symplectic geometry and dissipative differential operators abstract everitt and markus characterized the domains of self adjoint differential operators in terms of lagrangian subspaces of complex symplectic spaces in this paper we define dissipative and strictly dissipative subspaces for complex symplectic spaces and characterize the domains of dissipative and strictly dissipative differential operators in terms of these subspaces

global existence and blow up of solutions to a nonlocal quasilinear degenerate parabolic system abstract this paper investigates the properties of nonnegative solutions of a quasilinear degenerate parabolic system u t div u p 2 u a v x t d x v t div v q 2 v b u x t d x with zero dirichlet boundary conditions in a smooth bounded domain r n n 1 where p q 2 1 and a b 0 are positive constants under appropriate hypotheses we first establish the local existence and uniqueness of solutions then we show that whether or not the solution blows up in finite time depends on the initial data and the relations between and p 1 q 1 in the special case of q 1 and p 1 we conclude that the solution exists globally if p 1 d x q 1 d x 1 a b while if p 1 d x q 1 d x 1 a b then the solution blows up in finite time here x and x denote the unique solution of the following elliptic problem div p 2 1 in x 0 and div q 2 1 in x 0 respectively

an efficient algorithm for solving integro differential equations system in this paper an application of he s homotopy perturbation hpm method is applied to solve of system of integro differential equations the results reveal that the hpm is very effective and simple

global existence and l estimates of solution for doubly nonlinear parabolic equation the existence uniqueness and l estimate of the weak solution to the initial boundary value problem for the doubly nonlinear parabolic equationut div u r u m 2 u au fx t uin 0 with zero boundary condition in a bounded domain rn are established in particular the behavior of the solution as t 0 and t is investigated

large time behaviour of the doubly nonlinear porous medium equation abstract the main goal of this paper is to study the asymptotic behaviour of nonnegative solutions of the general porous medium equation in the slow diffusion case as well as in the limit case when diffusion loses its slow character we show that these solutions approach the unique solution of the associated steady state problem when t goes to infinity

some existence and nonexistence theorems for solutions of degenerate parabolic equations abstract the initial and boundary value problem for the degenerate parabolic equation v t v f v in the cylinder 0 r n bounded for a certain class of point functions satisfying v 0 e g v v m sign v is considered in the case that f v sign v c 1 v the equation has a global time solution the same is true for 1 provided the measure of is sufficiently small in the case that f v v is nondecreasing a condition is given on the initial state v x 0 which implies that the solution must blow up in finite time the existence of such initial states is discussed

application of homotopy perturbation method for solving eighth order boundary value problems homotopy perturbation method is applied to the numerical solution for solving eighth order boundary value problems comparison of the result obtained by the present method with that obtained by modified decomposition method m mesrovic the modified decomposition method for eighth order boundary value problems appl math comput 2006 doi 10 1016 j amc 2006 11 015 reveals that the present method is very effective and convenient

smoothing effects of the initial boundary value problem for logarithmic type quasilinear parabolic equations abstract we give existence theorems of global solutions in l l o c 0 w 0 1 to the initial boundary value problem for quasilinear degenerate parabolic equations of the form u t div u 2 u 0 where the class of v 2 includes the logarithmic case u 2 log 1 u 2 for a typical example we assume that the initial data belong to w 0 1 p 0 p 0 2 or l r r 1 and we derive precise estimates for u t near t 0

self adjoint domains symplectic geometry and limit circle solutions abstract everitt and markus characterized the domains of self adjoint operator realizations of very general even and odd order symmetric ordinary differential equations in terms of lagrangian subspaces of symplectic spaces recently for the even order case with real coefficients wang sun and zettl constructed limit circle lc solutions and hao wang sun and zettl characterized the self adjoint domains in terms of lc solutions these lc solutions are higher order analogues of the celebrated titchmarsh weyl limit circle solutions in the second order case this lc characterization has been used to obtain information about the discrete continuous and essential spectra of these operators in this paper we investigate the connection between these two very different kinds of characterizations and thus add the methods of symplectic geometry to the techniques available for the investigation of the spectrum of self adjoint operators

