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

numerical solution of linear volterra integral equations of the second kind with sharp gradients collocation methods are a well developed approach for the numerical solution of smooth and weakly singular volterra integral equations in this paper we extend these methods through the use of partitioned quadrature based on the qualocation framework to allow the efficient numerical solution of linear scalar volterra integral equations of the second kind with smooth kernels containing sharp gradients in this case the standard collocation methods may lose computational efficiency despite the smoothness of the kernel we illustrate how the qualocation framework can allow one to focus computational effort where necessary through improved quadrature approximations while keeping the solution approximation fixed the computational performance improvement introduced by our new method is examined through several test examples the final example we consider is the original problem that motivated this work the problem of calculating the probability density associated with a continuous time random walk in three dimensions that may be killed at a fixed lattice site to demonstrate how separating the solution approximation from quadrature approximation may improve computational performance we also compare our new method to several existing gregory sinc and global spectral methods where quadrature approximation and solution approximation are coupled

positive and conservative schemes for mass action kinetics abstract some implicit schemes for the discretization of mass action kinetics are presented and discussed although all the schemes are classical the solution procedure takes great advantage of the peculiar structure of the ode all the schemes maintain linear first integrals such as atomic mass conservation the main features of the first order scheme is positivity preservation for arbitrarily large time steps while for the higher order schemes there is numerical evidence that this property is maintained with fairly large time steps for the first order scheme the existence of a nonnegative solution of the algebraic system arising in the advancing step is proved for arbitrarily large time steps while uniqueness is proved for limited time steps finally an efficient procedure is presented for solving the nonlinear systems involved in the time step these systems are solved without the use of jacobian matrices or their approximations but by the repeated inversion of m matrices a procedure which is both easy and fast considerably simplifying and accelerating the computer implementation of the schemes

reaction diffusion master equation diffusion limited reactions and singular potentials to model biochemical systems in which both noise in the chemical reaction process and spatial movement of molecules is important both the reaction diffusion master equation rdme and smoluchowski diffusion limited reaction sdlr partial differential equation pde models have been used in previous work we showed that the solution to the rdme may be interpreted as an asymptotic approximation in the reaction radius to the solution of the sdlr pde s a isaacson siam j appl math 70 77 2009 the approximation was shown to be divergent in the limit that the lattice spacing in the rdme approached zero in this work we expand upon these results for the special case of the two molecule annihilation reaction text a text b ensuremath rightarrow ensuremath emptyset we first introduce a third stochastic reaction diffusion pde model that incorporates a pseudopotential based bimolecular reaction mechanism the solution to the pseudopotential model is then shown to be an asymptotic approximation to the solution of the sdlr pde for small reaction radii we next illustrate how the rdme may be obtained by a formal discretization of the pseudopotential model motivating why the rdme is itself an asymptotic approximation of the sdlr pde finally we give a more detailed numerical analysis of the difference between solutions to the rdme and sdlr pde models as a function of both the reaction radius and the lattice spacing in the rdme

a class of compact boundary value methods applied to semi linear reaction diffusion equations abstract this paper deals with a class of compact boundary value methods cbvms for solving semi linear reaction diffusion equations slres the presented cbvms are constructed by combining a fourth order compact difference method cdm with the p order boundary value methods bvms where the former is for the spatial discretization and the latter for temporal discretization it is proven under some suitable conditions that the cbvms are locally stable and uniquely solvable and have fourth order accuracy in space and p order accuracy in time the computational effectiveness and accuracy of cbvms are further testified by applying the methods to the fisher equation besides these research we also extend the cbvms to solve the two component coupled system of slres the numerical experiment shows that the extended cbvms are effective and can arrive at the high precision

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

resolution of a controversy surrounding the kirchhoff approach and the small perturbation method in rough surface scattering theory it is shown that the small amplitude limit of the mean backscatter cross section for a perfectly conducting random rough surface calculated from the first two terms in the iterative series solution of the surface current integral equation gives the same results as the small perturbation method this demonstration resolves a long standing controversy in rough surface scattering theory

