volterra discrete equations summability of the fundamental matrix conditions are proven which assure the summability of the first difference of the fundamental matrix of nonconvolution volterra discrete equations these conditions are applied to the stability analysis of some linear methods for solving volterra integral equations of nonconvolution type

block boundary value methods for solving volterra integral and integro differential equations reducible quadrature rules generated by boundary value methods are considered in block version and applied to solve the second kind volterra integral equations and volterra integro differential equations these extended block boundary value methods are shown to possess both excellent stability properties and high accuracy for volterra type equations numerical experiments are presented and the efficiency accuracy and stability of the schemes are confirmed

linear multistep methods for volterra integral and integro differential equations a general class of linear multistep methods is presented for numerically solving firstand second kind volterra integral equations and volterra integro differential equations these so called vlm methods which include the well known direct quadrature methods allow for a unified treatment of the problems of consistency and convergence and have an analogue in linear multistep methods for odes as treated in any textbook on computational methods in ordinary differential equations general consistency and convergence results are presented and proved in an appendix together with results of numerical experiments which support the theory

on nilpotency of matrices over antirings abstract an antiring is a semiring which is zerosumfree i e a b 0 implies a b 0 for any a b in this semiring in this paper we study the nilpotency of matrices over commutative antirings we first provide some properties and characterizations of the nilpotent matrices in terms of principal permanental minors main diagonals and permanental adjoint matrices when a family of matrices are simultaneously considered we establish some characterizations of the simultaneous nilpotence for a family of matrices

benchmark fluid flow problems for continuous simulation languages abstract two benchmark problems for continuous simulation languages are discussed the use of the advanced continuous simulation language acsl and the sparse ordinary differential equation solver dstpgt which has been incorporated into acsl are discussed for the solution of these fluid flow problems the one dimensional incompressible navier stokes partial differential equations are discretized spatially using the method of pseudo characteristics the resulting sparse system of ordinary differential equations is then solved using the method of lines a continuous space discrete time solution is also given in order to illustrate the use of the dstpgt special event detection mechanism root finding in acsl the discussions illustrate several important considerations related to the solution of complex fluid problems or more generally to sparse systems and or systems requiring the detection and processing of special events

bases in semimodules over commutative semirings abstract in this paper the bases of a semimodule over a commutative semiring r are investigated some properties and characterizations of the bases are discussed and some equivalent conditions for a basis to be a free basis in a finitely generated free semimodule over r are given the different possible cardinalities for a basis in a finitely generated free semimodule over r are considered and some equivalent descriptions are obtained for a commutative semiring r satisfying the property that any two bases for a finitely generated free r semimodule have the same cardinality partial results obtained in the paper develop and generalize the corresponding results for commutative join semirings and for commutative zerosumfree semirings

existence and approximate lp and continuous solutions of nonlinear integral equations of the hammerstein and volterra types in this paper we consider a compact interval a b a positive real number p 1 and give different existence results for l p a b and c a b solutions of some nonlinear integral equations of the hammerstein and volterra types the main ingredients of our existence results are the shaefer s and schauder s fixed point theorems combined with a general version of gronwall s inequality moreover we give a numerical method for the approximation of the solutions of the volterra hammerstein integral equations finally to illustrate the results of this work we provide the reader with some numerical examples

matrices over semirings introducing the concept of semi invertibility of square matrices over semirings some sufficient conditions for semi invertibility of square matrices over various kinds of semirings are obtained also a necessary and sufficient condition for semi invertibility of square matrices over semirings is obtained

a multistep legendre gauss spectral collocation method for nonlinear volterra integral equations we introduce a multistep legendre gauss spectral collocation method for the nonlinear volterra integral equations of the second kind this method is easy to implement and possesses high order accuracy in addition it is very suitable for long time calculations we also derive the optimal convergence of the hp version of the multistep collocation method under the l 2 norm numerical experiments confirm the theoretical expectations

