newton interpolation in fejer and chebyshev points soit une courbe de jordan dans le plan complexe et soit l ensemble compact borne par soit f une fonction analytique sur l approximation de f sur par un polynome p de degre inferieur a n et qui interpole f en n points de est etudiee

asymptotic behavior of solutions for the damped wave equation with slowly decaying data abstract we consider the cauchy problem for the damped wave equation u t t u u t u 1 u t x r r n and the heat equation t 1 t x r r n if the data is small and slowly decays likely c 1 1 x k n 0 k 1 then the critical exponent is c k 1 2 k n for the semilinear heat equation in this paper it is shown that in the supercritical case there exists a unique time global solution to the cauchy problem for the semilinear heat equation in any dimensional space r n whose asymptotic profile is given by 0 t x r n e x y 2 4 t 4 t n 2 c 1 1 y 2 k n 2 d y provided that the data 0 satisfies lim x x k n 0 x c 1 0 even in the semilinear damped wave equation in the supercritical case a time global solution u with the data u u t 0 x u 0 u 1 x is shown in low dimensional spaces r n n 1 2 3 to have the same asymptotic profile 0 t x provided that lim x x k n u 0 u 1 x c 1 0 those proofs are given by elementary estimates on the explicit formulas of solutions

asymptotic behavior and decay estimates of the solutions for a nonlinear parabolic equation with exponential nonlinearity we consider a nonlinear parabolic equation with an exponential nonlinearity which is critical with respect to the growth of the nonlinearity and the regularity of the initial data after showing the equivalence of the notions of weak and mild solutions we derive decay estimates and the asymptotic behavior for small global in time solutions

streamwise upwind algorithm for computing unsteady transonic flows past oscillating wings a streamwise upwind algorithm has been extended to compute unsteady flows with a moving grid system and applied to compute flows over oscillating wings at transonic mach numbers comparisons have been made between results obtained from this upwind algorithm using both temporally nonconservative and conservativeimplicit methods with the results obtained from a central difference method and also with experimental data the efficiency and practicality of the temporally nonconservative implicit solver are illustrated results to show the robustness and accuracy of the upwind method for unsteady computations compared to the central difference method are presented

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

implicit time stepping methods for the navier stokes equations a novel two factor approximate factorization for three dimensional flows is described the method uses a preconditioned conjugate gradient solver for one of the factors and this solution is based on a method developed for two dimensional flows this two dimensional method is described and its performance is summarized for steady and unsteady flows the two factor method is then given larger time steps can be used with this method than is possible with the more conventional three factor method test results for the transonic steady flow over the onera m6 wing are given

the behavior of the life span for solutions tout δu a x upinrd abstract let t denote the life span of the positive bounded solution u x t to the equation u t u a u u p in r d and u x 0 x where 0 a x c r d 0 x c b r d p 1 and 0 is a parameter depending on a p and d it is possible that t for 0 sufficiently small or that t 0 in which case lim 0 t it is always true that lim t 0 in this paper we investigate the asymptotic behavior of t as 0 in the case that t 0 and as in all cases the asymptotic order depends heavily on a p and d in the case that 0 whereas in the case that it depends only on whether there exists an x 0 with a x 0 x 0 0 or whether the supports of a and are separated by a positive distance

a numerical method to compute the dissolution of second phases in ternary alloys abstract dissolution of stoichiometric multi component particles in ternary alloys is an important process occurring during the heat treatment of as cast aluminium alloys prior to hot extrusion a mathematical model is proposed to describe such a process in this model an equation is given to determine the position of the particle interface in time using two diffusion equations which are coupled by nonlinear boundary conditions at the interface some results concerning existence uniqueness and monotonicity are given furthermore for an unbounded domain an analytical approximation is derived the main part of this work is the development of a numerical solution method finite differences are used on a grid which changes in time the discretization of the boundary conditions is important to obtain an accurate solution the resulting nonlinear algebraic system is solved by the newton raphson method numerical experiments illustrate the accuracy of the numerical method the numerical solution is compared with the analytical approximation

threshold solutions and sharp transitions for nonautonomous parabolic equations on mathbb r n this paper is devoted to a class of nonautonomous parabolic equations of the form ut u f t u on mathbb r n we consider a monotone one parameter family of initial data with compact support such that for small values of the parameter the corresponding solutions decay to zero whereas for large values they exhibit a different behavior either blowup in finite time or locally uniform convergence to a positive constant steady state we are interested in the set of intermediate values of the parameter for which neither of these behaviors occurs we refer to such values as threshold values and to the corresponding solutions as threshold solutions we prove that the transition from decay to the other behavior is sharp there is just one threshold value we also describe the behavior of the threshold solution it is global bounded and asymptotically symmetric in the sense that all its limit profiles as t are radially symmetric about the same center our proofs rely on parabolic liouville theorems asymptotic symmetry results for nonlinear parabolic equations and theorems on exponential separation and principal floquet bundles for linear parabolic equations

