an algorithmic method for showing existence of nontrivial non classical symmetries of partial differential equations without solving determining equations abstract in this paper based on differential characteristic set theory and the associated algorithm also called wu s method an algorithmic method is presented to decide on the existence of a nontrivial non classical symmetry of a given partial differential equation without solving the corresponding nonlinear determining system the theory and algorithm give a partial answer for the open problem posed by p a clarkson and e l mansfield in 21 on non classical symmetries of partial differential equations as applications of our algorithm non classical symmetries and corresponding invariant solutions are found for several evolution equations

first order compatibility for a 2 1 dimensional diffusion equation a class of first order partial differential equations is obtained compatible with a 2 1 dimensional linear diffusion equation with a nonlinear source term we will show that if the source term is quasilinear then compatible equations are quasilinear furthermore we obtain diffusion equations with source terms that admit non quasilinear compatible equations

a cg fft approach to the solution of a stress velocity formulation of three dimensional elastic scattering problems in this paper we introduce a conjugate gradient fast fourier transform cg fft scheme for the numerical solution of the integral equation formulating three dimensional elastic scattering problems the formulation is in terms of the stress tensor and particle velocities as the unknown field variables in contrast with the formulation based on particle displacements this approach leads to integral representations that do not involve derivatives of the unknown fields thus resulting in simplified and more stable numerics the numerical procedure is based on suitable quadrature formulas that provide second order accurate approximations while retaining the convolution nature of the relevant integrals that make them amenable to efficient evaluation via ffts the scheme is further improved through the introduction of approximation based pre conditioners that are shown to accelerate the convergence of the cg iterations numerical results are presented that demonstrate the accuracy and efficiency of the proposed methodology

symmetry properties of a system of euler type equations for magnetized plasmas we provide all lie point symmetries of a system of partial differential equations of great interest to plasma physics we deduce the corresponding invariant solutions and discuss their physical interpretation we also consider some of its conditional and partial symmetries and obtain other interesting solutions finally we point out that although the system can be cast in divergence form and admits conserved currents it does not admit potential symmetries

regularized integral equation methods for elastic scattering problems in three dimensions abstract this paper presents novel methodologies for the numerical simulation of scattering of elastic waves by both closed and open surfaces in three dimensional space the proposed approach utilizes new integral formulations as well as an extension to the elastic context of the efficient high order singular integration methods 13 introduced recently for the acoustic case in order to obtain formulations leading to iterative solvers gmres which converge in small numbers of iterations we investigate theoretically and computationally the character of the spectra of various operators associated with the elastic wave calderon relation including some of their possible compositions and combinations in particular by relying on the fact that the eigenvalues of the composite operator ns are bounded away from zero and infinity new uniquely solvable low gmres iteration integral formulation for the closed surface case are presented the introduction of corresponding low gmres iteration equations for the open surface equations additionally requires for both spectral quality as well as accuracy and efficiency use of weighted versions of the classical integral operators to match the singularity of the unknown density at edges several numerical examples demonstrate the accuracy and efficiency of the proposed methodology

generalized conditional symmetries of evolution equations we analyze the relationship of generalized conditional symmetries of evolution equations to the formal compatibility and passivity of systems of differential equations as well as to systems of vector fields in involution earlier results on the connection between generalized conditional invariance and generalized reduction of evolution equations are revisited this leads to a no go theorem on determining equations for operators of generalized conditional symmetry it is also shown that up to certain equivalences there exists a one to one correspondence between generalized conditional symmetries of an evolution equation and parametric families of its solutions

a boundedness criterion for generalized calderon zygmund operators in 1965 a p calderon showed the l2 boundedness of the so called first calderon commutator this is one of the first examples of a non convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the hilbert transform these properties together with the l2 boundedness imply the lp boundedness for all p s in 1 oo many operators in analysis such as certain classes of pseudo differential operators and the cauchy integral operator on a curve are associated with kernels having these properties for these operators one of the major questions is if they are bounded on l2 we are going to give necessary and sufficient conditions for such an operator to be bounded on l2 they are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in bmo in the case of the aforementioned first commutator this can be checked by an integration by parts in the first section we present some basic notions and state the theorem which is proved in sections 2 and 3 in section 4 we show how to recover some classical results in sections 5 and 6 we construct a functional calculus for small perturbations of a and in section 7 we show a connection between the theory of calderon zygmund operators and kato s conjecture it is a pleasure to express our thanks to r r coifman and y meyer for suggesting many elegant simplifications in our proofs and most of the applications we also wish to thank stephen semmes for several pertinent remarks

variable weak hardy spaces and their applications abstract let p r n 0 be a variable exponent function satisfying the globally log holder continuous condition in this article the authors first introduce the variable weak hardy space on r n wh p r n via the radial grand maximal function and then establish its radial or non tangential maximal function characterizations moreover the authors also obtain various equivalent characterizations of wh p r n respectively by means of atoms molecules the lusin area function the littlewood paley g function or g function as an application the authors establish the boundedness of convolutional type and non convolutional order calderon zygmund operators from h p r n to wh p r n including the critical case when p n n or when p n n where p ess inf x r n p x

