a boundary value problem for second order fuzzy differential equations abstract in this paper we interpret a two point boundary value problem for a second order fuzzy differential equation by using a generalized differentiability concept we present a new concept of solutions and utilizing the generalized differentiability we investigate the problem of finding new solutions some examples are provided for which the new solutions are found

a new cloud based hp finite element method a hybrid computational method for solving boundary value problems is introduced which combines features of the meshless hp cloud methods with features of conventional finite elements the method admits straightforward nonuniform hp type approximations easy implementation of essential boundary conditions is robust under severe distortions of the mesh and can deliver exponential rates of convergence results of numerical experiments are presented

on the existence of solutions to periodic boundary value problems for fuzzy linear differential equations in this work we provide sufficient conditions which guarantee the existence of solutions to periodic boundary value problems for first order linear fuzzy differential equations by using generalized differentiability and switching points in comparison with some previous works we consider equations whose coefficient may change its sign a finite number of times in the interval of interest we also study the existence of solutions which are crisp or real at the switching points where the diameter of the level sets changes from nonincreasing to nondecreasing character

the generalized method of moments for electromagnetic boundary integral equations the generalized method of moments gmm is a partition of unity based technique for solving electromagnetic and acoustic boundary integral equations past work on gmm for electromagnetics was confined to geometries modeled by piecewise flat tessellations and suffered from spurious internal line charges in the present article we redesign the gmm scheme and demonstrate its ability to model scattering from pec scatterers composed of mixtures of smooth and non smooth geometrical features furthermore we demonstrate that because the partition of unity provides both functional and effective geometrical continuity between patches gmm permits mixtures of local geometry descriptions and approximation function spaces with significantly more freedom than traditional moment methods

two point boundary value problems of undamped uncertain dynamical systems in this paper we study two point boundary value problems of a class of uncertain dynamical systems i e undamped uncertain dynamical systems we introduce the concept of large solutions and small solutions for two point boundary value problems of undamped uncertain dynamical systems and under some conditions establish the existence and uniqueness of large solutions and small solutions and settle the relationship between large solutions and small solutions

characterization of compact subsets of fuzzy sets a characterization of compact subsets is presented for the metric space of normal fuzzy convex fuzzy sets on the base space rn the metric for which is the supremum over the hausdorff distances between corresponding level sets it is shown that a closed subset is compact if and only if it is uniformly support bounded and the corresponding set of support functions is equileftcontinuous in the membership grade variable

intuitive formulation of discontinuous galerkin surface integral equations for electromagnetic scattering problems an intuitive formulation of discontinuous galerkin surface integral equation dg sie method is proposed in this paper which is established by performing singularity extraction and singling out the infinitely large term in the entries of the impedance matrices associated with the sies which are popular in solving electromagnetic scattering problems nonoverlapping trial and test functions defined on triangles are employed without requirements of the surface current normal continuity across their internal boundary contours the proposed sie dg formulation is very flexible and is validated by examples of perfect electrically conducting targets in both conformal and nonconformal meshes

existence of global solutions to nonlinear fuzzy volterra integro differential equations abstract in this paper we introduce new solutions for fuzzy differential equations as mixed solutions and prove the existence and uniqueness of global solutions for fuzzy initial value problems involving integro differential operators of volterra type one example is also given by applying mixed solution concept to fuzzy linear differential equations for obtaining their global solutions

error estimation in the approximation of the solution of nonlinear fuzzy fredholm integral equations using a fixed point technique the sequence of successive approximations and a recent quadrature formula for fuzzy number valued functions it is constructed a numerical method for the solution of nonlinear fuzzy fredholm integral equations moreover the error estimate of the method and a criterion to stop the corresponding algorithm are given

singular higher order divergence conforming bases of additive kind and moments method applications to 3d sharp wedge structures we present new subsectional singular divergence conforming vector bases that incorporate the edge conditions for conducting wedges the bases are of additive kind because obtained by incrementing the regular polynomial vector bases with other subsectional basis sets that model the singular behavior of the unknown vector field in the wedge neighborhood singular bases of this kind complete to arbitrarily high order are described in a unified and consistent manner for curved quadrilateral and triangular elements the higher order basis functions are obtained as the product of lowest order functions and silvester lagrange interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties are discussed and these bases are proved to be fully compatible with the standard high order regular vector bases used in adjacent elements our singular bases guarantee normal continuity along the edges of the elements allowing for the discontinuity of tangential components adequate modelling of the divergence and removal of spurious solutions these singular high order bases provide more accurate and efficient numerical solutions of surface integral problems several test case problems are considered in the paper thereby obtaining highly accurate numerical results for the current and charge density induced on 3d sharp wedge structures the results are compared with other solutions when available and confirm the faster convergence of these bases on wedge problems

