an invariant imbedding method for computation of eigenlengths of singular two point boundary value problems in this article we have described an invariant imbedding method for calculating the smallest eigenlength of a singular tpbvp with the singularity at the origin the invariant imbedding yields a first order nonlinear equation called a riccati equation and also gives the initial conditions at the origin for this equation with the aid of theorem 8 in section 3 we numerically integrate the riccati equation to blowup which gives our computed eigenlength in closing we would like to comment on the numerical merits of the integration to blowup technique on the basis of the examples presented it appears that this technique combined with the available numerical integrators with variable step size is capable of producing accurate results the feature of a variable step size is essential as the value of z approaches the actual eigenlength however it is desirable to have a priori estimate or bounds of the eigenlength similar to those of boland and nelson 2 for the nonsingular case the singular system however presents difficulties due to the lack of sign conditions on the coefficient matrices in obtaining such bounds hopefully an investigation of the matrix r z will yield these results

an implicit fourth order compact finite difference scheme for one dimensional burgers equation abstract a fourth order compact finite difference method is proposed in this paper to solve one dimensional burgers equation the newly proposed method is based on the hopf cole transformation which transforms the original nonlinear burgers equation into a linear heat equation and transforms the dirichlet boundary condition into the robin boundary condition the linear heat equation is then solved by an implicit fourth order compact finite difference scheme a compact fourth order formula is also developed to approximate the robin boundary conditions while the initial condition for the heat equation is approximated using simpson s rule to maintain overall fourth order accuracy numerical experiments have been conducted to demonstrate the efficiency and high order accuracy of this method the numerical results also show that the method is unconditionally stable as there is no constraint on time step size

why pre tensioning stiffens cable systems it is proved that the potential energy of any pin bar assembly with totally tensioned members possesses strict minimum independently of the assembly topology geometry and magnitudes of member forces

on the efficient numerical solution of systems of second order boundary value problems an explicit reduction is given of the centered euler scheme for a first order two dimensional system producing an accurate difference approximation for a second order scalar equation all of the advantages of the centered euler formula are retained except for the condition number results of numerical experiments are presented

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

compact finite difference method for american option pricing a compact finite difference method is designed to obtain quick and accurate solutions to partial differential equation problems the problem of pricing an american option can be cast as a partial differential equation using the compact finite difference method this problem can be recast as an ordinary differential equation initial value problem the complicating factor for american options is the existence of an optimal exercise boundary which is jointly determined with the value of the option in this article we develop three ways of combining compact finite difference methods for american option price on a single asset with methods for dealing with this optimal exercise boundary compact finite difference method one uses the implicit condition that solutions of the transformed partial differential equation be nonnegative to detect the optimal exercise value this method is very fast and accurate even when the spatial step size h is large h 0 1 compact difference method two must solve an algebraic nonlinear equation obtained by pantazopoulos 1998 at every time step this method can obtain second order accuracy for space x and requires a moderate amount of time comparable with that required by the crank nicolson projected successive over relaxation method compact finite difference method three refines the free boundary value by a method developed by barone adesi and lugano the saga of the american put 2003 and this method can obtain high accuracy for space x the last two of these three methods are convergent moreover all the three methods work for both short term and long term options through comparison with existing popular methods by numerical experiments our work shows that compact finite difference methods provide an exciting new tool for american option pricing

compact finite difference method for the fractional diffusion equation high order compact finite difference scheme for solving one dimensional fractional diffusion equation is considered in this paper after approximating the second order derivative with respect to space by the compact finite difference we use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the local truncation error and discuss the stability using the fourier method then we prove that the compact finite difference scheme converges with the spatial accuracy of fourth order using matrix analysis numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

