a numerical study of some radial basis function based solution methods for elliptic pdes abstract during the last decade three main variations have been proposed for solving elliptic pdes by means of collocation with radial basis functions rbfs in this study we have implemented them for infinitely smooth rbfs and then compared them across the full range of values for the shape parameter of the rbfs this was made possible by a recently discovered numerical procedure that bypasses the ill conditioning which has previously limited the range that could be used for this parameter we find that the best values for it often fall outside the range that was previously available we have also looked at piecewise smooth versus infinitely smooth rbfs and found that for pde applications with smooth solutions the infinitely smooth rbfs are preferable mainly because they lead to higher accuracy in a comparison of rbf based methods against two standard techniques a second order finite difference method and a pseudospectral method the former gave a much superior accuracy

a chebyshev approximation for solving nonlinear integral equations of hammerstein type abstract a numerical method for solving fredholm volterra hammerstein integral equations is presented this method is based on replacement of the unknown function by truncated series of well known chebyshev expansion of functions the quadrature formula which we use to calculate integral terms can be estimated by fast fourier transform fft the numerical examples and the number of operations show the advantages of this method to some other usual methods

new approach for numerical solution of hammerstein integral equations in this paper we use a numerical procedure for solving a class of nonlinear integral equations of the hammerstein type using the globally defined sinc basis functions properties of the sinc procedure are utilized to reduce the computation of the hammerstein equations to some algebraic equations we used some numerical examples to illustrate the accuracy and the implementation of the method

efficient chebyshev collocation methods for solving optimal control problems governed by volterra integral equations the main purpose of this work is to provide efficient chebyshev collocation methods for solving optimal control problems ocps governed by volterra integral equations the basic principle of our approach is to approximate the state and control using the chebyshev polynomials and collocate the dynamic constraints at the chebyshev type points furthermore we present an exact efficient and stable approach for computing the associated chebyshev integration matrices numerical results on benchmark ocps demonstrate the spectral rate of convergence for the proposed methods

a faster combinatorial approximation algorithm for scheduling unrelated parallel machines we consider the problem of scheduling n independent jobs on m unrelated parallel machines without preemption job i takes processing time p i j on machine j and the total time used by a machine is the sum of the processing times for the jobs assigned to it the objective is to minimize makespan the best known approximation algorithms for this problem compute an optimum fractional solution and then use rounding techniques to get an integral 2 approximation in this paper we present a combinatorial approximation algorithm that matches this approximation quality it is much simpler than the previously known algorithms and its running time is better this is the first time that a combinatorial algorithm always beats the interior point approach for this problem our algorithm is a generic minimum cost flow algorithm without any complex enhancements tailored to handle unsplittable flow it pushes unsplittable jobs through a two layered bipartite generalized network defined by the scheduling problem in our analysis we take advantage from addressing the approximation problem directly in particular we replace the classical technique of solving the lp relaxation and rounding afterwards by a completely integral approach we feel that this approach will be helpful also for other applications

approximation algorithms for scheduling unrelated parallel machines we consider the following scheduling problem there arem parallel machines andn independent jobs each job is to be assigned to one of the machines the processing of jobj on machinei requires timep ij the objective is to find a schedule that minimizes the makespan our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum we also present a polynomial approximation scheme for the case that the number of machines is fixed both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations in particular we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints in contrast to our main result we prove that no polynomial algorithm can achieve a worst case ratio less than 3 2 unlessp np we finally obtain a complexity classification for all special cases with a fixed number of processing times

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

existence of solutions to nonlinear hammerstein integral equations and applications abstract in this paper we study the existence and multiplicity of solutions of the operator equation k f u u in the real hilbert space l 2 g under certain conditions on the linear operator k we establish the conditions on f which are able to guarantee that the operator equation has at least one solution a unique solution and infinitely many solutions respectively the monotone operator principle and the critical point theory are employed to discuss this problem respectively in argument quadratic root operator k 1 2 and its properties play an important role as an application we investigate the existence and multiplicity of solutions to fourth order boundary value problems for ordinary differential equations with two parameters and give some new existence results of solutions

an efficient method for the numerical solution of hammerstein mixed vf integral equations on 2d irregular domains abstract in this work we adapt the implicity linear collocation method for solving nonlinear two dimensional mixed volterra fredholm integral equations of hammerstein type by using imq rbfs on a non rectangular domain the proposed method is mesh free and it is independent of the geometry of domain convergence analysis of the proposed method together with some benchmark examples is provided which support its reliability and numerical stability

