witten s lectures on crumpling abstract crumpling is a distortion brought about by a strong compression of a surface in which energy condenses in extremely small regions rather than being stored uniformly scaling laws describe how much energy goes into how little space as crumpling of a sheet proceeds a network of ridges and vertices develops the precise nature of these geometrical singularities depends on the stretching and bending characteristics it is shown that for two dimensional surfaces in three dimensional space energy mainly condenses in stretching ridges the scaling arguments are put on a firm basis by using the mathematical description of an elastic membrane through the von karman equations and the universal properties of ridges are discussed from this theory one can understand that crumpling confers strength finally it is sketched how energy condensation and its scaling laws depend on the dimensions of the confined sheet and embedding space

solving boundary value problems integral and integro differential equations using gegenbauer integration matrices we introduce a hybrid gegenbauer ultraspherical integration method hgim for solving boundary value problems bvps integral and integro differential equations the proposed approach recasts the original problems into their integral formulations which are then discretized into linear systems of algebraic equations using gegenbauer integration matrices gims the resulting linear systems are well conditioned and can be easily solved using standard linear system solvers a study on the error bounds of the proposed method is presented and the spectral convergence is proven for two point bvps tpbvps comparisons with other competitive methods in the recent literature are included the proposed method results in an efficient algorithm and spectral accuracy is verified using eight test examples addressing the aforementioned classes of problems the proposed method can be applied on a broad range of mathematical problems while producing highly accurate results the developed numerical scheme provides a viable alternative to other solution methods when high order approximations are required using only a relatively small number of solution nodes

study on a new nonlinear parametric excitation equation stability and bifurcation anstract the parameter stability and global bifurcations of a strong nonlinear system with parametric excitation and external excitations are investigated in detail using the method of multiple scales the nonlinear system is transformed to the averaged equation the parameter stability of solution in the case of principal parametric resonance is developed based on the averaged equation the continuation algorithm is utilized to analyze the detailed bifurcation scenario as the parameter f0 is varied the results indicate that there exist two limit points and neutral saddle points finally a series of branching points were obtained by changing the parameters f0 and

a smooth approximation for the solution of a fourth order boundary value problem based on nonpolynomial splines we develop a method of second order for the continuous approximation of the solution of a two point boundary value problem involving a fourth order differential equation via mixed spline functions consisting of a polynomial part of degree three and a trigonometric part it will be shown that the angular frequency k of the trigonometric part can be used to raise the order of accuracy of the obtained scheme two numerical examples are given to sustain our autonomous parametric spline method and to compare it with a quintic polynomial spline method of second order

the use of cubic splines in the numerical solution of a system of second order boundary value problems abstract we use uniform cubic polynomial splines to develop some consistency relations which are then used to develop a numerical method for computing smooth approximations to the solution and its derivatives for a system of second order boundary value problems associated with obstacle unilateral and contact problems we show that the present method gives approximations which are better than those produced by other collocation finite difference and spline methods

solution of sixth order boundary value problems using non polynomial spline technique abstract non polynomial spline is used for the numerical solutions of the 12th order linear special case boundary value problems the end conditions are derived for the definition of spline siddiqi and twizell s s siddiqi e h twizell spline solutions of linear twelfth order boundary value problems int j comput math 78 1998 345 362 presented the solutions of 12th order boundary value problems using 12th degree spline where some unexpected results for the solution and its derivatives were obtained near the boundaries of the interval no such situation is observed in this method near the boundaries of the interval and the results are better in the whole interval out of three two examples compared with that considered by siddiqi and twizell 1998 show that the method developed in the paper is more efficient

numerical treatment of sth order boundary value problems by solving second kind fredholm integral equations a nystrom method is proposed for solving fredholm integral equations equivalent to boundary value problems of order s with complete differential equations the stability and the convergence of the proposed procedure are proved some numerical examples are provided in order to illustrate the accuracy of the method and to compare the procedure with some other ones given in the literature