conditional lie backlund symmetries and solutions to n 1 dimensional nonlinear diffusion equations this paper discusses a class of n 1 dimensional nonlinear diffusion equations with source term which arises in nonlinear shear flows of non newtonian fluids it is shown that some radially symmetric equations admit certain types of conditional lie backlund symmetries as a result exact solutions to the resulting equations are obtained those solutions extend the known ones such as instantaneous source solutions of the porous medium equation with absorption term the phenomena of extinction and blow up and behavior to many of the solutions are described

boundary integral equation methods for the elastic and thermoelastic waves in three dimensions in this paper we consider the boundary integral equation bie methods for solving the exterior neumann boundary value problems of elastic and thermoelastic waves in three dimensions based on the fredholm integral equations of the first kind the innovative contribution of this work lies in the proposal of the new regularized formulations for the hyper singular boundary integral operators bio associated with the time harmonic elastic and thermoelastic wave equations with the help of the new regularized formulations we only need to compute the integrals with weak singularities at most in the corresponding variational forms of the boundary integral equations the accuracy of the regularized formulations is demonstrated through numerical examples using the galerkin boundary element method bem

boundary integral equations for thin bodies summary a boundary integral equation formulation for thin bodies which uses cbie conventional bie only is well known to be degenerate a mixed formulation for a thin rigid scatterer which combines cbie and hbie hypersingular bie is motivated by examining the discretized form of the integral equations and this formulation is shown to be nondegenerate for thin non rigid inclusion problems a near singular integration procedure useful for singular integrals as well is presented finally numerical examples for acoustic wave scattering from rigid and soft scatterers are presented

potential symmetries and direct reduction methods of order two for partial differential equations written in conservative form a remarkable link between potential symmetries and direct reduction methods of order two is enlightened

numerical modeling of acoustic radiation and scattering from thin bodies using a cauchy principal integral equation in this paper the helmholtz integral equation and its normal derivative are used for the numerical solution of acoustic radiation and scattering from thin bodies a regularized form of the normal derivative integral equation originally derived by maue is adopted to calculate the pressure jump across the thin body this regularized normal derivative integral equation converges in the cauchy principal value sense rather than only in the finite part sense the cauchy principal value integral can be further transformed into an integral that converges in the normal sense the c0 continuous isoparametric elements are used in the formulation so that the numerical model is compatible with other analysis tools such as the finite element method collocation points are placed inside each element to insure a unique normal direction and the c1 continuity condition the pressure jump is forced to be zero at the knife edge where the 1 r 1 2 singularity of the tangential velocities is modeled by the quarter point tec

conditional lie backlund symmetries and differential constraints for inhomogeneous nonlinear diffusion equations due to linear determining equations abstract the method of linear determining equations is developed to study conditional lie backlund symmetries for evolution equations which is more general than the classical determining equations for lie s generators as an application of this approach the complete classification is presented for the inhomogeneous nonlinear diffusion equations which admit the second order and third order conditional lie backlund symmetries several examples are given to illustrate the corresponding symmetry reductions due to the compatibility of the invariant conditions and the governing equations

asymptotic symmetries and asymptotically symmetric solutions of partial differential equations symmetry methods for differential equations are a powerful tool to attack nonlinear problems in particular for determining solutions with given symmetries to nonlinear pdes since in real applications one is often interested in solutions which are asymptotically symmetric we propose here an approach to asymptotic symmetry based on the methods of lie theory we adopt translate in geometric language and develop the renormalization group approach recently proposed by bricmont and kupiainen 1992 for the ginzburg landau equation

reduction operators of burgers equation abstract the solution of the problem on reduction operators and nonclassical reductions of the burgers equation is systematically treated and completed a new proof of the theorem on the special no go case of regular reduction operators is presented and the representation of the coefficients of operators in terms of solutions of the initial equation is constructed for this case all possible nonclassical reductions of the burgers equation to single ordinary differential equations are exhaustively described any lie reduction of the burgers equation proves to be equivalent via the hopf cole transformation to a parameterized family of lie reductions of the linear heat equation

on the modeling of narrow gaps using the standard boundary element method numerical methods based on the helmholtz integral equation are well suited for solving acoustic scattering and diffraction problems at relatively low frequencies however it is well known that the standard method becomes degenerate if the objects that disturb the sound field are very thin this paper makes use of a standard axisymmetric helmholtz integral equation formulation and its boundary element method bem implementation to study the behavior of the method on two test cases a thin rigid disk of variable thickness and two rigid cylinders separated by a gap of variable width both problems give rise to the same kind of degeneracy in the method and modified formulations have been proposed to overcome this difficulty however such techniques are better suited for the so called thin body problem than for the reciprocal narrow gap problem and only the first is usually dealt with in the literature a simple integration technique that can extend the range of thicknesses widths tractable by the otherwise unmodified standard formulation is presented and tested this technique is valid for both cases the modeling of acoustic transducers like sound intensity probes and condenser microphones has motivated this work although the proposed technique has a wider range of applications

symmetry reductions and exact solutions of a class of nonlinear heat equations abstract classical and nonclassical symmetries of the nonlinear heat equation ut uxx f u are considered the method of differential grobner bases is used both to find the conditions on f u under which symmetries other than the trivial spatial and temporal translational symmetries exist and to solve the determining equations for the infinitesimals a catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of the nonlinear heat equation for cubic f u in terms of the roots of f u 0