an equivalence lemma for a class of fuzzy implicit integro differential equations abstract in the current paper an equivalence lemma is proposed which converts a second order implicit form of nonlinear fuzzy volterra integro differential equation to four different kinds of nonlinear integral equations based on kind of gh differentiability of the first and second derivatives finally some examples are given to illustrate the theoretical results

improved predictor corrector method for solving fuzzy initial value problems in this paper an improved predictor corrector ipc method is proposed to solve the fuzzy initial value problem the ipc method is generated by combining an explicit three step method and an implicit two step method the methods are compared with the methods discussed in t allahviranloo n ahmady e ahmady numerical solution of fuzzy differential equations by predictor corrector method information sciences 177 7 2007 1633 1647 and are shown to be more accurate the convergence and stability of the proposed methods are also presented in detail these methods are illustrated by solving some examples

numerical solution of fuzzy differential equations by predictor corrector method in this paper three numerical methods to solve the fuzzy ordinary differential equations are discussed these methods are adams bashforth adams moulton and predictor corrector predictor corrector is obtained by combining adams bashforth and adams moulton methods convergence and stability of the proposed methods are also proved in detail in addition these methods are illustrated by solving two fuzzy cauchy problems

generalized debye sources based efie solver on subdivision surfaces the electric field integral equation is a well known workhorse for obtaining fields scattered by a perfect electric conducting object as a result the nuances and challenges of solving this equation have been examined for a while two recent papers motivate the effort presented in this paper unlike traditional work that uses equivalent currents defined on surfaces recent research proposes a technique that results in well conditioned systems by employing generalized debye sources gds as unknowns in a complementary effort some of us developed a method that exploits the same representation for both the geometry subdivision surface representations and functions defined on the geometry also known as isogeometric analysis iga the challenge in generalizing gds method to a discretized geometry is the complexity of the intermediate operators however thanks to our earlier work on subdivision surfaces the additional smoothness of geometric representation permits discretizing these intermediate operations in this paper we employ both ideas to present a well conditioned gds electric field integral equation here the intermediate surface laplacian is well discretized by using subdivision basis likewise using subdivision basis to represent the sources results in an efficient and accurate iga framework numerous results are presented to demonstrate the efficacy of the approach

some topological properties of solutions to fuzzy differential systems in this paper the initial value problems of fuzzy differential systems are investigated by using a kind of new methods at first the concept of relatively open sets is introduced then some basic topological properties of solutions for the initial value problems of fuzzy differential systems are obtained

monotone method for fuzzy differential equations we develop the monotone iterative technique to approximate the extremal solutions for the initial value problem relative to a fuzzy differential equation in a fuzzy functional interval to our procedure it is crucial to have at our disposal fuzzy comparison results and several properties of sets of fuzzy functions and fuzzy numbers including a compactness result in spaces of fuzzy functions which are useful from the point of view of applications

the cauchy problem for continuous fuzzy differential equations abstract we prove a version of the classical peano s theorem for the initial value problem for a fuzzy differential equation in the metric space of normal fuzzy convex sets with distance given by the maximum of the hausdorff distances between level sets

integral equation based domain decomposition method for solving electromagnetic wave scattering from non penetrable objects the integral equation ie method is commonly utilized to model time harmonic electromagnetic em problems one of the greatest challenges in its applications arises in the solution of the resulting ill conditioned matrix equation we introduce a new domain decomposition method ddm for the ie solution of em wave scattering from non penetrable objects the proposed method is a non overlapping non conformal ddm and it provides a computationally efficient and effective preconditioner for the ie matrix equations moreover the proposed approach is very suitable for dealing with multi scale electromagnetic problems since each sub domain has its own characteristics length and will be meshed independently furthermore for each sub domain we are free to choose the most effective ie sub domain solver based on its local geometrical features and electromagnetic characteristics additionally the multilevel fast multi pole algorithm mlfma is utilized to accelerate the computations of couplings between sub domains numerical results demonstrate that the proposed method yields rapid convergence in the outer krylov iterative solution process finally simulations of several large scale examples testify to the effectiveness and robustness of the proposed ie based ddm

initial and boundary value problems for fuzzy differential equations we prove a wintner type result for fuzzy ivps and a superlinear type result for fuzzy bvps both results rely on a generalized schauder theorem in metric spaces

