the mixed problem in lipschitz domains with general decompositions of the boundary this paper continues the study of the mixed problem for the laplacian we consider a bounded lipschitz domain rn n 2 with boundary that is decomposed as d n d and n disjoint we let denote the boundary of d relative to and impose conditions on the dimension and shape of and the sets n and d under these geometric criteria we show that there exists p0 1 depending on the domain such that for p in the interval 1 p0 the mixed problem with neumann data in the space lp n and dirichlet data in the sobolev space w 1 p d has a unique solution with the non tangential maximal function of the gradient of the solution in lp we also obtain results for p 1 when the dirichlet and neumann data comes from hardy spaces and a result when the boundary data comes from weighted sobolev spaces

on the asymptotic stability ofθ methods for delay differential equations stability regions of methods for the linear delay differential test equations begin gathered y t py t qy t tau t 0 hfill y t varphi t t in tau 0 hfill end gathered where is a positive constant are presented in the case thatp andq are real constant coefficients necessary and sufficient conditions on the stepsize for the stability of a method are obtained furthermore whenp andq are complex coefficients sufficient conditions for the stability of the methods are also given

the solution of coupled burgers equations using adomian pade technique abstract the purpose of this study is to implement adm pade madm pade technique which is a combination of adomian decomposition method modified adomian decomposition method and pade approximation for solving homogeneous inhomogeneous two dimensional parabolic equation also we apply adm pade technique to solve coupled burgers equations the results obtained by adm pade madm pade techniques are compared to those obtained by using adm madm alone the numerical results demonstrate that adm pade madm pade technique gives the approximate solution with faster convergence rate and higher accuracy than using adm madm

the l p regularity problem on lipschitz domains this paper contains two results on the l p regularity problem on lipschitz domains for second order elliptic systems and 1 2 the solvability of the l p regularity problem is equivalent to a weak reverse holder condition with exponent p

restarted adomian method for system of nonlinear volterra integral equations in this paper we generalize restarted adomian method as a new method based on standard adomian method for solving system of nonlinear volterra integral equations

the theoretical foundation of the adomian method abstract in this paper we work out the theoretical basis of the adomian method this powerful method that g adomian has developed in the beginning of the 1980 s supplies analytical approximations to the solution of many kinds of equations thanks to our decomposition theory we justify the practical method and we generalize the convergence results obtained by y cherruault afterward we explain how to solve rigorously equations with linear nonlinear or composed operator and also equations with several terms or with a second member

weak maximum principle for biharmonic equations in quasiconvex lipschitz domains abstract in dimension two or three the weak maximum principle for biharmonic equation is valid in any bounded lipschitz domains in higher dimensions greater than three it was only known that the weak maximum principle holds in convex domains or c 1 domains and may fail in general lipschitz domains in this paper we prove the weak maximum principle in higher dimensions in quasiconvex lipschitz domains which is a sharp condition in some sense and recovers both convex and c 1 domains

a bilinear estimate for biharmonic functions in lipschitz domains we show that a bilinear estimate for biharmonic functions in a lipschitz domain is equivalent to the solvability of the dirichlet problem for the biharmonic equation in as a result we prove that for any given bounded lipschitz domain in mathbb r d and 1 q the solvability of the l q dirichlet problem for 2 u 0 in with boundary data in wa 1 q is equivalent to that of the l p regularity problem for 2 u 0 in with boundary data in wa 2 p where frac 1 p frac 1 q 1 this duality relation together with known results on the dirichlet problem allows us to solve the l p regularity problem for d 4 and p in certain ranges

layer potentials and boundary value problems for elliptic systems in lipschitz domains various layer potential operators are constructed for general elliptic systems of partial differential equations with constant coefficients these operators are used to study the boundary value problems for these systems for a bounded lipschitz domain in rn n 3 it is shown that if the coefficients of the system satisfy the legendre hadamard condition and a symmetry condition then there exists a unique solution u of the system that solves the dirichlet problem for given data g in lp for p in a neighborhood of 2 also the oblique derivative problem for the system is studied furthermore some regularity results for the dirichlet problem are obtained

preservation of hopf bifurcation for neutral delay differential equations by θ methods this paper considers the preservation of hopf bifurcation of some neural delay differential equations by q method by analyzing the dynamics of the numerical discrete system derived by q method we show that q method could inherit the hopf bifurcation and the asymptotical stability for sufficiently small stepsize h 1 m where m is a positive integer in particular for q 1 2 the result holds for any stepsize h 1 m furthermore the stability of the closed invariant curve is established finally some numerical examples are illustrated to support the analytic results

stability properties of numerical methods for solving delay differential equations abstract stability properties of numerical methods for delay differential equations are considered some suitable definitions for the stability of the numerical methods are included and runge kutta type methods satisfying these properties are tested on a numerical example

stability analysis of numerical methods for delay differential equations this paper deals with the stability analysis of step by step methods for the numerical solution of delay differential equations we focus on the behaviour of such methods when they are applied to the linear testproblemu t u t u t with 0 and complex a general theorem is presented which can be used to obtain complete characterizations of the stability regions of these methods

the stability of difference formulas for delay differential equations a new simple geometric technique is presented for analyzing the stability of difference formulas for the model delay differential equation y t py t qy t 8 where p and q are complex constants and the delay 8 is a positive constant the technique is based on the argument principle and directly relates the region of absolute stability for ordinary differential equations corresponding to the py t term with the region corresponding to the delay term qy t 8 a sufficient condition for stability is that these regions be disjoint the technique is used to show that for each a stable a a stable or stiffly stable linear multistep formula for ordinary differential equations there is a correspond ing linear multistep formula for delay differential equations with analogous stability properties the analogy does not extend however to a stable one step formulas