deforming mesh for computational aeroelasticity using a nonlinear elastic boundary element method a nonlinear elastic boundary element method nbem approach is developed as an innovative deforming mesh generator for computational aeroelastic simulation the computational fluid dynamics cfd mesh is assumed to be embedded in an infinite nonlinear elastic medium of a hardening material leading to the formulation of a pseudononlinear elastostatic problem whereas the cfd surface mesh is treated as a boundary element model and the cfd flowfield grid as domain sample points the nbem approach solves navier s equations using a particular solution scheme that removes the requirement of the domain integral in the conventional nbem formulation the nbem approach has a unified feature that is applicable to all mesh systems including unstructured multiblock structured and overset grids an optimization strategy is employed to determine the optimum hardening material properties by minimizing the mesh distortion in the viscous region where grid orthogonality must be preserved three test cases are performed to demonstrate the robustness and effectiveness of the nbem approach for deforming mesh generation

optimum shape design for unsteady flows with time accurate continuous and discrete adjoint methods this paper presents an adjoint method for the optimal control of unsteady flows the goal is to develop the continuous and discrete unsteady adjoint equations and their corresponding boundary conditions for the time accurate method first this paper presents the complete formulation of the time dependent optimal design problem second we present the time accurate unsteady continuous and discrete adjoint equations third we present results that demonstrate the application of the theory to a two dimensional oscillating airfoil the results are compared with a multipoint approach to illustrate the added benefit of performing full unsteady optimization

life span of solutions for a semilinear parabolic problem with small diffusion abstract this paper is concerned with the initial boundary value problem formula where p 1 e 0 is a bounded domain in rn and is a continuous function on it is shown that the blowup time t e of the solution of this problem satisfies t e 1 p 1 1 p as e 0 moreover when the maximum of x is attained at one point we determine the higher order term of t e which reflects the pointedness of the peak of the proof is based on a careful construction of super and subsolutions

fast and accurate computation of gauss legendre and gauss jacobi quadrature nodes and weights an efficient algorithm for the accurate computation of gauss legendre and gauss jacobi quadrature nodes and weights is presented the algorithm is based on newton s root finding method with initial guesses and function evaluations computed via asymptotic formulae the n point quadrature rule is computed in mathcal o n operations to an accuracy of essentially double precision for any n geq 100

asymptotic behavior of solutions of quasilinear parabolic equations with supercritical nonlinearity abstract the cauchy problem p u t u m u p in r n 0 t u x 0 u 0 x in r n is considered where 0 p m 1 and u 0 x 0 is a bounded continuous radially symmetric function in r n we choose p in some open interval p s p p with p s m n 2 n 2 such that a peaking solution incomplete blow up solution of p exists denote the solution of p by u we show that if u 0 x is nonincreasing in large r x and decays slowly u 0 x o x as x 2 p m then u is classified into one of the next three types according to the value as follows there exists 1 0 such that i u blows up completely in finite time if 1 ii u blows up incompletely in finite time and u t l r n o t 1 p 1 as t if 1 iii u does not blow up in finite time and u t l r n o t 1 p 1 as t if 0 1

barycentric hermite interpolation let z 1 ldots z k be distinct grid points if f k 0 is the prescribed value of a function at the grid point z k and f k r the prescribed value of the r th derivative for 1 leq r leq n k 1 the hermite interpolant is the unique polynomial of degree n 1 n n 1 cdots n k which interpolates the prescribed function values and function derivatives we obtain another derivation of a method for hermite interpolation recently proposed by butcher et al numer algorithms 56 2011 pp 319 347 one advantage of our derivation is that it leads to an efficient method for updating the barycentric weights if an additional derivative is prescribed at one of the interpolation points we show how to update the barycentric coefficients using only mathcal o n operations even in the context of confluent newton series a comparably efficient and general method to update the coefficients appears not to be known if the method is properly implemented it computes the barycentric wei

error bounds for approximation in chebyshev points this paper improves error bounds for gauss clenshaw curtis and fejer s first quadrature by using new error estimates for polynomial interpolation in chebyshev points we also derive convergence rates of chebyshev interpolation polynomials of the first and second kind for numerical evaluation of highly oscillatory integrals preliminary numerical results show that the improved error bounds are reasonably sharp

existence of many positive solutions of semilinear elliptic equations on annulus abstract we study the existence of many nonradial positive solutions in an annulus of rn it is an improvement of results in c v coffman a nonlinear boundary value problem with many positive solutions j differential equations 54 1984 429 437 an annulus is invariant under many group actions and these group actions act naturaly on the functional and leave the functional invariant therefore when we look for critical points of the functional we can restrict the functional to the manifold consisting of all fixed points under the group action we are going to use different group actions to obtain critical points and furthermore we can distinguish them

lifespan of solutions to the damped wave equation with a critical nonlinearity abstract in the present paper we study a lifespan of solutions to the cauchy problem for semilinear damped wave equations dw t 2 u u t u f u t x 0 t e r n u 0 x e u 0 x x r n t u 0 x e u 1 x x r n where n 1 f u u p 1 u or u p p 1 e 0 is a small parameter and u 0 u 1 is a given initial data the main purpose of this paper is to prove that if the nonlinear term is f u u p and the nonlinear power is the fujita critical exponent p p f 1 2 n then the upper estimate to the lifespan is estimated by t e exp c e p for all e 0 1 and suitable data u 0 u 1 without any restriction on the spatial dimension our proof is based on a test function method utilized by zhang 35 we also prove a sharp lower estimate of the lifespan t e to dw in the critical case p p f