plane wave interpolation in direct collocation boundary element method for radiation and wave scattering numerical aspects and applications the classical boundary element formulation for the helmholtz equation is rehearsed and its limitations with respect to the number of variables needed to model a wavelength are explained a new type of interpolation for the potential is then described in which the usual boundary element shape functions are modified by the inclusion of a set of plane waves propagating in a range of directions this is termed the plane wave basis boundary element method the modifications needed to the classical procedures in terms of integration of the element matrices and location of collocation points are described the well known singular value decomposition solution technique which is adopted here for the solution of the system matrix equation in its complex form is briefly outlined the conditioning of the system matrix is analysed for a simple radiation problem the corresponding diffraction problem is also analysed and results are compared with analytical and classical boundary element solutions the chief method is adopted to enhance the quality of the solution particularly in the vicinity of irregular frequencies the plane wave basis boundary element method is then applied to two problems scattering of plane waves by an elliptical cylinder and the multiple circular cylinder plane wave scattering problem in both cases results are compared with analytical solutions the results clearly demonstrate that the new method is considerably more efficient than the classical approach for a given number of degrees of freedom the frequency for which accurate results can be obtained using the new technique can be up to three or four times higher than that of the classical method this makes the method a powerful new addition to our tools for tackling high frequency radiation and scattering problems

an o 1 integration scheme for three dimensional surface scattering problems we present an accurate method of o 1 complexity with respect to frequency i e a method that to achieve a prescribed error tolerance requires a bounded computational cost for arbitrarily high frequencies for the computation of singular oscillatory integrals arising in the boundary integral formulation of problems of acoustic scattering by surfaces in three dimensional space like the two dimensional counterpart of this algorithm which we introduced recently and which is applicable to scattering by curves in the plane the present method is based on a combination of two main elements 1 a high frequency ansatz for the unknown density in a boundary integral formulation of the problem and 2 an extension of the ideas of the method of stationary phase to allow for o 1 high order accurate integration of oscillatory functions the techniques we introduce to implement an efficient o 1 integrator in the present three dimensional context differ significantly from those used in the earlier two dimensional algorithm in particular in the present text we introduce an efficient canonical hybrid analytic numerical algorithm which in addition to allowing for integration of oscillatory functions around both singular points and points of stationary phase can handle the significant difficulty that arises as singular points and one or more stationary points approach each other within the two dimensional scattering surface we include numerical results illustrating the behavior of the integration algorithm on sound soft spheres with diameters of up to 5000 wavelengths in such cases for a single integral the algorithm yields accuracies of the order of three digits in computational times of less than two seconds in a preliminary full scattering simulation we present a solution with two digits of accuracy in the surface density was obtained in about three hours running time in a single 1 5ghz amd athlon processor for a sphere of 500 wavelengths in diameter

arbitrarily smooth generalized finite element approximations this paper presents a procedure to build ck k arbitrarily large generalized finite element fe shape functions defined on non structured finite element meshes the functions have the same support as corresponding global fe lagrangian shape functions meshes with both convex and non convex clouds set of elements sharing a vertex node can be used the so called r functions are used to build ck fe based partition of unity functions with non convex support a technique to combine c0 lagrangian fe shape functions with the proposed ck partition of unity is presented the technique allows the use of ck generalized fe shape functions in parts of the computational domain where their high smoothness is required as in the case of problems with distributional boundary conditions and the less computationally demanding c0 generalized fe shape functions elsewhere in the domain a linear elasticity problem with a concentrated moment is solved using the proposed ck generalized fe method higher order distributional boundary conditions can also be handled by the method a detailed convergence analysis is presented for this class of problems as well as for problems in energy space the integrability of the functions using standard gauss legendre rules is also investigated

hierarchal vector boundary elements and p adaption for 3 d electromagnetic scattering this paper describes the development of new vector boundary elements for solving electromagnetic em scattering problems the new elements are suitable for the magnetic field integral equation mfie electrical field integral equation efie or the combination field integral equation cfie the basis functions are assigned to the edges of an element rather than to its nodes the new element guarantees the continuity of the normal component of the surface current across element edges furthermore the basis functions are hierarchical from linear to higher order which enables one to use the new elements in a p adaption scheme

a class of linear differential dynamical systems with fuzzy initial condition this paper investigates linear first order fuzzy differential dynamical systems where the initial condition is described by a fuzzy number we use a complex number representation of the a level sets of the fuzzy system and prove theorems that provide the solutions under such representation which is applicable to practical computations and also has some implications for theory then the paper shows some properties of the two dimensional dynamical systems and their phase portraits are described by means of examples there may be a significant difference between the solutions according to whether the matrix is nonnegative or not finally the paper points out future research on the fuzzy dynamical systems

on fuzzy boundary value problems the statement that a two point boundary value problem of fuzzy differential equation is equivalent to a fuzzy integral equation was pointed out by lakshmikantham et al and o regan et al recently bede gave a counterexample to show that this statement does not hold and he also argued that in many cases two point boundary value problems have no solutions under a new structure and certain conditions we show that a two point boundary value problem is equivalent to a fuzzy integral equation we also prove the existence of solutions to the two point boundary value problem in some sense this is an amendment to results of lakshmikantham et al and o regan et al and it is an answer to one of bede s problems