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

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

numerical solution of integral equations by means of the sinc collocation method based on the double exponential transformation numerical solution of linear integral equations by means of the sinc collocation method based on the double exponential transformation abbreviated the de transformation is considered we first apply the method to the volterra integral equation of the second kind and then to the volterra equation of the first kind this method is also applied to the fredholm integral equation of the second kind for the volterra equations we employed a formula for numerical indefinite integration developed by muhammad and mori obtained by applying the de transformation incorporated into the sinc expansion of the integrand while for the fredholm equation we employed the conventional de transformation for definite integrals an error analysis of the method is given and in every case a convergence rate of o exp cn logn for the error is established where n is a parameter representing the number of terms of the sinc expansion also the condition of the matrix of the main system of linear equations is watched through an estimate of the condition number returned by the program numerical examples show the convergence rate mentioned above and confirm the high efficiency of the present method

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

constructing projections on sums and intersections abstract in this note we list several formulas for computing orthogonal projections onto the linear sum and intersection of two subspaces of c n in terms of the projections on the individual subspaces

a numerical approach to the solution of abel integral equations of the second kind with nonsmooth solution abstract in this paper we study the feasibility of numerical solutions analogous to those methods introduced by bownds 1979 1982 and bownds and wood 1976 1979 for volterra integral equations of the second kind with weakly singular kernels

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

numerical treatment of o d es the theory ofa methods almost all commonly used methods for o d es and their most miscellaneous compositions area methods i e they can be reduced toz o z j az j 1 h x j 1 z j 1 z j h z j s a s s j 1 1 m this paper presents a general theory fora methods and discusses its practical consequences an analysis of local discretization error l d e accumulation results in a general order criterium and reveals which part of the l d e effectively influences the global error this facilitates the comparison of methods and generalizes considerably the concept of error constants it is shown as a consequence that the global error cannot be safely controlled by the size of the l d e and that the conventional error control may fail in important cases furthermore butcher s effective order methods the concept of nordsieck forms and gear s interpretation of lineark step schemes as relaxation methods are generalized the stability of step changing is shortly discussed

block boundary value methods for the solution of ordinary differential equations block boundary value methods are considered and applied to solve initial value problems results on their convergence and stability properties are presented and their intermediate position between the class of multistep and runge kutta methods is established

on the relations between b2v ms and runge kutta collocation methods the principal aim of this paper is a rigorous analysis of the relations between block boundary value methods b 2v ms with minimal blocksize defined over a suitable nonuniform finer mesh and well known runge kutta collocation methods moreover a further aspect that will be briefly investigated is the construction of an extended finer mesh for building b 2v ms with nonminimal blocksize some advantages that may arise from the use of the so obtained methods will be also discussed

a note on the use of the neumann expansion in calculating the scatter from rough surfaces the neumann expansion has been used to compute the solutions of the magnetic field integral equation mfie for two dimensional perfectly conducting gaussian rough surfaces for surfaces whose roughness is of a similar order to the incident wavelength it is shown that the expansion may diverge rapidly the rate of convergence is compared with the conjugate gradient cg method whose convergence is sure when it converges the neumann expansion convergence is more rapid it is concluded that the neumann expansion is not suitable without qualification as a numerical solution to the rough surface mfie moreover the failure of the neumann expansion of the solution of the discrete representation of the mfie provides strong evidence that the use of the neumann expansion as a formal solution to the mfie is open to doubt

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

product integration methods for solving a system of nonlinear volterra integral equations in this paper the technique of subtracting out singularities is used to derive explicit and implicit product euler schemes with order one convergence and a product trapezoidal scheme with order two convergence for a system of volterra integral equations with a weakly singular kernel the convergence proofs of the numerical schemes are presented these are nonstandard since the nonlinear function involved in the integral equation system does not satisfy a global lipschitz condition

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

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

pouzet runge kutta chebyshev method for volterra integral equations of the second kind we construct a stabilized runge kutta method of pouzet type pouzet runge kutta chebyshev method prkc for nonlinear stiff volterra integral equations of the second kind the prkc method is an explicit second order scheme which possesses extended stability region due to the first kind chebyshev polynomial used to establish the stability function

adaptive fast and oblivious convolution in evolution equations with memory to approximate convolutions which occur in evolution equations with memory terms a variable step size algorithm is presented for which advancing n steps requires only o n log n operations and o log n active memory in place of o n 2 operations and o n memory for a direct implementation a basic feature of the fast algorithm is the reduction via contour integral representations to differential equations which are solved numerically with adaptive step sizes rather than the kernel itself its laplace transform is used in the algorithm the algorithm is illustrated on three examples a blowup example originating from a schrodinger equation with concentrated nonlinearity chemical reactions with inhibited diffusion and viscoelasticity with a fractional order constitutive law

