restart could optimize the probability of success in a bernoulli trial the recently noticed ability of restart to reduce the expected completion time of first passage processes allows appealing opportunities for performance improvement in a variety of settings however complex stochastic processes often exhibit several possible scenarios of completion which are not equally desirable in terms of efficiency here we show that restart may have profound consequences on the splitting probabilities of a bernoulli like first passage process i e of a process which can end with one of two outcomes particularly intriguing in this respect is the class of problems where a carefully adjusted restart mechanism maximizes the probability that the process will complete in a desired way we reveal the universal aspects of this kind of optimal behavior by applying the general approach recently proposed for the problem of first passage under restart

fast fourier galerkin methods for solving singular boundary integral equations numerical integration and precondition we develop a fast fully discrete fourier galerkin method for solving a class of singular boundary integral equations we prove that the number of multiplications used in generating the compressed matrix is o nlog 3n and the solution of the proposed method preserves the optimal convergence order o n t where n is the order of the fourier basis functions used in the method and t denotes the degree of regularity of the exact solution moreover we propose a preconditioning which ensures the numerical stability when solving the preconditioned linear system numerical examples are presented to confirm the theoretical estimates and to demonstrate the approximation accuracy and computational efficiency of the proposed algorithm

transfer of energy to high frequencies in the cubic defocusing nonlinear schrodinger equation we consider the cubic defocusing nonlinear schrodinger equa tion on the two dimensional torus we exhibit smooth solutions for which the support of the conserved energy moves to higher fourier modes this behavior is quantified by the growth of higher sobolev norms given any 1 k 1 s 1 we construct smooth initial data u0 withu0 h s k at some time t this growth occurs despite the hamiltonian s bound onu t h 1 and despite the conservation of the quantityu t l2

the solvability and explicit solutions of two integral equations via generalized convolutions abstract this paper presents the necessary and sufficient conditions for the solvability of two integral equations of convolution type the first equation generalizes from integral equations with the gaussian kernel and the second one contains the toeplitz plus hankel kernels furthermore the paper shows that the normed rings on l 1 r d are constructed by using the obtained convolutions and an arbitrary hermite function and appropriate linear combination of those functions are the weight function of four generalized convolutions associating f and f the open question about hermitian weight function of generalized convolution is posed at the end of the paper

reproducing kernel method for solving fredholm integro differential equations with weakly singularity singular integral equations sies are often encountered in certain contact and fracture problems in solid mechanics numerical methods for solving sies have been the focus of much research including reproducing kernel methods however there are no reports on reproducing kernel methods for solving differential integral equations with weakly singular kernels we developed a reproducing kernel method for solving fredholm integro differential equations with weakly singular kernels in reproducing kernel hilbert space this involves changing a weakly singular kernel to a logarithm kernel to a kalman kernel weak singularity is removed by applying a smooth transformation to the kalman kernel solution representations are obtained in reproducing kernel hilbert space numerical experiments show that our reproducing kernel method is efficient

a new efficient method for cases of the singular integral equation of the first kind various cases of cauchy type singular integral equation of the first kind occur rather frequently in mathematical physics and possess very unusual properties these equations are usually difficult to solve analytically and it is required to obtain approximate solutions this paper investigates the numerical solution of various cases of cauchy type singular integral equations using reproducing kernel hilbert space rkhs method the solution u x is represented in the form of a series in the reproducing kernel space afterwards the n term approximate solution u n x is obtained and it is proved to converge to the exact solution u x the major advantage of the method is that it can produce good globally smooth approximate solutions moreover in this paper an efficient error estimation of the rkhs method is introduced finally numerical experiments show that our reproducing kernel method is efficient

analytic expression for the mean time to absorption for a random walker on the sierpinski gasket ii the eigenvalue spectrum we continue the study of a particle atom molecule undergoing an unbiased random walk on the sierpinski gasket and obtain for the gasket and tower the eigenvalue spectrum of the associated stochastic master equation analytic expressions for recurrence relations among the eigenvalues are derived the recurrence relations obtained are compared with those determined for two euclidean lattices the closed chain with an absorbing site and a finite chain with an absorbing site at one end we check and confirm the internal consistency between the smallest eigenvalue and the mean walklength in each of the cases studied attention is drawn to the relevance of the results obtained to a problem of electron transfer in proteins