aeroelastic solutions using the nonlinear frequency domain method in this work both fully implicit time domain and nonlinear frequency domain methods are used to compute aeroelastic solutions in transonic flows specifically flutter boundaries for a two dimensional naca64a010 airfoil are calculated and compared with preexisting numerical results the second order backward difference time accurate scheme will serve as our numerical benchmark when determining the computational efficiency of the nonlinear frequency domain method comparable results between the time domain and nonlinear frequency domain approaches will further justify the nonlinear frequency domain method as an efficient alternative to the study of unsteady periodic problems a temporal resolution study will establish the required number of modes or time steps per period required for unsteady transonic flows

life span of positive solutions for a semilinear heat equation with general non decaying initial data abstract we prove upper bounds on the life span of positive solutions for a semilinear heat equation for non decaying initial data it is well known that the solutions blow up in finite time we give two types estimates of the life span in terms of the limiting values of the initial data in space

blow up and global existence of solutions to an inhomogeneous parabolic system abstract we study the critical phenomenon of a weakly coupled inhomogeneous parabolic system which is largely unexplored even though the homogeneous problem has been widely studied the conditions on the powers of the nonlinear terms are new compared with the well known homogeneous case the method appears to be different from the standard ones in the literature an application to the lame emden system is also given

fujita type phenomenon of nonlinear coupled nonlocal diffusion system abstract this paper studies a cauchy problem for a nonlinear coupled nonlocal diffusion system u t j u u v p v t j v v u q we first deal with the critical fujita exponent and then determine the second critical exponent to characterize the critical space decay rate of initial data in the co existence region of global and non global solutions it turns out that the results are coincident with the classical heat system we not only generalize the results in the scalar case obtained by j garcia melian and f quiros 2010 6 to the case of a system but also delete the monotonicity condition on the kernel function j the key techniques are modifying the rescaled test function method with a new test function rather than the eigenfunction of the nonlocal operator to deal with blow up of the solution and selecting a suitable banach space via the estimates obtained by j terra and n wolanski 2011 13 to obtain the global solutions

direct solution of two dimensional navier stokes equations for static aeroelasticity problems a new method has been developed to calculate the steady flow and structural deformations for fluid structure interaction problems the discretized fluid dynamic and structural equations are regarded as a single set of coupled nonlinear algebraic equations the equilibrium solution is directly obtained using newton s method the governing equations used for the fluid flow are the two dimensional navier stokes equations and a finite element model is used to represent the structure this paper describes the analytical method and presents sample calculations demonstrating the technique the results show rapid convergence and good agreement with experimental data nomenclature bw half bandwidth of tangent matrix d set of structural displacement variables e set of governing equations for the aeroelastic system f flux vector in x direction g flux vector in y direction h distance from nozzle throat to centerline number of fluid dynamic control volumes in streamwise direction j number of fluid dynamic control volumes in cross stream direction l reference length total length of nozzle n total number of variables in system p pressure pe exit static pressure pt total pressure r system residual r magnitude of residual r radius at nozzle throat t tangent matrix system jacobian t wall thickness u set of fluid dynamic variables x set of variables for the aeroelastic system ax change in system variables during a newton iteration x streamwise coordinate y cross stream coordinate

life span and a new critical exponent for a degenerate parabolic equation abstract in this paper we consider the positive solution of the cauchy problem for the equation u t u p u u q p 1 q 1 and give a secondary critical exponent of the behavior of initial value at infinity for the existence of global and nonglobal solutions of the cauchy problem furthermore the life span of solutions are also studied

first order system least squares fosls for coupled fluid elastic problems mathematical models for the mechanical coupling between a moving fluid and an elastic solid are inherently nonlinear because the shape of the eulerian fluid domain is not known a priori it is at least partially determined by the displacement of the elastic solid in this paper a first order system least squares finite element formulation is used to solve the nonlinear system of model equations using different iteration techniques including an approach where the equations are fully coupled and two other approaches in which the equations are decoupled the discrete linear system of equations is solved using an algebraic multigrid solver as a preconditioner for a conjugate gradient iteration the numerical results show that the approach is optimal in the sense that computational cost is proportional to the degrees of freedom the results also show that the choice of iteration method from fully coupled to fully decoupled does not significantly effect computational cost but it does influence the error in the solution

a variational finite element method for stationary nonlinear fluid solid interaction we consider the problem of the interaction of a stationary viscous fluid with an elastic solid that undergoes large displacement the fluid is modeled by the stationary incompressible navier stokes equations in an eulerian frame of reference while a lagrangian reference frame and large displacement small strain theory is used for the solid a variational formulation of the problem is developed that ensures satisfaction of continuity of interface tractions and velocities the variational formulation is approximated by a galerkin finite element method yielding a system of nonlinear algebraic equations in unknown fluid velocities and pressures and solid displacements a newton like method is introduced for solution of the discrete system the method employs a modified jacobian that enables decomposition into separate fluid and solid subdomains this domain decomposition avoids possible ill conditioning of the jacobian as well as the need to compute and store geometric coupling terms between fluid and interface shape the capability of the methodology is illustrated by solution of a problem of the flow induced large displacements of an elastic infinite cylinder