characterization of domains of self adjoint ordinary differential operators article history the gkn glazman krein naimark theorem characterizes all self adjoint realizations of linear symmetric formally self adjoint ordinary differential equations in terms of maximal domain functions these functions depend on the coefficients and this dependence is implicit and complicated in the regular case an explicit characterization in terms of two point boundary conditions can be given in the singular case when the deficiency index d is maximal the gkn characterization can be made more explicit by replacing the maximal domain functions by a solution basis for any real or complex value of the spectral parameter i n the much more difficult intermediate cases not all solutions contribute to the singular self adjoint conditions in 1986 sun found a representation of the self adjoint singular conditions in terms of certain solutions for nonreal values of in this paper we give a representation in terms of certain solutions for real this leads to a classification of solutions as limit point lp or limit circle lc in analogy with the celebrated weyl classification in the second order case the lc solutions contribute to the singular boundary conditions the lp solutions do not the advantage of using real is not only because it is in general easier to find explicit solutions but more importantly it yields information about the spectrum

interior and boundary continuity of weak solutions of degenerate parabolic equations in this paper we consider degenerate parabolic equations of the form bf u t diva x t u u b x t u u d 0 where a and b are respectively vector and scalar valued baire functions defined on u x r1 x r where u is an open subset of rn x t the functions a and b are subject to natural structural inequalities sufficiently general conditions are allowed on the relation b c r1 x rl so that the porus medium equation and the model for the two phase stefan problem can be considered the main result of the paper is that weak solutions of are continuous throughout u in the event that u q x 0 t where q is an open set of r it is also shown that a weak solution is continuous at xo to c aq x 0 t provided xo is a regular point for the laplacian on u 0 introduction in this paper we consider the problem of continuity of weak solutions of parabolic equations of the form at 1 tf3 u diva x t u ux b x t u ux 3 0 where a and b are respectively vector and scalar valued baire functions defined on u x r x rn and u is an open subset of r n x t throughout this paper u will take the form qt q x 0 t where q is an open bounded subset of rn the functions a and b are required to satisfy the following structural inequalities ia x t u p i co p 12 c1 x t i u 12 c2 x t where ao bo and c0 are constants with ao and bo nonnegative and c0 0 the remaining coefficients are nonnegative measurable functions which for simplicity of exposition we will assume to be bounded thus we assume there is a nonnegative constant k such that 3 a x t k b1 x t k ci x t k our results remain valid if the coefficients lie in those appropriate lebesgue classes which have been considered by previous authors e g lsu t throughout we received by the editors march 28 1981 1980 mathematics subject classification primary 35k10 35k15 secondary 35k20 35k65 research supported in part by a grant from the national science foundation d1982 american mathematical society 0002 9947 81 0000 0376 05 00

application of he s homotopy perturbation method for nth order integro differential equations abstract in this paper he s homotopy perturbation method is proposed to solve n th order integro differential equations the results reveal that the method is very effective and simple

a necessary and sufficient condition for finite speed of propagation in the theory of doubly nonlinear degenerate parabolic equations a degenerate parabolic partial differential equation with a time derivative and first and second order derivatives with respect to one spatial variable is studied the coefficients in the equation depend nonlinearly on both the unknown and the first spatial derivative of a function of the unknown the equation is said to display finite speed of propagation if a non negative weak solution which has bounded support with respect to the spatial variable at some initial time also possesses this property at later times a criterion on the coefficients in the equation which is both necessary and sufficient for the occurrence of this phenomenon is established according to whether or not the criterion holds weak travelling wave solutions or weak travelling wave strict subsolutions of the equation are constructed and used to prove the main theorem via a comparison principle applications to special cases are provided

on the cauchy problem for the evolution p laplacian equations with gradient term and source abstract in this paper the authors study the equation u t div d u p 2 d u u q 1 u d u l in r n with p 2 we first prove that for 1 l p 1 the solution exists at least for a short time then for p 2 l p 1 the existence and nonexistence of global in time solutions are studied in various situations