an accurate boundary element method for the exterior elastic scattering problem in two dimensions this paper is concerned with a galerkin boundary element method solving the two dimensional exterior elastic wave scattering problem the original problem is first reduced to the so called burtonmiller 1 boundary integral formulation and essential mathematical features of its variational form are discussed in numerical implementations a newly derived and analytically accurate regularization formula 2 is employed for the numerical evaluation of hyper singular boundary integral operator a new computational approach is employed based on the series expansions of hankel functions for the computation of weakly singular boundary integral operators during the reduction of corresponding galerkin equations into a discrete linear system the effectiveness of proposed numerical methods is demonstrated using several numerical examples a new and accurate regularization formula which replaces the weak form of hypersingular boundary integral operator by a coupling of several weakly singular boundary integrals is given based on the series expansions of hankel functions several special integrals all weakly singular boundary integrals are evaluated exactly

a unified boundary element method for the analysis of sound and shell like structure interactions i formulation and verification a unified boundary element method bem is developed in this paper to model both the exterior acoustic field and the elastic shell like structure in a coupled analysis the conventional boundary integral equation bie for three dimensional 3d elastodynamics is applied to thin shell like structures which can have arbitrary shapes and small thicknesses the nearly singular integrals existing in the bie when applied to thin bodies are transformed to nonsingular line integrals and are evaluated accurately and efficiently for the exterior 3d acoustic domain the burton and miller composite bie formulation is employed to overcome the fictitious eigenfrequency difficulty fed and the thin shape breakdown tsb conforming c0 quadratic elements are employed in the discretization of the two sets of bies the developed bie formulations are valid for both radiation and scattering problems and for all wave numbers numerical examples using spherical and cylindrical shells including nonuniform thickness and nond

partial lie point symmetries of differential equations when we consider a differential equation 0 whose set of solutions is a lie point exact symmetry of this is a lie point invertible transformation t such that t i e such that any solution to 0 is transformed into a generally different solution to the same equation here we define partial symmetries of 0 as lie point invertible transformations t such that there is a non empty subset such that t i e such that there is a subset of solutions to 0 which are transformed into one another we discuss how to determine both partial symmetries and the invariant set and show that our procedure is effective by means of concrete examples we also discuss relations with conditional symmetries and how our discussion applies to the special case of dynamical systems our discussion will focus on continuous lie point partial symmetries but our approach would also be suitable for more general classes of transformations the discussion is indeed extended to partial generalized or lie backlund symmetries along the same lines and in the appendix we will discuss the case of discrete partial symmetries

non classical symmetries and the singular manifold method revisited the connection between the singular manifold method painleve expansions truncated at the constant term and symmetry reductions of two members of a family of cahn hilliard equations is considered the conjecture that similarity information for a nonlinear partial differential equation may always be fully recovered from the singular manifold method is violated for these equations and is thus shown to be invalid in general given that several earlier examples demonstrate the connection between the two techniques in some cases it now becomes necessary to establish when such a relationship exists a question related to a deeper understanding of painleve analysis this issue is also briefly discussed

on the weak symmetry groups of partial differential equations abstract we define an algorithm to characterize all weak symmetry groups of a partial differential equation new invariant solutions to the fokker planck equation are found

a boundary integral formulation for thin walled shapes of revolution much of the recent literature on boundary elements has addressed the practical limitations of the method one issue under current scrutiny appears to be the inapplicability of formulations based on the helmholtz integral to free flooded shapes characterized by kh 1 where k and h are the acoustic wavenumber and the object s cross sectional thickness dimension the present article develops and applies a modal boundary integral technique especially tailored to thin geometries of revolution the problems chosen for its demonstration are cases of acoustic diffraction by an open ended cylindrical duct containing a sound source where the scattering wall s outer and inner surfaces are an infinitesimal distance apart and are respectively rigid and either rigid or compliant the study concludes that general shapes with kh 1 or kh 0 should be treatable by at least one established numerical approach after some analytical reinterpretation

a direct boundary element method for acoustic radiation and scattering from mixed regular and thin bodies in this paper the direct boundary element method is applied to acoustic radiation and scattering from mixed regular and thin bodies a typical application problem involves a regular body with thin fins either attached to or in the neighborhood of the body it is shown that the mixed body integral formulation requires interactions between the conventional helmholtz integral equation and the hypersingular thin body integral equation if at least one regular body is involved in a mixed body environment the direct integral formulation possesses the well known nonuniqueness difficulty this nonuniqueness difficulty is overcome by either the chief method or the burton and miller method since both the thin body integral equation and the burton and miller equation contain a hypersingular integral a regularized form recently derived by krishnasamy et al asme j appl mech 57 404 414 1990 is adopted in this paper numerical results show excellent agreement with a multidomain boundary element solution and a point source solution

a fast and high order method for the three dimensional elastic wave scattering problem in this paper we present a fast and high order boundary integral equation method for the elastic scattering by three dimensional large penetrable obstacles the algorithm extends the method introduced in 5 for the acoustic surface scattering to the fully elastic case in our algorithm high order accuracy is achieved through the use of the partition of unity and a semi classical treatment of relevant singular integrals the computational efficiency associated with the nonsingular integrals is attained by the method of equivalent source representations on a cartesian grid and fast fourier transform fft the resulting algorithm computes one matrix vector product associated with the discretization of the integral equation with o n 4 3 log n operations and it shows algebraic convergence several numerical experiments are provided to demonstrate the efficiency of the method