spreading vanishing dichotomy in a diffusive logistic model with a free boundary ii abstract we study the diffusive logistic equation with a free boundary in higher space dimensions and heterogeneous environment such a model may be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front for simplicity we assume that the environment and the solution are radially symmetric in the special case of one space dimension and homogeneous environment this free boundary problem was investigated in du and lin 2010 10 we prove that the spreading vanishing dichotomy established in du and lin 2010 10 still holds in the more general and ecologically realistic setting considered here moreover when spreading occurs we obtain best possible upper and lower bounds for the spreading speed of the expanding front when the environment is asymptotically homogeneous at infinity these two bounds coincide our results indicate that the asymptotic spreading speed determined by this model does not depend on the spatial dimension

goal oriented error estimation and adaptivity for fluid structure interaction using exact linearized adjoints we develop duality based a posteriori error estimates for functional outputs of solutions of steady fluid structure interaction problems the crucial complication in obtaining these estimates pertains to the derivation of the coupled dual exact linearized adjoint problem owing to the free boundary character of fluid structure interaction we present two approaches to derive the dual problem in the domain map linearization approach the fluid subproblem is first transformed to a fixed reference domain after which one essentially linearizes with respect to the domain transformation map in the shape linearization approach fluid unknowns are fixed in the current configuration and a very weak formulation of the fluid subproblem is then linearized using shape derivative techniques we show that the dual problems correspond to coupled fluid structure problems with nonstandard coupling conditions furthermore we present numerical experiments that demonstrate the consistency of the dual based error estimates and their usefulness in goal oriented adaptive mesh refinement

life span of solutions for a semilinear heat equation with initial data having positive limit inferior at infinity abstract we present a new upper bound of the life span of positive solutions of a semilinear heat equation for initial data having positive limit inferior at space infinity the upper bound is expressed by the data in limit inferior not in every direction but around a specific direction it is also shown that the minimal time blow up occurs when initial data attains its maximum at space infinity

on the cauchy problem for a reaction diffusion equation with a singular nonlinearity abstract we consider the following cauchy problem with a singular nonlinearity p u t u u x r n t 0 0 u t 0 c lb r n with n 3 and having a positive lower bound we find some conditions on the initial value such that the local solutions of p vanish in finite time meanwhile we obtain optimal conditions on for global existence and study the large time behavior of those global solutions in particular we prove that if 0 and n 3 x u s x 2 1 n 2 2 1 1 1 x 2 1 where u s is a singular equilibrium of p and 1 then p has a unique global classical solution u with u u s and u x t 1 1 1 1 1 1 1 t 1 1 on the other hand the structure of positive radial solutions of the steady state of p is studied and some interesting properties of the positive solutions are obtained moreover the stability and weakly asymptotic stability of the positive radial solutions of the steady state of p are also discussed

a fully eulerian formulation for fluid structure interaction problems in this work we present a fully eulerian framework for fluid structure interaction fsi problems coupling the incompressible navier stokes equations with a hyperelastic solid the fully eulerian framework is a monolithic implicit variational formulation for the coupled problem in contrast to the well established arbitrary lagrangian eulerian ale coordinates the fully eulerian framework formulates both subproblems fluid and solid in eulerian coordinates this concept circumvents various difficulties connected to ale coordinates since no artificial domain mapping is used the formulation is an interface capturing method and uses an extension of the solid s deformation the initial point set to detect the interface location by construction very large deformation as well as topology changes like contact of the solid to the domain boundary or other solid parts are possible

a feti based domain decomposition technique for time dependent first order systems based on a dae approach we present a novel partitioned coupling algorithm to solve first order time dependent non linear problems e g transient heat conduction the spatial domain is partitioned into a set of totally disconnected subdomains the continuity conditions at the interface are modeled using a dual schur formulation where the lagrange multipliers represent the interface fluxes or the reaction forces that are required to maintain the continuity conditions the interface equations along with the subdomain equations lead to a system of differential algebraic equations daes for the resulting equations a numerical algorithm is developed which includes choosing appropriate constraint stabilization techniques the algorithm first solves for the interface lagrange multipliers which are subsequently used to advance the solution in the subdomains the proposed coupling algorithm enables arbitrary numeric schemes to be coupled with different time steps i e it allows subcycling in each subdomain this implies that existing software and numerical techniques can be used to solve each subdomain separately the coupling algorithm can also be applied to multiple subdomains and is suitable for parallel computers we present examples showing the feasibility of the proposed coupling algorithm copyright 2008 john wiley sons ltd

improvements on lower bounds for the blow up time under local nonlinear neumann conditions abstract this paper studies the heat equation u t u in a bounded domain r n n 2 with positive initial data and a local nonlinear neumann boundary condition the normal derivative u n u q on partial boundary 1 for some q 1 while u n 0 on the other part we investigate the lower bound of the blow up time t of u in several aspects first t is proved to be at least of order q 1 1 as q 1 since the existing upper bound is of order q 1 1 this result is sharp secondly if is convex and 1 denotes the surface area of 1 then t is shown to be at least of order 1 1 n 1 for n 3 and 1 1 ln 1 1 for n 2 as 1 0 while the previous result is 1 for any 1 n 1 finally we generalize the results for convex domains to the domains with only local convexity near 1

existence and nonexistence of global solutions for a semilinear reaction diffusion system abstract this paper is concerned with the blow up and global existence of nonnegative solutions to the following cauchy problem u t u v p t 0 x r n v t v a x u q t 0 x r n u x 0 u 0 x v x 0 v 0 x x r n where the constants p q 0 and a x 0 is on the order x m as x m r the fujita critical exponent is determined when m 0 and some results of global existence of solution under some assumptions when m 0 are also obtained the results extend those in escobedo and herrero 1991 9 and indicate that m affects the fujita critical exponent