numerical approximation to a class of nonlinear hybrid system with distributed delay via block boundary value methods abstract this paper deals with the numerical approximation to a class of nonlinear hybrid system with distributed delay hsdd by combining block boundary value methods bbvms and reducible quadrature rules based on the underlying boundary value methods bvms a kind of extended bbvms are proposed for solving nonlinear hsdd under some suitable conditions an extended bbvm is proved to be uniquely solvable globally stable and convergent of order p where p is consistency order of the used underlying bvm the presented numerical experiments further illustrate the obtained theoretical results and computational effectiveness of the extended bbvms

convergence and stability of boundary value methods for ordinary differential equations a usual way to approximate the solution of initial value problems for ordinary differential equations is the use of a linear multistep formula if the formula has k steps k values are needed to obtain the discrete solution the continuous problem provides only the initial value it is customary to impose the additional k 1 conditions at the successive k 1 initial points however the class of methods obtained in this way suffers from heavy limitations summarized by the two dahlquist barriers it is also possible to impose the additional conditions at different grid points for example some conditions can be imposed at the initial points and the remaining ones at the final points the obtained methods called boundary value methods bvms do not have barriers whatsoever in this paper the question of convergence of bvms is discussed along with the linear stability theory some numerical examples on stiff test problems are also presented

a new theoretical approach to runge kutta methods runge kutta rk methods are treated here as linear multistage methods within the author s concept of a methods as a consequence the traditional difference in treatment of rk methods and linear multistep methods vanishes in particular it is not necessary to consider elementary differentials it is shown how the order of a rk method depends on the error constants of its stages and the conditions nonlinear equations for order p are generated from a very simple recursion

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

convergence and stability of multistep methods solving nonlinear initial value problems convergence and stability of initial and boundary value multistep methods are analyzed for a class of nonlinear problems satisfying a one sided lipschitz condition the linear multistep methods are recast to handle the numerical solution globally on the time interval this allows us to use the theory of toeplitz matrices to show that under suitable assumptions global properties of the exact solution are preserved by its numerical approximation in particular a new concept of stability which avoids the unpleasant passage to one leg methods is introduced

incorporating diffusion in complex geometries into stochastic chemical kinetics simulations a method is developed for incorporating diffusion of chemicals in complex geometries into stochastic chemical kinetics simulations systems are modeled using the reaction diffusion master equation with jump rates for diffusive motion between mesh cells calculated from the discretization weights of an embedded boundary method since diffusive jumps between cells are treated as first order reactions individual realizations of the stochastic process can be created by the gillespie method numerical convergence results for the underlying embedded boundary method and for the stochastic reaction diffusion method are presented in two dimensions a two dimensional model of transcription translation and nuclear membrane transport in eukaryotic cells is presented to demonstrate the feasibility of the method in studying cell wide biological processes

boundary value techniques for initial value problems in ordinary differential equations the numerical solution of initial value problems in ordinary differential equations by means of boundary value techniques is considered we discuss a finite difference method which was already investigated by fox in 1954 and fox and mitchell in 1957 hereby we concentrate on explaining the fundamentals of the method because for initial value problems the boundary value method seems to be fairly unknown we further propose and discuss new jalerkin methods for initial value problems along the lines of the boundary value approach

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

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

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

the validity of the perturbation approximation for rough surface scattering using a gaussian roughness spectrum the validity of the perturbation approximation for rough surface scattering is examined 1 by comparison with exact results obtained by solving an integral equation and 2 through comparison of low order perturbation predictions with higher order predictions the pressure release boundary condition is assumed and the field quantity calculated is the bistatic scattering cross section a gaussian roughness spectrum is used and the surfaces have height variations in only one direction it is found in general that the condition kh 1 k is the acoustic wavenumber h is the rms surface height is insufficient to guarantee the accuracy of first order or higher order perturbation theory when the surface correlation length l becomes too large or too small with h held fixed higher order perturbation terms can make larger contributions to the scattering cross section than lower order terms an explanation for this result is given the regions of validity for low order perturbation theory are also given th