solving high order ordinary differential equations with radial basis function networks this paper is concerned with the application of radial basis function networks rbfns for numerical solution of high order ordinary differential equations odes two unsymmetric rbf collocation schemes namely the usual direct approach based on a differentiation process and the proposed indirect approach based on an integration process are developed to solve high order odes directly and the latter is found to be considerably superior to the former good accuracy and high rate of convergence are obtained with the proposed indirect method

o n algorithms for discrete n point approximation by quasi convex functions this article considers the problem of approximating a function defined on a finite set of n 1 points by quasi convex functions defined there and constructs linear time o n algorithms for computation of optimal solutions

numerical solution of integral equations by using combination of spline collocation method and lagrange interpolation abstract the numerical solutions to the nonlinear integral equations of the hammerstein type y t f t 0 1 k t s g s y s d s t 0 1 with using b splines functions are investigated an interpolation method based on b splines functions combined with a new collocation method is presented finally for showing efficiency of the method we give some numerical examples

solution of nonlinear volterra fredholm hammerstein integral equations via rationalized haar functions abstract rationalized haar functions are developed to approximate of the nonlinear volterra fredholm hammerstein integral equations the properties of rationalized haar functions are first presented and the operational matrix of integration together with the product operational matrix 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

application of global optimization and radial basis functions to numerical solutions of weakly singular volterra integral equations a novel approach to the numerical solution of weakly singular volterra integral equa tions is presented using the cm multiquadric mq radial basis function rbf expansion rather than the more traditional finite difference finite element or polynomial spline schemes to avoid the collocation procedure that is usually ill conditioned we used a global minimization procedure combined with the method of successive approximations that utilized a small finite set of mq basis functions accurate solutions of weakly singular volterra integral equations are obtained with the minimal number of mq basis functions the expansion and optimization procedure was terminated whenever the global errors were less than 5 lo 2002 elsevier science ltd all rights reserved

a new spectral meshless radial point interpolation smrpi method for the two dimensional fredholm integral equations on general domains with error analysis in this article we present a numerical method to solve two dimensional fredholm integral equations of the second kind on general domains the method utilizes meshless and spectral collocation techniques but it is not traditional meshless collocation method the point interpolation method with the help of strictly positive definite radial basis functions is used to construct shape functions as a basis functions in the frame of spectral collocation methods these basis functions shape functions have kronecker delta function property since the proposed method is meshless it does not need any domain element and so it is independent of the geometry of the domain the method reduces the solution of the two dimensional integral equation to the solution of a linear system of algebraic equations convergence analysis with error estimates are given with full discussion furthermore some numerical examples are presented to show the validity and efficiency of smrpi

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

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

adaptive numerical approach based upon chebyshev operational vector for nonlinear volterra integral equations and its convergence analysis abstract in the current paper we present a direct numerical scheme to approximate a second kind nonlinear volterra integral equations nvies the scheme is based upon shifted chebyshev polynomials and its operational matrices which eventually leads to the sparsity of the coefficients matrix of obtained system the main idea of the proposed approach is based on a useful property of chebyshev polynomials that yields to construct a new operational vector this vector eliminates any requirement of using projection methods and also enhances the accuracy vs other methods applied projection methods the constructive technique and the convergence analysis of this approach under the l w 2 norm are also described numerical experiments and comparisons confirm the applicability and the validity of the presented scheme

numerical solution of the nonlinear fredholm integral equations by positive definite functions a numerical method for solving the nonlinear fredholom integral equations is presented the method is based on interpolation by radial basis functions rbf to approximate the solution of the fredholm nonlinear integral equations several examples are given and numerical examples are presented to demonstrate the validity and applicability of the method

bernoulli polynomials for the numerical solution of some classes of linear and nonlinear integral equations a new operational matrix for integration of bernoulli polynomials is introduced by using this new operational matrix of integration and the so called collocation method linear volterra and nonlinear volterra fredholm hammerstein integral equations are reduced to systems of algebraic equations with unknown bernoulli coefficients some error estimations are provided and illustrative examples are also included to demonstrate the efficiency and applicability of the technique a new operational matrix for integration of bernoulli polynomials is introduced a collocation method is used to reduce equations to systems of algebraic equations some error bounds and convergence results are provided for the proposed method the efficiency and applicability of the technique are checked by some examples