solving two point boundary value problems using combined homotopy perturbation method and green s function method in this paper an algorithm is proposed for the solution of second order boundary value problems with two point boundary conditions the green s function method is applied first to transform the ordinary differential equation into an equivalent integral one which has already satisfied the boundary conditions and then the homotopy perturbation method is used to the resulting equation to construct the numerical solution for such problems numerical examples demonstrate the efficiency and reliability of the algorithm developed it is quite accurate and readily implemented for both linear and nonlinear differential equations with homogeneous and nonhomogeneous boundary conditions furthermore the lower order approximation is of higher accuracy for most cases some other extended applications of this algorithm are also exhibited

iterative method and convergence analysis for a kind of mixed nonlinear volterra fredholm integral equation in this paper the iterative method is presented for numerically solving the nonlinear volterra fredholm integral equation first considering some conditions on k i and l i i 1 2 of the integral equation then we define the equidistance collocation points and the integral part of equation is discretized by newton cotes integration formula finally the approximate solutions of integral equation are obtain by iterative method the convergence analysis of the integral equation is given some numerical examples are given to illustrate the accuracy and dependability of the method

continuous spectrum and square integrable solutions of differential operators with intermediate deficiency index we explore the connection between square integrable solutions for real values of the spectral parameter and the continuous spectrum of self adjoint ordinary differential operators with arbitrary deficiency index d we show that if for all in an open interval i there are d of linearly independent square integrable solutions then for every extension of dmin the point spectrum is nowhere dense in i and there is a self adjoint extension of smin which has no continuous spectrum in i this analysis is based on our construction of limit point lp and limit circle lc solutions obtained recently in an earlier paper

blow up phenomena for polytropic equation with inhomogeneous density and source the subject of this investigation is the blow up phenomena of the positive solutions of the mixed problem for the one dimensional polytropic filtration equation with inhomogeneous density and source it is shown that under certain conditions on the nonlinearities and data blow up will occur at some finite time note that the technique applied for the proof does not use the zel dovich kompaneets barenblatt solutions since the construction of such type of function is more complicated in our case therefore we obtain a result by multiplying on a special factor which has convenient properties in particular by choosing the parameters of the factor and using the properties of the solution we obtain the inequality which allows us to show the blow up phenomena

the classification of self adjoint boundary conditions separated coupled and mixed there are three basic types of self adjoint regular and singular boundary conditions separated coupled and mixed for even order problems with real coefficients one regular endpoint and arbitrary deficiency index d we give a construction for each type and determine the number of possible conditions of each type under the assumption that there are d linearly independent square integrable solutions for some real value of the spectral parameter in the separated case our construction yields non real conditions for all orders greater than two it is well known that no such conditions exist in the second order case our construction gives a direct alternative to the recent construction of everitt and markus which uses the theory of symplectic spaces we believe our construction will prove useful in the spectral analysis of these operators and in obtaining canonical forms of self adjoint boundary conditions such forms are known only in the second order i e sturm liouville case even for regular problems of order four no such forms are available

the asymptotic behavior of solutions for a class of doubly degenerate nonlinear parabolic equations abstract by using the morse interaction technique supposing that the uniqueness of the barenblatt type solution is true the paper studies the large time asymptotic behavior of solutions for the doubly degenerate parabolic equation u t div d u m p 2 d u m d u m p 1 u q with initial condition u x 0 u 0 x here the exponents m p p 1 q satisfy p p 1 p 1 q m p 1 1 p 1 m 1

study of weak solutions for parabolic equations with nonstandard growth conditions abstract the authors of this paper study the existence and uniqueness of weak solutions of the initial and boundary value problem for u t div u d 0 u p x t 2 u f x t localization property of weak solutions is also discussed

