solution of boundary value problems for integro differential equations by using differential transform method in this study we extended the differential transform method dtm to solve the integro differential equations new theorems for the transformation of integrals are introduced and prooved the commonly encountered linear and nonlinear integro differential equations that appear in literature are solved as an illustration for the efficiency of the method some numerical results are also given to demonstrate the superiority of the method to other common techniques the results we obtained by using this technique is much more accurate compared to the existing ones

primes in intervals of bounded length in april 2013 yitang zhang proved the existence of a finite bound b such that there are infinitely many pairs of distinct primes which differ by no more than b this is a massive breakthrough makes the twin prime conjecture look highly plausible which can be re interpreted as the conjecture that one can take b 2 and his work helps us to better understand other delicate questions about prime numbers that had previously seemed intractable the original purpose of this talk was to discuss zhang s extraordinary work putting it in its context in analytic number theory and to sketch a proof of his theorem zhang had even proved the result with b 70 000 000 moreover a co operative team polymath8 collaborating only on line had been able to lower the value of b to 4680 not only had they been more careful in several difficult arguments in zhang s original paper they had also developed zhang s techniques to be both more powerful and to allow a much simpler proof indeed the proof of zhang s theorem that will be given in the write up of this talk is based on these developments in november inspired by zhang s extraordinary breakthrough james maynard dramatically slashed this bound to 600 by a substantially easier method both maynard and terry tao who had independently developed the same idea were able to extend their proofs to show that for any given integer m 1 there exists a bound bm such that there are infinitely many intervals of length bm containing at least m distinct primes we will also prove this much stronger result herein even showing that one can take bm e 8m if zhang s method is combined with the maynard tao set up then it appears that the bound can be further reduced to 576 if all of these techniques could be pushed to their limit then we would obtain b b2 12 so new ideas are still needed to have a feasible plan for proving the twin prime conjecture the article will be split into two parts the first half which appears here we will introduce the work of zhang polymath8 maynard and tao and explain their arguments that allow them to prove their spectacular results as we will discuss zhang s main novel contribution is an estimate for primes in relatively short arithmetic progressions the second half of this article sketches a proof of this result the bulletin article will contain full details of this extraordinary work part 1 primes in short intervals

parameter uniform numerical method for singularly perturbed turning point problems exhibiting boundary layers this article presents a numerical method to solve singularly perturbed turning point problems exhibiting two exponential boundary layers classical finite difference schemes do not yield parameter uniform convergent results on a uniform mesh in general robust computational techniques for boundary layers chapman hall london crc press boca raton fl 2000 in order to overcome this difficulty we propose an appropriate piecewise uniform shishkin mesh and apply the classical finite difference schemes on this mesh error estimates are derived by decomposing the solution into smooth and singular components the present method is layer resolving as well as parameter uniform convergent numerical examples are presented to show the applicability and efficiency of the method

small gaps between primes we introduce a renement of the gpy sieve method for studying prime k tuples and small gaps between primes this renement avoids previous limitations of the method and allows us to show that for each k the prime k tuples conjecture holds for a positive proportion of admissible k tuples in particular lim infn pn m pn 1 for every integer m we also show that lim inf pn 1 pn 600 and if we assume the elliott halberstam conjecture that lim infn pn 1 pn 12 and lim infn pn 2 pn 600

a novel semi analytical approach for solving nonlinear volterra integro differential equations in this work we present an efficient semi analytical method based on the taylor series for solving nonlinear volterra integro differential equations namely the differential transform method dtm the dtm provides a recursive relation for the coefficients of the taylor series that is derived from the given equations we provide a new recursive relation for the nonlinear volterra integro differential equations with complex nonlinear kernels since the dtm is based on the taylor series it is difficult to obtain accurate approximate solutions in a large domain to overcome this difficulty the standard dtm is applied in each subdomain called the multistage differential transform method msdtm we also present an convergence analysis for the proposed method to demonstrate the efficiency of the proposed method several numerical examples are performed and support the results in our analysis

numerical solution of integro differential equations by using cas wavelet operational matrix of integration the cas wavelet operational matrix p of integration is first presented and a general procedure to generate this matrix p is given cas wavelet approximating method is then utilized to reduce the integro differential equation to the algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique

rationalized haar functions method for solving fredholm and volterra integral equations this paper presents a computational technique for fredholm integral equation of the second kind and volterra integral equation of the second kind the method is based upon haar functions approximation properties of rationalized haar functions are first presented the operational matrix of integration together with product operational matrix and newton cotes nodes are utilized to reduce the computation of integral equations into some algebraic equations the method is computationally attractive and applications are demonstrated through illustrative examples

