbance equations that provides an asymptotic description of the mach reflection of weak shock waves we develop a new numerical scheme to solve the equations in self similar coordinates and use local grid refinement to resolve the solution in the reflection region the solutions contain a remarkably complex structure there is a sequence of triple points and tiny supersonic patches immediately behind the leading triple point that is formed by the reflection of weak shocks and expansion waves between the sonic line and the mach shock an expansion fan originates at each triple point thus resolving the von neumann paradox of weak shock reflection these numerical solutions raise the question of whether there is an infinite sequence of triple points in an inviscid weak shock mach reflection

transonic shock in a nozzle i 2d case in this paper we establish the existence and uniqueness of a transonic shock for the steady flow through a general two dimensional nozzle with variable sections the flow is governed by the inviscid potential equation and is supersonic upstream has no flow boundary conditions on the nozzle walls and an appropriate boundary condition at the exit of the exhaust section the transonic shock is a free boundary dividing two regions of c 1 1 0 flow in the nozzle the potential equation is hyperbolic upstream where the flow is supersonic and elliptic in the downstream subsonic region in particular our results show that there exists a solution to the corresponding free boundary problem such that the equation is always subsonic in the downstream region of the nozzle when the pressure in the exit of the exhaustion section is appropriately larger than that in the entry this problem is motivated by the conjecture of courant and friedrichs on the transonic phenomena in a nozzle 10 furthermore the stability of the transonic shock is also proven when the upstream supersonic flow is a small steady perturbation for the uniform supersonic flow and the corresponding pressure at the exit has a small perturbation the main ingredients of our analysis are a generalized hodograph transformation and multiplier methods for elliptic equation with mixed boundary conditions and comer singularities

free boundary and mixed boundary value problems for quasilinear elliptic equations tangential oblique derivative and degenerate dirichlet boundary problems abstract we present holder estimates and holder gradient estimates for a class of free boundary problems with tangential oblique derivative boundary conditions provided the oblique vector does not vanish at any point on the boundary we also establish the existence result for a general class of quasilinear degenerate problems of this type including nonlinear wave systems and the unsteady transonic small disturbance equation

a proof of existence of perturbed steady transonic shocks via a free boundary problem we prove the existence of a solution of a free boundary problem for the transonic small disturbance equation the free boundary is the position of a transonic shock dividing two regions of smooth flow assuming inviscid irrotational flow as modeled by the transonic small disturbance equation the equation is hyperbolic upstream where the flow is supersonic and elliptic in the downstream subsonic region to study the stability of a uniform planar transonic shock we consider perturbation by a steady c1 upstream disturbance if the upstream disturbance is small in a c1 sense then there is a steady solution in which the downstream flow and the transonic shock are holder continuous perturbations of the uniform configuration this result provides a new use of inviscid perturbation techniques to demonstrate in two dimensions the stability of transonic shock waves of the type that appear for example over the wing of an airplane along an airfoil or as bow shocks in a flow with a supersonic free stream velocity 2000 john wiley sons inc

stability of the e h type regular shock refraction abstract in this paper the refraction of shocks on the interface for 2 d steady compressible flow is discussed particularly the e h type regular refraction is defined and its global stability of the wave structure is proved the 2 d steady potential flow equation is employed to describe the motion of the fluid the stability of the e h type regular refraction can be reduced to a free boundary problem of a nonlinear mixed type equation in an unbounded domain the corresponding linearized problem is a generalized tricomi problem of the lavrentiev bitsadze mixed type equation and it can be reduced to a nonlocal boundary value problem of an elliptic system the later is finally solved by establishing the bijection of a corresponding nonlocal operator in a weighted holder space the result obtained in this paper develops the theory of nonlinear mixed type equations and gives an application to a significant physical problem