forced periodic vibration of unsymmetric piecewise linear systems the forced steady response of a single degree of freedom system involving a large non linearity represented by unsymmetric piecewise linear stiffness is determined by a harmonic balance newton raphson method with the application of the fast fourier transformation fft algorithm all possible subharmonic harmonic and superharmonic responses are sought the responses of two systems involving subharmonic and superharmonic dominant motions are calculated by the newly developed method and compared with previously published findings the results obtained by using this method reveal the details of the response of the systems more efficiently than previous methods

adapted numerical methods for advection reaction diffusion problems generating periodic wavefronts abstract the paper presents an adapted numerical integration for advection reaction diffusion problems the numerical scheme exploiting the a priori knowledge of the qualitative behaviour of the solution gains advantages in terms of efficiency and accuracy with respect to classic schemes already known in literature the adaptation is here carried out through the so called trigonometrical fitting technique for the discretization in space giving rise to a system of odes whose vector field contains both stiff and non stiff terms due to this mixed nature of the vector field an implicit explicit imex method is here employed for the integration in time based on the first order forward backward euler method the coefficients of the method here introduced rely on unknown parameters which have to be properly estimated in this work such an estimate is performed by minimizing the leading term of the local truncation error the effectiveness of this problem oriented approach is shown through a rigorous theoretical analysis and some numerical experiments

high order accuracy nonpolynomial spline solutions for 2μth order two point boundary value problems sixth order accurate method based on quintic nonpolynomial spline function for the numerical solution of 2 th order two point bvps where is a positive integer is presented by transforming the problem into a system of second order two point bvps the nonpolynomial spline function is used to drive some consistency relations for computing approximation to the solution of this problem the proposed approach gives better approximations than existing polynomial spline and finite difference methods and has a lower computational cost convergence analysis of the proposed method is discussed numerical examples are included to illustrate the practical usefulness of our method

five diagonal finite difference methods based on mixed type interpolation for a certain fourth order two point boundary value problem abstract modified finite difference methods based on mixed type interpolation are derived and applied for fourth order linear ordinary differential equations subject to a special kind of boundary conditions in this paper we present and analyse methods of orders 2 4 and 6 several numerical experiments are performed to illustrate the theoretical background

boundary layer analysis of the ridge singularity in a thin plate large deformations of thin elastic plates and shells present a formidable problem in continuum mechanics which is generally intractable except by numerical methods conventional approaches break down in the limit of small plate thickness due to appearance of discontinuities in the solution which require boundary layer treatment we examine a simple case of a plate bent by forces exerted along its boundary so as to create a sharp crease in the limit of infinitely small thickness we find a separable boundary layer solution of the von karman plate equations which is valid along the ridge line we confirm a scaling argument t a witten and hao li europhys lett 23 51 1993 that asserts that the ridge possesses a characteristic radius of curvature r given by the thickness of the sheet h and the length of the ridge x it viz r sim h 1 3 x 2 3 the elastic energy of the ridge scales as e sim kappa x h 1 3 where kappa is the bending modulus of the sheet we determine the dependence of these quantities on the dihedral angle of the ridge pi 2 alpha for all angles r sim alpha 4 3 and e sim alpha 7 3 the framework developed in this paper is suitable for determination of other properties of ridges such as their interaction or behavior under various types of loading we expect these results to have broad importance in describing forced crumpling of thin sheets

solution of the system of fourth order boundary value problems using non polynomial spline technique non polynomial spline is used for the numerical solutions of a system of fourth order boundary value problems associated with obstacle unilateral and contact problems in addition to the boundary conditions corresponding to the second derivatives the boundary conditions corresponding to the first derivatives are also considered two examples one for each case of boundary conditions are considered for the numerical illustration of the method developed it has been observed that the method developed in this paper is better than the other collocation finite difference and spline methods

