interpolation for runge kutta methods runge kutta methods provide a popular way to solve the initial value problem for a system of ordinary differential equations in contrast to the adams methods there is no natural way to approximate the solution between mesh points a way to accomplish this is proposed which is applicable to some important formulas its theoretical support is much better than that of interpolation in the popular variable order variable step adams codes

higher order method for singular boundary value problems by using spline function abstract in problems of physics and engineering we often come across singular boundary value problems which cannot be solved by the usual numerical methods special methods for solving such problems have been developed these methods lead to banded systems linear and non linear depending upon the nature of boundary value problem in this paper we discuss the construction of a spline function for a class of singular two point boundary value problems x x u f x u 0 x 1 u 0 a u 1 b five point finite difference method using the above splines is obtained it is shown to be order h4 convergent for all 0 1 the method is illustrated computationally

he s variational iteration method for treating nonlinear singular boundary value problems this paper applies he s variational iteration method for solving nonlinear singular boundary value problems the solution process is illustrated and various physically relevant results are obtained comparison of the obtained results with exact solutions shows that the method used is an effective and highly promising method for treating various classes of both linear and nonlinear singular boundary value problems

a compact finite difference method for a general class of nonlinear singular boundary value problems with neumann and robin boundary conditions abstract in this paper we develop and analyze a high order compact finite difference method cfdm for solving a general class of two point nonlinear singular boundary value problems with neumann and robin boundary conditions arising in various physical models convergence result of this method is established through matrix analysis approach to illustrate the applicability and accuracy of the method we consider nine numerical examples including heat conduction in the human head equilibrium of isothermal gas sphere oxygen diffusion in a spherical cell and reaction diffusion process in a spherical permeable catalyst it is shown that the computational order of convergence of the proposed cfdm is four the obtained results are compared with those obtained by other existing numerical methods

a computational multilevel approach for solving 2d navier stokes equations over non matching grids a multilevel approach with parallel implementation is developed for obtaining fast solutions of the navier stokes equations solved on domains with non matching grids the method relies on computing solutions over different subdomains with different multigrid levels by using multiple processors a local vanka type relaxation operator for the multigrid solution of the navier stokes system allows solutions to be computed at the element level the natural implementation on a multiprocessor architecture results in a straightforward and flexible algorithm numerical computations are presented using benchmark applications in order to support the method parallelization is discussed to achieve proper accuracy load balancing and computational efficiency between different processors 2005 elsevier b v all rights reserved

diffusion problems with a mixed nonlinear boundary condition abstract existence results are established for a class of singular diffusion problems with a general nonlinear boundary condition the proofs are based on a priori bounds for all solutions and a shooting technique applied to a corresponding initial value problem various uniqueness results are obtained by assuming either the variables in the diffusion coefficient or in the source term are separable

a three point finite difference method for a class of singular two point boundary value problems a three point finite difference method based on uniform mesh for solving the singular two point boundary value problems y 1 xy f x y 0 0 x 1 y 0 0 y 1 a has been derived under quite general conditions on f and f and f y 4 we show that our present method provides o h2 convergent approximations the method is illustrated by three numerical examples two linear and one non linear

a fourth order finite difference method based on non uniform mesh for a class of singular two point boundary value problems abstract we present a three point finite difference method based on non uniform mesh for a class of singular two point boundary value problem x y f x y y 0 a y 1 b 0 we show that the method based on non uniform mesh provides o h4 convergent approximations this method is illustrated by two numerical examples one is linear and the other is non linear

the monotone iterative method and zeros of bessel functions for nonlinear singular derivative dependent bvp in the presence of upper and lower solutions abstract in this paper we consider a class of nonlinear singular boundary value problems x y x x f x y x x y x 0 0 x 1 y 0 y 1 0 for 1 we assume that the source function f x y x y is lipschitz in x y and one sided lipschitz in y the initial approximations are an upper solution u 0 x and a lower solution v 0 x which can be ordered in one way v 0 x u 0 x or the other u 0 x v 0 x we propose an iterative scheme and establish the existence of solutions bounded by v 0 and u 0 and allow f y to take both positive and negative values the method is constructive in nature and can be used to generate solutions of the nonlinear singular boundary value problems

