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

extremal estrada indices of the weighted trees with fixed total weight sum abstract let t n m be the set of positively weighted trees of order n with the fixed total weight sum m where all the weights are integers and m n 1 3 among t n m the weighted trees with the maximal and the second maximal estrada indices are derived for m n 1 3 and m n 1 7 respectively

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

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

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

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

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

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

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

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

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

bicyclic signed graphs with minimal and second minimal energy abstract let s g be a signed graph of order n and size m where g is its underlying graph and is the sign function a connected signed graph s is said to be bicyclic if m n 1 in this paper we provide characterizations of bicyclic signed graphs with minimal and second minimal energy we notice that there are gaps in the proofs of the main results in j zhang h kan 2014 32 and j zhang b zhou 2005 31 which deal with minimal energy of bicyclic graphs and we plug these gaps as a consequence of the results obtained for bicyclic signed graphs

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

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

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

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

directed recursion models for fractal growth fractals constructed by recursive processes are introduced to model growth phenomena these fractals are simultaneously directed and self similar in analogy with patterns growing under diffusion limited conditions the multifractal nature of the harmonic measure associated with laplacian interfaces is qualitatively interpreted using the models calculation of the largest singularity exponent allows one to make conclusions about the behaviour of diffusion limited aggregates

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 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

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 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

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

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

phase transitions in scheidegger and eden networks we study multifractal scaling and phase transitions in scheidegger and eden networks in the plane on several lattices the horton constants and are found not to depend on the lattice the scaling exponent in the integrated area distribution function is found to be consistent with the relation the exponent defines a phase transition point in the multifractal spectrum of the area distribution the approach to this phase transition is slow and controlled by where m is the number of lattice points for the scheidegger model we are able to calculate the exact probability distribution p a for small areas a and thus to study the finite size scaling

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

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

power law distribution of discharge in ideal networks data for several river networks show algebraically decaying distribution functions for the mean annual discharge and the dissipation rate rodriguez iturbe et al 1992 we derive a relation between the exponent a in the integrated mean annual discharge distribution p q q q and the topological dimension dt of the network 1 1 dt using the experimentally determined value dt 1 8 tarboton et al 1988 we find 0 45 in good agreement with the data of rodriguez iturbe et al 1992 for the random model shreve 1967 1969 smart and werner 1974 we find an exponent of 1 2

graph energy change due to edge deletion the energy of a graph is the sum of the singular values of its adjacency matrix we are interested in how the energy of a graph changes when edges are deleted examples show that all cases are possible increased decreased unchanged our goal is to find possible graph theoretical descriptions and to provide an infinite family of graphs for each case the main tool is a singular value inequality for complementary submatrices and its equality case

edge addition singular values and energy of graphs and matrices the energy of a graph matrix is the sum of the absolute values of its eigenvalues we investigate the result of duplicating removing an edge to the energy of a graph we also deal with the problem that which graphs g have the property that if the edges of g are covered by some subgraphs then the energy of g does not exceed the sum of the subgraphs energies the problems are addressed in the general setting of energy of matrices which leads us to consider the singular values too among the other results it is shown that the energy of a complete multipartite graph increases if a new edge added or an old edge is deleted

the energy of graphs and matrices abstract given a complex m n matrix a we index its singular values as 1 a 2 a and call the value e a 1 a 2 a the energy of a thereby extending the concept of graph energy introduced by gutman let 2 m n a be an m n nonnegative matrix with maximum entry and a 1 n extending previous results of koolen and moulton for graphs we prove that e a a 1 m n m 1 a 2 2 a 1 2 m n n m m 2 furthermore if a is any nonconstant matrix then e a 1 a a 2 2 1 2 a 2 a finally we note that wigner s semicircle law implies that e g 4 3 o 1 n 3 2 for almost all graphs g

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

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

the energy change of weighted graphs the energy of an edge weighted graph is the sum of the absolute values of the eigenvalues of its weighted adjacency matrix we study how the energy of a weighted graph changes when the weights change we give some sufficient conditions so that the energy of a weighted graph increases when the positive weight increases we also characterize some classes of weighted graphs satisfying these sufficient conditions

applications of a theorem by ky fan in the theory of graph energy abstract the energy of a graph g is equal to the sum of the absolute values of the eigenvalues of g which in turn is equal to the sum of the singular values of the adjacency matrix of g let x y and z be matrices such that x y z the ky fan theorem establishes an inequality between the sum of the singular values of z and the sum of the sum of the singular values of x and y this theorem is applied in the theory of graph energy resulting in several new inequalities as well as new proofs of some earlier known inequalities

the normalized incidence energy of a graph abstract the energy of a graph g is the sum of the singular values of its adjacency matrix it is a graph invariant used in mathematical chemistry the normalized incidence energy of the graph g denoted by nie g is defined as the sum of the singular values of its normalized incidence matrix we find some upper and lower bounds and determine the coulson integral formula for nie g based on the integral formula we give a way to compare the normalized incidence energies moreover we show a relation between normalized incidence energy and randic energy