linear time algorithms for convex and monotone approximation abstract this article presents algorithms of linear time complexity o n for computation of optimal solutions to the two problems of convex and monotone approximation where data points are approximated respectively by convex and monotone nondecreasing functions on a grid of n 1 points for the convex approximation case the algorithms are based on a linear programming approach which exploits the structure of matrices involved and uses a special pivoting procedure to obtain a maximal optimal solution analogously in the monotone approximation case the algorithms compute maximal and minimal optimal solutions which also enclose any other optimal solution between them computational results are presented

a new computational method for volterra fredholm integral equations abstract the main purpose of this article is to demonstrate the use of the adomian decomposition method for mixed nonlinear volterra fredholm integral equations a bound is also given for the adomian decomposition series finally numerical examples are presented to illustrate the implementation and accuracy of the decomposition method

numerical solution of a class of mixed two dimensional nonlinear volterra fredholm integral equations using multiquadric radial basis functions an efficient numerical method is proposed for solving the mixed nonlinear volterra fredholm two dimensional integral equations ies using two dimensional radial basis functions rbfs this method is based on interpolation by radial basis functions including multiquadrics mqs using legendre gauss lobatto nodes and weights also a theorem is proved for convergence analysis some numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the proposed method

a meshless discrete collocation method for the numerical solution of singular logarithmic boundary integral equations utilizing radial basis functions abstract the main intention of the current paper is to describe a scheme for the numerical solution of boundary integral equations of the second kind with logarithmic singular kernels these types of integral equations result from boundary value problems of laplace s equations with linear robin boundary conditions the method approximates the solution using the radial basis function rbf expansion with polynomial precision in the discrete collocation method the collocation method for solving logarithmic boundary integral equations encounters more difficulties for computing the singular integrals which cannot be approximated by the classical quadrature formulae to overcome this problem we utilize the non uniform composite gauss legendre integration rule and employ it to estimate the singular logarithm integrals appeared in the method since the scheme is based on the use of scattered points spread on the analyzed domain and does not need any domain elements we can call it as the meshless discrete collocation method the new algorithm is successful and easy to solve various types of boundary integral equations with singular kernels we also provide the error estimate of the proposed method the efficiency and accuracy of the new approach are illustrated by some numerical examples

a meshfree method for the numerical solution of the rlw equation in this paper we present a meshfree technique for the numerical solution of the regularized long wave rlw equation this approach is based on a global collocation method using the radial basis functions rbfs different kinds of rbfs are used for this purpose accuracy of the new method is tested in terms of l 2 and l error norms in case of non availability of the exact solution performance of the new method is compared with existing methods stability analysis of the method is established propagation of single and double solitary waves wave undulation and conservation properties of mass energy and momentum of the rlw equation are discussed

continuous time collocation methods for volterra fredholm integral equations integral equations of mixed volterra fredholm type arise in various physical and biological problems in the present paper we study continuous time collocation time discretization and their global and discrete convergence properties

numerical solution of the higher order linear fredholm integro differential difference equation with variable coefficients the main aim of this paper is to apply the legendre polynomials for the solution of the linear fredholm integro differential difference equation of high order this equation is usually difficult to solve analytically our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted legendre polynomials with unknown coefficients the operational matrices of delay and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted legendre polynomials illustrative examples are included to demonstrate the validity and applicability of the presented technique and a comparison is made with existing results

a pseudospectral method for hammerstein equations abstract this paper considers hammerstein equations formula and presents a pseudospectral method for solving them numerically the method is applied to approximate the solution not to the equation in its original form but rather to an equivalent equation z t g t y t the function z is approximated by a polynomial z n of degree n with coefficients determined by collocation at the legendre gauss lobatto nodes we define the desire approximation to y by formula and establish that under suitable conditions lim n y n t y t uniformly in t advantages of the proposed method compared with existing methods in the literature are discussed an illustrative example is included to show the accuracy of the proposed method

a numerical solution of the nonlinear controlled duffing oscillator by radial basis functions in this research a new numerical method is applied to investigate the nonlinear controlled duffing oscillator this method is based on the radial basis functions rbfs to approximate the solution of the optimal control problem by using the collocation method we apply legendre gauss lobatto points for rbfs center nodes in order to use the numerical integration method more easily then the method of lagrange multipliers is used to obtain the optimum of the problems for this purpose different applications of rbfs are used the differential and integral expressions which arise in the dynamic systems the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients illustrative examples are included to demonstrate the validity and applicability of the technique