the radon transform between monogenic and generalized slice monogenic functions in bures et al elements of quaternionic analysis and radon transform 2009 the authors describe a link between holomorphic functions depending on a parameter and monogenic functions defined on mathbb r n 1 using the radon and dual radon transforms the main aim of this paper is to further develop this approach in fact the radon transform for functions with values in the clifford algebra mathbb r n is mapping solutions of the generalized cauchy riemann equation i e monogenic functions to a parametric family of holomorphic functions with values in mathbb r n and analogously the dual radon transform is mapping parametric families of holomorphic functions as above to monogenic functions the parametric families of holomorphic functions considered in the paper can be viewed as a generalization of the so called slice monogenic functions an important part of the problem solved in the paper is to find a suitable definition of the function spaces serving as the domain and the target of both integral transforms

numerical solution of systems of cauchy singular integral equations with constant coefficients abstract this paper deals with the numerical solution of a class of systems of cauchy singular integral equations with constant coefficients the proposed procedure consists of two basic steps the first one is to consider a modified problem equivalent to the original one under suitable conditions the second one is to approximate its solution by means of a vector of polynomial functions such array is constructed by applying a quadrature type method based on gaussian rules that leads to solve a determined and well conditioned linear system the convergence and stability of the method are proved in weighted l 2 spaces some numerical tests are also shown

evanescent continuous time random walks we study how an evanescence process affects the number of distinct sites visited by a continuous time random walker in one dimension we distinguish two very different cases namely when evanescence can only occur concurrently with a jump and when evanescence can occur at any time the first is characteristic of trapping processes on a lattice whereas the second is associated with spontaneous death processes such as radioactive decay in both of these situations we consider three different forms of the waiting time distribution between jumps namely exponential long tailed and ultraslow

non normal type singular integral differential equations by riemann hilbert approach abstract this article deals with one class of singular integral differential equations of non normal type with convolution and cauchy principal value integral in class 0 by using fourier transform this classes of equations are transformed into a riemann boundary value problem with nodal point and we prove the existence of solutions and the noether theory for the equations for such equations we propose one method different from classical one and we obtain the general solutions and the conditions of solvability all cases about the behaviors of the solution are considered at nodal points therefore the theory of integral equations and the classical riemann boundary value problems is extended further

on the consistency of numerical solutions of singular integral equations of the second kind with negative index using jacobi polynomials abstract singular integral equations of second kind with negative index possess bounded solutions whenever the forcing function satisfies the orthogonality conditions numerically the discretization of the integrals via gauss jacobi quadrature does not ensure the preservation of a consistent system it is shown that for gauss jacobi quadrature the overdetermined system of equations is consistent if and only if the discretized compatibility condition is satisfied the solutions of inconsistent systems via generalized inverses are considered error estimates are established which are shown to be independent of which equation is excluded in the solution of the overdetermined system

two integral operators in clifford analysis article history segal bargmann space f2 c n and monogenic fock space m2 r n 1 are introduced first then with the help of exponential functions in clifford analysis two integral operators are defined to connect f2 c n and m2 r n 1 together the corresponding integral properties are studied in detail

the solvability and explicit solutions of singular integral differential equations of non normal type via riemann hilbert problem abstract in this article we study non normal type singular integral differential equations involving convolution kernel and cauchy kernel in class 0 and we prove the existence of solutions and the noether conditions by using theory of fourier analysis the equations under consideration are transformed into boundary value problems for analytic functions i e riemann hilbert problems with discontinuous coefficients for such problems we propose one method different from classical one and we obtain the analytic solutions and the conditions of noether solvability at nodal points all cases as regards the regularity and the asymptotic behaviors of the solution are discussed in detail in particular we generalize the case of normal type discussed in the literature li 2017 7 8 2020 2019 to the case of non normal type

asymptotic solution for first and second order linear volterra integro differential equations with convolution kernels this paper addresses the problem of finding an asymptotic solution for first and second order integro differential equations containing an arbitrary kernel by evaluating the corresponding inverse laplace and fourier transforms the aim of the paper is to go beyond the tauberian theorem in the case of integral differential equations which are widely used by the scientific community the results are applied to the convolute form of the lindblad equation setting generic conditions on the kernel in such a way as to generate a positive definite density matrix and show that the structure of the eigenvalues of the correspondent liouvillian operator plays a crucial role in determining the positivity of the density matrix