combining the higher order method of moments with geometric modeling by nurbs surfaces this paper introduces the nonuniform rational b spline nurbs surfaces to improve the geometric modeling of the higher order method of moments mom the electric field integral equation efie is discretized by the hierarchical higher order basis functions and converted to a matrix equation then the elements of the impedance matrix are efficiently evaluated by a new set of formulas the bistatic radar cross sections rcs obtained by this new technique are compared with those obtained by the commonly used higher order mom the example of a cylinder and a missile shows excellent accuracy of the nurbs surfaces and that of the resultant rcs moreover this new technique can fully exploit the flexibility of the higher order basis functions when the surface is highly curved whereas the commonly used higher order mom can not

two point boundary value problems of uncertain dynamical systems abstract recent research shows that in many cases there are no solutions periodic solutions etc to fuzzy two point boundary value problems in the sense of h derivatives to overcome the disadvantage of h derivatives in studying fuzzy two point boundary value problems in this paper we study fuzzy two point boundary value problems in the sense of differential inclusions i e two point boundary value problems of uncertain dynamical systems we obtain the existence and uniqueness of solutions and the existence of periodic solutions to two point boundary value problems of uncertain dynamical systems after introducing the concept of big solutions to two point boundary value problems of undamped uncertain dynamical systems we establish the existence and uniqueness of big solutions discuss the relationship between solutions and big solutions and further get a conclusion concerning the solutions of two point boundary value problems of uncertain dynamical systems

numerical methods forfuzzy differential inclusions in this paper a numerical method for solving fuzzy differential inclusions is considered fuzzy reachable set can be approximated by proposed method with complete error analysis which is discussed in detail the method is illustrated by solving some linear and nonlinear fuzzy initial value problems

on interactive fuzzy boundary value problems abstract in this paper we use the concept of interactivity between fuzzy numbers for the solution to a linear fuzzy boundary value problem fbvp we show that a solution of a fbvp with non interactive fuzzy numbers as boundary values can be obtained by the zadeh s extension principle in addition we show it is possible to obtain a fuzzy solution by means of the extension principle based on joint possibility distributions for the case where the boundary values are given by interactive fuzzy numbers examples of fbvps with both cases interactive and non interactive are presented also from arithmetic operations for linearly correlated fuzzy numbers we compare our solutions to the one proposed by gasilov et al we conclude that the fuzzy solution in the interactive case when the boundary values are linearly correlated fuzzy numbers is contained in the fuzzy solution for the non interactive case finally we present the fuzzy solution for a nonlinear fbvp with gaussian fuzzy numbers as boundary values

towards fuzzy differential calculus part 1 integration of fuzzy mappings abstract this paper deals with fuzzy set valued mappings of the real line and more particularly focuses on mappings from the real line to the set of convex normal fuzzy sets of the real line these mappings can also be viewed as fuzzy relations using zadeh s extension principle the integral of such fuzzy mappings over a crisp interval is defined provided a special analytical representation of the fuzzy mapping the practical computation of such an integral is shown to be easy practically speaking it yields the fuzzy surface of a fuzzily bounded area

estimating population size by recapture sampling a design called recapture debugging is proposed to get extra information to estimate the original number of faults n in a reliability system the system is observed for a fixed amount of time r and the failures are assumed to follow an exponential law with an unknown common parameter a the maximum likelihood and moment estimators of n and a are derived and compared with the existing estimators the recapture sampling provides substantially more information and improved estimators compared to the standard experiment if r is not small significance tests for testing hypotheses about n and a are developed finally for testing the system one can consider r to be determined sequentially and we present a suitable stopping rule

efficient surface integral equation using hierarchical vector bases for complex em scattering problems a set of hierarchical divergence conforming vector basis functions based on curved triangular patches is presented for method of moment mom solutions of surface integral equations in this paper the higher order method of combined field integral equation cfie solves perfect electric conductor electromagnetic scattering problems the higher order electric magnetic current combined field integral equation jmcfie formulations solves dielectric objects and their combination is able to efficiently analyzes the scattering of hybrid pec dielectric objects the expressions of the divergence conforming hierarchical basis functions up to order 3 5 are reported in this paper the multilevel fast multipole algorithm mlfma is then employed to reduce the memory requirements and computational complexity numerical experiments indicate that the proposed hierarchical vector basis functions can provide well conditioned linear system for iterative solution