the collocation method for hammerstein equations by daubechies wavelets the numerical solutions to the nonlinear integral equations of hammerstein typey t f t email protected 01k t s g s y s ds email protected 0 1 with using daubechies wavelets are investigated a general kernel scheme basing on daubechies wavelets combined with a collocation method is presented the approach of creating daubechies interval wavelets and their main properties are briefly mentioned also we present an algorithm for computing of daubechies wavelets in collocation points the rate of approximation solution converging to the exact solution is given finally we also give some numerical examples for showing efficiency of the method

on the numerical solutions of fredholm volterra integral equation toeplitz matrix method and the product nystrom method are described for mixed fredholm volterra singular integral equation of the second kind the results are compared with the exact solution of the integral equation the error of each method is calculated

an o n algorithm for least squares quasi convex approximation this article is concerned with the approximation problem of fitting n data points by a quasi convex function using the least squares distance function an algorithm of o n worst case time complexity for computing a best fit is developed this problem arises in the context of curve fitting or statistical estimation

pseudospectral legendre based optimal computation of nonlinear constrained variational problems it is well known that spectrally accurate solution can be maintained if the grids on which a nonlinear physical problem is to be solved must be obtained by spectrally accurate techniques in this paper the pseudospectral legendre method for general nonlinear smooth and nonsmooth constrained problems of the calculus of variations is studied the technique is based on spectral collocation methods in which the trajectory x t is approximated by the nth degree interpolating polynomial using legendre gauss lobatto points as the collocation points and lagrange polynomials as trial functions the integral involved in the formulation of the problem is approximated based on legendre gauss lobatto integration rule thereby reducing the problem to a nonlinear programming one to which existing well developed algorithms may be applied the method is easy to implement and yields very accurate results illustrative examples are included to confirm the convergence of the pseudospectral legendre method moreover a numerical experiment on a nonsmooth problem indicates that by applying a smoothing filter procedure to the pseudospectral legendre approximation one can recover the nonsmooth solution within spectral accuracy

numerical solution of nonlinear volterra fredholm hammerstein integral equations via collocation method based on radial basis functions abstract a numerical technique based on the spectral method is presented for the solution of nonlinear volterra fredholm hammerstein integral equations this method is a combination of collocation method and radial basis functions rbfs with the differentiation process drbf using zeros of the shifted legendre polynomial as the collocation points different applications of rbfs are used for this purpose the integral involved in the formulation of the problems are approximated based on legendre gauss lobatto integration rule the results of numerical experiments are compared with the analytical solution in illustrative examples to confirm the accuracy and efficiency of the presented scheme

computing eigenmodes ofelliptic operators using radial basis functions radial basis function rbf approximations have been successfully used to solve boundary value problems numerically we show that rbfs can also be used to compute eigenmodes of elliptic operators particular attention is given to the laplacian operator in two dimensions including techniques to avoid degradation of the solution near the boundaries for regions with corner singularities special functions must be added to the basis to maintain good convergence numerical results compare favorably to basic finite element methods

solving system of volterra fredholm integral equations with bernstein polynomials and hybrid bernstein block pulse functions this work approximates the unknown functions based on the bernstein polynomials and hybrid bernstein block pulse functions in conjunction with the collocation method for the numerical solution of system of fredholm volterra integral equations in both methods the system of integral equations is approximated by the gauss quadrature formula with respect to the legendre weight function the proposed methods reduce the system of integral equations to a system of algebraic equations that can be easily solved by any usual numerical methods moreover the convergence analysis of these algorithms will be shown by preparing some theorems numerical experiments are presented to show the superiority and efficiency of proposed methods in comparison with some other well known methods

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

best quasi convex uniform approximation une fonction reelle bornee g sur 0 1 est quasiconvexe si on a g x max g s g t pour tous x s t tels que 0 s x s t 1 soit k l ensemble de ces fonctions quasiconvexes le probleme considere est celui de la meilleure approximation uniforme quasi convexe d une fonction continue f sur 0 1 c a d le probleme de trouver g k telle que f g inf f g g k

he s variational iteration method for solving nonlinear mixed volterra fredholm integral equations in this paper he s variational iteration method is used to give the approximate solution of nonlinear mixed volterra fredholm integral equations the method constructs a convergent sequence of functions which approximates the exact solution with few iterations numerical results and a comparison with the exact solution are given which reveal its efficiency