a uniform mesh finite difference method for a class of singular two point boundary value problems a new finite difference method based on uniform mesh is given for the weakly singular two point boundary value problem x alpha y f x y y 0 a y 1 b 0 alpha 1 under quite general conditions on f and f we show that our present method based on uniform mesh provides o h 2 convergent approximations for all alpha in 0 1 our method is based on one evaluation of f and for alpha 0 it reduces to the classical second order method for y f x y

a novel numerical approach and its convergence for numerical solution of nonlinear doubly singular boundary value problems this paper deals with two new recursive numerical schemes for the solution of a class of doubly singular two point boundary value problems with different types of boundary conditions we first transform the original problem into an equivalent integral equation to overcome the singular behaviour at the origin and then establish recursive schemes by employing homotopy perturbation method for the solution of the resultant equation we establish the convergence results of the new numerical methods four numerical examples are considered to demonstrate the efficiency and accuracy of the proposed methods the present methods are shown their advantage over some existing methods chawla and katti 1982 pandey and singh 2007 pandey and singh 2004 caglar et al 2009 kumar and aziz 2004 khuri and sayfy 2010 developed for the solution of singular two point boundary value problems moreover the new methods are used to obtain the approximate solutions of the singular boundary value problems arising in various physical problems

multigrid for the mortar finite element method a multigrid technique for uniformly preconditioning linear systems arising from a mortar finite element discretization of second order elliptic boundary value problems is described and analyzed these problems are posed on domains partitioned into subdomains each of which is independently triangulated in a multilevel fashion the multilevel mortar finite element spaces based on such triangulations which need not align across subdomain interfaces are in general not nested suitable grid transfer operators and smoothers are developed which lead to a variable v cycle preconditioner resulting in a uniformly preconditioned algebraic system computational results illustrating the theory are also presented

the optimal convergence rate of the p version of the finite element method optimal error estimates for the p version of the finite element method in two dimensions are proven for the case when u in h k omega or u has singularities induced by the corners of the domain the case of nonhomogenous essential boundary conditions is also analyzed

a non conforming computational methodology for modeling coupled problems a variety of engineering applications that involve coupling different physical phenomena often require detailed finite element analysis to be carried out over complex domains often such analysis may be accomplished by dividing the global domain into several local subdomains over each of which a local model can be analyzed independently the global solution can then be constructed by suitably piecing together local solutions from these individually modeled subdomains however during the assembly it is often too cumbersome or even infeasible to coordinate the meshes over separate subdomains one must therefore employ non conforming techniques to accomplish such modeling in this paper we develop a non conforming computational methodology via the mortar finite element method to solve a coupled problem where we are interested in determining the effects of temperature variations on a given flow or the transfer of heat within the flow using this method the solution over different subdomains with different multigrid levels is efficiently computed our numerical results clearly suggest that the proposed methodology is robust and stable 2005 elsevier ltd all rights reserved

taylor series method for solving a class of nonlinear singular boundary value problems arising in applied science abstract taylor series method is applied to a class of nonlinear singular boundary value problems without requiring any specific technique in handling the singularity at the origin for the problems with exponential nonlinearity a variable transformation is introduced to accelerate the convergence of the solution by converting this exponential nonlinearity into a polynomial nonlinearity the efficiency and applicability of the algorithm are assessed and tested on several problems arising in applied science pade approximants are used to verify that the series solutions to these problems converge within the boundaries an error analysis is presented and the results show that the analytical approximate solutions obtained are highly accurate

toward a universal h p adaptive finite element strategy part 1 constrained approximation and data structure abstract this is the first of a series of papers devoted to the development of an adaptive finite element method to solve various boundary value problems arising in solid and fluid mechanics the paper deals with the general concepts of h p finite element methods constrained approximation and data structure and related issues several numerical examples illustrate the main ideas of the method