an iterative multistep kernel based method for nonlinear volterra integral and integro differential equations of fractional order abstract an iterative multistep kernel based method is proposed for the nonlinear volterra integral equations and nonlinear volterra integro differential equations of fractional order which can produce reliable globally smooth numerical solutions an error estimate of the positive definite kernel interpolation and the convergence and error analysis of the proposed iterative scheme are investigated here we focused on positive definite radial basis kernels and further a new and applicable shape parameter selection strategy is proposed the proposed multi step method set up and solve several small local problems instead of a single large problem which makes it suitable for problems with long time simulations in order to show the efficiency and versatility of the proposed method some numerical experiments are reported the comparison of the numerical results with the analytical solutions and the best reported results in the literature confirm the good accuracy and applicability of the proposed method

on simultaneously nilpotent fuzzy matrices abstract nilpotent fuzzy matrices play a crucial role in the study of fuzzy matrices in this paper we shall extend the nilpotence to the notion of simultaneous nilpotence for a finite set of fuzzy matrices the notion of simultaneous nilpotence relates to the infinite products of a finite number of fuzzy matrices which converge to the zero matrix properties of the simultaneous nilpotence will be established in the study of consecutive powers of a fuzzy matrix a controllable fuzzy matrix can be characterized by an associated nilpotent fuzzy matrix in this paper we propose the notion of simultaneously controllable fuzzy matrices which can be thought of as a generalization of the notion of controllable fuzzy matrices similar to the nilpotent characterization for a controllable fuzzy matrix the simultaneously controllable fuzzy matrices can be characterized by a finite set of associated simultaneously nilpotent fuzzy matrices

high performance parallel numerical methods for volterra equations with weakly singular kernels non stationary discrete time waveform relaxation methods for abel systems of volterra integral equations using fractional linear multistep formulae are introduced fully parallel discrete waveform relaxation methods having an optimal convergence rate are constructed a significant expression of the error is proved which allows us to estimate the number of iterations needed to satisfy a prescribed tolerance and allows us to identify the problems where the optimal methods offer the best performance the numerical experiments confirm the theoretical expectations

boundary value methods for volterra integral and integro differential equations abstract new and effective quadrature rules generated by boundary value methods are introduced we employ the introduced quadrature rules to construct quadrature methods for the second kind volterra integral equations and volterra integro differential equations these methods are shown to be effective and possess excellent convergence properties the nonlinear multigrid method is applied to solve the discrete systems derived from the introduced numerical scheme numerical simulations are presented and confirm the efficiency and accuracy of the methods

algebraically stable and implementable runge kutta methods of high order there are three interesting properties of methods for stiff ordinary differential equations order stability and efficiency of implementation this paper constructs runge kutta methods of orders 5 and 6 which possess these properties to a high extent we further classify all algebraically stable methods of an arbitrary order and give various relationships between contractivity and order of implicit methods

fast numerical solution of weakly singular volterra integral equations abstract we give a fortran program for the numerical solution of abel volterra convolution equations of the second and first kind we use a fractional linear multistep method bdf4 1 2 and use fast fourier transform techniques to exploit the convolution structure a detailed description of the computation of the quadrature weights using fft is given numerical examples are included

the exponential homomorphism for the second syntomic cohomology abstract let ok be a complete discrete valuation ring with perfect residue field f of characteristic p 3 and m a free filtered dieudonne module the second relative syntomic cohomology h2 ok f s m 2 with coefficients in m is an object related to arithmetic geometry such as the albanese kernel in this article we study h2 ok f s m 2 by constructing the exponential homomorphism we determine the structure of h2 ok f s m 2 under the assumption that m is of hodge witt type and that the absolute ramification index of ok is prime to p