solving singular integral equations using gaussian quadrature and overdetermined systems abstract for singular integral equations of the cauchy type on an open interval a bounded solution exists only if an additional condition is satisfied in direct methods of solution when the integrals are replaced by quadratures it is not obvious that the resulting discrete system of equations will be consistent it is shown that for gauss chebyshev quadrature the overdetermined system of equations is consistent if and only if the discrete analog of the compatibility condition is satisfied the behavior of solutions to inconsistent systems is examined

an approximate numerical method for solving cauchy singular integral equations composed of multiple implicit parameter functions with unknown integral limits in contact mechanics abstract cauchy singular integral equations are widely used in physics and mathematics especially in solid contact mechanics the solution of cauchy singular integral equations composed of explicit functions has already been achieved in existing research however when dealing with the contact problem between two solid bodies with irregular surfaces described by implicit parametric functions difficulties arise when trying to solve the cauchy singular integral because it is composed of multiple implicit parameter functions moreover the integral limits are unknown and constrained by physical characteristics to solve this kind of problem an approximate calculation method with high accuracy will be provided in this paper specifically based on the quadrature method and taking the constraint function of the boundary as the convergence criterion both the integral limits satisfying the physical characteristics condition and the solution of the cauchy singular integral equations composed of multiple parameter functions can be derived by an iterative method finally five different examples are calculated using the new method and the absolute errors between the approximate values provided by the new method and the true values are analysed

aspects of variational arguments in the theory of elasticity fact and folklore abstract an account of the use of the calculus of variations in elastostatics is given variations of both the independent and dependent variables are used to obtain the equations of balance and natural boundary conditions the meaning and use of the necessary transversality conditions for movable internal or external boundaries are examined satisfaction of the transversality conditions is shown to provide immediate access to the forces that act on such boundaries through a virtual work type argument misconceptions concerning the general form of lagrangian densities for which the field equations are identically satisfied are corrected the complete characterization of such lagrangian densities is shown to lead to methods for realization of a wide class of traction boundary conditions as natural boundary conditions and to a significantly richer class of invariance transformations and associated conservation laws

solving the generalized regularized long wave equation on the basis of a reproducing kernel space on the basis of a reproducing kernel space an iterative algorithm for solving the generalized regularized long wave equation is presented the analytical solution in the reproducing kernel space is shown in a series form and the approximate solution u n is constructed by truncating the series to n terms the convergence of u n to the analytical solution is also proved results obtained by the proposed method imply that it can be considered as a simple and accurate method for solving such evolution equations

cauchy singular integral equations in spaces of continuous functions and methods for their numerical solution cauchy singular integral equations on an interval are studied in weighted spaces of continuous functions convergence rates for a general method for their numerical solution are proved in weighted uniform norms applications of these results to the collocation method and the quadrature method are given

generalized convolution type singular integral equations in this paper we study one class of generalized convolution type singular integral equations in class 0 such equations are turned into complete singular integral equations with nodal points and further turned into boundary value problems for analytic function with discontinuous coefficients by fourier transforms for such equations we will propose one method different from classical one and obtain the general solutions and their conditions of solvability in class 0 thus this paper generalizes the theory of classical equations of convolution type

exact calculations of first passage quantities on recursive networks we present general methods to exactly calculate mean first passage quantities on self similar networks defined recursively in particular we calculate the mean first passage time and the splitting probabilities associated to a source and one or several targets averaged quantities over a given set of sources e g same connectivity nodes are also derived the exact estimate of such quantities highlights the dependency of first passage processes with respect to the source target distance which has recently revealed to be a key parameter in characterizing transport in complex media we explicitly perform calculations for different classes of recursive networks finitely ramified fractals scale free trans fractals nonfractals mixtures between fractals and nonfractals nondecimable hierarchical graphs of arbitrary size our approach unifies and significantly extends the available results in the field

the inverse radon transform and the fundamental solution of the hyperbolic dirac equation in this paper the fundamental solution of the dirac equation in an m dimensional hyperbolic space is constructed the radon transform allowing a reduction from the original problem in m dimensions to a two dimensional problem is essential to this construction in case of an odd dimension m 3 the hyperbolic fundamental solution is found as a modulated version of the eulidean one