an o n algorithm for weighted least squares regression by integer quasi convex and unimodal or umbrella functions the problem of fitting n data points by an integer quasi convex also quasi concave umbrella or unimodal function using the weighted least squares distance function is considered an algorithm of linear time o n worst case complexity and thus optimal is constructed for computing a best fit this problem arises in the context of curve fitting or statistical estimation

a reliable treatment for mixed volterra fredholm integral equations the main goal of this paper is to demonstrate the use of the modified decomposition method for mixed nonlinear volterra fredholm integral equations the modified method combined with the noise terms phenomena may provide the exact solution by using two iterations only two numerical illustrations are given to show the pertinent features of the technique

a new collocation type method for hammerstein integral equations on presente une methode du type de collocation pour la resolution numerique des equations integrales d hammerstein on etablit la convergence de l approximation desiree vers la solution exacte

taylor polynomial solutions of nonlinear volterra fredholm integral equations the kanwal and liu method for the solution of fredholm integral equations is applied to certain nonlinear volterra fredholm integral equations in addition examples that illustrate the pertinent features of the method are presented and the results of study are discussed

on the numerical solution of nonlinear volterra fredholm integral equations by collocation methods particular cases of nonlinear mixed volterra fredholm integral equations of the second kind arise in the mathematical modeling of the spatio temporal development of an epidemic this paper is concerned with the numerical solution of general integral equations of this type by continuous time and discrete time spline collocation methods its focus is on the derivation and analysis of methods of high order of convergence

adomian s method for hammerstein integral equations arising from chemical reactor theory an ordinary differential equation with a parameter in the boundary conditions describes the steady state in an adiabatic tubular chemical reactor in this paper the problem is considered as a hammerstein integral equation and solutions are obtained using adomian s decomposition method

quasi convex optimization given a bounded real function f defined on a closed bounded real interval i the problem is to find a quasi convex function f so as to minimize the supremum of f s f s for all s in i over the class of all quasi convex functions f on i this article obtains optimal solutions to the problem and derives their properties this problem arises in the context of curve fitting or estimation

numerical solution for the weakly singular fredholm integro differential equations using legendre multiwavelets an effective method based upon legendre multiwavelets is proposed for the solution of fredholm weakly singular integro differential equations the properties of legendre multiwavelets are first given and their operational matrices of integral are constructed these wavelets are utilized to reduce the solution of the given integro differential equation to the solution of a sparse linear system of algebraic equations in order to save memory requirement and computational time a threshold procedure is applied to obtain the solution to this system of algebraic equations through numerical examples performance of the present method is investigated concerning the convergence and the sparseness of the resulted matrix equation

new algorithms for the numerical solution of nonlinear fredholm and volterra integral equations using haar wavelets two new algorithms based on haar wavelets are proposed the first algorithm is proposed for the numerical solution of nonlinear fredholm integral equations of the second kind and the second for the numerical solution of nonlinear volterra integral equations of the second kind these methods are designed to exploit the special characteristics of haar wavelets in both one and two dimensions formulae for calculating haar coefficients without solving the system of equations have been derived these formulae are then used in the proposed numerical methods in contrast to other numerical methods the advantage of our method is that it does not involve any intermediate numerical technique for evaluation of the integral present in integral equations the methods are validated on test problems and numerical results are compared with those from existing methods in the literature

numerical solution of the two dimensional fredholm integral equations using gaussian radial basis function in this paper we introduce a numerical method for the solution of two dimensional fredholm integral equations the method is based on interpolation by gaussian radial basis function based on legendre gauss lobatto nodes and weights numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the method

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

application of the multiquadric method for numerical solution of elliptic partial differential equations abstract we have used the multiquadric mq approximation scheme for the solution of elliptic partial differential equations with dirichlet and or neumann boundary conditions the scheme has the advantage of using the data points in arbitrary locations with an arbitrary ordering two dimensional laplace poisson and biharmonic equations describing the various physical processes have been taken as the test examples the agreement is found to be very good between the computed and exact solutions the method also provides an excellent approximation with a curved boundary

the parameter r2 in multiquadric interpolation the multiquadric mq method is an effective bivariate interpolant to three dimensional data xi yi zi where the xi yi are arbitrarily located in the plane the accuracy of the mq method is dependent on a user defined parameter r2 and most practitioners select r2 based upon the number of data points and the locations of the xi yi in the plane we observe that the optimal value of r2 is a strong function of the zi and that it is essentially independent of both the number and the locationsof the xi yi points contrary to some opinions we observe that the mq method can effectively interpolate to track data that is data sampled densely along tracks in the plane together with several other observations we present an algorithm that generally yields an effective value for r2