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

higher order interpolatory vector bases for computational electromagnetics low order vector basis functions compatible with the nedelec 1980 representations are widely used for electromagnetic field problems higher order functions are receiving wider application but their development is hampered by the complex procedures used to generate them and lack of a consistent notation for both elements and bases in this paper fully interpolatory higher order vector basis functions of the nedelec type are defined in a unified and consistent manner for the most common element shapes it is shown that these functions can be obtained as the product of zeroth order nedelec representations and interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties of the vector functions are discussed and expressions for the vector functions of arbitrary polynomial order are presented sample numerical results confirm the faster convergence of the higher order functions

periodic problems of first order uncertain dynamical systems it is well known that there are no periodic solutions to fuzzy differential equations in the sense of hukuhara derivatives i e h derivatives so fuzzy differential equations in the sense of h derivatives cannot be used to describe periodic phenomena in the real world to overcome this disadvantage this paper studies periodic problems of first order fuzzy differential equations in the sense of differential inclusions i e periodic problems of first order uncertain dynamical systems and obtains the existence of periodic solutions for first order uncertain dynamical systems

hybrid metric dynamical systems with impulses abstract combining the concepts of hybrid dynamical system and metric dynamical system we introduce the concept of hybrid metric dynamical system hmds having as base space a metric space we give a comparison result for hmds we also establish the concept of hmds with impulses present a comparison result and study the stability of a solution

on the convergence of fuzzy sets three kind of convergences are defined on a space of fuzzy sets relationships between these convergences are studied some comments on the fixed point property are represented

a likelihood based stopping rule for recapture debugging summary consideration is given to determining when all the faults in a reliability system have been detected assuming the use of the recapture debugging procedure introduced by nayak 1988 a stopping rule based on the likelihood ratio is proposed compared to the stopping rule suggested by nayak this likelihood based rule makes better use of the available information and for a given error level yields a small reduction in the average time taken to reach a decision a generalization is suggested for the situation where the faults in the software can be categorized into two or more classes between which detection rates are permitted to differ nayak 1988 presents a modification of the standard jelinski moranda model for the debugging of computer software in recapture debugging a detected error is not only corrected as jelinski moranda 1972 assumed but also a counter is added to note each subsequent occasion on which this area of the software is accessed initially there are n errors present and occurrences of any particular error are assumed to follow a homogeneous poisson process the n poisson processes have a common rate a and the errors are accessed independently of each other one weakness of the standard jelinski moranda model is that the information about n can only be partially disentangled from that about a the model is slow to distinguish between slow detection of a large number of errors and the relatively faster detection of a much smaller number by noting when each detected error would have recurred recapture debugging provides additional information about the rate a and this helps to reduce the effect of this confounding recapture debugging employs in a software reliability context methodology originally derived for estimating the size of animal populations a similar reapplication in the case of capturerecapture methodology is given by duran wiorkowski 1981 who describe a hypergeometric model attributed to an unpublished ibm report by h d mills mills s procedure is to insert deliberately a known number of additional errors called seeded errors into the software estimation of the number of indigenous errors already present then proceeds by using the hypergeometric distribution for the number of seeded errors observed this approach requires the major assumption that the distribution of the detection time for a seeded error is the same as that for an indigenous error recapture debugging avoids this difficulty

comparison results for fuzzy differential equations in this paper we present several comparison results for the solutions of fuzzy differential equations by using the hukuhara derivative these results constitute the extension to the fuzzy context of some comparison results for the solutions of linear differential equations besides some side results about the hukuhara difference and the partial orderings are given we illustrate the applicability of the new results by showing several examples

oscillation property for fuzzy delay differential equations in this paper we obtain an oscillation criterion of solutions and the existence of nonoscillatory solutions to a class of fuzzy delay differential equation fdde of second order some examples are given to illustrate the main theorems

generalized method of moments a boundary integral framework for adaptive analysis of acoustic scattering boundary integral equations bies find applications in problems ranging from sonar to medical diagnostics the two ingredients of the bie solution technique are 1 representation of the domain and 2 design of approximation spaces to represent physical quantities on the domain these in concert affect accuracy and convergence of the simulation this paper presents a framework that permits the development of a scheme for refinement of size and order in both geometry and function representations further this permits flexibility in the types of basis functions that can be used capabilities of the proposed framework are shown via a number of numerical examples

periodic boundary value problems for impulsive fuzzy differential equations the peculiarity of the hukuhara derivative makes it impossible to find periodic solutions for fuzzy differential equations with the exception of very restrictive situations in this work we consider a boundary value problem associated with an impulsive fuzzy differential equation and approximate the extremal solutions in a fuzzy functional interval using the monotone method fuzzy comparison results are useful in our procedure and the expression of the solution for some impulsive periodic linear differential problems is also provided