primes in tuples i we introduce a method for showing that there exist prime numbers which are very close together the method depends on the level of distribution of primes in arithmetic progressions assuming the elliott halberstam conjecture we prove that there are infinitely often primes differing by 16 or less even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart unconditionally we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing that is lim inf n pn 1 pn log pn log 0 we will quantify this result further in a later paper

numerical solution of functional differential integral and integro differential equations this paper describes a numerical method based on lagrange interpolation and chebyshev interpolation to treat functional integral equations of volterra type and fredholm type also the method can be extended to functional differential and integro differential equations various numerical examples are treated

compact finite difference method for integro differential equations in this paper we give sixth order compact finite difference formula for second order integro differential equations ide with different boundary conditions and both of error estimates and numerical experiments confirm our compact finite difference method can get fifth order of accuracy we also adjust compact finite difference method for first order ide and a system of ide and give numerical experiments for them our algorithm even can solve nonlinear ide and unsplit kernel of ide the most advantages of compact finite difference method for ide are that it obtains high order of accuracy while the time complexity to solve the matrix equations after we use compact finite difference method on ide is o n and it can solve very general case of ide

richardson extrapolation for a singularly perturbed turning point problem with exponential boundary layers we consider an interior turning point problem with two exponential boundary layers its discretisation on a piecewise uniform shishkin mesh yields a scheme which is uniformly convergent measured in the discrete maximum norm of almost order one richardson extrapolation improves the accuracy to o n 2 ln 2 n both can be proved under the assumption e c n 1

an algorithm for solving the high order nonlinear volterra fredholm integro differential equation with mechanization abstract in this paper by using the theories and methods of mathematics analysis and computer algebra a reliable algorithm for solving high order nonlinear volterra fredholm integro differential equations was established and a new maple procedure voltfredproc was established too the results of the examples indicated that the procedure voltfredproc of taylor polynomial method is simple and effective and could provide an accuracy approximate solution or exact solution of the high order nonlinear volterra fredholm integro differential equation this would be useful for solving integro differential equation integral equations and ordinary differential equation

numerical solution of linear integro differential equation by using sine cosine wavelets abstract we use the continuous sine cosine wavelets on the interval 0 1 to solve the linear integro differential equation to do so we construct the quadrature formulae for the calculation of inner products of any functions which are required in the approximation for the integro differential equation then we reduced the integro differential equation to the solution of linear algebraic equations

a simple taylor series expansion method for a class of second kind integral equations a simple yet effective taylor series expansion method is presented for a class of fredholm integral equations of the second kind with smooth and weakly singular kernels the equations studied in this paper arise in a number of applications e g potential theory radiative equilibrium radiative heat transfer and electrostatics the approach leads to an approximate solution of the integral equation which can be expressed explicitly in a simple closed form the approximate solution is of sufficient accuracy as illustrated by the numerical examples arising from radiative heat transfer and electrostatics

existence of solutions of boundary value problems with integral boundary conditions for second order impulsive integro differential equations in banach spaces this paper investigates the existence of solutions for a class of second order boundary value problems with integral boundary conditions of nonlinear impulsive integro differential equations in banach spaces the arguments are based upon the fixed point theorem of strict set contraction operators meanwhile an example is worked out to demonstrate the main results

a semi analytical approach to solve integro differential equations in this work we present a semi analytical method for solving integro differential equations under multi point or mixed boundary conditions the proposed method solves linear and nonlinear fredholmvolterra integro differential equations a convergence analysis of the proposed method is directly examined numerical examples are worked out to demonstrate the main results moreover proper graphs are provided to confirm the efficiency and the accuracy of the proposed scheme we show that with a few number of obtained approximating terms we achieve a high accuracy level of the obtained results however increasing the number of approximating terms yields a significant decrease of the error of the approximation the proposed method is very useful reliable and flexible for solving different kinds of integro differential equations

a new operational approach for numerical solution of generalized functional integro differential equations in this paper a class of linear and nonlinear functional integro differential equations are considered that can be found in the various fields of sciences such as stress strain states of materials motion of rigid bodies and models of polymer crystallization the operational collocation method with shifted jacobi polynomial bases is applied to approximate the solution of these equations in addition some theoretical results are given to simplify and reduce the computational costs finally some numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method

numerical simulation for the space fractional diffusion equations abstract this paper presents a novel semi analytical algorithm based on the chebyshev collocation method for the solution of space fractional diffusion equations the original fractional equation is transformed into a system of ordinary differential equations odes by the chebyshev collocation method a new semi analytical method is then used to approximate the solution of the resulting system to emphasize the reliability of the new scheme a convergence analysis is presented it is shown that highly accurate solutions can be achieved with relatively few approximating terms and absolute errors are rapidly decrease as the number of approximating terms is increased by presenting some numerical examples we show that the proposed method is a powerful and reliable algorithm for solving space fractional diffusion equations and can be extended to solve another space fractional partial differential equations comparison with other methods in the literature demonstrates that the proposed method is both efficient and accurate