makespan minimization on unrelated parallel machines with simple job intersection structure and bounded job assignments abstract let there be a set j of n jobs and a set m of m parallel machines where each job j j takes p i j z time units on machine m i and assume p i j implies job j j cannot be scheduled on machine m i in makespan minimization on unrelated parallel machines r c m a x the goal is to schedule each job non preemptively on a machine so as to minimize the makespan a job intersection graph g j j e j is an unweighted undirected graph where there is an edge j j j j e j if there is a machine m i such that both p i j and p i j in this paper we consider two variants of r c m a x where there are a small number of eligible jobs per machine first we prove that there are no approximation algorithms with approximation ratios less than 3 2 for r c m a x when restricted to instances with job intersection graphs belonging to some graph classes such as diamondless graphs and planar graphs unless p np second we match this lower bound by presenting a 3 2 approximation algorithm for r c m a x restricted to instances with diamondless job intersection graphs and furthermore show that when g j is triangle free r c m a x is solvable in polynomial time for r c m a x restricted to instances when every machine can process at most l jobs we give an approximation algorithm with approximation ratios 3 2 and 5 3 for l 3 and l 4 respectively a polynomial time algorithm when l 2 and prove that it is np hard to approximate the optimum solution within a factor less than 3 2 when l 3 in the special case where every p i j p j called the restricted assignment problem and there are only two job lengths p j we present a 2 1 l 1 approximation algorithm when l 3 in addition we give a 2 approximation algorithm for the so called unrelated graph balancing problem for l 3 in the case where j is partitioned into b sets called bags and no two jobs from the same bag can be scheduled on the same machine

an efficient algorithm for solving nonlinear volterra fredholm integral equations in this work we develop a new effective method for solving nonlinear volterra fredholm integral equation the existence of any e approximate solution is proved at the same time an effective method for obtaining the e approximate solution is established the final numerical examples illustrate that our approach is valid not only for weakly nonlinear problems but also for strongly nonlinear problems

a meshless method for solving nonlinear two dimensional integral equations of the second kind on non rectangular domains using radial basis functions with error analysis in this paper we present a numerical method for solving two dimensional nonlinear fredholm integral equations of the second kind on a non rectangular domain the method utilizes radial basis functions rbfs constructed on scattered points as a basis in the discrete collocation method the proposed scheme is meshless since it does not need any domain element and so it is independent of the geometry of the domain the method reduces the solution of the two dimensional nonlinear integral equation to the solution of a nonlinear system of algebraic equations error analysis is presented for this method finally numerical examples are included to show the validity and efficiency of the new technique

a meshless method on non fickian flows with mixing length growth in porous media based on radial basis functions a comparative study the present study aims to introduce a solution for parabolic integro differential equations arising in heat conduction in materials with memory which naturally occur in many applications two radial basis functions rbfs collocation schemes are employed for solving this equation the first method tested is an unsymmetric method and the second one which appears to be more efficient is a symmetric one the convergence of these two schemes is accelerated as we use the cartesian nodes as the center nodes

interpolating element free galerkin method for the regularized long wave equation and its error analysis abstract combining the simplified interpolating moving least squares imls method with the galerkin weak form an interpolating element free galerkin iefg method is presented for the numerical study of the regularized long wave rlw equation the trial function in the imls method has one less unknown coefficient than in the traditional mls approximation and the shape functions satisfy the interpolating property at nodes then the iefg method can apply the essential boundary conditions naturally and directly and has high computational accuracy and then this paper studies the error estimates of the iefg method for rlw equation the error estimate of the semi discrete iefg method for the rlw equation on the influence domain radius is first studied and then the convergence rate of the full discrete scheme of the method on the influence domain radius and time step is studied the theoretical results show that the iefg method for rlw equation has good convergence rate for the purpose of demonstration three selected numerical examples are given to demonstrate the method and the mathematical theory of this paper

numerical solution of the nonlinear klein gordon equation using radial basis functions the nonlinear klein gordon equation is used to model many nonlinear phenomena in this paper we propose a numerical scheme to solve the one dimensional nonlinear klein gordon equation with quadratic and cubic nonlinearity our scheme uses the collocation points and approximates the solution using thin plate splines tps radial basis functions rbf the implementation of the method is simple as finite difference methods the results of numerical experiments are presented and are compared with analytical solutions to confirm the good accuracy of the presented scheme