discontinuous generalized solutions of nonlinear nonconservative hyperbolic equations abstract nonlinear nonstationary problems arising in elastodynamics have naturally a nonconservative form for these systems it is not possible to define a weak solution according to distribution theory in order to define the concept of discontinuous solutions for these systems one is confronted with multiplications of distributions this is unavoidable from a physical viewpoint since discontinuous solutions are needed to represent shock waves the theory of generalized functions of the second author in which one may multiply arbitrary distributions gives a concept of generalized solutions that may be discontinuous functions existence of such global solutions of the cauchy problem for a system of two equations is proved by a compactness argument from a convergent numerical scheme in the case under consideration the generalized solutions are bounded variation functions the results thus obtained agree with the results expected by the engineers and lead to the development of new numerical schemes

the functional analytic foundation of colombeau algebras abstract colombeau algebras constitute a convenient framework for performing nonlinear operations like multiplication on schwartz distributions many variants and modifications of these algebras exist for various applications we present a functional analytic approach placing these algebras in a unifying hierarchy which clarifies their structural properties as well as their relation to each other

an innovative eigenvalue problem solver for free vibration of euler bernoulli beam by using the adomian decomposition method this paper deals with free vibration problems of euler bernoulli beam under various supporting conditions the technique we have used is based on applying the adomian decomposition method adm to our vibration problems doing some simple mathematical operations on the method we can obtain ith natural frequencies and mode shapes one at a time the computed results agree well with those analytical and numerical results given in the literature these results indicate that the present analysis is accurate and provides a unified and systematic procedure which is simple and more straightforward than the other modal analysis

products and compositions with the dirac delta function the need to define pointwise products and compositions with distributions is pointed out in the context of the problems of renormalisation junction conditions and curved shock waves earlier definitions are briefly reviewed and new definitions are proposed using non standard analysis basic properties are established and some products and compositions with the delta distribution are explicitly evaluated with these definitions the domain of validity of the nonlinear differential equations of classical field theory can be extended to include rankine hugoniot equations are derived from the euler equations an immediate application to quantum field theory is pointed out

denseness for norm attaining operator valued functions abstract in this note we offer a short constructive proof for hilbert spaces of lindenstrauss famous result on the denseness of norm attaining operators specifically we show given any a l h there is a sequence of rank 1 operators k n such that a k n is norm attaining for each n and k n converges in norm to zero we then apply our construction to establish denseness results for norm attaining operator valued functions

nonlinear generalized functions on manifolds in this work we adopt a new approach to the construction of a global theory of algebras of generalized functions on manifolds based on the concept of smoothing operators this produces a generaliz

multiplications of distributions in elasticity and hydrodynamics it is shown how a mathematical theory of generalized functions in which the multiplications of distributions appearing in nonlinear equations of physics make sense gives new formulas and new numerical results the new methods shown here are quite general but since each particular problem requires its own study this paper is limited to elasticity and hydrodynamics in elasticity hooke s law gives systems in a nonconservative form the study of shock waves for these systems gives nonclassical multiplications of distributions of the form y y heaviside function dirac mass at the origin using this new mathematical tool new formulas are obtained more generally new numerical schemes in a first step ambiguous results are obtained then the ambiguity is removed in hydrodynamics a formulation is obtained that has a nonconservative form and is at the basis of efficient new numerical schemes strictly speaking the reader is not assumed to know anything either on distributions or on elasticity and hydrodynamics since the basic equations are recalled all computations done in this paper are rigorous from the mathematical viewpoint

a note on vibrations of generally restrained beams exact frequency and normal mode shape expressions are derived in this note for generally restrained bernoulli euler beams with unsymmetrical translations and rotations at either end the eigenfrequencies and mode shape parameters are presented for a wide range of restraint parameters several degenerate cases have been studied and some of these have been compared with those available in the published literature it is believed that the results presented in this paper will be of use in design of beams shafts and piping under dynamic considerations