numerical analysis of singularly perturbed delay differential turning point problem abstract in this paper we describe a numerical method based on fitted operator finite difference scheme for the boundary value problems for singularly perturbed delay differential equations with turning point and mixed shifts similar boundary value problems are encountered while simulating several real life processes for instance first exit time problem in the modelling of neuronal variability a rigorous analysis is carried out to obtain priori estimates on the solution and its derivatives for the considered problem in the development of numerical methods for constructing an approximation to the solution of the problem a special type of mesh is generated to tackle the delay term along with the turning point then to develop robust numerical scheme and deal with the singularity because of the small parameter multiplying the highest order derivative term an exponential fitting parameter is used several numerical examples are presented to support the theory developed in the paper

the combined laplace transform adomian decomposition method for handling nonlinear volterra integro differential equations in this work a combined form of the laplace transform method with the adomian decomposition method is developed for analytic treatment of the nonlinear volterra integro differential equations the combined method is capable of handling both equations of the first and second kind illustrative examples will be examined to support the proposed analysis

a new iterative technique for solving nonlinear second order multi point boundary value problems abstract we present a semi analytical iterative method for solving nonlinear second order multi point boundary value problems to demonstrate the working of the method we consider a particular example of this class of problems in this example we demonstrate the accuracy and convergence of the method to the solution we demonstrate clearly that the method is accurate fast and has a reasonable order of convergence

comparison between the homotopy perturbation method and the sine cosine wavelet method for solving linear integro differential equations this paper compares the homotopy perturbation method with the sine cosine wavelet method for solving linear integro differential equations from the computational viewpoint the homotopy perturbation method is more efficient and easier than the sine cosine wavelet method

numerical methods for singular perturbation problems consider the two point boundary value problem for a stiff system of ordinary differential equations an adaptive method to solve these problems even when turning points are present is discussed

bounded gaps between primes it is proved that lim inf n 8 p n 1 p n 7 10 7 where p n is the n th prime our method is a refinement of the recent work of goldston pintz and yildirim on the small gaps between consecutive primes a major ingredient of the proof is a stronger version of the bombieri vinogradov theorem that is applicable when the moduli are free from large prime divisors only but it is adequate for our purpose

numerical solution of the system of fredholm integro differential equations by the tau method the tau method produces approximate polynomial solution of differential integral and integro differential equations see e l ortiz the tau method siam j numer anal 6 3 1969 480 492 e l ortiz h samara an operational approach to the tau method for the numerical solution of non linear differential equations computing 27 1981 15 25 s m hosseini s shahmorad a matrix formulation of the tau for fredholm and volterra linear integro differential equations the korean j comput appl math 9 2 2002 497 507 s m hosseini s shahmorad numerical solution of a class of integro differential equations by the tau method with an error estimation appl math comput 136 2003 559 570 in this paper we extend the tau method for the numerical solution of integro differential equations systems ides we also give a brief description of the structure of the tau program by the maple software an efficient error estimation of the numerical solution of the method is also introduced some examples are given to clarify the efficiency and high accuracy of the method

hybrid function method for solving fredholm and volterra integral equations of the second kind numerical solutions of fredholm and volterra integral equations of the second kind via hybrid functions are proposed in this paper based upon some useful properties of hybrid functions integration of the cross product a special product matrix and a related coefficient matrix with optimal order are applied to solve these integral equations the main characteristic of this technique is to convert an integral equation into an algebraic hence the solution procedures are either reduced or simplified accordingly the advantages of hybrid functions are that the values of n and m are adjustable as well as being able to yield more accurate numerical solutions than the piecewise constant orthogonal function for the solutions of integral equations we propose that the available optimal values of n and m can minimize the relative errors of the numerical solutions the high accuracy and the wide applicability of the hybrid function approach will be demonstrated with numerical examples the hybrid function method is superior to other piecewise constant orthogonal functions w f blyth r l may p widyaningsih volterra integral equations solved in fredholm form using walsh functions anziam j 45 e 2004 c269 c282 m h reihani z abadi rationalized haar functions method for solving fredholm and volterra integral equations j comp appl math 200 2007 12 20 for these problems

a new algorithm for integral of trigonometric functions with mechanization in this paper by using the theories of mathematical analysis the authors systematically studied the problem of integrating of trigonometric functions utilizing omnipotence expressions tan x2 t a new integral algorithm for the integral of trigonometric functions was established and a new maple program trigint for the calculation mechanizing for the integral of trigonometric functions was established trigint could evaluate not only the integral of trigonometric functions utilizing omnipotence expresssions correctly but also all the other kinds of functions the results indicated that the algorithm established in this paper had advantages of simple idea excellent property for operation and powerful competence this would be useful for some calculating problems of engineering mathematics