the lp boundary value problems on lipschitz domains abstract let be a bounded lipschitz domain in r n we develop a new approach to the invertibility on l p of the layer potentials associated with elliptic equations and systems in as a consequence for n 4 and 2 n 1 n 1 e p 2 where e 0 depends on we obtain the solvability of the l p neumann type boundary value problems for second order elliptic systems the analogous results for the biharmonic equation are also established

numerical stability of the t methods for systems of differential equations with several delay terms abstract consider the following two dimensional delay differential equation dde u t a 1 u t b 1 v t d 2 v t a 2 v t b 2 u t d 1 t t 0 with the initial conditions u t u 0 t t 0 d 1 t t 0 v t v 0 t t 0 d 2 t t 0 where a j b j c d j 0 and u 0 t 0 t are continuous and complex valued the author give the sufficient conditions for the asymptotic stability of equation 0 1 the numerical stabilities of the linear method and the one leg method are investigated by analysing their stabilities behaviour in the solutions of 0 1 it is shown that the linear method is gp stable if and only if 1 2 1 and that the one leg method is gp stable if and only if 1

the adomian decomposition method in turning point problems in this paper the decomposition method is applied to boundary value problems of ordinary differential equations with a parameter exhibiting turning points

necessary and sufficient conditions for the solvability of the l p dirichlet problem on lipschitz domains we study the homogeneous elliptic systems of order 2 ell with real constant coefficients on lipschitz domains in mathbb r n n ge 4 for any fixed p 2 we show that a reverse holder condition with exponent p is necessary and sufficient for the solvability of the dirichlet problem with boundary data in l p we also obtain a simple sufficient condition as a consequence we establish the solvability of the l p dirichlet problem for n ge 4 and 2 epsilon p frac 2 n 1 n 3 epsilon the range of p is known to be sharp if ell ge 2 and 4 le n le 2 ell 1 for the polyharmonic equation the sharp range of p is also found in the case n 6 7 if ell 2 and n 2 ell 2 if ell ge 3

stability in linear multistep methods for pure delay equations abstract the stability regions of linear multistep methods for pure delay equations are compared with the stability region of the delay equation itself a criterion is derived stating when the numerical stability region contains the analytical stability region this criterion yields an upper bound for the integration step conditional q stability these bounds are computed for the adams bashforth adams moulton and backward differentiation methods of orders 8 furthermore symmetric adams methods are considered which are shown to be unconditionally q stable finally the extended backward differentiation methods of cash are analysed

p stability properties of runge kutta methods for delay differential equations recently the author and others have studied a class of runge kutta methods for ddes these methods make use of certain constrained meshes which allow to get optimal order results in this paper we study their asymptotic stability properties using the test equation begin gathered y t ay t by t tau t 0 hfill y t g t for tau leqq t leqq 0 hfill end gathered wherea b 0 andg t is continuous and complex valued it is known thaty t 0 ast if b re a in particular we show that any astable one step collocation method for odes inherits the same property when it is applied to ddes with a constrained mesh i e it is p stable

stability analysis for delay differential equations with multidelays and numerical examples in this paper we are concerned with the asymptotic stability of the delay differential equation x t a 0 x t a k x t k where a 0 a k c d d are constant complex matrices and x t k x 1 t k1 x 2 t k2 x d t kd t kl 0 stand for n x d constant delays k 1 n i 1 d we obtain two criteria for stability through the evaluation of a harmonic function on the boundary of a certain region we also get similar results for the neutral delay differential equation x t lx t m i x t i n j x t j where l m i and nj e c d d are constant complex matrices and i j 0 stands for constant delays i 1 m j 1 n numerical examples on various circumstances are shown to check our results which are more general than those already reported

weighted maximal regularity estimates and solvability of non smooth elliptic systems i we develop new solvability methods for divergence form second order real and complex elliptic systems above lipschitz graphs with l 2 boundary data the coefficients a may depend on all

the regularity problem for second order elliptic operators with complex valued bounded measurable coefficients the present paper establishes a certain duality between the dirichlet and regularity problems for elliptic operators with t independent complex bounded measurable coefficients t being the transversal direction to the boundary to be precise we show that the dirichlet boundary value problem is solvable in l p subject to the square function and non tangential maximal function estimates if and only if the corresponding regularity problem is solvable in l p moreover the solutions admit layer potential representations in particular we prove that for any elliptic operator with t independent real possibly non symmetric coefficients there exists a p 1 such that the regularity problem is well posed in l p

solution of nonlinear equations by modified adomian decomposition method in this paper we modify the standard adomian method for solution of the nonlinear equation f x 0 four examples are presented and compared using standard and modified adomian methods

hardy spaces and the neumann problem in l p for laplace s equation in lipschitz domains on donne des resultats optimaux pour la resolubilite du probleme de neumann dans des domaines de lipschitz a donnees dans l p on obtient des resultats du point extreme correspondant pour des espaces de hardy

the l p regularity problem for the stokes system on lipschitz domains we study the lp regularity problem for the stationary stokes system on lipschitz domains for any p 2 we show that a weak reverse holder inequality with exponent p is both necessary and sufficient for the solvability of the regularity problem with data in w1 p lnp we also obtain the w1 p estimate u lp q lp f lp in a bounded lipschitz domain rd for solutions to the dirichlet problem u q div f div u 0 in and u 0 on where 1p 12 12d e