a priori l r error estimates for galerkin approximations to porous medium and fast diffusion equations galerkin approximations to solutions of a cauchy dirichlet problem governed by the generalized porous medium equation formulamath on bounded convex domains are considered the range of the parameter includes the fast diffusion case 1 2 using an euler finite difference approximation in time the semi discrete solution is shown to converge to the exact solution in l 0 t l norm with an error controlled by o t1 4 for 1 2 and o t1 2 for 2 for the fully discrete problem a global convergence rate of o t1 4 in l 2 0 t l norm is shown for the range 2n n 1 2 for 2 a rate of o t1 2 is shown in l 0 t l norm

on the initial boundary value problem for some quasilinear parabolic equations of divergence form abstract in this paper we give an existence theorem of global classical solution to the initial boundary value problem for the quasilinear parabolic equations of divergence form u t div u 2 u f u u x t where u 2 may not be bounded as u as an application the logarithmic type nonlinearity u 2 log 1 u 2 which is growing as u and degenerate at u 0 is considered under f 0

an unusual self adjoint linear partial differential operator hhin an american mathematical society memoir published in 2003 the authors everitt and markus apply their prior theory of symplectic algebra to the study of symmetric linear partial differential expressions and the generation of self adjoint differential operators in sobolev hilbert spaces in the case when the differential expression has smooth coefficients on the closure of a bounded open region in euclidean space and when the region has a smooth boundary this theory leads to the construction of certain self adjoint partial differential operators which cannot be defined by applying classical or generalized conditions on the boundary of the open region this present paper concerns the spectral properties of one of these unusual self adjoint operators sometimes called the harmonic operator the boundary value problems considered in the memoir see above and in this paper are called regular in that the cofficients of the differential expression do not have singularities within or on the boundary of the region also the region is bounded and has a smooth boundary under these and some additional technical conditions it is shown in the memoir and emphasized in this present paper that all the self adjoint operators considered are explicitly determined on their domains by the partial differential expression this property makes a remarkable comparison with the case of symmetric ordinary differential expressions in the regular ordinary case the spectrum of all the self adjoint operators is discrete in that it consists of a countable number of eigenvalues with no finite point of accumulation and each eigenvalue is of finite multiplicity thus the essential spectrum of all these operators is empty this spectral property extends to the present partial differential case for the classical dirichlet and neumann operators but not to the harmonic operator it is shown in this paper that the harmonic operator has an eigenvalue of infinite multiplicity at the origin of the complex spectral plane thus the essential spectrum of this operator is not empty both the weak and strong formulations of the harmonic boundary value problem are considered these two formulations are shown to be equivalent in the final section of the paper examples are considered which show that the harmonic operator defined by the methods of symplectic algebra has a domain that cannot be determined by applying either classical or generalized local conditions on the boundary of the region

extinction and positivity for the evolution p laplacian equation with l1 initial value abstract the aims of this paper are to discuss extinction and positivity for the evolution p laplacian equation u t x u x p 2 u x with initial value u 0 x l 1 and zero boundary value conditions where p 1 in particular the necessary and sufficient condition for extinction is obtained

homotopy perturbation technique abstract the homotopy perturbation technique does not depend upon a small parameter in the equation by the homotopy technique in topology a homotopy is constructed with an imbedding parameter p 0 1 which is considered as a small parameter some examples are given the approximations obtained by the proposed method are uniformly valid not only for small parameters but also for very large parameters

maximum estimates for generalized forchheimer flows in heterogeneous porous media this article continues the study in 4 of generalized forchheimer flows in heterogeneous porous media such flows are used to account for deviations from darcy s law in heterogeneous media the derived nonlinear partial differential equation for the pressure can be singular and degenerate in the spatial variables in addition to being degenerate for large pressure gradient here we obtain the estimates for the l l norms of the pressure and its time derivative in terms of the initial and the time dependent boundary data they are established by implementing de giorgi moser s iteration in the context of weighted norms with the weights specifically defined by the forchheimer equation s coefficient functions with these weights we prove suitable weighted parabolic poincare sobolev inequalities and use them to facilitate the iteration moreover local in time l l bounds are combined with uniform gronwall type energy inequalities to obtain long time l l estimates