the design and analysis of the generalized finite element method abstract in this paper we introduce the generalized finite element method gfem as a combination of the classical finite element method fem and the partition of unity method pum the standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data e g the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point etc the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space in this way the local approximability afforded by the special functions is included in the approximation while maintaining the existing infrastructure of finite element codes the major features of the gfem are 1 the essential boundary conditions can be imposed exactly as in the standard fem unlike other partition of unity based methods where this is a major issue 2 the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods and 3 linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver the power of the gfem for solving problems in domains with complex geometry with less error and less computer resources than the standard fem is illustrated by numerical examples

a class of linear differential dynamical systems with fuzzy matrices abstract this paper investigates the first order linear fuzzy differential dynamical systems with fuzzy matrices we use a complex number representation of the level sets of the fuzzy system and obtain the solution by employing such representation it is applicable to practical computations and has also some implications for the theory of fuzzy differential equations we then present some properties of the 2 dimensional dynamical systems and their phase portraits some examples are considered to show the richness of the theory and we can clearly see that new behaviors appear we finally present some conclusions and new directions for further research in the area of fuzzy dynamical systems

a mixed potential mlfma for higher order moment methods with application to the generalized method of moments the application of the multilevel fast multipole algorithm mlfma to higher order moment method discretizations is a continuing open problem herein we present a point based mixed potential variant of the mlfma algorithm that exhibits an mlfma tree of arbitrary height with no restriction related to basis function support size efficient nearfield precomputation and that maintains favorable scaling for any mixture of low and high order bases the flexibility of the algorithm is also leveraged to accelerate the precomputation of algebraic preconditioners we demonstrate the method through application to the generalized method of moments gmm a recently introduced moment method discretization capable of combining both low and high order bases and geometries in the same simulation however the method may be used to accelerate other higher order moment methods as well

higher order frequency domain computational electromagnetics a review of the higher order computational electromagnetics cem for antenna wireless and microwave engineering applications is presented higher order cem techniques use current field basis functions of higher orders defined on large e g on the order of a wavelength in each dimension curvilinear geometrical elements which greatly reduces the number of unknowns for a given problem the paper reviews all major surface volume integral and differential equation electromagnetic formulations within a higher order computational framework focusing on frequency domain solutions with a systematic and unified review of generalized curved parametric quadrilateral triangular hexahedral and tetrahedral elements and various types of higher order hierarchical and interpolatory vector basis functions in both divergence and curl conforming arrangements a large number of actual higher order techniques representing various combinations of formulations elements bases and solution procedures are identified and discussed the examples demonstrate the accuracy efficiency and versatility of higher order techniques and their advantages over low order discretizations the most important one being a much faster higher order convergence of the solution it is demonstrated that both components of the higher order modeling namely higher order geometrical modeling and higher order current field modeling are essential for accurate and efficient cem analysis of general antenna scattering and microwave structures

an interior penalty method for the generalized method of moments the generalized method of moments gmm is a technique to discretize integral equations that permits integration of different types of basis functions as well as different geometric descriptions using a partition of unity framework while accuracy and efficacy of the method have been demonstrated the integration quadratures required to compute the inner products are often high as they have to respect spatial variation of the integrand to overcome this problem we introduce an interior penalty function method within the gmm framework the penalty formulation yields solutions that are smooth and accurate both in surface currents and fields with significantly lower numerical quadrature orders than would be required for the uncompensated operator to demonstrate and analyze the method we conduct an analytical and numerical investigation of the properties of the penalty method applied to the 2 d text te z electric field integral equation efie operator

numerical solutions of the nonlinear fuzzy hammerstein volterra delay integral equations in this paper an iterative numerical method that solves nonlinear fuzzy hammerstein volterra integral equations with constant delay is developed by using the error estimates a practical stopping criterion of the algorithm is obtained the convergence and the numerical stability of the method is proved and its accuracy is illustrated on three numerical examples the study of this integral equation is important because it has as a particular case the fuzzy variant of a mathematical model from epidemiology

a note on two point boundary value problems associated with non linear fuzzy differential equations in recent papers lakshmikantham et al and o regan et al assert that two point boundary value problem for a fuzzy differential equation is equivalent to an integral equation and the proofs of the main results in the above cited papers use this equivalence we prove by a counterexample that by using h differentiability or hukuhara differentiability this assertion does not hold moreover it is proved that a large class of two point boundary value problems have no solutions at all under h differentiability or hukuhara differentiability