a free boundary problem for a quasi linear degenerate elliptic equation regular reflection of weak shocks we prove the existence of a solution to the weak regular reflection problem for the unsteady transonic small disturbance utsd model for shock reflection by a wedge in weak regular reflection the state immediately behind the reflected shock is supersonic and constant the flow becomes subsonic further downstream the equation in self similar coordinates is degenerate at the sonic line the reflected shock becomes transonic and begins to curve there its position is the solution to a free boundary problem for the degenerate equation using the rankine hugoniot conditions along the reflected shock we derive an evolution equation for the transonic shock and an oblique derivative boundary condition at the unknown shock position by regularizing the degenerate problem we construct uniform bounds we apply local compactness arguments to extract a limit that solves the problem the solution is smooth in the interior and continuous up to the degenerate boundary this work completes a stage in our program to construct self similar solutions of two dimensional riemann problems in a series of papers we developed techniques for solving the degenerate elliptic equations that arise in self similar reductions of hyperbolic conservation laws in other papers especially in joint work with gary lieberman we developed techniques for solving free boundary problems of the type that arise from rankine hugoniot relations for the first time in this paper we combine these approaches and show that they are compatible although our construction is limited to a finite part of the unbounded subsonic region it suggests that this approach has the potential to solve a variety of problems in weak shock reflection including mach and von neumann reflection in the utsd equation and the analogous problems for the unsteady full potential equation 2002 john wiley sons inc

multistep collocation methods for volterra integro differential equations multistep collocation methods for volterra integro differential equations are derived and analyzed they increase the order of convergence of classical one step collocation methods at the same computational cost the numerical stability analysis is carried out and classes of a 0 stable methods are provided numerical experiments confirm theoretical expectations and make comparisons with the one step collocation methods

existence and stability of perturbed transonic shocks for compressible steady potential flows abstract we establish an existence theorem for perturbed transonic steady potential flows where the boundary of the subsonic region consists of a transonic shock and thus the governing equation becomes a free boundary problem we establish gradient estimates and use the fixed point theorem of caristi s which does not require a compactness of the map to establish the existence result we provide new techniques to handle the gradients and show that the flow is strictly elliptic inside of the subsonic region

on transonic shocks in a conic divergent nozzle with axi symmetric exit pressures abstract in this paper we establish the existence and stability of a 3 d transonic shock solution to the full steady compressible euler system in a class of de laval nozzles with a conic divergent part when a given variable axi symmetric exit pressure lies in a suitable scope thus for this class of nozzles we have solved such a transonic shock problem in the axi symmetric case described by courant and friedrichs 1948 in section 147 of 8 given the appropriately large exit pressure p e x if the upstream flow is still supersonic behind the throat of the nozzle then at a certain place in the diverging part of the nozzle a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes p e x

multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type we are concerned with the existence and stability of multidimensional transonic shocks for the euler equations for steady potential compressible fluids the euler equations consisting of the conservation law of mass and the bernoulli law for the velocity can be written as the following second order nonlinear equation of mixed elliptic hyperbolic type for the velocity potential r r 1 1 div d d 0 where the density function q is

a parallel fifth order algorithm for the numerical solution of volterra equations an iterative algorithm based upon the use of a nonpurely polynomial extension of the standard newton cotes quadrature rules is constructed to yield the numerical solution of volterra type integral equations with nonsingular kernel with an accuracy of fifth order in the distance between grid points this algorithm which is easily parallelizable for a small number of processors is illustrated and analysed by means of three test equations

constraints on possible singularities for the unsteady transonic small disturbance utsd equations we discuss the singular behavior of solutions to two dimensional general secondorder uniformly elliptic equations in divergence form with bounded measurable coefficients and discontinuous dirichlet data along a portion of a lipschitz boundary we show that the conjugate to the solution develops a singularity that is at least logarithmic along the boundary at the points of discontinuity in the boundary data a problem like this arises in the study of self similar solutions to the hyperbolic conservation laws in two space dimensions given by the unsteady transonic small disturbance utsd flow equations these solutions model the reflection of a weak shock wave upon a thin wedge in the regime where the von neumann paradox applies the present result provides a step in the direction of understanding the nature of the solutions to the utsd equations near a triple point it shows that the flow behind the triple point cannot be strictly subsonic under some mild assumptions on the solutions c 1999 john wiley sons inc