resonant overlap phenomena in stochastic layers of non linear hamiltonian systems with periodical excitations an accurate standard map approach instead of the approximate standard map approach is developed for the analytical prediction of the onset of a new resonant overlap forming a new resonant separatrix web in the stochastic layer this accurate approach applies to the twin well duffing oscillator for illustration of the resonant overlap phenomena in stochastic layers the excitation strengths are obtained analytically for the onset of the specified resonant separatrix webs in the stochastic layer this accurate approach gives a very good prediction compared to numerical results for the lower order resonant overlap when excitation strength is very weak the accurate and approximate standard map methods are in good agreement and the accurate one is applicable to non linear systems with strong excitations however the further improvement should be completed for a more accurate analytical prediction of the onset of a new resonant overlap in stochastic layers of non linear hamiltonian systems with periodical excitations

chaotic motion of van der pol mathieu duffing system under bounded noise parametric excitation abstract the chaotic behavior of van der pol mathieu duffing oscillator under bounded noise is investigated by using random melnikov technique a mean square criterion is used to detect the necessary conditions for chaotic motion of this stochastic system the results show that the threshold of bounded noise amplitude for the onset of chaos in this system increases as the intensity of the noise in frequency increases which is further verified by the maximal lyapunov exponents of the system the effect of bounded noise on poincare map is also investigated in addition the numerical simulation of the maximal lyapunov exponents

extended exponentially fitted interpolation formulas for oscillatory functions we consider exponentially fitted interpolation formulas that use not only the pointwise values of an w dependent function f but also of its derivatives up to the nth order at the two boundary nodes of a closed interval a b the function f is of the form f x f 1 x cos wx f 2 x sin wx email protected a b where the functions f 1 and f 2 are smooth on the interval as n increases numerical results about the errors for the formulas are illustrated and compared some properties of the coefficients of the formulas at the two nodes are presented and extended to the case with three nodes we investigate if the properties of the coefficients can be generalised to the cases with more than three nodes finally some other numerical results regarding the errors are given

a nystrom method for solving 2sth order boundary value problems a nystrom method is proposed for solving fredholm integral equations equivalent to special boundary value problems of order 2s the stability and the convergence of the proposed procedure is proved some numerical examples are provided in order to illustrate the accuracy of the method

compensation of gaussian curvature in developable cones is local we use the angular deficit scheme v borrelli f cazals and j m morvan comput aided geom des 20 319 2003 to determine the distribution of gaussian curvature in developable cones d cones e cerda s chaieb f melo and l mahadevan nature london 401 46 1999 numerically these d cones are formed by pushing a thin elastic sheet into a circular container negative gaussian curvatures are identified at the rim where the sheet touches the container around the rim there are two narrow bands with positive gaussian curvatures the integral of the negative gaussian curvature near the rim is almost completely compensated by that of the two adjacent bands this suggests that the gauss bonnet theorem which constrains the integral of gaussian curvature globally does not explain the spontaneous curvature cancellation phenomenon t liang and t a witten phys rev e 73 046604 2006 the locality of the compensation seems to increase for decreasing d cone thickness the angular deficit scheme also provides a way to confirm the curvature cancellation phenomenon

period one motions of a mechanical oscillator with periodically time varying piecewise nonlinear stiffness the dynamic behavior of a piecewise nonlinear mechanical oscillator with parametric and external excitations is investigated the viscously damped single degree of freedom oscillator is subjected to a periodically time varying piecewise nonlinear restoring function defined by a clearance surrounded by continuously nonlinear quadratic and cubic regions typical applications represented by this oscillator are highlighted a multi term harmonic balance formulation is used in conjunction with discrete fourier transforms and a parametric continuation scheme to determine steady state period 1 motions of the system due to both parametric and external excitations the accuracy of the analytical solutions is demonstrated by comparing them to direct numerical integration solutions floquet theory is applied to determine the stability of the steady state harmonic balance solutions at the end detailed parametric studies are presented to quantify the combined influence of clearance quadratic and cubic nonlinearities within reasonable ranges of all other system parameters a comparison between time varying and time invariant systems is also provided to demonstrate the influence of the parametric and external excitations on a piecewise nonlinear system