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

further development of vector generalized finite element method and its hybridization with boundary integrals recently vector generalized finite element method vgfem was introduced for the solution of the vector helmholtz equation and its applicability was validated for canonical problems vgfem uses a local helmholtz decomposition to construct basis functions in overlapping local domains of some canonical shape while using a canonical shape for local domains adds flexibility to the method one needs to provide information regarding boundaries of domains inhomogeneities the need for surface information proves to be a bottleneck in using the method for a larger class of problems this paper is targeted towards overcoming these deficiencies here we will introduce the modifications to this method that permit interfacing with arbitrarily shaped local domains to facilitate interfacing with existing meshing software integrate this method with boundary integrals and provide a framework for studying dispersion as will be apparent the hybridization of the method with boundary integrals is not a simple adaptation of existing methods onto the vgfem framework likewise dispersion analysis is nontrivial due to the overlapping nature of vgfem basis functions a range of practical problems has been analyzed within the presented framework and results are compared either against measurements or existing fem data to validate the presented methodology

solution method for a boundary value problem with fuzzy forcing function in this paper we present a new approach to a non homogeneous fuzzy boundary value problem we consider a linear differential equation with real coefficients but with a fuzzy forcing function and fuzzy boundary values we assume that the forcing function is a triangular fuzzy function unlike previous studies we look for a solution that is a fuzzy set of real functions not a fuzzy valued function each of these real functions satisfies the boundary value problem with some membership degree we have developed a method that finds this solution and demonstrated its effectiveness using a test example to show that the approach can be extended to other types of fuzzy numbers we extended it to the trapezoidal case for a particular example we used the product t norm to demonstrate how a new solution type can be obtained

towards fuzzy differential calculus part 2 integration on fuzzy intervals abstract the problem of integrating mappings over fuzzy domains has already received some attention in the literature however there is not a unique approach but several definitions have been proposed each carrying its own meaning this paper tries to emphasize the relationships and the differences between the various points of view

towards fuzzy differential calculus part 3 differentiation this paper deals successively with differentiation of ordinary functions at a fuzzy point and with differentiation of fuzzy valued functions at a nonfuzzy point in both cases the concept of differentiation is extended and the problem of the linearity is addressed in the second case practical methods of computation are presented in practical situations the available knowledge about mappings may be fuzzy it is the motivation of this work

set valued and fuzzy stochastic differential equations driven by semimartingales abstract in the paper we present set valued and fuzzy stochastic integrals with respect to semimartingale integrators as well as their main properties then we study the existence of solutions to set valued and fuzzy set valued stochastic differential equations driven by semimartingales the stability of solutions is also established

ulam type stability of first order linear impulsive fuzzy differential equations abstract in this paper we consider ulam stability concepts for first order linear impulsive fuzzy differential equations which generalize standard definitions for impulsive differential equations sufficient conditions are established to guarantee that our equations are ulam stable numerically examples are also provided to illustrate our results

a peano theorem for fuzzy differential equations with evolving membership grade abstract the peano theorem on the existence without possible uniqueness of solutions has been a perplexing problem in the theory of fuzzy differential equations the difficulty appears to be due to the standard use of the supremum metric defined by the supremum over the hausdorff metric between the level sets of the fuzzy sets another may have been the classical formulation of fuzzy differential equations in terms of the hukuhara derivative of the level sets here a peano theorem is established for fuzzy differential equations formulated in a recent paper by the authors by combining hullermeier s suggestion of defining fuzzy differential equations at each level set via differential inclusions with aubin s morphological equations which allow non local set evolution a major difference from previous publications is the use of the endograph metric end essentially the hausdorff metric between the endographs in r n 0 1 of fuzzy sets instead of the supremum metric another is that the membership grades of the fuzzy sets are also allowed to evolve under the fuzzy differential equations the result applies for a very general class of fuzzy sets without additional assumptions of fuzzy convexity compact supports or even normality

higher order hierarchical legendre basis functions for electromagnetic modeling this paper presents a new hierarchical basis of arbitrary order for integral equations solved with the method of moments mom the basis is derived from orthogonal legendre polynomials which are modified to impose continuity of vector quantities between neighboring elements while maintaining most of their desirable features expressions are presented for wire surface and volume elements but emphasis is given to the surface elements in this case the new hierarchical basis leads to a near orthogonal expansion of the unknown surface current and implicitly an orthogonal expansion of the surface charge in addition all higher order terms in the expansion have two vanishing moments in contrast to existing formulations these properties allow the use of very high order basis functions without introducing ill conditioning of the resulting mom matrix numerical results confirm that the condition number of the mom matrix obtained with this new basis is much lower than existing higher order interpolatory and hierarchical basis functions as a consequence of the excellent condition numbers we demonstrate that even very high order mom systems e g tenth order can be solved efficiently with an iterative solver in relatively few iterations