order conditions for numerical methods for partitioned ordinary differential equations motivated by the consideration of runge kutta formulas for partitioned systems the theory of p series is studied this theory yields the general structure of the order conditions for numerical methods for partitioned systems and in addition for nystrom methods fory f y y for rosenbrock type methods with inexact jacobian w methods it is a direct generalization of the theory of butcher series 7 8 in a later publication the theory ofp series will be used for the derivation of order conditions for runge kutta type methods for volterra integral equations 1

invertible and nilpotent matrices over antirings abstract in this paper we characterize invertible matrices over an arbitrary commutative antiring s with 1 and find the structure of gl n s we find the number of nilpotent matrices over an entire commutative finite antiring we prove that every nilpotent n n matrix over an entire antiring can be written as a sum of log 2 n square zero matrices and also find the necessary number of square zero summands for an arbitrary trace zero matrix to be expressible as their sum

piecewise homotopy perturbation method for solving linear and nonlinear weakly singular vie of second kind the purpose of this paper is to obtain the approximation solution of linear and strong nonlinear weakly singular volterra integral equation of the second kind especially for such a situation that the equation is of nonsmooth solution and the situation that the problem is a strong nonlinear problem for this purpose we firstly make a transform to the equation such that the solution of the new equation is as smooth as we like through modifying homotopy perturbation method an algorithm is successfully established to solve the linear and nonlinear weakly singular volterra integral equation of the second kind and the convergence of the algorithm is proved strictly comparisons are made between our method and other methods and the results reveal that the modified homotopy perturbation is effective

runge kutta theory for volterra integrodifferential equations the present paper develops the theory of general runge kutta methods for volterra integrodifferential equations the local order is characterized in terms of the coefficients of the method we investigate the global convergence of mixed and extended runge kutta methods and give results on asymptotic error expansions in a further section we construct examples of methods up to order 4

p adic elliptic polylogarithm p adic eisenstein series and katz measure the specializations of the motivic elliptic polylogarithm on the universal elliptic curve to the modular curve are referred to as eisenstein classes in this paper we prove that the syntomic realization of the eisenstein classes restricted to the ordinary locus of the modular curve may be expressed using p adic eisenstein series of negative weight which are p adic modular forms defined using the two variable p adic measure with values in p adic modular forms constructed by katz the motivation of our research is the p adic beilinson conjecture formulated by perrin riou

generalized fuzzy h bi ideals and h quasi ideals of hemirings a new kind of generalized fuzzy h ideals of a hemiring namely the set membership variant set membership variantlogical orq fuzzy h bi ideal resp h quasi ideal is studied and the relationships between these generalized fuzzy h ideals are described some characterization theorems of prime and semiprime set membership variant set membership variantlogical orq fuzzy h bi ideals resp h quasi ideals of a hemiring are also given in particular we show that the h hemiregular hemirings and h intra hemiregular hemirings can be described by using some of their generalized fuzzy h ideals finally the implication based fuzzy h bi ideals resp h quasi ideals of a hemiring are considered

numerical methods for treating problems of viscoelastic isotropic solid deformation mathematical models for treating problems of linear viscoelasticity involving hereditary constitutive relations for compressible solids are discussed and their discretization using finite element methods in space together with quadrature rules in time to treat the hereditary integrals is described the range of applicability of this type of formulation is reviewed in the context of geometric and constitutive linearity nonlinearity and the limitations imposed by the availability of physical data are discussed one of the above models is a volterra integral equation of the second kind in this when the kernel is separable an established technique due to goursat 1933 can be exploited to reformulate the problem as a system of ordinary differential equations this approach will be described for the special case of a linear viscoelastic isotropic homogeneous synchronous constant poisson s ratio solid this method results in a complete decoupling of the space and time dependence in this case the problem can be solved at each time level by solving first a problem of linear elasticity and then a system of ordinary differential equations for each point in the spatial mesh at which the viscoelastic displacements are required the advantages disadvantages and limitations offered by this and various other schemes for solving problems of viscoelasticity as outlined below are discussed