solvability of singular integro differential equations via riemann hilbert problem abstract the article is to study singular integro differential equations involving convolutional operators and cauchy integral operators via riemann hilbert problem to do this we adopt a new approach through fourier transform on l 2 subspace which is holder continuous with a certain decay at infinity the fourier transform converts the equations into a riemann hilbert problem with holder continuous coefficients and with nodal points which allows us to construct the general solutions

existence of solutions of functional integral equations of convolution type using a new construction of a measure of noncompactness on l p r let l p r denote the space of lebesgue integrable functions on r with the standard norm x p 0 x t p d t 1 p first we define a new measure of noncompactness on the spaces l p r 1 p in addition we study the existence of entire solutions for a class of nonlinear functional integral equations of convolution type using darbo s fixed point theorem which is associated with the new measure of noncompactness we provide some examples to demonstrate that our results are applicable whereas the previous results are not

a collocation scheme for a certain cauchy singular integral equation based on the superconvergence analysis in this paper we investigate the composite midpoint rule for the evaluation of cauchy principal value integral in an interval and place the key point on its pointwise superconvergence phenomenon the error expansion of the rule is obtained which shows that the superconvergence phenomenon occurs at the points of each subinterval whose local coordinate is the zeros of some function then by applying the midpoint rule to approximate the cauchy principal value integral and choosing the superconvergence points as the collocation points we obtain a collocation scheme for solving a certain cauchy singular integral equation the more interesting thing is that the coefficient matrix of the resulting linear system possesses some good properties from which we obtain an optimal error estimate finally some numerical examples are provided to validate the theoretical analysis

determining mean first passage time on a class of treelike regular fractals relatively general techniques for computing mean first passage time mfpt of random walks on networks with a specific property are very useful since a universal method for calculating mfpt on general graphs is not available because of their complexity and diversity in this paper we present techniques for explicitly determining the partial mean first passage time pmfpt i e the average of mfpts to a given target averaged over all possible starting positions and the entire mean first passage time emfpt which is the average of mfpts over all pairs of nodes on regular treelike fractals we describe the processes with a family of regular fractals with treelike structure the proposed fractals include the t fractal and the peano basin fractal as their special cases we provide a formula for mfpt between two directly connected nodes in general trees on the basis of which we derive an exact expression for pmfpt to the central node in the fractals moreover we give a technique for calculating emfpt which is based on the relationship between characteristic polynomials of the fractals at different generations and avoids the computation of eigenvalues of the characteristic polynomials making use of the proposed methods we obtain analytically the closed form solutions to pmfpt and emfpt on the fractals and show how they scale with the number of nodes in addition to exhibit the generality of our methods we also apply them to the vicsek fractals and the iterative scale free fractal tree and recover the results previously obtained

clenshaw curtis type quadrature rule for hypersingular integrals with highly oscillatory kernels abstract the clenshaw curtis type quadrature rule is proposed for the numerical evaluation of the hypersingular integrals with highly oscillatory kernels and weak singularities at the end points for any smooth functions g x based on the fast hermite interpolation this paper provides a stable recurrence relation for these modified moments convergence rates with respect to the frequency k and the number of interpolation points n are considered these theoretical results and high accuracy of the presented algorithm are illustrated by some numerical examples

first passage method for the study of the efficiency of a two channel reaction on a lattice we study the efficiency of a two channel reaction between two walkers on a finite one dimensional periodic lattice the walkers perform a combination of synchronous and asynchronous jumps on the lattice and react instantaneously when they meet at the same site first channel or upon position exchange second channel we develop a method based on a conditional first passage problem to obtain exact results for the mean number of time steps needed for the reaction to take place as well as for higher order moments previous results obtained in the framework of a difference equation approach are fully confirmed including the existence of a parity effect for even lattices the maximum efficiency corresponds to a mixture of synchronous events and a small amount of asynchronous events while for odd lattices the reaction time is minimized by a purely synchronous process we provide an intuitive explanation for this behavior in addition we give explicit expressions for the variance of the reaction time the latter displays a similar even odd behavior suggesting that the parity effect extends to higher order moments 2005 the american physical society