similarity reductions of partial differential equations the author determine the relation between two methods of reduction the similarity reduction method using nonclassical symmetry groups and the direct approach used by clarkson and kruskal 1989 they prove that the solutions which are obtained by similarity in correspondence to nonclassical groups constitute a larger family than the one obtained by the method of clarkson and kruskal the two procedures are equivalent only if the generators zeta x t u tau x t u eta x t u of the nonclassical groups are such that zeta tau is independent of u to explain these results he proves the existence of families of solutions of the burgers equation which are found by means of nonclassical symmetry reduction and which cannot be determined via the general reduction form of clarkson and kruskal

on the relation between standard and μ symmetries for pdes we give a geometrical interpretation of the notion of prolongations of vector fields and of the related concept of symmetry for partial differential equations extending to pdes the notion of symmetry for odes we give in particular a result concerning the relationship between symmetries and standard exact symmetries the notion is also extended to the case of conditional and partial symmetries and we analyse the relation between local symmetries and nonlocal standard symmetries

reduction operators of linear second order parabolic equations the reduction operators i e the operators of nonclassical conditional symmetry of 1 1 dimensional second order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary differential ones are exhaustively described this problem proves to be equivalent in some sense to solving initial equations the no go result is extended to the investigation of point transformations admissible transformations equivalence transformations lie symmetries and lie reductions of the determining equations for the nonclassical symmetries transformations linearizing the determining equations are obtained in the general case and under different additional constraints a nontrivial example illustrating applications of reduction operators to finding exact solutions of equations from the class under consideration is presented an observed connection between reduction operators and darboux transformations is discussed

lie and conditional symmetries of the three component diffusive lotka volterra system lie and q conditional symmetries of the classical three component diffusive lotka volterra system in the case of one space variable are studied the group classification problems for finding lie symmetries and q conditional symmetries of the first type are completely solved notably non lie symmetries q conditional symmetry operators for a multi component nonlinear reaction diffusion system are constructed for the first time the results are compared with those derived for the two component diffusive lotka volterra system the conditional symmetry obtained for the non lie reduction of the three component system used for modeling competition between three species in population dynamics is applied and the relevant exact solutions are found particularly the exact solution describing different scenarios of competition between three species is constructed

scattering of elastic waves from thin shapes in three dimensions using the composite boundary integral equation formulation in this paper the composite boundary integral equation bie formulation is applied to scattering of elastic waves from thin shapes with small but finite thickness open cracks or thin voids thin inclusions thin layer interfaces etc which are modeled with two surfaces this composite bie formulation which is an extension of the burton and miller s formulation for acoustic waves uses a linear combination of the conventional bie and the hypersingular bie for thin shapes the conventional bie as well as the hypersingular bie will degenerate or nearly degenerate if they are applied individually on the two surfaces the composite bie formulation however will not degenerate for such problems as demonstrated in this paper nearly singular and hypersingular integrals which arise in problems involving thin shapes modeled with two surfaces are transformed into sums of weakly singular integrals and nonsingular line integrals thus no finer mesh is needed to compute these nearly singular integrals

group invariant solutions of differential equations we introduce the concept of a weak symmetry group of a system of partial differential equations that generalizes the nonclassical method introduced by bluman and cole for finding group invariant

a new form of the hypersingular boundary integral equation for 3 d acoustics and its implementation with c0 boundary elements abstract an improved weakly singular form of the hypersingular boundary integral equation hbie for 3 d acoustic wave problems is presented in this paper compared with the weakly singular form of the hbie published earlier y j liu and f j rizzo a weakly singular form of the hypersingular boundary integral equation applied to 3 d acoustic wave problems comput methods appl mech engrg 96 1992 271 287 this new form involves only tangential derivatives of the density function and thus its discretization using the boundary element method bem is easier to perform instead of using nonconforming and c1 continuous boundary elements advocated earlier c0 continuous conforming quadratic elements are employed in the discretization of this weakly singular form of the hbie some justifications on using c0 elements for hbies are provided to reflect the current views on this crucial issue for hbies it is postulated that the original c1 continuity requirement for the density function in the analytical hbie formulation can be relaxed to piecewise c1 continuity in the numerical implementation of the weakly singular forms of the hbie numerical examples of both scattering and radiation problems clearly demonstrate the accuracy and versatility of the new weakly singular form of the hbie for 3 d acoustics

simple waves and invariant solutions of quasilinear systems the solutions of a system of homogeneous first order quasilinear evolution equations for which the propagation velocity depends only on the field value are characterized by their invariance under a particular type of flow all simple wave solutions are shown to have this property the local images of simple wave solutions are the integral curves of a system of ordinary differential equations on the field space the model for barotropic compressible flow is an example of a system that has invariant solutions of higher rank a solution of this system is invariant under the flow of the convective derivative if and only if it is divergence free the evolution of initial data is shown to be divergence free if and only if its differential is nilpotent