rankin eisenstein classes for modular forms in this paper we make a systematic study of certain motivic cohomology classes rankin eisenstein classes attached to the rankin selberg convolution of two modular forms of weight 2 the main result is the computation of the p adic syntomic regulators of these classes as a consequence we prove many cases of the perrin riou conjecture for rankin selberg convolutions of cusp forms

a parallel algorithm for large systems of volterra integral equations of abel type a significative number of recent applications require numerical solution of large systems of abel volterra integral equations here we propose a parallel algorithm to numerically solve a class of these systems designed for a distributed memory mimd architecture in order to achieve a good efficiency we employ a fully parallel and fast convergent waveform relaxation wr method and evaluate the lag term by using fft techniques to accelerate the convergence of the wr method and to best exploit the parallel architecture we develop special strategies the performances of the resulting code nswr4 are illustrated on some examples

syntomic cohomology as a p adic absolute hodge cohomology the purpose of this paper is to interpret rigid syntomic cohomology defined by amnon besser bes as a p adic absolute hodge cohomology this is a p adic analogue of a work of beilinson be1 which interprets beilinson deligne cohomology in terms of absolute hodge cohomology in the process we will define a theory of p adic absolute hodge cohomology with coefficients which may be interpreted as a generalization of rigid syntomic cohomology to the case with coefficients

variable transformations in the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels extensions to fredholm integral equations abstract a number of techniques that use variable transformations in numerical integration have been developed recently cf sidi numerical integration iv h brass g hammerlin eds birkhauser basel 1993 pp 359 373 laurie j comput appl math 66 1996 337 344 the use of these transformations resulted in increasing the order of convergence of the trapezoidal and the midpoint quadrature rule in this paper the application of variable transformation techniques of sidi and laurie type to the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels is considered since the transformations are such that the end points of integration need not be used as mesh points the methods introduced can be used for vie with both continuous and weakly singular kernel in a uniform way the methods have also the advantages of simplicity of application and of achieving high order of convergence the application of the idea to fredholm integral equations with continuous and weakly singular equations is also considered numerical results are included and they verify the expected increased order of convergence they were obtained by using the trapezoidal formula for the evaluation of the transformed integrals

runge kutta theory for volterra integral equations of the second kind the present paper develops the theory of general runge kutta methods for volterra integral equations of the second kind the order conditions are derived by using the theory of p series which for our problem reduces to the theory of v series these results are then applied to two special classes of runge kutta methods introduced by pouzet and by bel tyukov

superconvergence of collocation methods for volterra and abel integral equations of the second kind this paper deals with the question of the attainable order of convergence in the numerical solution of volterra and abel integral equations by collocation methods in certain piecewise polynomial spaces and which are based on suitable interpolatory quadrature for the resulting moment integrals the use of a nonlinear variation of constants formula for the representation of the error function in terms of the defect allows for a unified treatment of equations with continuous and weakly singular kernels

on simultaneously nilpotent antiring matrices antirings are an important type of semirings which generalize boolean algebra fuzzy algebra distributive lattice and incline in this paper we study the issue of nilpotent antiring matrices provide some properties and characterizations of the simultaneous nilpotence for a finite number of antiring matrices present some methods for calculating the simultaneously nilpotent index of a finite number of antiring matrices partial results obtained in this paper generalize and develop the corresponding ones on nilpotent antiring matrices and on simultaneously nilpotent fuzzy matrices lattice matrices

mathematical programming methods in the numerical solution of volterra integral and integro differential equations with weakly singular kernel abstract nonlinear volterra integral and integro differential equations with weakly singular kernel are considered and solved numerically using nonlinear mathematical programming methods based on minimax approximations in both cases polynomial and multiquadric approximation are used

inversion of matrices over a commutative semiring it is a well known consequence of the elementary theory of vector spaces that if a and b are n by n matrices over a field or even a skew field such that ab 1 then ba 1 this result remains true for matrices over a commutative ring however it is not in general true for matrices over noncommutatives rings in this paper we show that if a and b are n by n matrices over a commutative semiring then the equation ab 1 implies ba 1 we give two proofs one algebraic in nature the other more combinatorial both proofs use a generalization of the familiar product law for determinants over a commutative semiring