a note on existence uniqueness results for a class of doubly singular boundary value problems abstract in this paper we establish the existence and uniqueness of solution for nonlinear singular boundary value problem p x y x q x f x y for 0 x b and lim x 0 p x y x 0 1 y b 1 p b y b 1 here p 0 0 and also q is not assumed to be continuous at 0 so the problem may be doubly singular in this note we discuss the cases when y 0 0 at the singular end is not possible but lim x 0 p x y x 0 regions for multiple solutions have also been determined

monotone method for singular bvp in the presence of upper and lower solutions in this paper we examine existence of monotone approximations of solutions of singular boundary value problem p x y x q x f x y py for 0 0 p x y x 0 a 1y b email protected 1p b y b email protected 1 under quite general conditions on f x y py we show that solution of the singular two point boundary value problem is unique here p 0 0 q x is allowed to have integrable singularity at x 0 and we do not assume 0 bdsp s

a fourth order spline method for singular two point boundary value problems this paper describes a new fourth order spline approximation method for the class of singular two point boundary value problems y frac alpha x y f x y 0 0 x 1 y 0 0 y 1 a for all alpha geqslant 1

a fourth order finite difference method for a class of singular two point boundary value problems a method is described based on a non uniform mesh for the class of singular two point boundary value problems x ay f x y 0

on the convergence of finite difference methods for weakly regular singular boundary value problems the second order finite difference methods m 1 based on a non uniform mesh and m 2 based on an uniform mesh developed by chawla and katti finite difference methods and their convergence for a class of singular two point boundary value problems numer math 39 1982 341 350 for weakly regular singular boundary value problems p x y f x y 0

stability and convergence of nonconforming hp finite element methods the stability and convergence of nonconforming hp finite element methods in par ticular the mortar finite element method and its variants are established based on a new stability measure for these methods using a generalized eigenvalue analysis estimates for this measure are computed numerically our numerical results demonstrate that these nonconforming methods prove to be good candidates for hp implementation and also behave ss well as conforming finite element methods the discussion here ls primarily in two dimensions but some extensions to three dimensions

a quartic b spline for second order singular boundary value problems second order singular boundary value problems are treated using quartic fourth degree b spline approximations the values of the coefficients c i are chosen via optimization error analysis is discussed the efficiency of the method is illustrated with several problems from the literature

haar wavelet collocation method for lane emden equations with dirichlet neumann and neumann robin boundary conditions abstract in this paper we present a numerically stable algorithm based on the haar wavelet collocation method hwcm for numerical solution of a class of lane emden equation with dirichlet neumann and neumann robin type boundary conditions arising in various physical models the haar wavelet collocation method uses simple box functions and hence the formulation of the numerical algorithm is straightforward unlike other methods the hwcm can be applied easily and accurately to the neumann type boundary conditions the advantage of the haar wavelet method is its efficiency and simple applicability for a variety of boundary conditions the haar wavelet collocation method is utilized to reduce the lane emden equation with boundary conditions to a system of algebraic equations then the newton s method is employed for numerical solutions nine examples of the lane emden equation with various types of boundary conditions are provided to demonstrate the accuracy and efficiency of the hwcm the numerical results are compared with the results obtained by other techniques and the exact solution

a nonconforming finite element method for fluid structure interaction problems in this paper we develop a nonconforming finite element methodology using a three field formulation to analyze a fluid structure interaction problem the methodology is used to couple a lagrangian model describing the structure with the arbitrary lagrangian eulerian strategy used to describe the fluid in order to simulate a full unsteady physical phenomenon consistency error estimates are obtained which show that the numerical scheme employed yields a first order approximation for the solution to the fluid structure interaction problem finally we present a discrete energy estimate to demonstrate the stablity of the proposed method 2005 elsevier b v all rights reserved