a semilinear pseudo parabolic equation with initial data non rarefied at abstract in this paper we study the cauchy problem for a semilinear pseudo parabolic equation with initial data non rarefied at our interest lies in the discussion of the effect of the non rarefied factors on the bow up phenomenon and the life span of the solution

hair triggered instability of radial steady states spread and extinction in semilinear heat equations we first study the initial value problem for a general semilinear heat equation we prove that every bounded nonconstant radial steady state is unstable if the spatial dimension is low n 10 or if the steady state is flat enough at infinity the solution of the heat equation either becomes unbounded as t approaches the lifespan or eventually stays above or below another bounded radial steady state depending on if the initial value is above or below the first steady state moreover the second steady state must be a constant if n 10 using this instability result we then prove that every nonconstant radial steady state of the generalized fisher equation is a hair trigger for two kinds of dynamical behavior extinction and spreading we also prove more criteria on initial values for these types of behavior similar results for a reaction diffusion system modeling an isothermal autocatalytic chemical reaction are also obtained

unsteady aerodynamic and aeroelastic calculations for wings using euler equations a procedure to solve simultaneously the euler flow equations and modal structural equations of motion is presented for computing aeroelastic responses of wings the euler flow equations are solved by a finite difference scheme with dynamic grids the coupled aeroelastic equations of motion are solved using the linear acceleration method the aeroelastic configuration adaptive dynamic grids are time accurately generated using the aeroelastically deformed shape of the wing the unsteady flow calculations are validated with the experiment both for a semi infinite wing and a wall mounted cantilever rectangular wing aeroelastic responses are computed for a rectangular wing using the modal data generated by the finite element method the robustness of the present approach in computing unsteady flows and aeroelastic responses that are beyond the capability of earlier approaches using potential equations are demonstrated

non uniqueness of solutions to the cauchy problem for semilinear heat equations with singular initial data the cauchy problem for semilinear heat equations with singular initial data is studied where n 2 0 is a parameter and a 0 a 0 we show that when p n 2 n and n 2 p n 2 there exists a positive constant such that the problem has two positive self similar solutions and with if and no positive self similar solutions if furthermore for each fixed and in l rn as 0 where w0 is a non unique solution to the problem with zero initial data which is constructed by haraux and weissler

asymptotically self similar global solutions of a general semilinear heat equation we consider the general nonlinear heat equation partial t u delta u a u p 1 1 u g u u 0 varphi on 0 infty times i r n where a in i r p 1 1 2 n and g satisfies certain growth conditions we prove the existence of global solutions for small initial data with respect to a norm which is related to the structure of the equation we also prove that some of those global solutions are asymptotic for large time to self similar solutions of the single power nonlinear heat equation i e with g equiv 0

self similar subsolutions and blowup for nonlinear parabolic equations abstract for a wide class of nonlinear parabolic equations of the formut u f u u we prove the nonexistence of global solutions for large initial data we also estimate the maximal existence time to do so we use a method of comparison with suitable blowing up self similar subsolutions as a consequence we improve several known results onut u up on generalized burgers equations and on other semilinear equations this method can also apply to degenerate equations of porous medium type and provides a unified treatment for a large class of problems both semilinear and quasilinear

a new partitioned staggered scheme for flexible multibody interactions with strong inertial effects abstract in generic flexible multibody interaction problems the differences in the mass damping and stiffness of the bodies can be very large of particular interest is a small change in the higher mass structure which triggers relatively large inertial effects on the lower mass structure in this work a new partitioned staggered time integration scheme is developed and its applicability is extended to the problems involving low mass ratios in coupled multibody problems the equation of motion of body with larger inertia is integrated by implicit newmark method whereas the smaller inertia is solved using a robust self starting explicit integration procedure the coupled formulation includes explicit correction terms that adjusts the amount of interfacial velocity which plays a key role in the stability and accuracy of the simulations a wide range of mass 10 6 10 1 damping and stiffness 10 3 10 3 ratios are chosen to assess the stability and accuracy characteristics of the proposed scheme the closed form expressions for the coupling parameter have been constructed for both matching and non matching time stepping through the godunov ryabenkii normal mode analysis the stability of the proposed method is observed as a function of the relative properties and temporal discretisation of the coupled system the time step of the heavier body significantly influences the stability more than the lighter body to maintain the staggering error minimal an optimal range of the coupling parameter is identified the error computed with uniform variation in the time step revealed that the present method is more accurate than the existing methods the partitioned method is an energy preserving integration method for the dynamic analysis of multibody systems as a potential application of this scheme flexible multibody problems in the field of offshore engineering viz wave energy converter and floater mooring systems are presented and validated with experimental measurements