a rational basis for second kind abel integral equations abstract a rational basis set is constructed using 1 1 rational interpolation the basis is used in a product integration method for solving second kind abel integral equations criteria are established for determining the amount of bias to be used in the basis set when the integral equation has a nonsmooth solution several test problems are solved to illustrate the performance of the approximation method

accurate evaluation of european and american options under the cgmy process a finite difference method for integro differential equations arising from le vy driven asset processes in finance is discussed the equations are discretized in space by the collocation method and in time by an explicit backward differentiation formula the discretization is shown to be second order accurate for a relevant parameter range determining the degree of the singularity in the le vy measure the singularity is dealt with by means of an integration by parts technique an application of the fast fourier transform gives the overall amount of work o n t n log n rendering the method fast

numerical solution of integro differential equations arising from singular boundary value problems abstract we consider the numerical solution of a singular boundary value problem on the half line for a second order nonlinear ordinary differential equation due to the fact that the nonlinear differential equation has a singularity at the origin and the boundary value problem is posed on an unbounded domain the proposed approaches are complex and require a considerable computational effort in the present paper we describe an alternative approach based on the reduction of the original problem to an integro differential equation though the problem is posed on the half line we just need to approximate the solution on a finite interval by analyzing the behavior of the numerical approximation on this interval we identify the solution that satisfies the prescribed boundary condition although the numerical algorithm is much simpler than the ones proposed before it provides accurate approximations we illustrate the proposed methods with some numerical examples

product integration for volterra integral equations of the second kind with weakly singular kernels we introduce a new numerical approach for solving volterra integral equations of the second kind when the kernel contains a mild singularity we give a convergence result we also present numerical examples which show the performance and efficiency of our method

a discrete time version for models of population dynamics in the presence of an infection we present a set of difference equations which represents the discrete counterpart of a large class of continuous model concerning the dynamics of an infection in an organism or in a host population the limiting behavior of the discrete model is studied and a threshold parameter playing the role of the basic reproduction number is derived

order of convergence of one step methods for volterra integral equations of the second kind nice proofs of convergence and asymptotic expansions are known for one step methods for ordinary differential equations it is shown that these proofs can be generalized in a natural way to extended one step methods for volterra integral equations of the second kind furthermore the convergence of mixed one step methods is investigated for both types general volterra runge kutta methods are considered as examples

inner products on semimodules over a commutative semiring abstract in this paper the inner products on the semimodules over a commutative semiring are investigated some characterizations for inner products and for orthogonal sets in the semimodules are given and standard orthogonal basis and orthogonal subsemimodule are discussed in particular an equivalent description is obtained for a semifield s satisfying the property that every standard orthogonal set in a finitely generated semimodule m over s can be extended to a standard orthogonal basis for m also the analogue of the kronecker capelli theorem is proved to be valid for a matrix equation over a commutative semiring partial results obtained in this paper generalize and develop corresponding results for semilinear spaces of n dimensional vectors over commutative zerosumfree semirings

a note on the relative efficiency of adams methods abstract in several pieces of the most effective mathematical software for non stiff and mildly stiff initial value problems in ordinary differential equations the classical adams methods are implemented in a particular mode one reason for the choice of mode is sometimes explained in terms of the associated scaled absolute stability regions for the methods in this note it is shown that the same choice of mode remains appropriate from this point of view when certain families of generalized adams type methods are considered

natural continuous extensions of runge kutta methods for volterra integral equations of the second kind and their applications we consider a very general class of runge kutta methods for the numerical solution of volterra integral equations of the second kind which includes as special cases all the more important methods which have been considered in the literature the main purpose of this paper is to define and prove the existence of the natural continuous extensions nce s of runge kutta methods i e piecewise polynomial functions which extend the approximation at the grid points to the whole interval of integration the particular properties required of the nce s allow us to construct the tail approximations which are quite efficient in terms of kernel evaluations