runge kutta software with defect control four boundary value odes a popular approach to the numerical solution of boundary value ode problems involves the use of collocation methods such methods can be naturally implemented so as to provide a continuous approximation to the solution over the entire problem interval on the other hand several authors have suggested as an alternative certain subclasses of the implicit runge kutta formulas known as mono implicit runge kutta mirk formulas which can be implemented at a lower cost per step than the collocation methods these latter formulas do not have a natural implementation that provides a continuous approximation to the solution rather only a discrete approximation at certain points within the problem interval is obtained however recent work in the area of initial value problems has demonstrated the possibility of generating inexpensive interpolants for any explicit runge kutta formula these ideas have recently been extended to develop continuous extensions of the mirk formulas in this paper we describe our investigation of the use of continuous mirk formulas in the numerical solution of boundary value ode problems a primary thrust of this investigation is to consider defect control based on the continuous mirk formulas as an alternative to the standard use of global error control as the basis for termination and mesh redistribution criteria

cubic spline for a class of non linear singular boundary value problems arising in physiology abstract cubic spline method to analyze a class of non linear singular boundary value problems defined by y x m x y f x y y 0 0 y 1 y 1 is presented the quesilinearization technique is used to reduce the given non linear problem to a sequence of linear problems the resulting set of differential equations are modified at the singular point and are treated by using cubic spline for finding the numerical solution the numerical method is tested for its efficiency by considering two examples from physiology

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

on the convergence of a finite difference method for a class of singular boundary value problems arising in physiology using chawla s identity bit 29 1989 566 a finite difference method based on uniform mesh is described for a class of singular boundary value problems p x y p x f x y 0 x 1 y 0 a y 1 y 1 or y 0 0 y 1 y 1 with p x xb0 g x b0 0 and it is shown that the method is of second order accuracy under quite general conditions on p x and f x y this work also extends the method developed by chawla et al bit 26 1986 326 for p x xb0 b0 1 to a general class of function p x xb0 g x b0 0 numerical examples for general function p x verify the order of the convergence of the method and two physiological problems have also been solved

a uniform mesh finite difference method for a class of singular two point boundary value problems abstract a method is described based on a uniform mesh for the class of singular two point boundary value problems x y f x y 0 x 1 y 0 a y 1 b it is shown to be order h 4 convergent for all 0 1

a fourth order finite difference method based on uniform mesh for singular two point boundary value problems we present a new fourth order finite difference method based on uniform mesh for the weakly singular two point boundary value problem x y f x y 0 x 1 y 0 a y 1 b 0 1 our method provides o h4 convergent approximations for all 0 1 for 0 it reduces to the well known fourth order method of numerov for y x y

integral methods for computing solutions of a class of singular two point boundary value problems in this paper we establish three iteration methods to compute solutions for a class of weakly singular two point boundary value problems x y f x y where x 0 1 and 2 we obtain the sufficient conditions for existence of a unique solution on r finally we given some numerical examples

tage iterative algorithm and nonpolynomial spline basis for the solution of nonlinear singular second order ordinary differential equations abstract in the present paper we discuss three point difference method based on nonpolynomial spline basis for the second order ordinary differential equation difference schemes are derived for linear and nonlinear case and are used to solve via two parameter alternating group explicit iterative algorithm the schemes have a fourth and second order of uniform convergence for the choice of the parameters involved in the method computational results are presented comparing the two methods in terms of accuracy and execution times the results indicate the advantage of using parallel implementation of the new method

a spectral element methodology tuned to parallel implementations abstract we present in this paper a modification of the conforming spectral element method for the approximation of the poisson equation this new procedure allows the implementation on a parallel machine in an easier and faster way the resulting method is nonconforming but the numerical analysis shows that the error between the exact solution and the discrete solution is of the same asymptotic order as the error of the original conforming method

fourth and fifth order scaled rungs kutta algorithms for treating dense output new runge kutta algorithms are developed which determine the solution of a system of ordinary differential equations at any point within a given integration step as well as at the end of each step

pointwise solution bounds for a class of singular diffusion problems in physiology a numerical technique for obtaining pointwise bounds for the solution of a class of nonlinear boundary value problems in physiology is presented simple analytic bounding functions are obtained using an integral representation for the solution the computations are performed in interval arithmetic thus obtaining lower and upper bounds simultaneously the oxygen diffusion problem in spherical cells and a nonlinear heat conduction model of the human head are presented as illustrative examples for these examples the present technique is computationally more efficient than the existing ones in that it yields sharper bounds with fewer integration steps