global supersonic conic shock wave for the steady supersonic flow past a cone polytropic gas abstract in this paper we establish the global existence and stability of a steady conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body as long as the vertex angle is less than a critical value the flow is assumed to be polytropic isentropic and described by a steady potential equation based on the delicate asymptotic expansion of the background solution one can verify that the boundary conditions on the shock and the conic surface satisfy the dissipative property from this property by use of the reflected characteristics method and the special form of the shock equation we show that the conic shock attached at the vertex of the cone exists globally in the whole space when the speed of the supersonic coming flow is appropriately large on the other hand we remove the smallness restriction on the sharp vertex angle in order to establish the global existence of a shock or a global weak solution moreover our proof approach is different from that in shuxing chen zhouping xin huicheng yin global shock wave for the supersonic flow past a perturbed cone comm math phys 228 2002 47 84 and zhouping xin huicheng yin global multidimensional shock wave for the steady supersonic flow past a three dimensional curved cone anal appl 4 2 2006 101 132

steady transonic shocks and free boundary problems in infinite cylinders for the euler equations we establish the existence and stability of multidimensional transonic shocks hyperbolic elliptic shocks for the euler equations for steady compressible potential fluids in infinite cylinders the euler equations consisting of the conservation law of mass and the bernoulli law for velocity can be written as a second order nonlinear equation of mixed elliptic hyperbolic type for the velocity potential the transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem the free boundary is the location of the multidimensional transonic shock which divides two regions of c1 flow in the infinite cylinder and the equation is hyperbolic in the upstream region where the c1 perturbed flow is supersonic we develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase and the free boundary is c1 provided that the hyperbolic phase is close in c1 to a uniform flow we further prove that if the steady perturbation of the hyperbolic phase is c2 the free boundary is c2 and stable under the steady perturbation 2003 wiley periodicals inc

subsonic solutions for compressible transonic potential flows abstract we establish an existence theorem for transonic isentropic potential flows where the subsonic region is bounded by the sonic line and thus the governing equation may become degenerate on the boundary partly or entirely it has been conjectured by experiments and numerical studies that the self similar multidimensional flow changes its type namely hyperbolic far from the origin supersonic region and elliptic near the origin subsonic region furthermore the potential equation has a different nonlinearity compared to other transonic problems such as the unsteady transonic small disturbance equation the nonlinear wave equation and the pressure gradient equation namely the coefficients of the potential equation depend on the gradients while others are independent of the gradients we provide techniques to handle the gradients establish interior and boundary gradient estimates for the potential flow in a convex region and answer the conjecture that is the flow is strictly elliptic and the region is subsonic

mach configuration in pseudo stationary compressible flow when a shock front in unsteady compressible flow is moving along an inclined ramp the incident shock will be reflected by the ramp as it moves forward in the process of reflection various reflected wave pattern may occur depending on the slope of the ramp and the strength of the incident shock 14 23 this problem is one of the most important multidimensional prototype problems in gas dynamics generally if the angle of inclination is larger than a critical value then the reflected shock starting from the intersection of the incident shock with the ramp forms a smooth bubble which expands when time goes on such a reflection is called regular reflection on the other hand if the angle of inclination is small then the intersection of the incident shock and the reflected shock will not touch the ramp and an additional shock front called mach stem appears which connects the ramp with the intersection of the incident shock and the reflected shock in this case three shock fronts meet at one point and form a triple shock configuration an important fact is that the structure solely containing three shock fronts separating the neighborhood of the triple intersection into three zones with different continuous states does not exist 14 27 many physical experiments indicate that there is a slip line issuing from the triple intersection such a local wave pattern is called mach configuration or mach structure correspondingly the whole process of reflection is called mach reflection because of the complexity of the whole picture of mach reflection it is natural and necessary to start work on getting a clear understanding of the local mach configuration mach reflection also occurs in various other cases for compressible flow for instance in the study on oblique shock reflection in stationary compressible flow von neumann indicated that see 14 25 if the incident shock hit a wall with an angle larger than a critical value then the reflection must be mach reflection many physical experiments approximate analysis and numerical computations verified the appearance of mach configuration and a tremendous amount of literature is devoted to study such problems see 3 4 11 19 20 24 and the references therein however for the rigorous proof on the stability of mach configuration we can only mention a related result on stationary flow in 13