continuity of solutions of a singular parabolic equation which is of parabolic divergence form since 3 0 a e however it is the solution u of 0 1 which will be continuous while the solution u of 0 2 need not be in certain physical applications u corresponds to temperature while u is the enthalpy the difficulty in the analysis of this equation written in the form 0 2 stems from the fact that c need not be bounded or bounded away from zero thus the equation may be of degenerate or singular parabolic type as particular cases of the equations covered by our results we mention the three model problems

new iterative methods for nonlinear equations by modified hpm a numerical method based on modified homotopy perturbation method hpm is proposed for solving nonlinear algebric equations it is shown that the proposed method has third order convergence to assess its validity and accuracy the method is applied to solve several test problems

a third order newton type method for nonlinear equations based on modified homotopy perturbation method in this paper we present a new iterative method for solving nonlinear algebric equations by using modified homotopy perturbation method we also discuss the convergence criteria of the present method to assess its validity and accuracy the method is applied to solve several test problems

numerical solution of volterra fredholm integral equations via modification of hat functions abstract in this paper a new numerical approach is developed for solving linear and nonlinear volterra fredholm integral equations the fundamental structure of the presented method is based on the modification of hat functions the properties of modification of hat functions mhfs are first presented after implementation of our scheme the solution of the main problem would be transformed into the solution of a system of linear or nonlinear algebraic equations also an error analysis is provided under several mild conditions in addition examples are presented to illustrate the pertinent features of the method and the results are discussed

the blow up phenomenon for degenerate parabolic equations with variable coefficients and nonlinear source abstract the cauchy problem for a degenerate parabolic equation with a source and variable coefficient of the form u t d i v x u m 1 d u 1 d u u p is studied global in time existence and nonexistence conditions are found for a solution to the cauchy problem exact estimates of a solution are obtained in the case of global solvability a sharp universal i e independent of the initial function estimate of a solution near the blow up time is obtained

the homotopy perturbation method for nonlinear oscillators with discontinuities in the paper the homotopy perturbation method is applied to the nonlinear oscillators with discontinuities only one iteration leads to high accuracy of the solutions

the fisher kpp problem with doubly nonlinear fast diffusion abstract the famous fisher kpp reaction diffusion model combines linear diffusion with the typical fisher kpp reaction term and appears in a number of relevant applications it is remarkable as a mathematical model since in the case of linear diffusion it possesses a family of travelling waves that describe the asymptotic behaviour of a wide class of solutions 0 u x t 1 of the problem posed in the real line the existence of propagation wave with finite speed has been confirmed in the cases of slow and pseudo linear doubly nonlinear diffusion too see audrito and vazquez 2016 we investigate here the corresponding theory with fast doubly nonlinear diffusion and we find that general solutions show a non tw asymptotic behaviour and exponential propagation in space for large times finally we prove precise bounds for the level sets of general solutions even when we work in with spacial dimension n 1 in particular taking spatial logarithmic scale we show that the location of the positive level sets is approximately linear for large times this represents a strong departure from the linear case in which the location of the level sets is not purely linear but presents the celebrated logarithmic deviation for large times

support properties of solutions to a degenerate equation with absorption and variable density abstract in this paper we study the properties of solutions to a degenerate parabolic equation with variable density and absorption we first obtain a critical exponent which distinguishes the localization of solutions from the positivity of them when positivity prevails we obtain the other critical exponent with respect to the decay of the variable density which separates the global existence of interfaces from the disappearance of them moreover the long time behavior of interfaces is characterized

newton like iterative methods for solving system of non linear equations abstract homotopy perturbation method hpm is applied to construct a new family of newton like iterative methods for solving system of non linear equations comparison of the result obtained by the present method with newton raphson method reveals that the accuracy and fast convergence of the new method