on the use of lagrange multipliers in domain decomposition for solving elliptic problems the primal hybrid method for solving second order elliptic equa tions is extended from finite element approximations to general bases varia tional techniques are used to show convergence of approximations to the so lution of the homogeneous dirichlet problem for selfadjoint equations error estimates are obtained and examples are given

numerical solution of troesch s problem by simple shooting method this paper describes a simple and efficient approach to the troesch s problem in this approach the hyperbolic nonlinear term in the equation is first converted into polynomial nonlinear terms by variable transformation and a simple shooting method is then used directly to solve this transformed problem the calculated results are in excellent agreement with those obtained by other analytical and numerical methods

hp submeshing via non conforming finite element methods abstract non conformity in the hp version can involve incompatibility in both the degrees and the meshes between adjoining subdomains in this paper we show how the mortar finite element method m0 and two new variants m1 m2 can be used to join together such incompatible hp sub discretizations our results show optimality of the resulting non conforming method for various h p and hp discretizations including the case of exponential hp convergence over geometric meshes we also present numerical results for the lagrange multiplier when the method is implemented via a mixed method three dimensional considerations suggest that our methods m1 m2 are easier to generalize to arbitrary meshes than m0

existence uniqueness and stability for a class of diffusion problems abstract singular boundary value problems are used to model diffusion of a specimen in a nonhomogeneous environment existence and uniqueness results are obtained by establishing a priori bounds for all solutions and then applying a shooting technique to appropriate initial value problems asymptotic stability results are obtained by constructing an appropriate lyapunov function and deriving differential inequalities to be satisfied by the parameters of the problem

a new high order numerical method for solving singular two point boundary value problems abstract recently goh et al 2012 28 proposed a numerical technique based on quartic b spline collocation for solving a class of singular boundary value problems sbvp with neumann and dirichlet boundary conditions bc this method is only fourth order accurate in this paper we propose an optimal numerical technique for solving a more general class of nonlinear sbvp subject to neumann and robin bc the method is based on high order perturbation of the problem under consideration the convergence of the proposed method is analyzed to demonstrate the applicability and efficiency of the method we consider four numerical examples three of which arise in various physical models in applied science and engineering a comparison with other available numerical solutions has been carried out to justify the advantage of the proposed technique numerical result reveals that the proposed method is sixth order convergent which in turn is two orders of magnitude larger than in goh et al 2012 28

a linearisation method for non linear singular boundary value problems this paper presents a novel numerical approach for the solution of a class of nonlinear singular boundary value problems arising in physiology the approach is based on a new application of the successive linearisation method slm three illustrative examples are presented to demonstrate the effectiveness of the proposed method the new approach is found to give accurate results comparable to results in the literature found using existing numerical methods

existence uniqueness results for a class of singular boundary value problems ii abstract in this paper we establish existence uniqueness of solution of a class of singular boundary value problem p x y x q x f x y for 0 x b and y 0 a 1 y b 1 y b 1 where p x satisfies i p x 0 in 0 b ii p x c 1 0 r and for some r b iii x p x p x is analytic in z z r and q x satisfies i q x 0 in 0 b ii q x l 1 0 b and for some r b iii x 2 q x p x is analytic in z z r with quite general conditions on f x y region for multiple solutions have also been determined

high order interpolants for solutionsof two point boundary value problems using mirk methods in this paper high order interpolants are presented for constructing continuous solutionsto a system of two point boundary value differential equations between widely spaced but accurate dependent variable values these interpolants are local and symmetric requiring data only within a single mesh interval and they require a small number of right hand side evaluations of the defining ode system to achieve the required order of accuracy internal derivative information in the mono implicit runge kutta formulae is exploited to reduce the number of additional right hand side evaluations necessary to define the interpolant to the required order of accuracy when the underlying ode system is second order very economical and accurate interpolants are found all of the interpolants are suitable for grid refinement algorithms in automatic adaptive two point boundary value packages such as twpbvp