on the asymptotic convergence of collocation methods we prove quasioptimal and optimal order estimates in various sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines the analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications equations to which the analysis is applied include fredholm integral equations of the second kind certain first kind fredholm equations singular integral equations involving cauchy kernels a variety of integro differential equations and two point boundary value problems for ordinary differential equations the error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard galerkin methods we compare the collocation method with a standard galerkin method using splines of the same degree showing that the galerkin method is quasioptimal in a sobolev space of lower index and furnishes optimal order approximation for a range of sobolev indices containing and extending below that for the collocation method and so the standard galerkin method achieves higher rates of convergence

monte carlo simulations of large scale problems of random rough surface scattering and applications to grazing incidence with the bmia canonical grid method scattering of a te incident wave from a perfectly conducting one dimensional random rough surface is studied with the banded matrix iterative approach canonical grid bmia cag method the bmia cag is an improvement over the previous bmia the key idea of bmia cag is that outside the near field interaction the rest of the interactions can be translated to a canonical grid by taylor series expansion the use of a flat surface as a canonical grid for a rough surface facilitates the use of the fast fourier transform for nonnear field interaction the method can be used for monte carlo simulations of random rough surface problems with a large surface length including all the coherent wave interactions within the entire surface we illustrate results up to a surface length of 2500 wavelengths with 25000 surface unknowns the method is also applied to study scattering from random rough surfaces at near grazing incidence the numerical examples illustrate the importance of using a large surface length for some backscattering problems

estimating the error of the classic runge kutta formula abstract a high quality error estimator is presented for the classic four stage fourth order runge kutta formula that is free when using this estimator the solution can be approximated accurately throughout the step at no additional cost

affine projections this paper presents a constructive method for projecting a vector onto an affine subspace of a vector space it also provides formulas for projecting onto the intersection and sums of such subspaces

the numerical solution of weakly singular volterra integral equations by collocation on graded meshes since the solution of a second kind volterra integral equation with weakly singular kernel has in general unbounded derivatives at the left endpoint of the interval of integration its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates in this paper we investigate the convergence rates with respect to graded meshes and we discuss the problem of how to select the quadrature formulas to obtain the fully discretized collocation equation

piecewise polynomial collocation methods for linear volterra integro differential equations with weakly singular kernels in the first part of this paper we study the regularity properties of solutions of linear volterra integro differential equations with weakly singular or other nonsmooth kernels we then use these results in the analysis of two piecewise polynomial collocation methods for solving such equations numerically the main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of local superconvergence at the collocation points

projection method and the distance between two linear varieties abstract some important results involving projections and subspaces are generalized for linear varieties namely the anderson duffin formula for intersecting and non intersecting linear varieties and the projecting equation parallelism skewness and distances between linear varieties are considered as well including both the characterization of the minimal pair of two linear varieties and the exhibition of such a minimal pair formulae for the end varieties through projectors and moore penrose inverses are presented as well

numerical methods for ordinary differential equations in the 20th century abstract numerical methods for the solution of initial value problems in ordinary differential equations made enormous progress during the 20th century for several reasons the first reasons lie in the impetus that was given to the subject in the concluding years of the previous century by the seminal papers of bashforth and adams for linear multistep methods and runge for runge kutta methods other reasons which of course apply to numerical analysis in general are in the invention of electronic computers half way through the century and the needs in mathematical modelling of efficient numerical algorithms as an alternative to classical methods of applied mathematics this survey paper follows many of the main strands in the developments of these methods both for general problems stiff systems and for many of the special problem types that have been gaining in significance as the century draws to an end

numerical methods for the lotka mckendrick s equation the lotka mckendrick s model is a well known model which describes the evolution in time of the age structure of a population in this paper we consider this linear model and discuss a range of methods for its numerical solution we take advantage of different analytical approaches to the system to design different numerical methods and compare them with already existing algorithms in particular we set up some algorithms inspired by the approach based on volterra integral equations and we also consider a direct approach based on the nonlinear system that describes the evolution of the age profile of the population

using runge kutta method for numerical solution of the system of volterra integral equation in this paper we introduce some of general methods of runge kutta they are using for numerical solution of volterra integral equations of the second kind h brunner p j vander houwen the numerical solution of volterra equation cwi monograph 3 north holland amsterdam 1986 p w sharp j h verner extended explicit bel tyukov pairs of orders 4 and volterra integral equations of the second kind appl numer math from author s homepage we first introduce pouzet volterra runge kutta methods implicit and explicit forms pvrk and then explain about some extended explicit bel tyukov pairs ebvrk each pair uses six stages and consists of on order 3 formula completely embedded in order 4 formula in numerical examples this method uses for solving a system of volterra integral equations and comparison with iterative methods as comput math appl 37 1999 1 nonlinear stud 6 1 1999