symmetric galerkin boundary formulations employing curved elements accounts of the symmetric galerkin approach to boundary element analysis bea have recently been published this paper attempts to add to the understanding of this method by addressing a series of fundamental issues associated with its potential computational efficiency a new symmetric galerkin theoretical formulation for both the harmonic heat conduction and the biharmonic elasticity problem that employs regularized singular and hypersingular boundary integral equations bies is presented the novel use of regularized bies in the galerkin context is shown to allow straightforward incorporation of curved isoparametric elements a symmetric reusable intrinsic sample point risp numerical integration algorithm is shown to produce a galerkin i e double integration strategy that is competitive with its counterpart i e singular integration procedure in the collocation bea approach when the time saved in the symmetric equation solution phase is also taken into account this new formulation is shown to be capable of employing hypersingular bies while obviating the requirement of c 1 continuity a fact that allows the employment of the popular continuous element technology the behavior of the symmetric galerkin bea method with regard to both direct and iterative equation solution operations is also addressed a series of example problems are presented to quantify the performance of this symmetric approach relative to the more conventional unsymmetric bea in terms of both accuracy and efficiency it is concluded that appropriate implementations of the symmetric galerkin approach to bea indeed have the potential to be competitive with if not superior to collocation based bea for large scale problems

a p extension of the symmetric boundary element method abstract this paper describes an implementation of the symmetric galerkin boundary element method for plane potential and elasticity problems on domains with polygonal boundaries the method makes use of the hyper singular boundary integral equations for the normal derivatives of the unknown function we discuss an integration technique which is sufficiently robust to handle p extensions on geometrically graded meshes computer times required compare favorably with corresponding finite element methods we present numerical results obtained by a uniform p extension for various mixed boundary value problems we demonstrate that the same convergence rates can be obtained by the symmetric variational boundary element method as with p extensions of the finite element method on the basis of the p extension process a reliable error estimate based on extrapolation from a sequence of p levels can be obtained this is a unique feature of the variational methods as compared to the classical collocation approach the error estimate remains valid even for boundary element analysis employing substructures and mixed discretizations with boundary elements and finite elements

new molecular characterizations of anisotropic musielak orlicz hardy spaces and their applications abstract let r n 0 0 be an anisotropic growth function and a a general expansive matrix on r n let h a r n be the anisotropic musielak orlicz hardy space associated with a defined via the non tangential grand maximal function in this article the authors establish a new molecular characterization of h a r n via its known atomic characterizations which positively answer a question mentioned by li et al 2016 34 as an application the authors obtain a criterion on the boundedness of linear operators on h a r n which further implies the boundedness of integral anisotropic calderon zygmund operators on h a r n in addition the boundedness of these operators from h a r n to the musielak orlicz space l r n is also presented

a remarkable class of non classical symmetries of the steady two dimensional boundary layer equations we study the steady two dimensional boundary layer equations in the flat and axisymmetric case to show that some similarity reductions already found in the literature with ad hoc new methods of reduction are indeed invariant solutions under the action of non classical symmetries as introduced by bluman and cole moreover we show that the celebrated von mises transformation that reduces the two dimensional flat boundary layer equations to a second order evolution equation is indeed a backlund transformation related to a non classical symmetry

nystrom method for elastic wave scattering by three dimensional obstacles nystrom method is developed to solve for boundary integral equations bie s for elastic wave scattering by three dimensional obstacles to generate the matrix equation from a bie nystrom method applies a quadrature rule to the integrations of smooth integrands over a discretized element directly and chooses the values of the unknown function at quadrature points as the system s unknowns to be solved this leads to a simple procedure to form the off diagonal entries of matrix by simply evaluating the integrands without numerical integrations for the diagonal or near diagonal entries corresponding to the integrals over a singular or near singular element where the kernels are singular or near singular we develop a systematic singularity treatment technique known as the local correction scheme based on the linear approximation of elements the scheme differs from the singularity regularization or subtraction technique used in the boundary element method bem it applies the series expansion of scalar green s function to the kernels and derives analytical solutions for the strongly singular integrals under the cauchy principal value like cpv like sense since the approach avoids the need for reformulating the bie for singularity removal in bem and solves for the somigliana s equation directly it is easy to implement and efficient in calculation numerical examples are used to demonstrate its robustness

analysis of shell like structures by the boundary element method based on 3 d elasticity formulation and verification in this paper the boundary element method bem for 3 d elastostatic problems is studied for the analysis of shell or shell like structures it is shown that the conventional boundary integral equation cbie for 3 d elasticity does not degenerate when applied to shell like structures contrary to the case when it is applied to crack like problems where it does degenerate due to the closeness of the two crack surfaces the treatment of the nearly singular integrals which is a crucial step in the applications of bies to thin shapes is presented in detail to verify the theory numerical examples of spherical and ellipsoidal vessels are presented using the bem approach developed in this paper it is found that the system of equations using the cbie is well conditioned for all the thickness studied for the vessels the advantages disadvantages and potential applications of the proposed bem approach to shell like structures as compared with the fem regarding modelling and accuracy are discussed in the last section applications of this bem approach to shell like structures with non uniform thickness stiffeners and layers will be reported in a subsequent paper 1998 john wiley sons ltd

evolution equations invariant surface conditions and functional separation of variables abstract this paper is devoted to a discussion of the reduction methods for evolution equations based on invariant surface conditions related to functional separation of variables the relationship of these methods with nonclassical and weak point symmetries is stressed applications to diffusion equations with an inhomogeneous reaction term or with saturating dissipation are provided