a high order numerical method and its convergence for time fractional fourth order partial differential equations abstract this paper is concerned with design and analysis of a high order numerical approach based on a uniform mesh to approximate the solution of time fractional fourth order partial differential equations in this approach we first approximate the time fractional derivative appearing in the governing equation by means of caputo s definition and then construct a sextic b spline collocation method for solving the resulting equation it is proved that the present method is unconditionally stable convergence analysis of the method is discussed we prove that the method is 2 th order convergence with respect to time variable and fourth order convergence with respect to space variable three test problems are considered to demonstrate the applicability and efficiency of the new method it is shown that the rate of convergence predicted theoretically is the same as that obtained experimentally numerical results have been compared with those reported previously in literature it is shown that our method yields more accurate results when compared to the existing methods

on the existence of solutions of a class of singular nonlinear two point boundary value problems we examine the existence of solutions of the class of singular nonlinear two point boundary value problems y a x y ffi f x y 0 l we show that for every a l a unique solution of the singular two point boundary value problem exists provided u kl where u sup 3f ay and k i is the first positive zero of jta l 2 fk j z is bessel s function of the first kind of order s interestingly k i is a monotonically increasing function of a values of k i for some values of a are tabulated

on the convergence of a fourth order method for a class of singular boundary value problems in the present paper we extend the fourth order method developed by chawla et al m m chawla r subramanian h l sathi a fourth order method for a singular two point boundary value problem bit 28 1988 88 97 to a class of singular boundary value problems p x y p x f x y 0 0 is a non negative function the order of accuracy of the method is established under quite general conditions on f x y and is also verified by one example the oxygen diffusion problem in a spherical cell and a nonlinear heat conduction model of a human head are presented as illustrative examples for these examples the results are in good agreement with existing ones

some nonoverlapping domain decomposition methods the purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving second order elliptic problems in two and three dimensions the methods under scrutiny fall into two major categories the substructuring type methods and the neumann neumann type methods the basic framework used for analysis is the parallel subspace correction method or additive schwarz method and other technical tools include local global and global local techniques the analyses for both two and three dimensional cases are carried out simultaneously some internal relationships between various algorithms are observed and several new variants of the algorithms are also derived

a stable high order interpolation scheme for superconvergent data a local collocation scheme is developed that yields stable high order accurate interpolants of discrete data arising from the numerical solution of a differential equation it should prove to be especially attractive for applications where data are superconvergent e g spline collocation at gauss points for simplicity the formulas are initially developed for a scalar equation but generalizations are later given for systems numerical examples are shown that illustrate the stability even for the case of highly nonuniform meshes which have proven difficult in prior studies

spline finite difference methods for singular two point boundary value problems in this paper we discuss the construction of a spline function for a class of singular two point boundary value problemx x u f x u u 0 a u 1 b 0 1 or 1 2 the boundary conditions may also be of the formu 0 0 u 1 b three point finite difference methods using the above splines are obtained for the solution of the boundary value problem these methods are of second order and are illustrated by four numerical examples

cubic spline polynomial for non linear singular two point boundary value problems we present a cubic spline polynomial for the solution of non linear singular two point boundary value problems the quesilinearization technique is used to reduce the non linear problem to a sequence of linear problems the resulting sets of differential equations are modified at the singular point and are treated by using cubic spline for finding the numerical solution the method is tested on three physical model problems from the literature

nonlinear boundary value problems for shallow membrane caps ii abstract suppose a shallow membrane cap with an undeformed shape described in cylindrical coordinates by z c 1 r where 0 r 1 and 1 is subjected to a uniform vertical pressure p if the resulting deformed shape is radially symmetric then under certain assumptions the radial stress sr satisfies the ordinary differential equation r 2 s r 3rs r 2 r 2y 2 2 vr 2 s r r 2 8s r 2 for 0 0 if the boundary stress s is specified or s r 1 1 v s r 1 if the boundary displacement is specified here v 0 v 1 a radially symmetric solution sr r positive for 0 s 1 4 v or 1 v 1 4 v the solution is unique in the case 4 3 if is fairly large it may happen that sr r 0 as r 0 in all other cases sr r has a positive limit as r 0 rather detailed information on the behavior of solutions sr r is provided conditions are obtained which guarantee monotonicity of sr in any case sr has at most one critical point and is monotone in some neighborhood of r 0 a computational algorithm making use of the qualitative behavior of sr is discussed and some numerical results are included