noncommuting graphs of matrices over semirings we calculate diameters and girths of commuting graphs of the set of all nilpotent matrices over a semiring the group of all invertible matrices over a semiring and the full matrix semiring

on nilpotent matrices over antirings in this paper the nilpotent matrices over commutative antirings are characterized in terms of principal permanental minors main diagonals and permanental adjoint matrices and a necessary and sufficient condition for a nilpotent matrix over a commutative antiring which has a given nilpotent index is obtained also a method for calculating the nilpotent index of any nilpotent matrix over a commutative entire antiring is given

on nilpotent fuzzy matrices abstract in this paper we study the issue of nilpotent fuzzy matrices we first provide some properties of nilpotent fuzzy matrices in terms of eigenvalues when a finite number of fuzzy matrices are simultaneously considered we establish some characterizations of the simultaneous nilpotence for a finite number of fuzzy matrices on the other hand it is well known that the max min algebra is one kind of lattice we shall extend the results on simultaneous nilpotence to matrices on a bounded distributive lattice

a survey of recent advances in the numerical treatment of volterra integral and integro differential equations abstract the present survey paper samples recent advances in the numerical analysis of volterra integral equations of the first and second kind and of integro differential equations including equations with weakly singular kernels except for some important earlier references the discussion focuses on the development which has taken place during the last dozen years a fairly extensive bibliography selected to be representative rather than comprehensive complements the paper

a rank theorem for vandermonde matrices we show that certain matrices built from vandermonde matrices are of full rank this result plays a key role in the construction of the limit theory of generic polynomials

strong superconvergence of the eulermaruyama method for linear stochastic volterra integral equations the eulermaruyama method is presented for linear stochastic volterra integral equations then the strong convergence property is analyzed for convolution kernels and general kernels respectively it is well known that for stochastic ordinary differential equations the strong convergence order of the eulermaruyama method is 12 however the strong superconvergence order of 1 is obtained for linear stochastic volterra integral equations with convolution kernels if the kernel k2 of the diffusion term satisfies k2 0 0 and this strong superconvergence property is inherited by linear stochastic volterra integral equations with general kernels if the kernel k2 of the diffusion term satisfies k2 t t 0 the theoretical results are illustrated by extensive numerical examples

a nonpolynomial collocation method for fractional terminal value problems in this paper we propose a nonpolynomial collocation method for solving a class of terminal or boundary value problems for differential equations of fractional order 0 1 the approach used is based on the equivalence between a problem of this type and a fredholm integral equation of a particular form taking into account the asymptotic behaviour of the solution of this problem we propose a nonpolynomial collocation method on a uniform mesh we study the order of convergence of the proposed algorithm and a result on optimal order of convergence is obtained in order to illustrate the theoretical results and the performance of the method we present several numerical examples

fft techniques in the numerical solution of convolution equations abstract we discuss the efficient use of fft routines to determine discrete convolutions required in the numerical solution of volterra and abel integral and integro differential equations the paper is based on a series of technical reports produced by the authors which include fortran 77 listings of computer codes these codes employ the fft routiens constructed for the nag library

a stability analysis of the trapezoidal method for volterra integral equations with completely positive kernels abstract the solution of the volterra integral equation with completely positive kernel y t 0t b t s y s ds u0 0t b t s g s ds t 0 is nonnegative and nonincreasing provided that g is nonincreasing and 0 g t u0 for any t 0 we prove that under some additional hypotheses this property is inherited by the solution of the recurrence relation resulting from applying the trapezoidal method to this equation