hardy space estimates for limited ranges of muckenhoupt weights abstract in this article we give a full necessary and sufficient set of conditions for a calderon zygmund operator to be bounded on weighted hardy spaces h w p where w is an muckenhoupt weight and 0 p in fact this result is new even when 1 p since it allows for h w p boundedness of an operator when 1 p q and w a q where it is possible that h w p l w p these singular integral results are achieved by proving littlewood paley stein square function type estimates from h w p into l w p for 0 p and a muckenhoupt weight w which are interesting results in their own right new techniques involving a weight invariant spaces are also used to prove the weighted estimates for calderon zygmund operators more precisely we prove the following bmo type weight invariance properties for a fixed s 0 the weighted sobolev bmo spaces i s b m o w coincide for all w a the weighted p type triebel lizorkin spaces f w s 2 coincide for all w a and these two classes of spaces coincide with each other as well all of which have comparable norms up to constants depending on an a p character of the weight w a

combined equivalent charge formulations and fast wavelet galerkin bem for 3 d electrostatic analysis we describe a new equivalent charge formulation ecf i e the combined ecf cecf for electrostatic analysis of structures consisting of conductors and dielectrics the cecf uses a weighted combination of the single and the adjoint double layer operators to account for the potential on the conductor dielectric surface and is found to have better conditioning than the standard ecf a perturbation approach is presented to insure that the capacitances are computed accurately even when the permittivity ratios of the dielectrics are vary large unlike the original perturbation approach this new approach uses only one system matrix with different right hand sides a wavelet galerkin boundary element method wgbem for solving the ecf and cecf is developed based on the new variable order wgbem introduced in j numer math 2004 12 3 233 254 the wavelets are directly constructed on the usual boundary element triangulation this enables the proposed wgbem to solve electrostatic problems in complicated geometries unstructured meshes and comparatively coarse discretizations the quasi vanishing moment wavelets introduced in comput methods appl mech engng 2008 197 4000 4006 are used to further reduce the memory and cpu time requirements several numerical examples show that the proposed cecf converges faster than the ecf and that the wgbem using both the ecf and cecf has almost linear complexity in solving large scale 3 d electrostatic problems moreover since the truncated non standard form is computed once and then stored the wgbem is very suitable for solving problems with multiple right hand sides like the perturbation approach in this paper copyright 2009 john wiley sons ltd

a dual bie approach for large scale modelling of 3 d electrostatic problems with the fast multipole boundary element method a dual boundary integral equation bie formulation is presented for the analysis of general 3 d electrostatic problems especially those involving thin structures this dual bie formulation uses a linear combination of the conventional bie and hypersingular bie on the entire boundary of a problem domain similar to crack problems in elasticity the conventional bie degenerates when the field outside a thin body is investigated such as the electrostatic field around a thin conducting plate the dual bie formulation however does not degenerate in such cases most importantly the dual bie is found to have better conditioning for the equations using the boundary element method bem compared with the conventional bie even for domains with regular shapes thus the dual bie is well suited for implementation with the fast multipole bem the fast multipole bem for the dual bie formulation is developed based on an adaptive fast multiple approach for the conventional bie several examples are studied with the fast multipole bem code including finite and infinite domain problems bulky and thin plate structures and simplified comb drive models having more than 440 thin beams with the total number of equations above 1 45 million and solved on a pc the numerical results clearly demonstrate that the dual bie is very effective in solving general 3 d electrostatic problems as well as special cases involving thin perfect conducting structures and that the adaptive fast multipole bem with the dual bie formulation is very efficient and promising in solving large scale electrostatic problems copyright 2007 john wiley sons ltd

calderon zygmund operators on herz type hardy spaces abstract we consider the boundedness of calderon zygmund operators from h k p q r n to h k p q r n where h k p q r n is the hardy space associated with the herz space k p q r n and h k p q r n is the local version of h k p q r n we show calderon s commutator is bounded from h k p q to h k p q

multilinear theory of strongly singular calderon zygmund operators and applications abstract the main purpose of this paper is to study the multilinear strongly singular calderon zygmund operator whose kernel is more singular near the diagonal than that of the standard multilinear calderon zygmund operator the sharp maximal estimate for this class of multilinear singular integrals is established and as applications its boundedness on product of weighted lebesgue spaces and product of variable exponent lebesgue spaces is obtained respectively moreover the endpoint estimates of the types l l b m o b m o b m o b m o and l m o l m o l m o are established for the multilinear strongly singular calderon zygmund operator these results improve the corresponding known ones for the standard multilinear calderon zygmund operator furthermore we remove the size condition assumption for the kernel of the multilinear strongly singular calderon zygmund operator which has been imposed in the literature for the case of the standard multilinear calderon zygmund operator to get the sharp maximal estimates see remarks 1 1 1 4 extra care is needed to deal with the mean oscillation over balls with small radius to overcome the stronger singularity in establishing such sharp maximal estimates and endpoint estimates