b spline collocation methods and their convergence for a class of nonlinear derivative dependent singular boundary value problems abstract this paper is concerned with the construction and convergence analysis of two b spline collocation methods for a class of nonlinear derivative dependent singular boundary value problems ddsbvp the first method is based on uniform mesh while the second method is based on non uniform mesh for the second method we use a grading function to construct the non uniform grid we prove that the method based on uniform mesh is of second order accuracy and the method based on non uniform mesh is of fourth order accuracy three nonlinear examples with derivative dependent source functions are considered to verify the performance and theoretical rate of convergence of present methods moreover we consider some special cases of the problem under consideration in order to compare our methods with other existing methods it is shown that our second method based on cubic b spline basis functions has the same order of convergence as quartic b spline collocation method 1 moreover our methods yield more accurate results and are computationally attractive than the methods developed in 1 2 3 4 5 6 7 8 the proposed methods are applied on three real life problems the first problem describes the distribution of radial stress on a rotationally shallow membrane cap the second problem arises in the study of thermal explosion in cylindrical vessel and the third problem arises in astronomy

correction method and extrapolation method for singular two point boundary value problems abstract in this paper we first discuss the constructions of three point finite difference approximation and a spline approximation for a class of singular two point boundary value problems x x u f x u u 0 0 u 1 a 1 the asymptotic error expansions of the numerical solutions of these problems are obtained from these asymptotic error expansions we find that the finite difference solution and the spline approximation solution approximate the exact solution from two sides so we can obtain correct solution of high order accuracy richardson s extrapolation can also be done and the accuracy of numerical solution can be improved greatly we also present numerical examples which show the performance and efficiency of our methods

wavelet galerkin method for solving nonlinear singular boundary value problems arising in physiology abstract in this study wavelet galerkin method has been developed for solving nonlinear singular boundary value problems associated with physiology science the quasilinearization technique is applied to reduce the given nonlinear problem to a sequence of linear problems we modify the resulting set of differential equations at the singular point then treat this set of boundary value problems by using compactly supported cubic b spline wavelets and galerkin method these wavelets are used as testing and weighting functions the method is computationally attractive and applications are demonstrated through illustrative examples

a difference method for singular two point boundary value problems a difference method based on uniform mesh for solving singular two point boundary value problems of the form x ay f x y 0

uniform hp convergence results for the mortar finite element method the mortar finite element is an example of a non conforming method which can be used to decompose and re compose a domain into sub domains without requiring compatibility between the meshes on the separate components we obtain stability and convergence results for this method that are uniform in terms of both the degree and the mesh used without assuming quasiuniformity for the meshes our results establish that the method is optimal when non quasiuniform h or hp methods are used such methods are essential in practice for good rates of convergence when the interface passes through a corner of the domain we also give an error estimate for when the p version is used numerical results for hp and hp mortar fems show that these methods behave as well as conforming fems an hp extension theorem is also proved

a new algorithm for calculating two dimensional differential transform of nonlinear functions a new algorithm for calculating the two dimensional differential transform of nonlinear functions is developed in this paper this new technique is illustrated by studying suitable forms of nonlinearity three strongly nonlinear partial differential equations are then solved by differential transform method to demonstrate the validity and applicability of the proposed algorithm the present framework offers a computationally easier approach to compute the transformed function for all forms of nonlinearity this gives the technique much wider applicability