solution of linear differential equations with fuzzy boundary values we investigate linear differential equations with boundary values expressed by fuzzy numbers in contrast to most approaches which search for a fuzzy valued function as the solution we search for a fuzzy set of real functions as the solution we define a real function as an element of the solution set if it satisfies the differential equation and its boundary values are in intervals determined by the corresponding fuzzy numbers the membership degree of the real function is defined as the lowest value among membership degrees of its boundary values in the corresponding fuzzy sets to find the fuzzy solution we use a method based on the properties of linear transformations we show that the fuzzy problem has a unique solution if the corresponding crisp problem has a unique solution we prove that if the boundary values are triangular fuzzy numbers then the value of the solution at a given time is also a triangular fuzzy number the defined solution is the same as one of the solutions obtained by zadeh s extension principle for a second order differential equation with constant coefficients the solution is expressed in analytical form examples are given to describe the proposed approach and to compare it to a method that uses the generalized hukuhara derivative which demonstrates the advantages of our method

stopping rules and estimation for recapture debugging with unequal failure rates summary a theoretical optimal stopping rule based on the minimization of the testing cost and expected penalty due to the unremoved bugs is derived without the equal failure rate assumption under nayak s 1988 recapture debugging procedure two adaptive stopping rules suitable for practical use are constructed and tested by simulation the estimation of the number of undetected bugs is also considered via remaining failure rate estimation

novel basis functions for the partition of unity boundary element method for helmholtz problems the bem is a popular technique for wave scattering problems given its inherent ability to deal with infinite domains in the last decade the partition of unity bem in which the approximation space is enriched with a linear combination of plane waves has been developed this significantly reduces the number of dofs required per wavelength it has been shown that the element ends are more susceptible to errors in the approximation than the mid element regions in this paper the authors propose that this is due to the use of a collocation approach in combination with a reduced order of continuity in the lagrangian shape function component of the basis functions it is demonstrated using numerical examples that choosing trigonometric shape functions rather than classical polynomial shape functions quadratic in this case provides accuracy benefits collocation schemes are investigated it is found that the somewhat arbitrary choice of collocating at equally spaced points about the surface of a scatterer is better than schemes based on the roots of polynomials or consideration of the fock domain

a novel meshless scheme for solving surface integral equations with flat integral domains numerical solutions for electromagnetic em integral equations rely on the discretization of integral domains and the use of meshes for geometric description meshing geometries is very tedious especially for complicated structures with many details tiny parts and geometric discontinuities corners or edges and remeshing could be required in many scenarios to reduce the costs of generating quality meshes meshless or mesh free methods were developed and they have been extensively studied in mechanical engineering though there are less obvious interests in em community the meshless methods employ discrete nodes to replace meshes in the description of geometries but the background meshes for integrations are still needed traditionally in this work we first address the traditional meshless scheme for solving em integral equations based on the moving least square mls approximation for unknown currents and the use of background meshes for integrations and then develop a novel meshless scheme by applying the green s lemma to the em surface integral equations with flat domains the novel scheme transforms a surface integral over a flat domain into a line integral along its boundaries when excluding a singular patch in the domain since only the domain boundaries are discretized and no background meshes are needed the scheme is truly meshless numerical examples for em scattering by flat surface objects are presented to demonstrate the effectiveness of the novel scheme

double higher order method of moments for surface integral equation modeling of metallic and dielectric antennas and scatterers a novel double higher order galerkin type method of moments based on higher order geometrical modeling and higher order current modeling is proposed for surface integral equation analysis of combined metallic and dielectric antennas and scatterers of arbitrary shapes the technique employs generalized curvilinear quadrilaterals of arbitrary geometrical orders for the approximation of geometry metallic and dielectric surfaces and hierarchical divergence conforming polynomial vector basis functions of arbitrary orders for the approximation of electric and magnetic surface currents within the elements the geometrical orders and current approximation orders of the elements are entirely independent from each other and can be combined independently for the best overall performance of the method in different applications the results obtained by the higher order technique are validated against the analytical solutions and the numerical results obtained by low order moment method techniques from literature the examples show excellent accuracy flexibility and efficiency of the new technique at modeling of both current variation and curvature and demonstrate advantages of large domain models using curved quadrilaterals of high geometrical orders with basis functions of high current approximation orders over commonly used small domain models and low order techniques the reduction in the number of unknowns is by an order of magnitude when compared to low order solutions

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

hukuhara derivative of the fuzzy expected value in this paper we establish sufficient conditions to guarantee the exchange of the fuzzy expected value for a fuzzy random variable depending on certain parameter and its hukuhara derivative with respect to the parameter main results are obtained for two different sets of general conditions one of them requiring measurability conditions with respect to the product space the fuzzy random variable is defined from and the second one just requiring conditions over the projections of the fuzzy random variables