transonic shocks for the full compressible euler system in a general two dimensional de laval nozzle in this paper we study the transonic shock problem for the full compressible euler system in a general two dimensional de laval nozzle as proposed in courant and friedrichs supersonic flow and shock waves interscience new york 1948 given the appropriately large exit pressure pe x if the upstream flow is still supersonic behind the throat of the nozzle then at a certain place in the diverging part of the nozzle a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes pe x we solve this problem completely for a general class of de laval nozzles whose divergent parts are small and arbitrary perturbations of divergent angular domains for the full steady compressible euler system the problem can be reduced to solve a nonlinear free boundary value problem for a mixed hyperbolic elliptic system one of the key ingredients in the analysis is to solve a nonlinear free boundary value problem in a weighted holder space with low regularities for a second order quasilinear elliptic equation with a free parameter the position of the shock curve at one wall of the nozzle and non local terms involving the trace on the shock of the first order derivatives of the unknown function

a generalization of the q exponential discounting function abstract the aim of this paper is to generalize the q exponential discounting function introduced by cajueiro 2006 1 using the hyperbolic function as a base the presented generalization has two aspects first we consider any discounting function f t and not just hyperbolic discounting second the value of the parameter q is extended to the joint interval 1 1 in this way we have found a family of discounting functions whose elements are subadditive or superadditive according to the value of q

a note on the relevance of the q exponential function in the context of intertemporal choices abstract this paper shows that the q exponential function well known in the deformed algebra inspired in the tsallis s nonextensive thermodynamics may be used to model discount functions in intertemporal choices which present the phenomenon known as increasing patience moreover we show that this insight may also be used to provide a measure of the degree of dynamic inconsistency presented in such decisions

on the instability problem of a 3 d transonic oblique shock wave abstract in this paper we are concerned with the instability problem of a 3 d transonic oblique shock wave for the steady supersonic flow past an infinitely long sharp wedge the flow is assumed to be isentropic and irrotational it was indicated on p 317 of 7 that if a steady supersonic flow comes from minus infinity and hits a sharp symmetric wedge then it follows from the rankine hugoniot conditions and the physical entropy condition that there possibly appears a weak shock or a strong shock attached at the edge of the sharp wedge which corresponds to a supersonic shock or a transonic shock respectively the question arises which of the two actually occurs it has frequently been stated that the strong one is unstable and that therefore only the weak one could occur however a convincing proof of this instability has apparently never been given the aim of this paper is to understand such a longstanding open question we will show that the attached 3 d transonic oblique shock problem is overdetermined with respect to the periodic perturbation which implies that the 3 d transonic shock is unstable in general

stability of transonic shock fronts in two dimensional euler systems we study the stability of stationary transonic shock fronts under two dimensional perturbation in gas dynamics the motion of the gas is described by the full euler system the system is hyperbolic ahead of the shock front and is a hyperbolic elliptic composed system behind the shock front the stability of the shock front and the downstream flow under two dimensional perturbation of the upstream flow can be reduced to a free boundary value problem of the hyperbolic elliptic composed system we develop a method to deal with boundary value problems for such systems the crucial point is to decompose the system to a canonical form in which the hyperbolic part and the elliptic part are only weakly coupled in their coefficients by several sophisticated iterative processes we establish the existence and uniqueness of the solution to the described free boundary value problem our result indicates the stability of the transonic shock front and the flow field behind the shock