high order closed newton cotes trigonometrically fitted formulae for the numerical solution of the schrodinger equation abstract in this paper we investigate the connection between closed newton cotes formulae trigonometrically fitted methods symplectic integrators and efficient integration of the schrodinger equation the study of multistep symplectic integrators is very poor although in the last decades several one step symplectic integrators have been produced based on symplectic geometry see the relevant literature and the references here in this paper we study the closed newton cotes formulae and we write them as symplectic multilayer structures based on the closed newton cotes formulae we also develop trigonometrically fitted symplectic methods an error analysis for the one dimensional schrodinger equation of the new developed methods and a comparison with previous developed methods is also given we apply the new symplectic schemes to the well known radial schrodinger equation in order to investigate the efficiency of the proposed method to these type of problems

chaotic motion in the resonant separatrix bands of a mathieu duffing oscillator with a twin well potential the m 1 resonant bands in the left and right potential wells are skew symmetric and the 2m 1 resonant bands of the large orbit motion are symmetric the analytical conditions for the onset and destruction of a resonant band are developed through the incremental energy approach the numerical predictions of such onset and destruction are also completed by the energy increment spectrum method the sub resonance interaction occurs for strong excitations which needs to be further investigated these results are applicable to the small and large orbit motions of post buckled structure under a parametric excitation

confined developable elastic surfaces cylinders cones and the elastica we consider two of the simplest problems associated with the packing of a naturally flat thin elastic sheet both problems involve packing the sheet into a hollow cylinder the first considers the partial contact of a cylindrically curved sheet with a cylindrical surface while the second considers the partial contact of a conically curved sheet with the edge of a cylindrical surface in each case we solve the free boundary problems to determine the shape response and stability of the confined surfaces in particular we show that an exact description of both the cylindrical and conical structures is given by solutions of the elastica equation allowing us to present a unified description of a large class of elastic developable surfaces this includes what is possibly the simplest example of strain localization occurring at a point and forming one of the constituent elements of a crumpled elastic sheet

transition curves for the quasi periodic mathieu equation in this work we investigate an extension of mathieu s equation the quasi periodic qp mathieu equation given by ddot psi delta eps cos t cos omega t psi 0 for small eps and irrational omega of interest is the generation of stability diagrams that identify the points or regions in the delta omega parameter plane for fixed eps for which all solutions of the qp mathieu equation are bounded numerical integration is employed to produce approximations to the true stability diagrams both directly and through contour plots oflyapunov exponents in addition we derive approximate analytic expressions for transition curves using two distinct techniques 1 a regular perturbation method under which transition curves delta delta omega eps are each expanded in powers of eps and 2 the method of harmonic balance utilizing hill s determinants both analytic methods deliver results in good agreement with those generated numerically in the sense that pre

nonlinear analysis of a geared rotor system supported by fluid film journal bearings abstract this paper presents a novel approach for modeling and analyzing a geared rotor bearing system including nonlinear forces in the gear set and the supporting fluid film journal bearings the rotordynamics system model has five degrees of freedom that define the transverse displacements of the shaft gear centerlines and the relative displacement of the gear tooth contact point the journal bearing nonlinear forces are obtained via a solution of reynolds equation for lubricant film pressure utilizing the finite element method co existing steady state autonomous and non autonomous responses are obtained in an accurate and computationally efficient manner utilizing the multiple shooting and continuation algorithms this yields the full manifolds of the multiple bifurcation system chaos is identified with maximum lyapunov exponents frequency spectra poincare attractors etc the results reveal a dependence of the gear set contact conditions and system nonlinear response characteristics i e jump co existing responses subharmonic resonances and chaos on the choice of journal bearing parameters the results also show that hopf bifurcations which occur along with oil whirl in a journal bearing system can be attenuated by increasing the gear torque