sharp upper bound for lifespan of solutions to some critical semilinear parabolic dispersive and hyperbolic equations via a test function method abstract this paper is concerned with the blowup phenomena for initial boundary value problem 0 1 t 2 u x t u x t a x t u x t u x t p x t c 0 t u x t 0 x t c 0 t u x 0 e f x x c t u x 0 e g x x c where c is a cone like domain in r n n 2 defined as c int r r n r 0 with a connected open set in s n 1 with smooth boundary if n 1 then we only consider two cases c 0 and c r here a x is a non zero coefficient of t u which could be complex valued and space dependent is a fixed constant and e 0 is a small parameter the constants 0 1 switch the parabolicity and hyperbolicity of the problem 0 1 the result proposes an argument for sharp upper lifespan estimates of solutions to 0 1 by a test function method based on mitidieri and pokhozhaev 2001 the crucial idea is to introduce an ordinary differential inequality with a parameter as a variable

goal oriented error estimation for fluid structure interaction problems in this work we present an adaptive finite element method for the numerical simulation of stationary fluid structure interaction problems the coupled system is given in a variational and monolithic arbitrary lagrangian eulerian framework we derive methods for goal oriented error estimation and mesh adaptation with the dual weighted residual method key to applying this error estimator is the underlying canonic variational formulation of the fluid structure interaction problem by mapping the flow problem to ale coordinates the developed method is applied to two and three dimensional stationary benchmark problems coupling the incompressible navier stokes equations with a nonlinear hyper elastic material law

blow up set for a semilinear heat equation with small diffusion abstract we consider the blow up problem for a semilinear heat equation t u u u p in 0 t u x t 0 on 0 t if u x 0 x 0 in where is a domain in r n n 1 0 p 1 and t 0 in this paper under suitable assumptions on we prove that if the family of the solutions u satisfies a uniform type i blow up estimate with respect to then the solution u blows up only near the maximum points of the initial datum for any sufficiently small 0 this is proved without any conditions on the exponent p and the domain such as n 2 p n 2 and the convexity of the domain

remark on upper bound for lifespan of solutions to semilinear evolution equations in a two dimensional exterior domain abstract in this paper we consider the following initial boundary value problem with the power type nonlinearity u p with 1 p 2 in a two dimensional exterior domain 0 1 t 2 u x t u x t e i t u x t u x t p x t 0 t u x t 0 x t 0 t u x 0 e f x x t u x 0 e g x x where is given by x r 2 x 1 2 2 c and 0 1 switches the parabolicity dispersivity and hyperbolicity remark that 2 1 2 n is well known as the fujita exponent if p 2 then there exists a small global in time solution of 0 1 for some initial data small enough see ikehata 11 and if p 2 then global in time solutions cannot exist for any positive initial data see ogawa takeda 22 and lai yin 14 the result is that for given initial data f g h 0 1 l 2 satisfying f g log x l 1 with some requirement the solution blows up at finite time and moreover the upper bound for lifespan of solutions to 0 1 is given as the following double exponential type when p 2 lifespan u exp exp c e 1 the crucial idea is to use test functions which approximates the harmonic function log x satisfying dirichlet boundary condition and the technique modified from 9

existence and nonexistence of global solutions of higher order parabolic problems with slow decay initial data weinberger 1 for a reference of the extensive literature on the extension of the classicalpaper by fujita in several directions we refer the reader to the survey papers by levine 10 and deng and levine 2 and to the monograph by mitidieri and pohozaev 11 the purpose of this paper is to study the problem of the existence and of the nonex istence of global solutions for problem 1 2 for initial data

blow up and asymptotic behavior of solutions for reaction diffusion equations with free boundaries abstract we study the blow up phenomena and the asymptotic behavior of time global solutions for reaction diffusion equations u t u x x a u u p a r and p 1 with free boundaries and initial data x we give a sharp threshold value a p 0 such that the solution blows up in finite time when vanishes i e u 0 as t when and when it converges as t to 0 or to an evenly decreasing positive stationary solution depending on whether a 0 or a 0 moreover when blow up happens we show that the blow up set is compact in the occupying domain of initial data and the free boundaries keep bounded

stability of finite elements under divergence constraints a finite dimensional stability test for checking velocity pressure finite element trial spaces is presented applications are made to a new class of element pairs proposed in this paper as well as to existing spaces

analysis on a coupled parabolic system with free boundary abstract the purpose of this paper is to investigate a parabolic problem with coupled reaction terms and free boundary including the existence uniqueness regularity and long time behavior of positive solutions firstly by the contraction mapping theorem we establish the local existence and uniqueness of positive solutions then we prove the solution blows up in finite time with large initial data by comparison principle and give more details for these large initial data by exhibiting an energy condition the simultaneous blow up result of the maximum of u and v is obtained and the blow up set of blow up solutions is a compact subset of 0 h 0 furthermore there is a global and fast solution which decays uniformly at an exponential rate if the initial datum is small while there is a global and slow solution provided that the initial datum is suitably large finally for initial data x x we obtain a trichotomy conclusion by considering the size of parameter

the critical exponent of a reaction diffusion equation on some lie groups we shall study the global existence and blow up of the following semilinear parabolic cauchy problem 1 1 hu h0u up x i aijxju tu up 0 in rn 0 u x 0 u0 x 0 in rn where aij aij x t are bounded measurable functions and the matrix a aij is uniformly elliptic xj j 1 k are smooth vector fields in rn and x j is the formal adjoint of xj basic assumptions i we assume that xj satisfy the hormander s condition for hypoellipticity i e the rank of the lie algebra generated by x1 xp equals n given two points x y rn d x y will be the x1 xp control distance x denotes the euclidean distance d x d x 0 b x r will represent the metric ball y rn d x y 6 r properties of the distance metric balls and the fundamental solutions of h0 have been studied extensively in nsw fes ks1 2 and sa ii there are positive constants q b c and b such that 1 2 b x r br lim x d x lim d x x