closed form solutions for frequency equation and mode shapes of elastically supported euler bernoulli beams abstract this paper is concerned with the problem of free vibration of an euler bernoulli beam supported with an arbitrary number of translational springs of different constant stiffnesses the natural frequencies and mode shapes are calculated by means of the green s function method seven different boundary conditions are considered at both ends of the beam fixed pinned sliding free translational spring rotational spring and combined translational rotational spring each boundary condition on one end is combined with all other boundary conditions on the other yielding a total of 49 possible cases it has been shown in this paper that a green s function for any set of boundary conditions can be represented as a linear combination of function coefficients calculated for homogeneous boundary conditions and also a new form of the general solution is formulated using this principle green s functions are calculated and tabulated for each of the 49 possible cases which in essence closes the question of analytical solutions for this particular kind of problem using green s functions the tabulated solutions were used as the starting point in the derivation of the general formulae and to provide a new insight into the underlying mathematical structure of the solutions formulated with green s functions this can be of practical interest to the end user who wishes to implement them in computer code the validity of the obtained solutions is demonstrated with examples whose solutions are compared to the results in the published literature and to the results obtained with fem

generalized solutions of a nonlinear parabolic equation with generalized functions as initial data abstract in h brezis a friedman nonlinear parabolic equations involving measures as initial conditions j math pure appl 9 1983 73 97 brezis and friedman prove that certain nonlinear parabolic equations with the measure as initial data have no solution however in j f colombeau m langlais generalized solutions of nonlinear parabolic equations with distributions as initial conditions j math anal appl 1990 186 196 colombeau and langlais prove that these equations have a unique solution even if the measure is substituted by any colombeau generalized function of compact support here we generalize colombeau and langlais result proving that we may take any generalized function as the initial data our approach relies on recent algebraic and topological developments of the theory of colombeau generalized functions and results from j aragona colombeau generalized functions on quasi regular sets publ math debrecen 2006 371 399

finite rank harmonic operator valued functions abstract the purpose of this paper is to characterize when a harmonic function with values in the finite rank operators on a hilbert space is expressible as a harmonic matrix valued function we show that harmonic function with values in the rank 1 normal operators is expressible as a harmonic matrix valued function we also prove that for any natural number n a harmonic function with values in the rank n non negative operators is expressible as a matrix valued function and we give examples showing that these decomposition theorems fail when various hypotheses are relaxed

whitney s extension theorem for generalized functions abstract if x is a subset of r n we define generalized functions on x as a direct generalization of c functions on x in whitney s sense and of generalized functions on x when x is open then we prove that if x is closed any generalized function on x may be extended as a generalized function on r n which is a whitney s extension theorem for generalized functions this result generalizes borel s theorem for generalized functions already proved by the same authors the proof is an adaptation of a classical proof

the elastoplastic shock problem as an example of the resolution of ambiguities in the multiplication of distributions one dimensional collisions between two isotropic solids in which the equations of physics lead to multiplications of distributions are considered based on this example a general physicomathematical method to be adapted to each particular case is proposed to resolve the ambiguity inherent in such products this can be achieved with the aid of a new mathematical theory of generalized functions which permits dealing with mathematical phenomena of a microscopic nature that govern products of distributions having singularities at the same point this tool has recently been applied in various situations in continuum mechanics in which the equations of physics lead to heuristic products of distributions one obtains new algebraic formulas in the simplest cases and new numerical schemes in more general cases the key to the resolution of ambiguities lies in more precise statements of the laws of physics than are permitted within distribution theory and have no analog in classical analysis so that in general a resolution cannot be obtained from formal calculations

a note on transverse vibrations of continuous beams subject to an axial force and carrying concentrated masses abstract polynomial co ordinate functions are used in order to obtain an approximate solution to the title problem the algorithmic procedure is quite simple leads to adequate engineering accuracy and can be easily extended to the case of several intermediate supports