cauchy problem for doubly nonlinear parabolic equations with initial data measures abstract we study the existence of solutions for the doubly nonlinear parabolic equation with strongly nonlinear source under optimal assumptions on the initial data which are assumed to be measures we also obtain some local regularity estimates for the supercritical case q m p 1 p n

numerical solution of linear volterra integral equations system of the second kind abstract there are several numerical approaches for solving linear volterra integral equations system of the second kind in this paper we have presented a method for numerical solution of linear volterra integral equations system based on the power series method the major advantage of which is being derivative free also this method reproduces the analytical solution when the exact solutions are polynomial the numerical results prove that the presented method is very effective and simple the software used for the numerical calculations in this study was matlab 7 4

a new family of iterative methods for solving system of nonlinear algebric equations abstract homotopy perturbation method hpm is applied to construct a new iterative method for solving system of nonlinear algebric equations comparison of the result obtained by the present method with that obtained by revised adomian decomposition method hossein jafari varsha daftardar gejji appl math comput 175 2006 1 7 reveals that the accuracy and fast convergence of the new method

the fisher kpp problem with doubly nonlinear diffusion abstract the famous fisher kpp reaction diffusion model combines linear diffusion with the typical kpp reaction term and appears in a number of relevant applications in biology and chemistry it is remarkable as a mathematical model since it possesses a family of travelling waves that describe the asymptotic behaviour of a large class solutions 0 u x t 1 of the problem posed in the real line the existence of propagation waves with finite speed has been confirmed in some related models and disproved in others we investigate here the corresponding theory when the linear diffusion is replaced by the slow doubly nonlinear diffusion and we find travelling waves that represent the wave propagation of more general solutions even when we extend the study to several space dimensions a similar study is performed in the critical case that we call pseudo linear i e when the operator is still nonlinear but has homogeneity one with respect to the classical model and the pseudo linear case the slow travelling waves exhibit free boundaries

a numerical approach for solving generalized abel type nonlinear differential equations in this paper a numerical power series algorithm which is based on the improved taylor matrix method is introduced for the approximate solution of abel type differential equations and also riccati differential equations the technique is defined and illustrated with some numerical examples the obtained results reveal that the method is very effective simple and valid high accuracy the method can be easily extended to other nonlinear equations

a complete characterization of nonlinear absorption for the evolution p laplacian equations to have positive or extinctive solutions this paper is concerned with long time behaviors of solutions to the initial boundary value problem of the evolution p laplacian equations with nonlinear absorption a long time behavior of solutions to these equations has been studied so far for particular types of nonlinear absorption term the purpose of this paper is to give a complete characterization of the nonlinear absorption term according to the parameter p in p laplacian operator and the growth of the nonlinear absorption term near the origin in order to determine whether the solution to the equations is extinctive or positive in addition we also give upper bounds for extinction times of solutions

some observations and remarks on differential operators generated by first order boundary value problems this paper deals with the study of the set of all self adjoint differential operators which are generated from first order linear ordinary boundary value problems these operators are defined on a weighted hilbert function space and are examined as an application of the result obtained by everitt and markus in their paper in 1997 an investigation is given so that first order self adjoint boundary value problems are transformed to a study of the nature of the spectrum of associated self adjoint operators however the analysis of this paper is restricted to consideration of conditions under which the spectral properties of these operators yield a discrete spectrum and consequently to the determination of conditions under which the construction of kramer analytic kernels from the above boundary value problems can be accomplished

a new matrix method for solving two dimensional time dependent diffusion equations with dirichlet boundary conditions abstract this paper is devoted to develop a new matrix scheme for solving two dimensional time dependent diffusion equations with dirichlet boundary conditions we first transform these equations into equivalent integro partial differential equations pdes such these integro pdes contain both of the initial and boundary conditions and can be solved numerically in a more appropriate manner subsequently all the existing known and unknown functions in the latter equations are approximated by bernoulli polynomials and operational matrices of differentiation and integration together with the completeness of these polynomials can be used to reduce the integro pdes into the associated algebraic generalized sylvester equations for solving these algebraic equations an efficient krylov subspace iterative method i e bicgstab is implemented two numerical examples are given to demonstrate the efficiency accuracy and versatility of the proposed method