a linear approximation for the regular reflection of a weak shock at a wedge satisfying sonic condition abstract the problem of shock reflection by a wedge which the flow is dominated by the unsteady potential flow equation is a important problem in weak regular reflection the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream the reflected shock is transonic its position is a free boundary for the unsteady potential equation which is degenerate at the sonic line in self similar coordinates applying the special partial hodograph transformation used in zhouping xin huicheng yin transonic shock in a nozzle i 2 d case comm pure appl math 57 2004 1 51 zhouping xin huicheng yin transonic shock in a nozzle ii 3 d case ims preprint 2003 we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain when the angle which between incident shock and wedge is small we can see that weak regular reflection as the disturbance of normal reflection as in shuxing chen linear approximation of shock reflection at a wedge with large angle comm partial differential equations 21 78 1996 1103 1118 by linearizing the resulted nonlinear equation and boundary conditions with above viewpoint we obtain a linear degenerate elliptic equation with mixed boundary conditions and a linear degenerate elliptic equation with oblique boundary conditions in a curved quadrilateral domain by means of elliptic regularization techniques delicate a priori estimate and compact arguments we show that the solution of linearized problem with oblique boundary conditions is smooth in the interior and lipschitz continuous up to the degenerate boundary

a global subsonic solution to an interacting transonic shock for the self similar nonlinear wave equation abstract we present the existence of the subsonic solution to a two dimensional riemann problem governed by a self similar nonlinear wave equation where the boundary of the subsonic region consists of a transonic shock and the sonic circle thus the governing equation becomes a free boundary problem on the transonic shock and degenerates on the sonic circle by utilizing the barrier methods and iterative methods we show the well posedness of the transonic shock in the entire subsonic region and thus establish the global solution this result does not rely on any smallness of riemann data

global transonic conic shock wave for the symmetrically perturbed supersonic flow past a cone in this paper we are concerned with the global existence and stability of a steady transonic conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body the flow is assumed to be polytropic isentropic and described by a steady potential equation theoretically as indicated in r courant k o friedrichs supersonic flow and shock waves interscience publishers inc new york 1948 it follows from the rankine hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the vertex of the sharp cone in terms of the different pressure states at infinity behind the shock surface which correspond to the supersonic shock and the transonic shock respectively in the references shuxing chen zhouping xin huicheng yin global shock wave for the supersonic flow past a perturbed cone comm math phys 228 2002 47 84 dacheng cui huicheng yin global conic shock wave for the steady supersonic flow past a cone polytropic case preprint 2006 dacheng cui huicheng yin global conic shock wave for the steady supersonic flow past a cone isothermal case pacific j math 233 2 2007 257 289 and zhouping xin huicheng yin global multidimensional shock wave for the steady supersonic flow past a three dimensional curved cone anal appl 4 2 2006 101 132 the authors have established the global existence and stability of a supersonic shock for the perturbed hypersonic incoming flow past a sharp cone when the pressure at infinity is appropriately smaller than that of the incoming flow at present for the supersonic symmetric incoming flow we will study the global transonic shock problem when the pressure at infinity is appropriately large

a singular multi dimensional piston problem in compressible flow abstract this paper concerns the multi dimensional piston problem which is a special initial boundary value problem of multi dimensional unsteady potential flow equation the problem is defined in a domain bounded by two conical surfaces one of them is shock whose location is also to be determined by introducing self similar coordinates the problem can be reduced to a free boundary value problem of an elliptic equation the existence of the problem is proved by using partial hodograph transformation and nonlinear alternating iteration the result also shows the stability of the structure of shock front in symmetric case under small perturbation

pitfalls in fast numerical solvers for fractional differential equations we consider the problem of implementing fast algorithms for the numerical solution of initial value problems of the form x t f t x t x 0 x0 where x is the derivative of x of order in the sense of caputo and 0 1 we review some of the existing methods and explain their respective strengths and weaknesses we identify and discuss potential problems in the development of generally applicable schemes

an elliptic problem arising from the unsteady transonic small disturbance equation abstract we prove a theorem on existence of a weak solution of the dirichlet problem for a quasilinear elliptic equation with a degeneracy on one part of the boundary the degeneracy is of a type keldysh type associated with singular behavior blow up of a derivative at the boundary we define an associated operator which is continuous pseudo monotone and coercive and show that a weak solution displaying singular behavior at the boundary exists