maximal lyapunov exponent and almost sure stability for stochastic mathieu duffing systems abstract the principal resonance of a mathieu duffing oscillator to random excitation is investigated the method of multiple scales is used to determine the equations of modulation of amplitude and phase the effects of damping detuning cubic term and magnitudes of random excitation are analyzed the explicit asymptotic formulas for the maximal lyapunov exponent is obtained the almost sure stability or instability of the stochastic mathieu duffing system depends on the sign of the maximal lyapunov exponent in the last part of the work the numerical results are obtained

a family of trigonometrically fitted partitioned runge kutta symplectic methods abstract we are presenting a family of trigonometrically fitted partitioned runge kutta symplectic methods of fourth order with six stages the solution of the one dimensional time independent schrodinger equation is considered by trigonometrically fitted symplectic integrators the schrodinger equation is first transformed into a hamiltonian canonical equation numerical results are obtained for the one dimensional harmonic oscillator and the exponential potential

non polynomial spline approach to the solution of a system of third order boundary value problems we use a quartic spline equivalent nonpolynomial spline functions to develop a numerical method for computing approximations to the solution of a system of third order boundary value problems associated with obstacle unilateral and contact problems we show that the present method gives approximations which are better than those produced by other collocation finite difference and spline methods a numerical example is given to illustrate the applicability and efficiency of the new method

confining thin elastic sheets and folding paper crumpling a sheet of paper leads to the formation of complex folding patterns over several length scales this can be understood on the basis of the interplay of a nonconvex elastic energy which favors locally isometric deformations and a small singular perturbation which penalizes high curvature based on three dimensional nonlinear elasticity and by using a combination of explicit constructions and general results from differential geometry we prove that in agreement with previous heuristic results in the physics literature the total energy per unit thickness of such folding patterns scales at most as the thickness of the sheet to the power 5 3 for the case of a single fold we also obtain a corresponding lower bound

subharmonic resonances of a mechanical oscillator with periodically time varying piecewise nonlinear stiffness abstract the subharmonic period 1 motions of a piecewise nonlinear pn mechanical oscillator having parametric and external excitations are investigated the system is formed by a viscously damped single degree of freedom oscillator subjected to a periodically time varying pn stiffness defined by a clearance surrounded by continuous forms of nonlinearity a multiterm harmonic balance formulation in conjunction with discrete fourier transforms is used to determine steady state period motions of the system near the parametric instability regions the accuracy of analytical solutions is verified through a comparison with direct numerical integration results a parametric study is also presented to demonstrate the combined influence of a clearance and of cubic nonlinearities on period motions within typical ranges of system and excitation parameters

analytical predictions of chaos in a non linear rod the resonant and chaotic conditions for non dampened non linear planar rods are developed through the chirikov criterion and the subharmonic bifurcation conditions for weakly dampened non linear planar rods are also presented through the melnikov method the analytical conditions are based on a simply supported geometrically non linear planar rod model with a specific single mode response but these conditions are applicable to geometrically non linear planar rods with different supports chaos and transient motion from chaos to periodic motion in the non linear rod are simulated through the approximate rod model and they are illustrated through the poincare mapping section

the optimal exponentially fitted numerov method for solving two point boundary value problems second order boundary value problems are solved by means of exponentially fitted numerov methods these methods which depend on a parameter can be constructed following a six step flow chart of ixaru and vanden berghe l gr ixaru g vanden berghe exponential fitting kluwer academic publishers dordrecht 2004 special attention is paid to the expression of the error term of such methods an algorithm concerning the choice of the best suited method and its parameter is discussed several numerical examples are given to sustain the theory

quartic spline method for solving fourth order obstacle boundary value problems in this paper we use uniform quartic polynomial splines to develop a new method which is used for computing approximations to the solution and its first second as well as third derivatives for a system of fourth order boundary value problems associated with obstacle unilateral and contact problems it is shown that the present method is of order two and gives approximations which are better than those produced by other collocation and finite difference methods numerical examples are presented to illustrate the applicability of the new method