a stochastic fourier transform approach to scattering from perfectly conducting randomly rough surfaces an exact alternative approach to the diagrammatic technique for treating scattering from rough surfaces is developed the magnetic field integral equation for the current induced on the rough perfectly conducting surface is multiplied by a fourier kernel involving all orders of surface height derivatives and their associated transform variables averages of this weighted equation are converted to convolations in the transform domain the result of this operation is a singular integral equation of the first kind of infinite dimensions because of the infinite number of height derivatives for the stochastic fourier transform of the current a procedure is developed for estimating the effects of ignoring one or more surface height derivatives in terms of the range of validity of the resulting approximate solution special limiting cases of very gently undulating surfaces and uniformly rough surfaces are examined new and illuminating results are obtained for the latter case

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

applications and numerical analysis of partial differential volterra equations a brief survey this article contains a concise survey of the numerical analysis of volterra equations and leads up to some recent results on a posteriori error estimation for finite element approximations

elements of a general theory of composite integration methods this contribution is a survey of results on general linear methods after an introduction into the concept of a methods which is a slight generalization of general linear methods and its main results rk methods are presented in the a method notation which offers the following advantages a the order conditions are obtained from a simple recursion without relying on rooted trees or elementary differentials b they appear in a transparent form as orthogonality relations in dependence of the error constants of the stages of the rk method c their orthogonal structure permits elegant existence proofs and yields new auxiliary conditions d the error accumulation within the stages of the method is completely understood the proof of the main result on rk methods theorem 5 is simplified

exponential fitted runge kutta methods of collocation type fixed or variable knot points two different classes of exponential fitted runge kutta collocation methods are considered methods with fixed points and methods with frequency dependent points for both cases we have obtained extensions of the classical two stage gauss radauiia and lobattoiiia methods numerical examples reveal important differences between both approaches

natural continuous extensions of runge kutta methods for volterra integrodifferential equations in this paper we deal with a very general class of runge kutta methods for the numerical solution of volterra integrodifferential equations our main contribution is the development of the theory of natural continuous extensions nces i e piecewice polynomial functions which interpolate the values given by the rk method at the mesh points the particular features of these nces allow us to construct tail approximations which are quite efficient since they require a minimal number of kernel evaluations

an improved method based on haar wavelets for numerical solution of nonlinear integral and integro differential equations of first and higher orders in this paper a novel technique is being formulated for the numerical solution of integral equations ies as well as integro differential equations ides of first and higher orders the present approach is an improved form of the haar wavelet methods aziz and siraj ul islam 2013 siraj ul islam et al 2013 the proposed modifications resulted in computational efficiency and simple applicability of the earlier methods aziz and siraj ul islam 2013 siraj ul islam et al 2013 in addition to this the new approach is being extended from ides of first order to ides of higher orders with initial and boundary conditions unlike the methods aziz and siraj ul islam 2013 siraj ul islam et al 2013 where the kernel function is being approximated by two dimensional haar wavelet the kernel function in the present case is being approximated by one dimensional haar wavelet the modified approach is easily extendable to higher order ides numerical examples are being included to show the accuracy and efficiency of the new method

starting methods for two step runge kutta methods of stage order 3 and order 6 jackiewicz and tracogna siam j numer anal 32 1995 1390 1427 proposed a general formulation of two step runge kutta tsrk methods using formulas for two step pairs of tsrk methods constructed in japan jiam 19 2002 227 248 jackiewicz and verner obtain results for order 8 pairs that fail to show this designated order hairer and wanner siam j numer anal 34 1997 2087 2089 identify the problem by using b series to formulate a complete set of order conditions for tsrk methods and emphasize that special starting methods are necessary for the first step of implementation they observe that for methods with stage order at least p 1 and design order p starting methods of order at least p are sufficient in this paper the more general challenge to provide correct starting values for methods of low stage order is met by showing how perturbed starting values should be selected for methods of order 6 and stage order 3 the approach is sufficiently general that it may and later will be provided for such methods of higher orders evidence of the accompanying improvement in the implementation of tsrk methods illustrates that carefully designed starting methods are essential for efficient production codes based on methods of low stage order