new conditional symmetries and exact solutions of reaction diffusion convection equations with exponential nonlinearities abstract a complete description of q conditional symmetries for a class of reaction diffusion convection equations with exponential diffusivities is derived it is shown that all the known results for reaction diffusion equations with exponential diffusivities follow as particular cases from those obtained here but not vice versa the symmetries obtained for constructing exact solutions of the relevant equations are successfully applied the exact solutions are compared with those found earlier by means of different approaches an application of the solutions for solving a boundary value problem arising in population dynamics is presented

symmetry reductions of partial differential equations related to singular manifold expansions it is shown that all the information yielded by the reduction methods of bluman and cole 1974 and clarkson and kruskal 1989 can be obtained using the singular manifold expansion the burgers equation modified korteweg de vries equation caudrey dodd gibbon equation and the fitzhugh nagumo equation are used as illustrative examples several new exact solutions are presented

solving the hypersingular boundary integral equation for the burton and miller formulation this paper presents an easy numerical implementation of the burton and miller bm formulation where the hypersingular helmholtz integral is regularized by identities from the associated laplace equation and thus needing only the evaluation of weakly singular integrals the helmholtz equation and its normal derivative are combined directly with combinations at edge or corner collocation nodes not used when the surface is not smooth the hypersingular operators arising in this process are regularized and then evaluated by an indirect procedure based on discretized versions of the calderon identities linking the integral operators for associated laplace problems the method is valid for acoustic radiation and scattering problems involving arbitrarily shaped three dimensional bodies unlike other approaches using direct evaluation of hypersingular integrals collocation points still coincide with mesh nodes as is usual when using conforming elements using higher order shape functions with the boundary element method model size kept fixed reduces the overall numerical integration effort while increasing the solution accuracy to reduce the condition number of the resulting bm formulation at low frequencies a regularized version ik k2 of the classical bm coupling factor i k is proposed comparisons with the combined helmholtz integral equation formulation method of schenck are made for four example configurations two of them featuring non smooth surfaces

all solutions of standard symmetric linear partial differential equations have classical lie symmetry abstract it is proven that every solution of any linear partial differential equation with an independent variable deforming classical lie point symmetry is invariant under some classical lie point symmetry this is true for any number of independent variables and for equations of any order higher than one although this result makes use of the infinite dimensional component of the lie symmetry algebra due to linear superposition it is shown that new similarity solutions previously thought not to be classical can be recovered prospectively by allowing symmetries to include superposition of similarity solutions already known from the finite part of the symmetry algebra this result applies to all constant coefficient equations and to many variable coefficient equations such as fokker planck equations

nonclassical symmetry reduction and riemann wave solutions abstract in this paper we employ the nonclassical symmetry method in order to obtain riemann multiple wave solutions of a system of first order quasilinear differential equations we show how to construct a lie module of vector fields which are symmetries of the system supplemented by certain first order differential constraints we demonstrate the usefulness of our approach on several examples

symmetry groups of differential difference equations and their compatibility abstract it is shown that the intrinsic determining equations of a given differential difference equation dde can be derived by the compatibility between the original equation and the intrinsic invariant surface condition the 2 1 dimensional toda lattice the special toda lattice and the dd kp equation serving as examples are used to illustrate this approach then backlund transformations of the 2 1 dimensional ddes including the special toda lattice the modified toda lattice and the dd kz equation are presented by using the non intrinsic direct method in addition the clarkson kruskal direct method is developed to find similarity reductions of the ddes

on time harmonic elastic wave analysis by the boundary element method for moderate to high frequencies abstract certain inherent difficulties in obtaining accurate boundary element solutions to problems of elastodynamics at moderate to high frequencies are examined some procedures for overcoming or compensating for these difficulties are described in addition results demonstrating the effectiveness of these procedures are presented finally the use of these procedures in solving transient problems via fourier transform inversion is illustrated

hypersingular boundary integral equations for radiation and scattering of elastic waves in three dimensions abstract a weakly singular form of the hypersingular boundary integral equation bie traction equation for 3 d elastic wave problems is developed in this paper all integrals involved are at most weakly singular and except for a stronger smoothness requirement on boundary elements regular quadrature and collocation procedures used for conventional bies are sufficient for the discretization of the original hypersingular bie this weakly singular form of the hypersingular bie is applied to the composite bie formulation which uses a linear combination of the conventional bie and the hypersingular bie to remove the fictitious eigenfrequencies existing in the conventional bie formulation for elastic wave problems numerical examples employing different types of boundary elements clearly demonstrate the effectiveness and efficiency of the developed formulation

on the existence of nonclassical symmetries of partial differential equations relations between determining equations for nonclassical symmetries of bluman and cole and integrability conditions of differential equations supplemented by invariance conditions are studied it is shown that every system of equations with p independent variables admits p dimensional groups of such symmetries nonclassical symmetries of the eikonal equation in a curved space are investigated they are related to shear free congruences of null geodesics

a boundary integral equation method for radiation and scattering of elastic waves in three dimensions etude du rayonnement et de la diffusion d ondes elastiques par des obstacles de forme arbitraire

solution of elastic scattering problems in linear acoustics using h p boundary element method abstract the work is a continuation of 1 a simplified 3d model problem reduced from the general class of elastic scattering problems is analyzed focusing attention on such practical details of the 3d implementation as a triangles based h p approximation and corresponding data structure integration of singular integrals and its effect on theoretically optimal rates of convergence a posteriori error estimation and sample adaptive strategies