scattering theory for the radial h 12 critical wave equation with a cubic convolution abstract in this paper we study the global well posedness and scattering for the wave equation with a cubic convolution t 2 u u x 3 u 2 u in dimensions d 4 we prove that if the radial solution u with life span i obeys u u t l t i h x 1 2 r d h x 1 2 r d then u is global and scatters by the strategy derived from concentration compactness we show that the proof of the global well posedness and scattering is reduced to disprove the existence of two scenarios soliton like solution and high to low frequency cascade making use of the no waste duhamel formula and double duhamel trick we deduce that these two scenarios enjoy the additional regularity by the bootstrap argument of 7 this together with virial analysis implies the energy of such two scenarios is zero and so we get a contradiction

optimal search strategies of space time coupled random walkers with finite lifetimes we present a simple paradigm for detection of an immobile target by a space time coupled random walker with a finite lifetime the motion of the walker is characterized by linear displacements at a fixed speed and exponentially distributed duration interrupted by random changes in the direction of motion and resumption of motion in the new direction with the same speed we call these walkers mortal creepers a mortal creeper may die at any time during its motion according to an exponential decay law characterized by a finite mean death rate m while still alive the creeper has a finite mean frequency of change of the direction of motion in particular we consider the efficiency of the target search process characterized by the probability that the creeper will eventually detect the target analytic results confirmed by numerical results show that there is an m dependent optimal frequency opt that maximizes the probability of eventual target detection we work primarily in one dimensional d 1 domains and examine the role of initial conditions and of finite domain sizes numerical results in d 2 domains confirm the existence of an optimal frequency of change of direction thereby suggesting that the observed effects are robust to changes in dimensionality in the d 1 case explicit expressions for the probability of target detection in the long time limit are given in the case of an infinite domain we compute the detection probability for arbitrary times and study its early and late time behavior we further consider the survival probability of the target in the presence of many independent creepers beginning their motion at the same location and at the same time we also consider a version of the standard target problem in which many creepers start at random locations at the same time

exploration and trapping of mortal random walkers exploration and trapping properties of random walkers that may evanesce at any time as they walk have seen very little treatment in the literature and yet a finite lifetime is a frequent occurrence and its effects on a number of random walk properties may be profound for instance whereas the average number of distinct sites visited by an immortal walker grows with time without bound that of a mortal walker may depending on dimensionality and rate of evanescence remain finite or keep growing with the passage of time this number can in turn be used to calculate other classic quantities such as the survival probability of a target surrounded by diffusing traps if the traps are immortal the survival probability will vanish with increasing time however if the traps are evanescent the target may be spared a certain death we analytically calculate a number of basic and broadly used quantities for evanescent random walkers

absolute time derivatives a four dimensional treatment of nonrelativistic spacetime gives a natural frame to deal with objective time derivatives in this framework some well known objective time derivatives of continuum mechanics appear as lie derivatives their coordinatized forms depend on the tensorial properties of the relevant physical quantities we calculate the particular forms of objective time derivatives for scalars vectors covectors and different second order tensors from the point of view of a rotating observer the relation of substantial material and objective time derivatives is treated

territory covered by n random walkers on fractal media the sierpinski gasket and the percolation aggregate we address the problem of evaluating the number s n t of distinct sites visited up to time t by n noninteracting random walkers all starting from the same origin in fractal media for a wide class of fractals of which the percolation cluster at criticality and the sierpinski gasket are typical examples we propose for large n and after the short time compact regime an asymptotic series for s n t analogous to that found for euclidean media s n t approximately s n t 1 delta here s n t is the number of sites volume inside a hypersphere of radius l ln n c 1 v where l is the root mean square chemical displacement of a single random walker and v and c determine how fast 1 gamma t l the probability that a given site at chemical distance l from the origin is visited by a single random walker by time t decays for large values of l l 1 gamma t l approximately exp c l l v for the fractals considered in this paper v d l w d l w 1 d l w being the chemical diffusion exponent the corrective term delta is expressed as a series in ln n n ln m ln n with n or 1 and 0 or m or n which is given explicitly up to n 2 this corrective term contributes substantially to the final value of s n t even for relatively large values of n

reproducing kernel method of solving singular integral equation with cosecant kernel in the paper a reproducing kernel method of solving singular integral equations sie with cosecant kernel is proposed for solving sie difficulties lie in its singular term in order to remove singular term of sie an equivalent transformation is made compared with known investigations its advantages are that the representation of exact solution is obtained in a reproducing kernel hilbert space and accuracy in numerical computation is higher on the other hand the representation of reproducing kernel becomes simple by improving the definition of traditional inner product and requirements for image space of operators are weakened comparing with traditional reproducing kernel method the final numerical experiments illustrate the method is efficient