order conditions for general linear methods we describe the derivation of order conditions without restrictions on stage order for general linear methods for ordinary differential equations this derivation is based on the extension of the albrecht approach proposed in the context of runge kutta and composite and linear cyclic methods this approach was generalized by jackiewicz and tracogna to two step runge kutta methods by jackiewicz and vermiglio to general linear methods with external stages of different orders and by garrappa to some classes of runge kutta methods for volterra integral equations with weakly singular kernels this leads to general order conditions for many special cases of general linear methods such as diagonally implicit multistage integration methods nordsieck methods and general linear methods with inherent runge kutta stability exact coefficients for several low order methods with some desirable stability properties are presented for illustration

fixed point method for solving nonlinear quadratic volterra integral equations we consider a paper of banas and sadarangani 2008 11 which deals with monotonicity properties of the superposition operator and their applications an application of the monotonicity properties is to study the solvability of a quadratic volterra integral equation in this paper we prepare an efficient numerical technique based on the fixed point method and quadrature rules to approximate a solution for quadratic volterra integral equation then convergence of numerical scheme is proved by some theorems and some numerical examples are given to show applicability and accuracy of the numerical method and guarantee the theoretical results

weak delay dependent stability of runge kutta methods for differential systems with distributed delays abstract this paper discusses the stability of runge kutta methods for differential systems with distributed delays weak delay dependent stability of pouzet type runge kutta methods is studied in this paper it is revealed that under suitable conditions runge kutta methods preserve weak delay dependent stability an algorithm is employed to check weak delay dependent stability of numerical solutions and the effectiveness of the derived theoretical results is demonstrated with numerical examples

a new numerical method for rough surface scattering calculations a new approach to solving the magnetic field integral equation mfie for the current induced on a infinite perfectly conducting rough surface is presented by splitting the propagator matrix into contributions from the left and from the right of the point of observation a second kind integral equation can be formed with a new born term and a new kernel following discretization of this new integral equation the unknown currents can be determined more rapidly and with significantly less storage requirements than conventional lu decomposition where the time saving factor is roughly n 3 where n is the number of current samples on the surface and the usual storage requirements associated with matrix inversion are eliminated while the new born term is usually adequate for scattered field calculations the new discretized integral equation can be iterated to amy desired accuracy with no apparent convergence problems results are presented for one dimensional rough surfaces with rms heights exceeding one wavelength and rms slopes exceeding 40 which illustrate the robustness of the new born term

numerical solution of linear quasistatic hereditary viscoelasticity problems we give a space time galerkin finite element discretization of the linear quasistatic compressible viscoelasticity problem as described by an elliptic partial differential equation with a volterra memory term the discretization consists of a continuous piecewise linear approximation in space with a discontinuous piecewise constant or linear approximation in time we derive an a priori maximum energy error estimate by exploiting galerkin orthogonality and the data stability of a related discrete backward problem illustrative numerical experiments are also included as also is a brief description of our first results on a posteriori error estimation this allows for adaptive control of the space mesh but not of the time step

the double exponential transformation in numerical analysis the double exponential transformation was first proposed by takahasi and mori in 1974 for the efficient evaluation of integrals of an analytic function with end point singularity afterwards this transformation was improved for the evaluation of oscillatory functions like fourier integrals recently it turned out that the double exponential transformation is useful not only for numerical integration but also for various kinds of sinc numerical methods the purpose of the present paper is to review the double exponential transformation in numerical integration and in a variety of sinc numerical methods

frequency evaluation for exponentially fitted runge kutta methods this paper can be seen as a further investigation of the frequency evaluation techniques which are very recently proposed by ixaru et al for exponentially fitted multistep algorithms for first order ordinary differential equations odes the question answered was how the frequencies should be tuned in order to have a maximal benefit from exponentially fitted methods in a previous paper this frequency evaluation algorithm was successfully applied in a direct way to a second order exponentially fitted runge kutta efrk method of collocation type but such a direct application is impossible for higher order efrk methods to overcome this difficulty we develop an efficient extension of ixaru s frequency evaluation algorithm for the exponentially fitted radauiia method of third order it is an adaption of ixaru s algorithm in the sense that instead being applied globally it is applied stagewise numerical experiments illustrate the properties of the developed algorithm

a general class of two step runge kutta methods for ordinary differential equations a general class of two step runge kutta methods that depend on stage values at two consecutive steps is studied these methods are special cases of general linear methods introduced by butcher and are quite efficient with respect to the number of function evaluations required for a given order general order conditions are derived using the approach proposed recently by albrecht and examples of methods are given up to the order 5 these methods can be divided into four classes that are appropriate for the numerical solution of nonstiff or stiff differential equations in sequential or parallel computing environments