diffraction of elastic waves by a penny shaped crack analytical and numerical results the diffraction of time harmonic stress waves by a penny shaped crack in an infinite elastic solid is an important problem in fracture mechanics and in the theory of the ultrasonic inspection of materials martin proc r soc lond a 378 263 1981 has proved that the corresponding linear boundary value problem has precisely one solution and that this solution can be constructed by solving a two dimensional fredholm integral equation of the second kind however this integral equation has a complicated matrix kernel and the components of its vector solution are coupled the main purpose of the present paper is to show how martin s integral equation can be explicitly solved in terms of a sequence of functions each of which satisfies a very simple scalar integral equation of the second kind this simplification may be made for any incident wave for an incident plane wave further simplifications are possible we show that the solution at an arbitrary angle of incidence can be derived from the solution at a particular angle of incidence namely grazing incidence the resulting computational procedure is especially attractive if only the stressintensity factors or the far field displacements are required finally we present some numerical results for the scattering of a p wave at normal incidence and an sv wave at oblique incidence and compare these with those of other authors

simulation of sound absorption in 2d thin elements using a coupled bem tbem formulation in the presence of fixed and moving 3d sources abstract modelling sound propagation in complex environments can be a difficult task in realistic applications the boundaries of scatterers present in the propagation domain can be partially absorbing and this must be accounted for in the numerical models this paper addresses the use of a frequency domain dual bem bem tbem formulation to model the propagation of sound generated by fixed and moving point loads in 2 5d configurations in the presence of very thin elements with partially absorbing surfaces the proposed approach is based on the concept of impedance boundary conditions and is applied in conjunction with a dual bem approach thereby allowing the definition of models in which only very compact descriptions of the propagation domain are required since a 2 5d formulation is used 3d responses can be computed as a discrete summation of simpler 2d solutions the formulation of the numerical methods used here bem tbem and dual bem are described together with the strategy devised by the authors to incorporate sound absorption the model is verified against closed form solutions and the calculations indicate that it can provide very accurate results several numerical applications involving fixed and moving 3d sources are described to illustrate the applicability and usefulness of the proposed approaches in the analysis of various scenarios

solving knife edge scattering problems using singular boundary elements a multidomain boundary element method bem is employed to study the scattering of time harmonic acoustic waves by knife edge structures in an acoustic field analysis shows that singularities exist in such structures and that the behavior of the acoustic pressure and its derivative near a knife edge resembles that of the displacement and stress near a crack tip in an elastic body singular elements are employed to model the behavior on the part of the boundary in the vicinity of the singularity the advantage of the singular elements is seen through several numerical examples involving mufflerlike structures containing baffles and one with a knife edge scattering in full space for a given level of mesh refinement both the near field and far field solutions are improved with the introduction of singular elements

singular reduction modules of differential equations the notion of singular reduction modules i e of singular modules of nonclassical conditional symmetry of differential equations is introduced it is shown that the derivation of nonclassical symmetries for differential equations can be improved by an in depth prior study of the associated singular modules of vector fields the form of differential functions and differential equations possessing parameterized families of singular modules is described up to point transformations singular cases of finding reduction modules are related to lowering the order of the corresponding reduced equations as examples singular reduction modules of evolution equations and second order quasi linear equations are studied reductions of differential equations to algebraic equations and to first order ordinary differential equations are considered in detail within the framework proposed and are related to previous no go results on nonclassical symmetries

contact symmetries and solutions by reduction of partial differential equations the correspondence between a contact symmetry of a second order pde e and a point symmetry of the equivalent first order system s is used to determine a class of solutions for e which may not always be invariant under contact transformation we name these solutions pseudo invariant because they are determined from a point symmetry of s via a reduction method invariant solutions under the related contact transformation are included in the family of pseudo invariant solutions

symmetries and conditional symmetries of differential difference equations two different methods of finding lie point symmetries of differential difference equations are presented and applied to the two dimensional toda lattice continuous symmetries are combined with discrete ones to obtain various reductions to lower dimensional equations in particular to differential equations of the delay type the concept of conditional symmetries is extended from purely differential to differential difference equations and shown to incorporate backlund transformations

potential symmetries and invariant solutions for the generalized one dimensional fokker planck equation abstract pucci and saccomandi j phys a 26 1993 681 have obtained invariant solutions for the fokker planck fp equation via potential symmetries herein we analyze the generalized one dimensional fp equation for various cases corresponding to the physically interesting situations through the application of the same technique some interesting outcomes of this study are the deductions of the new exact solutions that do not seem to have been reported in the literature

the nonclassical group analysis of the heat equation abstract the nonclassical method of reduction was devised originally by bluman and cole in 1969 to find new exact solutions of the heat equation much success has been had by many authors using the method to find new exact solutions of nonlinear equations of mathematical and physical significance however the defining equations for the nonclassical reductions of the heat equation itself have remained unsolved although particular solutions have been given recently arrigo goard and broadbridge showed that there are no nonclassical reduction solutions of constant coefficient linear equations that are not already classical lie symmetry reduction solutions their arguments leave open the problem of what is the general nonclassical group action and its effect on the relevant solution of the heat equation in this article both these problems are solved in the final section we use the methods developed to solve the remaining outstanding case of nonclassical reductions of burgers equation