some classes of equations of discrete type with harmonic singular operator and convolution in this paper we study four classes of discrete type equations with harmonic singular operator and convolution such equations are turned into boundary value problems for analytic function with discontinuous coefficients by discrete fourier transform the general solutions and the conditions of solvability are obtained in class h by our method thus this paper generalizes the theory of classical equations of convolution type

role of depletion on the dynamics of a diffusing forager we study the dynamics of a starving random walk in general spatial dimension d this model represents an idealized description for the fate of an unaware forager whose motion is not affected by the presence or absence of resources the forager depletes its environment by consuming resources and dies if it wanders too long without finding food in the exactly solvable case of one dimension we explicitly derive the average lifetime of the walk and the distribution for the number of distinct sites visited by the walk at the instant of starvation we also give a heuristic derivation for the averages of these two quantities we tackle the complex but ecologically relevant case of two dimensions by an approximation in which the depleted zone is assumed to always be circular and which grows incrementally each time the walk reaches the edge of this zone within this framework we derive a lower bound for the scaling of the average lifetime and number of distinct sites visited at starvation we also determine the asymptotic distribution of the number of distinct sites visited at starvation finally we solve the case of high spatial dimensions within a mean field approach

continua with microstructure modelled by the geometry of higher order contact abstract in the paper the finslerian geometry oriented model of the continuum with microstructure is formulated within the frame of newtonian eshelbian continuum mechanics based on the information characterizing a structure dependent evolution of state variables in this approach position and direction dependent deformation and strain measures are used to describe the motion of the continuum with microstructure at the macro and microlevel the variational arguments for a lagrangian functional defined on the finslerian bundle are used to derive dynamic balance laws boundary and transversality conditions for macro and microstresses of deformational and configurational type the dissipation inequality for the thermo inelastic deformation processes is formulated by the sufficiency condition of weierstrass type for the action integral the presented geometric technique is illustrated in the following examples the damage tensor identified with a measure of reduction of load carrying area elements caused by the development of microcracks or microvoids is defined on the tangent bundle using the lifting technique the macro micro constitutive equations and the associated phenomenological constitutive relations for the thermo inelastic processes are derived in terms of the free energy functional and a dissipation potential a strain induced crack propagation criterion defined by the difference between the strain energy release rate and the rate of the surface energy of the crack is formulated for the kinking of cracks

short time regime propagator in fractals the propagator p r t for fractals in the short time regime i e the probability of finding at distance r at time t a particle that diffuses in a fractal substrate when ensuremath xi r sqrt 2d t 1 d w ensuremath gg 1 is studied in order to elucidate its full functional form for finitely ramified fractals it is shown and for any other self similar media conjectured that the short time propagator is given by p r t ensuremath approx p 0 t ensuremath d s 2 ensuremath xi ensuremath alpha mathrm exp ensuremath c ensuremath xi ensuremath nu where ensuremath nu d w d w ensuremath 1 and ensuremath alpha ensuremath nu 2 ensuremath d f d f and d w being the fractal and random walk dimension of the medium respectively the value for ensuremath nu agrees with that generally accepted however our result for the as yet not well established value of ensuremath alpha differs from other recent proposals we have checked these various short time propagator proposals by comparing them to the short time propagator calculated numerically for the sierpinsky gasket our numerical results are precise enough to clearly support the validity of the short time propagator proposed here in particular the validity of our relation for ensuremath alpha and to rule out the others

a new efficient method with error analysis for solving the second kind fredholm integral equation with cauchy kernel the main objective of this paper is to give an efficient numerical method for the solution of the second kind fredholm integral equation with cauchy type kernel although numerical treatment of singular integral equations sies has been considered by many researchers and many numerical methods have been proposed for the solution of this equation strong singularity of the cauchy singular integral operator still remains as a challenge to numerical methods most of the previous methods rely on specific quadrature rule or suitable base functions for capturing the singularity here we focus on operator transformation and graded mesh in addition we study error analysis in this regard efficiency of reproducing kernel hilbert space rkhs method using smooth transformation on the graded mesh is improved and compared with some other numerical methods