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

compact finite difference schemes of sixth order for the helmholtz equation new compact finite difference schemes of sixth order are derived for the three dimensional helmholtz equation du k 2u f convergence characteristics and accuracy are compared and a truncation error analysis is presented for a broad range of k values

a compact finite difference scheme for the fractional sub diffusion equations in this paper a compact finite difference scheme for the fractional sub diffusion equations is derived after a transformation of the original problem the l1 discretization is applied for the time fractional part and fourth order accuracy compact approximation for the second order space derivative the unique solvability of the difference solution is discussed the stability and convergence of the finite difference scheme in maximum norm are proved using the energy method where a new inner product is introduced for the theoretical analysis the technique is quite novel and different from previous analytical methods finally a numerical example is provided to show the effectiveness and accuracy of the method

invariant imbedding applied to homogeneous two point boundary value problems with a singularity of the first kind it is shown that previous work of elder can be used to extend the version of invariant imbedding due to m r scott to homogeneous vector differential systems having a singularity of the first kind the boundary conditions considered consist of existence finite at the singularity and specified values for some subset of the dependent variables at a second point the important special case of a second order equation is discussed in some detail computational considerations are discussed and numerical examples are presented

optimization based stability analysis of structures under unilateral constraints this paper discusses an optimization based technique for determining the stability of a given equilibrium point of the unilaterally constrained structural system which is subjected to the static load we deal with the three problems in mechanics sharing the common mathematical properties i structures containing no compression cables ii frictionless contacts and iii elastic plastic trusses with non negative hardening it is shown that the stability of a given equilibrium point of these structures can be determined by solving a maximization problem of a convex function over a convex set on the basis of the difference of convex functions optimization we propose an algorithm to solve the stability determination problem at each iteration of which a second order cone programming problem is to be solved the problems presented are solved for various structures to determine the stability of given equilibrium points copyright 2008 john wiley sons ltd

minimum principle of complementary energy of cable networks by using second order cone programming abstract the minimum principle of complementary energy is established for cable networks involving only stress components as variables in geometrically nonlinear elasticity it is rather amazing that the complementary energy always attains minimum value at the equilibrium state irrespective of the stability of cable networks contrary to the fact that only the stationary principles have been presented for elastic trusses and continua even in the case of stable equilibrium state in order to show the strong duality between the minimization problems of total potential energy and complementary energy the convex formulations of these problems are investigated which can be embedded into a primal dual pair of second order cone programming problems the existence and uniqueness of solution are also investigated for the minimization problem of complementary energy

ideal spaces of banach algebras the ideal space mbox id a of a banach algebra a is studied as a bitopological space mbox id a tau u tau n where tau u is the weakest topology for which all the norm functions i to vert a i vert with a in a and i in mbox id a are upper semi continuous and tau n is the de groot dual of tau u when a is separable tau n vee tau u is either a compact metrizable topology or it is neither hausdorff nor first countable taf algebras are shown to exhibit the first type of behaviour applications to banach bundles which motivate the study and to pi banach algebras are given

b spline method for a class of singular two point boundary value problems using optimal grid in this paper we present b spline method for solving singular two point boundary value problems for a certain differential equation having singular coefficients the given singular boundary value problem is modified at the point of singularity the modified problem is then solved by b spline in the case of non linear problems we first linearize the equation using quasilinearization technique and the resulting problem is solved by b spline error estimates are obtained which enable a deferred correction to be made optimal grid has been generated using the method some numerical examples have been included to demonstrate the efficiency of the method

numerical solution of singular boundary value problems with logarithmic singularities by pade approximation and collocation methods in this paper we present a method using quintic b spline and pade approximation for numerically solving singular boundary value problems for certain ordinary differential equation having singular coefficient we use the modified quintic b spline to approximate the first coefficients in the taylor expansion on which all the other coefficients depend moreover we introduce the superposed pade approximation the resulting perturbed initial value problem is then efficiently treated

comparison of second and fourth order discretizations for multigrid poisson solvers we combine a compact high order difference approximation with multigrid v cycle algorithm to solve the two dimensional poisson equation with dirichlet boundary conditions this scheme along with several different orderings of grid space and projection operators is compared with the five point formula to show the dramatic improvement in computed accuracy on serial and vector machines

spectral synthesis for banach algebras ii this paper continues the study of spectral synthesis and the topologies tau infty and tau r on the ideal space of a banach algebra concentrating particularly on the class of haagerup tensor products of c algebras for this class it is shown that spectral synthesis is equivalent to the hausdorffness of tau infty under a weak extra condition spectral synthesis is shown to be equivalent to the hausdorffness of tau r