asymptotic analysis for singularly perturbed convection diffusion equations with a turning point turning points occur in many circumstances in fluid mechanics when the viscosity is small very complex phenomena can occur near turning points which are not yet well understood a model problem corresponding to a linear convection diffusion equation e g suitable linearization of the navier stokes or benard convection equations is considered our analysis shows the diversity and complexity of behaviors and boundary or interior layers which already appear for our equations simpler than the navier stokes or benard convection equations of course the diversity and complexity of these structures will have to be taken into consideration for the study of the nonlinear problems in our case at this stage the full theoretical asymptotic analysis is provided this study is totally new to the best of our knowledge numerical treatment and more complex problems will be considered elsewhere

variable mesh spline approximation method for solving singularly perturbed turning point problems having boundary layer s abstract a numerical method based on cubic spline with nonuniform grid is given for the singularly perturbed two point boundary value problems having turning point the scheme derived in this paper is second order accurate numerical examples are given to support the predicted theory

numerical study of singularly perturbed differential difference equation arising in the modeling of neuronal variability the objective of this paper is to present a numerical study of a class of boundary value problems of singularly perturbed differential difference equations spdde which arise in computational neuroscience in particular in the modeling of neuronal variability the mathematical modeling of the determination of the expected time for the generation of action potential in the nerve cells by random synaptic inputs in dendrites includes a general boundary value problem for singularly perturbed differential difference equation with shifts the problem considered in this paper exhibit turning point behavior which add to the complexity in the construction of numerical approximation to the solution of the problem as well as in obtaining theoretical estimates on the solution exponentially fitted finite difference scheme based on il in allen southwell fitting is used on a specially designed mesh some numerical examples are given to validate convergence and computational efficiency of the proposed numerical scheme effect of the shifts on the layer structure is illuminated for the considered examples

an analysis of a uniformly accurate difference method for a singular perturbation problem it will be proven that an exponential tridiagonal difference scheme when applied with a uniform mesh of size h to euxx b x ux f x for o o b and f smooth e in 0 1 and u o and u i given is uniformly second order accurate i e the maximum of the errors at the grid points is bounded by ch2 with the constant c independent of h and e this scheme was derived by el mistikawy and werle by a c patching of a pair of piecewise constant coefficient approximate differential equations across a common grid point the behavior of the approximate solution in between the grid points will be analyzed and some numerical results will also be given

numerical solution of linear fredholm integral equation by using hybrid taylor and block pulse functions in this paper we use a combination of taylor and block pulse functions on the interval 0 1 that is called hybrid functions to estimate the solution of a linear fredholm integral equation of the second kind we convert the integral equation to a system of linear equations and by using numerical examples we show our estimation have a good degree of accuracy

numerical solution of a class of integro differential equations by the tau method with an error estimation the tau method by construction produces approximate polynomial solutions of differential equations the purpose of this paper is to extend the tau method to the integro differential equations an efficient error estimation for the tau method is also introduced details of this method are presented and some numerical results along with estimated errors are given to clarify the method and its error estimator

singularly perturbed linear two point boundary value problems finding asymptotic solutions to two point boundary value problems for linear singularly perturbed second order ordinary differential equations is now quite well understood troublesome exceptions involving turning points known as problems of boundary layer resonance have been studied by many experts since the initial work of ackerberg and o malley stud appl math 49 1970 pp 277 295 this significant work is surveyed here closely related literature involves the canard phenomena new generally more geometric approaches including gevrey asymptotics and blowup promise improved understanding of such sensitive asymptotics and their generalizations

a priori estimates and analysis of a numerical method for a turning point problem bounds are obtained for the derivatives of the solution of a turning point problem these results suggest a modification of the el mistikawy werle finite difference scheme at the turning point a uniform error estimate is obtained for the resulting method and illustrative numerical results are given i introduction we will examine the following two point boundary value problem with dirichlet data at the endpoints l la ly eyx x p x yx x q x y x f x for i x 1

an efficient technique to find semi analytical solutions for higher order multi point boundary value problems abstract a new semi analytical algorithm is presented to solve general multi point boundary value problems this method can be applied on nth order linear nonlinear singular and nonsingular multi point boundary value problems mathematical base of the method is presented convergence of the method is proved also the algorithm is applied to solve multi point boundary value problems including nonlinear sixth order nonlinear singular second order five point boundary value problems and a singularly perturbed boundary value problem comparison results show that the new method works more accurate than the other methods