exact mean first passage time on the t graph we consider a simple random walk on the t fractal and we calculate the exact mean time taug to first reach the central node i0 the mean is performed over the set of possible walks from a given origin and over the set of starting points uniformly distributed throughout the sites of the graph except i0 by means of analytic techniques based on decimation procedures we find the explicit expression for taug as a function of the generation g and of the volume v of the underlying fractal our results agree with the asymptotic ones already known for diffusion on the t fractal and more generally they are consistent with the standard laws describing diffusion on low dimensional structures

condition numbers for singular integral equations in weighted l 2 spaces the convergence and stability of a discrete collocation method for cauchy singular integral equations in some weighted besov spaces are studied this numerical method results in solving a linear system in order to determine the unknown coefficients of the approximate solution the author proves that this linear system is well conditioned

representation of the exact solution and a stability analysis on the fredholm integral equation of the first kind in reproducing kernel space in this paper a new method is given in order to solve an ill posed problem on fredholm integral equation of the first kind the representation of the exact solution is given and the stability of the solution on fredholm integral equation of the first kind is discussed in the reproducing kernel space by the discussions a conclusion is obtained the stability problem is a well posed problem in the reproducing kernel space namely the measurement errors of the experimental data can not result in unbounded errors of the exact solution the numerical experiment shows that the new method given in the paper is valid

generalized hermite polynomials and the heat equation for dunkl operators based on the theory of dunkl operators this paper presents a general concept of multivariable hermite polynomials and hermite functions which are associated with finite reflection groups on n the definition and properties of these generalized hermite systems extend naturally those of their classical counterparts partial derivatives and the usual exponential kernel are here replaced by dunkl operators and the generalized exponential kernel k of the dunkl transform in the case of the symmetric group s n our setting includes the polynomial eigenfunctions of certain calogero sutherland type operators the second part of this paper is devoted to the heat equation associated with dunkl s laplacian as in the classical case the corresponding cauchy problem is governed by a positive one parameter semigroup this is assured by a maximum principle for the generalized laplacian the explicit solution to the cauchy problem involves again the kernel k which is on the way proven to be nonnegative for real arguments

nystrom method for cauchy singular integral equations with negative index in this paper the authors propose a nystrom method to approximate the solutions of cauchy singular integral equations with constant coefficients having a negative index they consider the equations in spaces of continuous functions with weighted uniform norm they prove the stability and the convergence of the method and show some numerical tests that confirm the error estimates

singular integral equations of convolution type with cauchy kernel in the class of exponentially increasing functions abstract in this paper we study some classes of generalized singular integral equations of convolution type with cauchy kernel in the class of exponentially increasing functions such equations can be transformed into riemann boundary value problems with two unknown functions on two parallel straight lines via fourier transformation the general solutions and the conditions of solvability are obtained by means of the classical boundary value theory of the theory of fourier analysis and of the principle of analytic continuation this paper will be of great significance for the study of improving and developing complex analysis integral equation and boundary value problem therefore the classic riemann boundary value problem is extended further

a unified procedure for construction of theories of deformable media ii generalized continua in contrast to classical theories discussed in the accompanying paper part i the present paper deals exclusively with generalized continua a term that refers to a body embedded in a euclidean three dimensional space with each of its material points endowed with additional kinematic structure such generalized continua are conveniently identified here by cosserat or directed continua in which the additional structure is represented by independent director fields and the material domain of the body manifold is classified according to the four categories a a three dimensional volume b a two dimensional surface c a one dimensional space curve and d a point in the derivation of a thermomechanical theory for each of these categories we use the same unified procedure as in part i but now the various energies that enter the balance of energy must be modified to include additional kinetic ingredients new theories of this kind have increasingly provided over the last three decades effective means of formulating and studying new behaviour of materials in both fluid and solid mechanics that were not previously possible by the classical theories our attention is first focused in the development of the theory for the first category and this is discussed in two stages first for a cosserat continuum with a single director and then for n directors our derived results include for the first time a derivation from the balance of energy of balances of director momenta and balances of entropies in the presence of more than one independent temperature field the treatments of categories b to d are somewhat shorter since their developments are formally similar to that of category a

solvability theory of convolution singular integral equations via riemann hilbert approach abstract in this paper we study the solvability for some classes of singular integral equations of convolution type with discontinuous property in class 0 though such equations can be solved by using the classical bekya regularization method we are to give a new sectionally jumping method that is they may be reduced to boundary value problems of holomorphic functions which can be systematically solved and easily discussed we obtain the general solutions and the conditions of their solvability thus this paper generalizes the classical theory of integral equations