uniform error estimates of operational quadrature methods for nonlinear convolution equations on the half line we study uniform error estimates of operational quadrature methods for nonlinear convolution equations on the half line equations of this kind arise in control engineering and diffusion problems the essential ingredients are the stability of the operational quadrature method in an l2 setting which is inherited from the continuous equation by its very construction and a theorem that says that the behavior of the linearized equations is the same in all lp spaces i p oo

the numerical solution of integral algebraic equations of index 1 by polynomial spline collocation methods in this paper we study polynomial spline collocation methods applied to a particular class of integral algebraic equations of volterra type we analyse mixed systems of second and first kind integral equations global convergence and local superconvergence results are established

on nilpotent incline matrices inclines are additively idempotent semirings in which products are less than or equal to either factor thus they generalize boolean algebra fuzzy algebra and distributive lattice this paper studies the nilpotent incline matrices in detail it is proved that an incline matrix is nilpotent if and only if it has index and the zero vector is its unique standard eigenvector the nilpotent matrices over an incline without nilpotent elements are characterized in terms of principal minors main diagonals nilpotent indices and adjoint matrices also some properties of the reduction of nilpotent matrices over an additively residuated incline without nilpotent elements are established the results obtained here generalize the corresponding ones on fuzzy matrices and lattice matrices shown in the references

numerical approximation to the fractional derivative operator in this paper we consider the numerical approximation of a alpha a by contour integral we are mainly interested to the case of a a representing the discretization of the first derivative by means of a backward differentiation formula and 0 alpha 1 0 1 the computation of the contour integral yields a rational approximation to a alpha a which can be used to define k k step formulas for the numerical integration of fractional differential equations

the non existence of certain a0 stable adams type correctors abstract the question of the existence of a0 stable adams type correctors is addressed in this paper a non existence relationship between step number and local order is established

theoretical and numerical analysis for volterra integro differential equations with ito integral under polynomially growth conditions abstract in this paper we theoretically and numerically deal with nonlinear volterra integro differential equations with ito integral under a one sided lipschitz condition and polynomially growth conditions it is proved that both the exact solutions and vector fields are bounded and satisfy a holder condition in the pth moment sense analogously the boundedness and holder condition in the pth moment sense are preserved by the semi implicit euler method for sufficiently small step size moreover by the local truncated errors we prove the strong convergence order 1 finally numerical simulations on stochastic control models and stochastic ginzburg landau equation illustrate our results

convolution quadrature and discretized operational calculus ii operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations singular integrals power and logarithmic singularities finite part integrals multiple timescale convolution volterra integral equations wiener hopf integral equations frequency domain conditions which determine the stability of such equations can be carried over to the discretization

on the nilpotent matrices over d01 lattice abstract in this paper we study nilpotent matrices over d 01 lattice a distributive lattice l is said to be d 01 lattice if 0 1 l and 0 x 1 for all x l a necessary and sufficient condition for a d 01 lattice matrix a to be a nilpotent matrix will be given we also establish some important properties of nilpotent matrices

stability of θ methods for delay integro differential equations stability of methods for delay integro differential equations dides is studied on the basis of the linear equation du dt u t u t k t t u d where k are complex numbers and is a constant delay it is shown that every a stable method possesses a similar stability property to p stability i e the method preserves the delay independent stability of the exact solution under the condition that k is real and h is an integer where h is a step size it is also shown that the method does not possess the same property if h is not an integer as a result no method can possess a similar stability property to gp stability with respect to dides

some extended explicit bel tyukov pairs for volterra integral equations of the second kind we derive and investigate a family of pairs of extended explicit bel tyukov runge kutta ebvrk formulas to treat volterra integral equations of the second kind each pair uses six stages and consists of an order 3 formula completely embedded in an order 4 formula we examine the structure of ebvrk pairs to facilitate their derivation and in particular we show that at least six stages are needed to form such pairs this design is motivated to allow a variable stepsize in order to satisfy a prescribed error tolerance with a minimum of computation some numerical examples illustrate the procedures