application of fibonacci collocation method for solving volterra fredholm integral equations in this paper a new matrix method based on fibonacci polynomials and collocation points is proposed for numerically solving the volterra fredholm integral equations in fact the approximate solution of the problem in the truncated fibonacci series form is obtained by this method also convergence analysis of the proposed method is provided under several mild conditions the reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments

taylor polynomial solutions of nonlinear volterra fredholm integral equations the kanwal and liu method for the solution of fredholm integral equations is applied to certain nonlinear volterra fredholm integral equations in addition examples that illustrate the pertinent features of the method are presented and the results of study are discussed

the sturm liouville group in several recent publications everitt and markus have developed a theory of ordinary and partial differential boundary value problems through their novel methods of complex symplectic algebra for instance in the case of the regular sturm liouville problem the corresponding boundary complex symplectic space has four dimensions and can be used to classify all the self adjoint boundary conditions by means of two dimensional lagrangian subspaces in this investigation the groups auto s of all symplectic automorphisms are analyzed for all finite dimensional complex symplectic spaces s say of dimension n 2 and these are demonstrated to be noncommutative connected lie groups of real dimension n2 this global topological analysis provides information on continuous families of canonical coordinates in s and their deformations

dissipative operators symplectic geometry and limit circle solutions for a general symmetric ordinary differential expression m and a positive weight function w the authors characterized the dissipative and strictly dissipative extensions of the minimal operator generated by m in terms of subspaces of a symplectic geometry space here we characterize these subspaces in terms of lc solutions namely we characterize the boundary conditions of the dissipative and strictly dissipative extensions by lc solutions in a hilbert space

global existence and blow up solutions and blow up estimates for a non local quasilinear degenerate parabolic system this paper deals with p laplacian systems ut div u p 2 u a u 1 x t v 1 x t dx x t 0 vt div v q 2 v b u 2 x t v 2 x t dx x t 0 with null dirichlet boundary conditions in a smooth bounded domain rn where p q 1 i i 0 i 1 2 and a b 0 are positive constants we first get the non existence result for a related elliptic systems of non increasing positive solutions secondly by using this non existence result blow up estimates for above p laplacian systems with the homogeneous dirichlet boundary value conditions are obtained under br x rn x 0 then under appropriate hypotheses we establish local theory of the solutions and obtain that the solutions either exists globally or blow up in finite time

the glazman krein naimark theory for a class of discrete hamiltonian systems abstract in this paper the glazman krein naimark theory for a class of discrete hamiltonian systems is developed a minimal and a maximal operators gkn sets and a boundary space for the system are introduced algebraic characterizations of the domains of self adjoint extensions of the minimal operator are given a close relationship between the domains of self adjoint extensions and the gkn sets is established it is shown that there exist one to one correspondences among the set of all the self adjoint extensions the set of all the d dimensional lagrangian subspaces of the boundary space and the set of all the complete lagrangian subspaces of the boundary space

gkn theory for linear hamiltonian systems abstract in this paper we give the definition of maximal and minimal operators for linear hamiltonian systems and investigate the relationship between the conjugate scalar product in a weighted hilbert space and the skew symmetric boundary form of the associated singular hamiltonian operator namely the one to one correspondence between the set of self adjoint extensions of the minimal operator and the set of lagrangian symplectic subspaces these results extend and improve the classical glazman krein naimark gkn theory for quasi differential operators

on a weyl titchmarsh theory for discrete symplectic systems on a half line recently bohner and sun introduced basic elements of a weyl titchmarsh theory into the study of discrete symplectic systems we extend this development through the introduction of weyl titchmarsh functions together with a preliminary study of their properties a limit point criterion is described and characterized green s function for the half line is introduced as a limit of such functions in the regular case and half line solutions obtained are seen to satisfy lambda dependent boundary conditions at infinity