generalized solutions of nonlinear parabolic equations with distributions as initial conditions abstract we consider the nonlinear parabolic equation u t u u 3 0 in q 0 t t 0 open set in r d d 1 2 with the boundary condition u x t 0 on 0 t and the initial condition u x 0 x in where is the dirac mass at the origin of r it is known that this problem has no weak solution in any known classical sense within the distribution theory using a theory of generalized functions we obtain existence uniqueness and consistence results which describe mathematically the behaviour of the solutions ue obtained with smooth initial conditions u e x 0 e x e d and e when e 0

a simple model for a symmetrical theory of generalized functions v some applications abstract the theory of generalized functions is shown to eliminate ambiguities in products which are unavoidable in distribution theory the physical examples are the velocity of shock waves and the ito stratonovich dilemma

vibration of beams with generally restrained boundary conditions using fourier series this work presents a method to find accurate vibration frequencies of beams with generally restrained boundary conditions using fourier series the suggested method is very convenient to find an accurate frequency parameter for beams with not only classical boundary conditions but also non classical boundary conditions restrained by rotational and translational springs numerical results for various degenerate cases are compared with existing results for natural frequency obtained by conventional analysis the results indicate that the present method can be used very effectively in the design of beams with various supporting conditions

an approximate method for one dimensional structures with strong nonlinear and nonhomogenous boundary conditions abstract there is a lack of analytic methods for vibration analysis of structures with strong nonlinear and non homogenous boundary conditions this paper proposes such a method a distinct feature of the method is to introduce the generalized coordinates for nonlinear and nonhomogeneous boundary conditions the total solution of the structure consists of a sum of a spatial particular solution for the boundary conditions and a field solution satisfying linear and homogeneous boundary conditions the field solution can be obtained in terms of the modal expansion the modal coefficients together with the generalized coordinates for the boundary conditions satisfy a set of nonlinear ordinary differential equations which are solved with the harmonic balance method two examples are presented to study the convergence and accuracy of the solution the first example is the longitudinal vibration of a bar subject to a base excitation on the left and restrained by a damper on the right the wave method and the differential quadrature method dqm are compared with the proposed method and are found in good agreement the second example is a beam with nonlinear torsional boundary absorbers the method shows a good accuracy to deal with the complex boundary conditions as compared with the differential quadrature element method dqem the proposed method is also compared with the multi scale method msm and shows an advantage to deal with strong nonlinearities the proposed method is highly promising for vibration analysis of complex structures with nonlinear and nonhomogeneous boundary conditions

harnack s theorem for harmonic compact operator valued functions abstract in this paper we show that harmonic compact operator valued functions are characterized by having harmonic diagonal matrix coefficients in any choice of basis we also give an example which shows that an operator valued function with values outside the compact operators can have harmonic diagonal matrix coefficients in any choice of basis without being a harmonic operator valued function we use our harmonic matrix coefficients characterization to establish a harnack s theorem for an increasing sequence of harmonic compact self adjoint operator valued functions and we show that this harnack s theorem need not hold when the compactness restriction is dropped

generalized solutions of parabolic initial value problems with discontinuous nonlinearities we proved recently that parabolic initial value problems with discontinuous nonlinearities have no unique weak solution in general but have a unique generalized solution in the sense of colombeau in this paper we study the relationship between generalized solutions and weak solutions

dynamic analysis of generally supported beams using fourier series the applicability of fourier series to the dynamic analysis of beams having arbitrary boundary conditions is explored for the simplicity of presenting the analysis procedure beams having zero end displacement are used in the discussion numerical results for free vibration and dynamic response based on the present analysis procedure are compared with existing results for natural frequencies and dynamic behavior obtained by the conventional modal analysis these results indicate that the present analysis is accurate and provides a unified and systematic procedure which is seemingly simpler and more straightforward than the conventional modal analysis