stiffly stable fourth order adams type methods abstract in this paper the two parameter family of order four step number five adams type methods is investigated methods are given from this family for which the stiffly stable parameters are optimized it is shown that the resulting methods are intermediate to the fourth order adams moulton and backward differentiation methods

continuous collocation approximations to solutions of first kind volterra equations in this paper we give necessary and sufficient conditions for convergence of continuous collocation approximations of solutions of first kind volterra integral equations the results close some longstanding gaps in the theory of polynomial spline collocation methods for such equations the convergence analysis is based on a runge kutta or ode approach

linearly implicit extrapolation methods for differential algebraic systems we consider the numerical solution of implicit differential equations in which the solution derivative appears multiplied by a solution dependent singular matrix we study extrapolation methods based on two linearly implicit euler discretizations their error behaviour is explained by perturbed asymptotic expansions

optimal convergence results of piecewise polynomial collocation solutions for integral algebraic equations of index 3 in this work we will consider piecewise polynomial collocation methods for a new class of integral algebraic equations of index 3 an analysis of the global convergence properties of the collocation methods is given and the optimal order of convergence results is established the performance of the proposed scheme is illustrated by means of two numerical examples and confirm that we can achieve the expected theoretical orders of convergence

stability of collocation at gaussian points symmetric runge kutta schemes are particularly useful for solving stiff two point boundary value problems such a stable schemes perform well in many cases but it is demonstrated that in some instances the stronger property of algebraic stability is required a characterization of symmetric algebraically stable runge kutta schemes is given the class of schemes thus defined turns out not to be very rich the only collocation schemes in it are those based on gauss points and other schemes in the class do not seem to offer any advantage over collocation at gaussian points

determinantal identities over commutative semirings abstract we present a development of determinantal identities over commutative semirings this includes a generalization of the cauchy binet and laplace theorems as well as results on compound matrices and adjoints it is further shown that laplace s theorem is a special case of the grade s adjoint identity

on the stability of volterra runge kutta methods we investigate the stability of extended runge kutta methods for volterra integral equations of the second kind as test equations we choose linear convolution equations with positive definite l 1 kernels our results are strongly connected with the contractivity of runge kutta methods for nonlinear dissipative differential equations

linear operators strongly preserving r potent matrices over semirings abstract we consider the problem of characterizing those linear operators l on the matrices over a semiring such that l x is idempotent if and only if x is complete characterizations are obtained for many semirings including the nonnegative reals the nonnegative integers the two element boolean algebra and the fuzzy scalars

a perspective on the numerical treatment of volterra equations abstract we discuss the properties and numerical treatment of various types of volterra and abel volterra integral and integro differential equations

on nilpotent matrices over distributive lattices abstract in this paper nilpotent matrices over distributive lattices are considered some properties and characterizations for nilpotent matrices are established and in particular a necessary and sufficient condition for an n n nilpotent matrix to have the nilpotent index n is given also some properties of the reduction of a nilpotent matrix are obtained partial results obtained in this paper are results on nilpotent fuzzy matrices by h hashimoto

on the powers of matrices over a distributive lattice abstract in this paper the index and the period for a lattice matrix are estimated some necessary and sufficent conditions for the convergence of the powers of a lattice matrix are obtained also some equivalent conditions for a nilpotent lattice matrix are discussed

pitfalls in fast numerical solvers for fractional differential equations we consider the problem of implementing fast algorithms for the numerical solution of initial value problems of the form x t f t x t x 0 x0 where x is the derivative of x of order in the sense of caputo and 0 1 we review some of the existing methods and explain their respective strengths and weaknesses we identify and discuss potential problems in the development of generally applicable schemes

fractional linear multistep methods for abel volterra integral equations of the second kind fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular volterra integral equations the proposed methods are convergent of the order of the underlying multistep method also in the generic case of solutions which are not smooth at the origin the stability properties stability region a stability a a stability are closely related to those of the underlying linear multistep method