a sixth order numerical method and its convergence for generalized black scholes pde abstract the main aim of this paper is to construct a new computational approach for the numerical solution of generalized black scholes equation in this approach the temporal variable is discretized using crank nicolson scheme and spatial variable is discretized using sextic b spline collocation method convergence analysis of the method is discussed the proposed method is proved to be stable and has second order convergence with respect to time variable and sixth order convergence with respect to space variable to illustrate the applicability and efficiency of the proposed method we consider some benchmark problems describing european call options numerical results verify the orders of convergence predicted by the analysis numerical results reveal that the present method provides better results as compared to some existing numerical methods based on b spline collocation approach

a numerical study for multiple solutions of a singular boundary value problem arising from laminar flow in a porous pipe with moving wall abstract this paper is concerned with multiple solutions of a singular nonlinear boundary value problem bvp on the interval 0 1 which arises in a study of the laminar flow in a porous pipe with an expanding or contracting wall for the singular nonlinear bvp the correct boundary conditions are derived to guarantee that its linearization has a unique smooth solution then a numerical technique is proposed to find all possible multiple solutions for the suction driven pipe flow with the expanding wall e g 2 we find a new solution numerically and classify it as a type vi solution the computed results agree well with what can be obtained by the bifurcation package auto in addition we also construct asymptotic solutions for a few cases of parameters which agree well with numerical solutions these serve as validations of our numerical results thus we believe that the numerical technique designed in the paper is reliable and may be further applied to solve a variety of nonlinear equations that arise from other flow problems

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

efficient mesh selection for collocation methods applied to singular bvps we describe a mesh selection strategy for the numerical solution of boundary value problems for singular ordinary differential equations this mesh adaptation procedure is implemented in our matlab code sbvp which is based on polynomial collocation we prove that under realistic assumptions our mesh selection strategy serves to approximately equidistribute the global error of the collocation solution thus enabling to reach prescribed tolerances efficiently moreover we demonstrate that this strategy yields a favorable performance of the code and compare its computational effort with other implementations of polynomial collocation

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 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

discrete invariant imbedding for singular boundary value problems with a regular singularity we describe a discrete invariant imbedding method for solving a two point boundary value problem in the interval 0 b for a linear second order ordinary differential equation with a singularity of the first kind at x 0 by employing the series expansion on 0 d where d is near the singularity we first replace it by a regular problem on some interval d b the discrete invariant imbedding method is then described to solve the problem over the reduced interval the stability analysis of the method is discussed some model problems are solved and the numerical results compared with those of other methods

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

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

numerical solution of singular boundary value problems by invariant imbedding abstract an invariant imbedding method is proposed for numerically solving singular boundary value problems with a regular singular point at one end of the interval the singular problem is first reduced to a regular boundary value problem by using series solution in the vicinity of the singular point to subtract out the singularity the regular boundary value problem is then solved by employing invariant imbedding method some numerical examples have been included to demonstrate the efficiency of the method

first order infinitesimal mechanisms abstract this paper discusses the analytical conditions under which a pin jointed assembly which has s independent states of self stress and m independent mechanisms tightens up when its mechanisms are excited a matrix algorithm is set up to distinguish between first order infinitesimal mechanisms which are associated with second order changes of bar length and higher order infinitesimal or finite mechanisms it is shown that in general this analysis requires the computation of s quadratic forms in m variables which can be easily computed from the states of self stress and mechanisms of the assembly if any linear combination of these quadratic forms is sign definite then the mechanisms are first order infinitesimal an efficient and general algorithm to investigate these quadratic forms is given the calculations required are illustrated for some simple examples many assemblies of practical relevance admit a single state of self stress s 1 and in this case the algorithm proposed is straightforward to implement this work is relevant to the analysis and design of pre stressed mechanisms such as cable systems and tensegrity frameworks