numerical solution of various cases of cauchy type singular integral equation abstract in this paper a robust but very simple numerical method is developed to solve various cases of cauchy type singular integral equation for this first bernstein polynomials are defined which are used for approximation of solution of the given singular integral equation then numerical method is introduced by using bernstein polynomials this ultimately leads to solution of a system of linear algebraic equations examples are illustrated to demonstrate simplicity of proposed method results are also compared with those present in literature to claim better efficiency of the method introduced

a fast algorithm for solving the generalized airfoil equation abstract the present paper is devoted to the quadrature method for cauchy type singular integral equations on 1 1 with an additional logarithmic singularity of the kernel an algorithm being based on this method is given whose realization only requires a computational work of o n ln n operations in order to determine n unknown coefficients of the approximate solution furthermore in a scale of weighted sobolev like spaces on 1 1 having analogous properties like the classical sobolev spaces on the unit circle the order of convergence is optimal

random walks with spontaneous emission on lattices with periodically distributed imperfect traps abstract we study random walks on d dimensional lattices with periodically distributed traps in which the walker has a finite probability per step of disappearing from the lattice and a finite probability of escaping from a trap general expressions are derived for the total probability that the walk ends in a trap and for the moments of the number of steps made before this happens if it does happen the analysis is extended to lattices with more types of traps and to a model where the trapping occurs during special steps finally the green s function at the origin g 0 z for a finite lattice with periodic boundary conditions which enters into the main expressions is studied more closely a generalization of an expression for g 0 1 for the square lattice given by montroll to values of z different from but close to 1 is derived

first passage properties of mortal random walks ballistic behavior effective reduction of dimensionality and scaling functions for hierarchical graphs we consider a mortal random walker on a family of hierarchical graphs in the presence of some trap sites the configuration comprising the graph the starting point of the walk and the locations of the trap sites is taken to be exactly self similar as one goes from one generation of the family to the next under these circumstances the total probability that the walker hits a trap is determined exactly as a function of the single step survival probability q of the mortal walker on the nth generation graph of the family this probability is shown to be given by the nth iterate of a certain scaling function or map q f q the properties of the map then determine in each case the behavior of the trapping probability the mean time to trapping the temporal scaling factor governing the random walk dimension on the graph and other related properties the formalism is illustrated for the cases of a linear hierarchical lattice and the sierpinski graphs in two and three euclidean dimensions we find an effective reduction of the random walk dimensionality due to the ballistic behavior of the surviving particles induced by the mortality constraint the relevance of this finding for experiments involving travel times of particles in diffusion decay systems is discussed

solving fredholm integro differential equations using reproducing kernel hilbert space method in this study the numerical solution of fredholm integro differential equation is discussed in a reproducing kernel hilbert space a reproducing kernel hilbert space is constructed in which the initial condition of the problem is satisfied the exact solution ux is represented in the form of series in the space w 2 2 a b in the mean time the n term approximate solution u n x is obtained and is proved to converge to the exact solution u x furthermore we present an iterative method for obtaining the solution in the space w 2 2 a b some examples are displayed to demonstrate the validity and applicability of the proposed method the numerical result indicates that the proposed method is straightforward to implement efficient and accurate for solving linear and nonlinear fredholm integro differential equations

analytic expression for the mean time to absorption for a random walker on the sierpinski gasket the exact analytic expression for the mean time to absorption or mean walk length for a particle performing a random walk on a finite sierpinski gasket with a trap at one vertex is found to be t n 3 n 5 n 1 4 5 n ensuremath 3 n 3 n 1 1 where n denotes the generation index of the gasket and the mean is over a set of starting points of the walk distributed uniformly over all the other sites of the gasket in terms of the number n n of sites on the gasket and the spectral dimension stackrel ifmmode tilde else fi d of the gasket the precise asymptotic behavior for large n n is t n ensuremath rightarrow 1 3 2n n 2 stackrel ifmmode tilde else fi d ensuremath sim n 1 464 this serves as a partial check on our result as it is a intermediate between the known results t ensuremath sim n 2 d 1 and t ensuremath sim n mathrm ln n d 2 for random walks on d dimensional euclidean lattices and b consistent with the known result for the asymptotic behavior of the mean number of distinct sites visited in a random walk on a fractal lattice