superconvergent interpolants for the collocation solution of boundary value ordinary differential equations a long standing open question associated with the use of collocation methods for boundary value ordinary differential equations is concerned with the development of a high order continuous solution approximation to augment the high order discrete solution approximation obtained at the mesh points which subdivide the problem interval it is well known that the use of collocation at gauss points leads to solution approximations at the mesh points for which the global error is o h2k where k is the number of collocation points used per subinterval and h is the subinterval size this discrete solution is said to be superconvergent the collocation solution also yields a c0 continuous solution approximation that has a global error of o hk 1 in this paper we show how to efficiently augment the superconvergent discrete collocation solution to obtain c1 continuous superconvergent interpolants whose global errors are o h2k the key ideas are to use the theoretical framework of continuous runge kutta schemes and to augment the collocation solution with inexpensive monoimplicit runge kutta stages specific schemes are derived for k 1 2 3 and 4 numerical results are provided to support the theoretical analysis

a difference scheme based on non uniform mesh for singular two point boundary value problems a difference scheme based on non uniform mesh for singular two point boundary value problems of the form y 1xy f x y 0 0

on a class of regular singular two point boundary value problems abstract existence of a unique solution for a class of regular singular two point boundary value problems p x y p x f x y 0 x b lim x 0 y x 0 y b b has been established wherep x satisfies i p x 0 on 0 b ii p x c1 0 r and for somer b iii xp x p x is analytic in z z xp x p x b 0 b 1 x b 0 1 and with quite general conditions onf x y these conditions onf x y are sharp which is seen through one example regions for multiple solutions have also been determined

the numerical solution of non linear singular boundary value problems arising in physiology a class of non linear singular ordinary differential equations is solved by a new method based on non polynomial cubic spline we use the quesilinearization technique to reduce the given non linear problem to a sequence of linear problems we modify the resulting set of differential equations at the singular point then treat this set of boundary value problems by using non polynomial cubic spline approximation the resulting system of algebraic equations is solved by using a tri diagonal solver computational results are provided to demonstrate the viability of the new method

a new finite difference method for a class of singular two point boundary value problems we present a new three point finite difference method based on uniform mesh for a class of singular two point boundary value problem x ay f x y y 0 a y 1 b 0 a 1 we show that the method based on uniform mesh provides o h 4 convergent approximations this method is illustrated by two numerical examples one is linear and another is non linear

on a class of weakly regular singular two point boundary value problems ii abstract existence of a unique solution of a class of weakly regular singular two point boundary value problems p x y p x f x y 0 x b lim x 0 y x 0 y b b has been established where p x satisfies i p x 0 on 0 b ii p x c 1 0 r and for some r b iii xp x p x is analytic in z z r with taylor expansion xp x p x b 0 b 1 x b 0 0 1 and with quite general conditions on f x y these conditions on f x y are sharp which is seen through one example regions for multiple solutions have also been determined

a second order spline finite difference method for singular two point boundary value problems a second order spline finite difference method based on a non uniform mesh for singular two point boundary value problems of the form x a x au f x u 0

finite difference methods and their convergence for a class of singular two point boundary value problems we discuss the construction of three point finite difference approximations and their convergence for the class of singular two point boundary value problems x y f x y y 0 a y 1 b 0 1 we first establish a certain identity based on general non uniform mesh from which various methods can be derived to obtain a method having order two for all 0 1 we investigate three possibilities by employing an appropriate non uniform mesh over 0 1 we obtain a methodm1 based on just one evaluation off for uniform mesh we obtain two methodsm2 andm3 each based on three evaluations off for 0 m1 andm2 both reduce to the classical second order method based on one evaluation off these three methods are investigated theiro h2 convergence established and illustrated by numerical examples

convergence analysis for derivative dependent fredholm hammerstein integral equations with green s kernel abstract in this article we consider a class of derivative dependent fredholm hammerstein integral equations i e the integral equation where the nonlinear function inside the integral sign is dependent on derivative and the kernel function is of green s type we propose the piecewise polynomial based galerkin and iterated galerkin methods to solve these type of derivative dependent fredholm hammerstein integral equations we discuss the convergence and error analysis of the proposed methods and also obtain the superconvergence results for iterated galerkin approximations some numerical results are given to illustrate this improvement