numerical pricing of options using high order compact finite difference schemes we consider high order compact hoc schemes for quasilinear parabolic partial differential equations to discretise the black scholes pde for the numerical pricing of european and american options we show that for the heat equation with smooth initial conditions the hoc schemes attain clear fourth order convergence but fail if non smooth payoff conditions are used to restore the fourth order convergence we use a grid stretching that concentrates grid nodes at the strike price for european options for an american option an efficient procedure is also described to compute the option price greeks and the optimal exercise curve comparisons with a fourth order non compact scheme are also done however fourth order convergence is not experienced with this strategy to improve the convergence rate for american options we discuss the use of a front fixing transformation with the hoc scheme we also show that the hoc scheme with grid stretching along the asset price dimension gives accurate numerical solutions for european options under stochastic volatility

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

cubic spline for solving singular two point boundary value problems in this paper we present another version of cubic spline method for numerically solving singular two point boundary value problems for certain ordinary differential equation having singular coefficients to remove the singularity we first use series solution in the vicinity of the singular point and the boundary condition at a point x d in the vicinity of the singularity is derived the resulting regular bvp is then efficiently treated by employing cubic spline for finding the numerical solution some examples have been included and comparison of the numerical results made with other methods

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

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

singular boundary value problems for odes this paper is concerned with the numerical solution of a system of ordinary differential equations odes y sy t f t y p on an interval 0 b subject to boundary conditions 0 g y 0 y b p the odes have a coefficient that is singular at t 0 but it is assumed that the boundary value problem bvp has a smooth solution some popular methods for bvps evaluate the odes at t 0 this paper deals with the practical issues of solving this class of singular bvps with such a method the bvp4c solver of matlab has been modified accordingly so that it can solve a class of singular bvps as effectively as it previously solved non singular bvps

a numerical method for singular two point boundary value problems via chebyshev economizition in this paper we present a numerical method for solving a two point boundary value problem in the interval 0 1 with regular singularity at x 0 by employing the chebyshev economizition on 0 d where d is near the singularity we first replace it by a regular problem on some interval d 1 the stable central difference method is then employed to solve the problem over the reduced interval some numerical results are presented to demonstrate the applicability of the method

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

design of software for odes we discuss the design of software that is easy to use for simple problems but still capable of solving complicated problems nowadays a design should accommodate related but fundamentally different tasks such as solving daes and ddes in addition to ivps and bvps for odes major issues are illustrated with the matlab ode suite and selected solvers written in fortran 90 and maple

higher order finite difference method for a class of singular boundary value problems in this paper a fourth order finite difference method for a class of singular boundary value problems is presented the original differential equation is modified at the singular point the fourth order finite difference method is then employed to solve the boundary value problem some model problems are solved and the numerical results are compared with exact solution

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

reformulation of boundary value problems into standard form boundary value problems in ode s arising in various applications are frequently not in the standard form required by the currently existing software however many problems can be converted to such a form thus enabling the practitioner to take advantage of the availability and reliability of this general purpose software here various conversion devices are surveyed

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

spectral synthesis and topologies on ideal spaces for banach algebras this paper continues the study of spectral synthesis and the topologies and r on the ideal space of a banach algebra concentrating on the class of banach algebras and in particular on l1 group algebras it is shown that if a group g is a finite extension of an abelian group then r is hausdorff on the ideal space of l1 g if and only if l1 g has spectral synthesis which in turn is equivalent to g being compact the result is applied to nilpotent groups fd groups and moore groups an example is given of a non compact non abelian group g for which l1 g has spectral synthesis it is also shown that if g is a non discrete group then r is not hausdorff on the ideal lattice of the fourier algebra a g

a sixth order compact finite difference scheme to the numerical solutions of burgers equation abstract a numerical solution of the one dimensional burgers equation is obtained using a sixth order compact finite difference method to achieve this a tridiagonal sixth order compact finite difference scheme in space and a low storage third order total variation diminishing runge kutta scheme in time have been combined the scheme is implemented to solve two test problems with known exact solutions comparisons of the computed results with exact solutions showed that the method is capable of achieving high accuracy and efficiency with minimal computational effort the present results are also seen to be more accurate than some available results given in the literature