blowup stability of solutions of the nonlinear heat equation with a large life span abstract we study the cauchy problem for the nonlinear heat equation u t u u p 1 u in r n the initial data is of the form u 0 where c 0 r n is fixed and 0 we first take 1 p p f where p f is the fujita critical exponent and c 0 r n l 1 r n with nonzero mean we show that u t blows up for small extending the h fujita blowup result for sign changing solutions next we consider 1 p p s where p s is the sobolev critical exponent and x decaying as x at infinity where p 1 2 we also prove that u t blows up when is small extending a result of t lee and w ni for both cases the solution enjoys some stable blowup properties for example there is single point blowup even if is not radial

nonexistence of solutions to parabolic differential inequalities with a potential on riemannian manifolds we are concerned with nonexistence results of nonnegative weak solutions for a class of quasilinear parabolic problems with a potential on complete noncompact riemannian manifolds in particular we highlight the interplay between the geometry of the underlying manifold the power nonlinearity and the behavior of the potential at infinity

o 1 computation of legendre polynomials and gauss legendre nodes and weights for parallel computing a self contained set of algorithms is proposed for the fast evaluation of legendre polynomials of arbitrary degree and argument in 1 1 more specifically the time required to evaluate any legendre polynomial regardless of argument and degree is bounded by a constant i e the complexity is mathcal o 1 the proposed algorithm also immediately yields an mathcal o 1 algorithm for computing an arbitrary gauss legendre quadrature node such a capability is crucial for efficiently performing certain parallel computations with high order legendre polynomials such as computing an integral in parallel by means of gauss legendre quadrature and the parallel evaluation of legendre series in order to achieve the mathcal o 1 complexity novel efficient asymptotic expansions are derived and used alongside known results a c implementation is available from the authors that includes the evaluation routines of the legendre polynomials and gauss legendre quadrature rules

the fast implementation of higher order hermite fejer interpolation the numerical computation of barycentric hermite interpolation may suffer from devastating inaccuracies in the absence of overflow or underflow in this paper we propose a new fast implementation of the second barycentric formula for higher order hermite fejer interpolation at gauss jacobi or jacobi gauss lobatto pointsystems our method relies upon the sturm liouville equations for jacobi polynomials whereby the barycentric weights can be efficiently evaluated and the overall computational complexity is linear with respect to the number of grids in particular the exponentially increasing common factors in the barycentric weights cancel out from the second barycentric formula which yields a superiorly stable method for computing the simplified barycentric weights

a characteristic of local existence for nonlinear fractional heat equations in lebesgue spaces abstract in this paper we consider the fractional heat equation u t 2 u f u with dirichlet conditions on the ball b r r d where 2 is the fractional laplacian f 0 0 is continuous and non decreasing we present the characterisations of f to ensure the equation has a local solution in l q b r provided that the non negative initial data u 0 l q b r for q 1 and 1 2 we show that the equation has a local solution in l q b r if and only if lim s sup s 1 q d f s and for q 1 and 1 2 if and only if 1 s 1 d f s d s where f s sup 1 t s f t t when lim s 0 f s s the same characterisations holds for the fractional heat equation on the whole space r d

linear behaviour of solutions of a superlinear heat equation we give sufficient conditions which guarantee that solutions of a superlinear heat equation decay to zero at the same rate as the solutions of the linear heat equation with the same initial data this improves a previous result of lee and ni by showing that this behaviour holds for a significantly larger and rather tightly defined class of solutions

global and nonglobal existence for a strongly coupled parabolic system on a general domain abstract we consider the parabolic system u t u f t u r v s v t v g t u q v s in 0 t where r n is either an unbounded or bounded domain and f g c 0 we find conditions for the global existence or nonglobal existence which are expressed in terms of the behavior of s t u 0 as t where u t s t u 0 is the solution of the linear problem u t u 0 u 0 u 0

vortical flow computations on swept flexible wings using navier stokes equations a procedure to couple the navier stokes solutions with modal structural equations of motion is presented for computing aeroelastic responses of swept flexible wings the navier stokes flow equations are solved by a finite difference scheme with dynamic grids the coupled aeroelastic equations of motion are solved using the linear acceleration method the configuration adaptive dynamic grids are time accurately generated using the aeroelastically deformed shape of the wing the calculations are compared with the experiment when available effects of flexibility sweep angle and pitch rate are demonstrated for flows with vortices

on the stability of the positive radial steady states for a semilinear cauchy problem abstract this paper is a contribution to the analysis of the cauchy problem 0 1 u t u i 1 k c i x l i u p i in r n 0 t u x 0 x in r n with initial function 0 the stability and asymptotic stability of the positive radial steady states which are positive solutions of 0 2 u i 1 k c i x l i u p i 0 have been discussed with in weighted banach spaces