theoretical analysis of the stability for extended trapezoidal rules abstract in this paper a rigorous analysis of the linear stability for two classes of methods both generalizing the trapezoidal rule is done the information on the coefficients of the methods useful for the analysis are obtained by an extensive use of properties of the pascal matrix some of which are derived in this work

a global convergence theorem for a class of parallel continuous explicit runge kutta methods and vanishing lag delay differential equations iterated continuous extensions ices are continuous explicit runge kutta methods developed for the numerical solution of evolutionary problems in ordinary and delay differential equations ddes ices have a particular role in the explicit solution of ddes with vanishing lags they may be regarded as parallel continuous explicit runge kutta pcerk methods as they allow one to take advantage of parallel architectures ices can also be related to a collocation method the purpose of this paper is to provide a theorem giving the global order of convergence for variable step implementations of ices applied to state dependent ddes with and without vanishing lags implications of the theory for the implementation of this class of methods are discussed and demonstrated the results establish that our approach allows the construction of pcerk methods whose order of convergence is restricted only by the continuity of the solution

error estimates for general linear methods for ordinary differential equations most methods used for the numerical solution of ordinary differential equations may be characterized by a pair of matrices a b in this article local error estimates for such methods are obtained as linear combinations of computed values this is often equivalent to constructing pairs of embedded methods although these local error estimates are obtained by assuming that exact solution values are used to start the step it is shown that this assumption is not required for certain estimates some hybrid methods have such computable estimates and also have simple step length change procedures

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

efficiency comparisons of methods for integrating odes abstract considerable work has been done comparing the relative efficiencies of linear multi step runge kutta and extrapolation methods within their respective classes however efficiency comparisons between these classes have been relegated to numerical case studies which can be misleading because they depend on a small set of test problems and particular implementations of the methods in this paper a unified way of comparing efficiency is presented that makes it possible to assess theoretically the relative efficiencies of methods from the different classes

quadrature formulas descending from bs hermite spline quasi interpolation two new classes of quadrature formulas associated to the bs boundary value methods are discussed the first is of lagrange type and is obtained by directly applying the bs methods to the integration problem formulated as a special cauchy problem the second descends from the related bs hermite quasi interpolation approach which produces a spline approximant from hermite data assigned on meshes with general distributions the second class formulas is also combined with suitable finite difference approximations of the necessary derivative values in order to define corresponding lagrange type formulas with the same accuracy

solving volterra integro differential equations by variable stepsize block bs methods properties and implementation techniques in this article block bs methods are considered for the numerical solution of volterra integro differential equations vides convergence and stability properties are analyzed a new matlab code for the solution of vides called videbs is presented numerical results using a variable stepsize implementation show the effectiveness of the proposed code

a new recurrence for computing runge kutta truncation error coefficients researchers investigating runge kutta methods use truncation error coefficients to derive formulas and assess their quality in this paper albrecht s recurrence for generating order conditions is refined to produce truncation error coefficients the refined recurrence provides a fast and relatively simple way of calculating truncation error coefficients of the albrecht expansion for general runge kutta methods

stability of runge kutta methods for delay integro differential equations we study stability of runge kutta rk methods for delay integro differential equations with a constant delay on the basis of the linear equation du dt lu t mu t k t t u d where l m k are constant complex matrices in particular we show that the same result as in the case k 0 koto bit 34 1994 262 267 holds for this test equation i e every a stable rk method preserves the delay independent stability of the exact solution whenever a step size of the form h m is used where m is a positive integer

a quadrature based approach to improving the collocation method the collocation method is a popular method for the approximate solution of boundary integral equations but typically does not achieve the high order of convergence reached by the galerkin method in appropriate negative norms in this paper a quadrature based method for improving upon the collocation method is proposed and developed in detail for a particular case that case involves operators with even symbol such as the logarithmic potential operating on 1 periodic functions a smooth spline trial space of odd degree with constant mesh spacingh 1 n and a quadrature rule with 2n points where ann point quadrature rule would be equivalent to the collocation method in this setting it is shown that a special quadrature rule which depends on the degree of the splines and the order of the operator can yield a maximum order of convergence two powers ofh higher than the collocation method