on solutions of some doubly nonlinear degenerate parabolic equations with absorption abstract the existence uniqueness regularity and behavior of solutions to the initial boundary value problem for the equation u t i 1 n x i u x i p 2 u x i u 0 with zero dirichlet boundary condition and with nonnegative initial condition are studied in a bounded domain in r n where 0 and p 1

cauchy problem for doubly degenerate parabolic equation with gradient source abstract we consider the cauchy problem for a doubly degenerate parabolic equation with a source term depending solely on the gradient a major result of this paper is to establish the local existence of solutions for this equation under optimal stipulated assumptions on the initial data

real parameter square integrable solutions and the spectrum of differential operators abstract we continue to investigate the connection between the spectrum of self adjoint ordinary differential operators with arbitrary deficiency index d and the number of linearly independent square integrable solutions for real values of the spectral parameter we show that if for all in an open interval i there are d linearly independent square integrable solutions then there is no continuous spectrum in i this for any self adjoint realization with boundary conditions which may be separated coupled or mixed the proof is based on a new characterization of self adjoint domains and on limit point lp and limit circle lc solutions established in an earlier paper

numerical solutions of the nonlinear volterra fredholm integral equations by using homotopy perturbation method abstract a numerical method for solving the nonlinear volterra fredholm integral equations is presented the method is based upon homotopy perturbation theory the results reveal that the homotopy perturbation method is very effective and simple and gives the exact solution

comparison of homotopy perturbation method and homotopy analysis method comparison of homotopy perturbation method hpm and homotopy analysis method is made revealing that the former is more powerful than the later furthermore the hpm is further developed in this paper by applying the modern perturbation methods

an efficient algorithm for solving nonlinear volterra fredholm integral equations in this work we develop a new effective method for solving nonlinear volterra fredholm integral equation the existence of any e approximate solution is proved at the same time an effective method for obtaining the e approximate solution is established the final numerical examples illustrate that our approach is valid not only for weakly nonlinear problems but also for strongly nonlinear problems

he s homotopy perturbation method an effective tool for solving nonlinear integral and integro differential equations in this paper we conduct a comparative study among he s homotopy perturbation method and three traditional methods for an analytic and approximate treatment of nonlinear integral and integro differential equations the proper implementation of he s homotopy perturbation method can extremely minimize the size of work if compared to existing traditional techniques the analysis is accompanied by examples that demonstrate the comparison and shows the pertinent features of the homotopy perturbation technique

iterative methods to nonlinear equations homotopy perturbation method is applied to construct a new iterative method for solving nonlinear algebraic equations the convergence of the new scheme is proposed an at least the cubic order of convergence is established several examples are presented and compared to other methods showing the accuracy and fast convergence of the new method

numerical solutions of the integral equations homotopy perturbation method and adomian s decomposition method in this paper a homotopy perturbation method is proposed to solve non singular integral equations comparisons are made between adomian s decomposition method and the proposed method it is shown adomian s decomposition method is a homotopy only finally by using homotopy perturbation method a new iterative scheme like adomian s decomposition method is proposed for solving the non singular integral equations of the first kind the results reveal that the proposed method is very effective and simple

on uniform and decay estimates for unbounded solutions of partial differential equations abstract we prove that if a function u satisfies certain integral estimates of energy type then it verifies estimates of the type u t l r c u 0 l r 0 0 t 1 t 0 where r and r 0 are exponents that appear in these integral estimates and 1 and 0 are positive constants that can be expressed in terms of these constants we will see that in some cases r r 0 supercontractive estimates and hence a regularizing effect on u appears while in other cases this improvement of regularity does not appear since r r 0 in any case we prove that the l r norm of u decays in time for t large since it verifies the previous estimate we show how to apply this result to obtain new estimates of this kind for the solutions of many nonlinear parabolic equations this new method allows also to explain the reason of the similar behavior of the solutions of very different parabolic problems finally we study sufficient conditions for extinction in finite time