conservative load projection and tracking for fluid structure problems the loose coupling of computational fluid dynamics and computational structural dynamics solvers introduces some problems related to the information transfer between the codes some techniques developed to solve the problems of the load transfer and interface surface tracking are presented the main criterion is to achieve conservation of total loads and total energy the load projection scheme is based on gaussian integration and fast interpolation algorithms for unstructured grids the surface tracking algorithm also based on interpolation is important for many applications including aeroelastic deformation of wings due to aerodynamic loads the methodologies not only improve present fluid structure interaction simulations but also increase the range of their applicability these techniques are of general character and can be used in other multidisciplinary applications as well

explicit barycentric weights for polynomial interpolation in the roots or extrema of classical orthogonal polynomials barycentric interpolation is arguably the method of choice for numerical polynomial interpolation in this paper we show that barycentric weights for the roots or extrema of classical families of orthogonal polynomials are expressible explicitly in terms of the nodes and weights of the corresponding gaussian quadrature rule based on the glaser liu rokhlin algorithm for gaussian quadrature this leads to an o n computational scheme for computing barycentric weights for the jacobi case known results on the barycentric weights of the chebyshev points are recovered as special cases and some new results such as the barycentric weights associated with the jacobi points and the gauss jacobi lobatto points are obtained we also show that the interpolants in the roots or extrema of classical orthogonal polynomials can be computed rapidly and stably by using their barycentric representations

on blow up at space infinity for semilinear heat equations abstract a nonnegative blowing up solution of the semilinear heat equation u t u u p with p 1 is considered when initial data u 0 satisfies lim x u 0 m 0 u 0 m and u 0 m it is shown that the solution blows up only at space infinity and that lim x u x t is the solution of the ordinary differential equation v t v p with v 0 m

the diffusive competition problem with a free boundary in heterogeneous time periodic environment abstract in this paper we consider the diffusive competition problem with a free boundary and sign changing intrinsic growth rate in heterogeneous time periodic environment consisting of an invasive species with density u and a native species with density v we assume that v undergoes diffusion and growth in r n and u exists initially in a ball b h 0 0 but invades into the environment with spreading front r h t the effect of the dispersal rate d 1 the initial occupying habitat h 0 the initial density u 0 of invasive species u and the parameter see 1 3 on the dynamics of this free boundary problem are studied a spreading vanishing dichotomy is obtained and some sufficient conditions for the invasive species spreading and vanishing are provided moreover when spreading of u happens some rough estimates of the spreading speed are also given

well posedness global existence and large time behavior for hardy henon parabolic equations in this paper we study the nonlinear parabolic equation tu u a x u u tu u a x u u t 0 x rn 0 n 1 a r 0 0 min 2 n and with initial value u 0 u 0 we establish local well posedness in lq rn lq rn and in c0 rn c0 rn in particular the value q n 2 q n 2 plays a critical role for 2 n 2 n we show the existence of global self similar solutions with initial values x x x 2 x x x 2 where l rn l rn is homogeneous of degree 00 and is sufficiently small we then prove that if x x x 2 x x x 2 for x x large then the solution is global and is asymptotic in the l l norm to a self similar solution of the nonlinear equation while if x x x x x x for x x large with 2 n 2 n then the solution is global but is asymptotic in the l l norm to et x x et x x the equation with more general potential tu u v x u u v x x l rn is also studied in particular for initial data x x x 2 x x x 2 x x large we show that the large time behavior is linear if vv is compactly supported near the origin while it is nonlinear if vv is compactly supported near infinity

global existence large time behavior and life span of solutions of a semilinear parabolic cauchy problem we investigate the behavior of the solution u x t of where 1 1 n 2 xi 2 is the laplace operator p 1 is a constant t 0 and is a nonnegative bounded continuous function in r n the main results are for the case when the initial value has polynomial decay near x assuming 1 x 75a with a 0 various questions of global in time existence and nonexistence large time behavior or life span of the solution u x t are answered in terms of simple conditions on a p and the space dimension n

a semilinear parabolic problem with variable reaction on a general domain abstract we are concerned with the parabolic equation u t u f t u p x in 0 t with homogeneous dirichlet boundary condition p c f c 0 and is either a bounded or an unbounded domain the initial data is considered in the space u 0 c 0 u 0 0 we find conditions that guarantee the global existence and the blow up in finite time of nonnegative solutions these conditions are given in terms of the asymptotic behavior of the solution of the homogeneous linear problem u t u 0

blowup for the heat equation with a noise term in this paper we study blow up of the equation u t u xx u gamma dot w tx where dot w tx is a two dimensional white noise field and where dirichlet boundary conditions are enforced it is known that if 3 2 then the solution exists for all time in this paper we show that if is much larger than 3 2 then the solution blows up in finite time with positive probability we prove this by considering how peaks in the solution propagate if a peak of high mass forms we rescale the equation and divide the mass of the peak into a collection of peaks of smaller mass and these peaks evolve almost independently in this way we compare the evolution ofu to a branching process large peaks are regarded as particles in this branching process offspring are peaks which are higher by some factor we show that the expected number of offspring is greater than one when is much larger than 3 2 and thus the branching process survives with positive probability corresponding to blowup in finite time

a new critical phenomenon for semilinear parabolic problems abstract we study the cauchy problem of the inhomogeneous semilinear parabolic equations u up ut w 0 onmn 0 with initial valueu0 0 wheremnis a riemannian manifold with possibly nonnegative ricci curvature there is an exponentp which is critical in the following sense when 1 p the problem has a global positive solution for somew w x 0 and someu0 0