cubic splines method for solving fourth order obstacle problems in this paper we develop a new cubic spline method for computing approximate solution of a system of fourth order boundary value problems associated with obstacle unilateral and contact problems it is shown that the present method is of order two and gives approximations which are better than those produced by some other collocation finite difference and spline methods numerical examples are presented to illustrate the applicability of the new method

exponentially fitted methods for solving time fractional nonlinear reaction diffusion equation abstract in this article we will develop a new numerical scheme with the second order in time and a class of fourth order or sixth order in space based on the exponential fitting techniques to approximate the nonlinear time fractional reaction diffusion equation with fixed order and distributed order derivatives these techniques depend on a parameter which will be used to annihilate the error and increase the order of accuracy the proposed methods are proved to be unconditionally stable and convergent by fourier analysis also the theoretical results and the effectiveness of the numerical scheme are confirmed by numerical test problems and a comparison with other methods is presented

quadratic non polynomial spline approach to the solution of a system of second order boundary value problems a quadratic non polynomial spline functions based method is developed to find approximations solution to a system of second order boundary value problems associated with obstacle unilateral and contact problems the present approach has less computational cost and gives better approximations than those produced by other collocation finite difference and spline methods convergence analysis of the method is discussed a numerical example is given to illustrate practical usefulness of the new method

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

scaling properties of randomly folded plastic sheets we study the scaling properties of randomly folded aluminum sheets of different thicknesses h and widths l we found that the fractal dimension d 2 30 0 01 and the force scaling exponent delta 0 21 0 02 are independent of the sheet thickness and close to those obtained in numerical simulations with a coarse grained model of triangulated self avoiding surfaces with bending and stretching rigidity so our findings suggest that finite bending rigidity and self avoidance play the predominant roles in the scaling behavior of randomly folded plastic sheets

exponentially fitted imex methods for advection diffusion problems the paper is devoted to the numerical solution of advection diffusion problems of boussinesq type by means of adapted numerical methods the adaptation occurs at two levels along space by suitably semidiscretizing the spatial derivatives through finite differences based on exponential fitting along time integration through an adapted imex method based on exponential fitting itself stability analysis is provided and numerical examples showing the effectiveness of the approach also in comparison with the classical one are given

numerical solution of reaction diffusion systems of λ ω type by trigonometrically fitted methods the numerical solution of reaction diffusion equations of type which are known to possess a one parameter family of periodic plane wave solutions is object of this paper due to the periodic character of such solutions a special purpose numerical integration is here proposed based on adapted finite differences the adaptation occurs at the level of the problem by a suitable spatial semi discretization based on trigonometrically fitted finite differences numerical experiments confirming the effectiveness of the approach are given

rim curvature anomaly in thin conical sheets revisited this paper revisits one of the puzzling behaviors in a developable cone d cone the shape obtained by pushing a thin sheet into a circular container of radius r by a distance the mean curvature was reported to vanish at the rim where the d cone is supported we investigate the ratio of the two principal curvatures versus sheet thickness h over a wider dynamic range than was used previously holding r and fixed instead of tending toward 1 as suggested by previous work the ratio scales as h r 1 3 thus the mean curvature does not vanish for very thin sheets as previously claimed moreover we find that the normalized rim profile of radial curvature in a d cone is identical to that in a c cone which is made by pushing a regular cone into a circular container in both c cones and d cones the ratio of the principal curvatures at the rim scales as r h 5 2 f yr 2 where f is the pushing force and y is the young s modulus scaling arguments and analytical solutions confirm the numerical results

a note on application of cubic splines to two point boundary value problems abstract two methods are proposed to solve a linear two point boundary value problem without eliminating the m i s from the system of equations

spline solutions of linear eighth order boundary value problems abstract linear eighth order boundary value problems special case are solved using polynomial splines of degree six the spline function values at the midknots of the interpolation interval and the corresponding values of the even order derivatives are related through consistency relations the algorithm developed approximates the solution and their higher order derivatives of differential equations four numerical illustrations are given to show the practical usefulness of the algorithm developed it is observed that this algorithm is second order convergent

exponentially fitted methods applied to fourth order boundary value problems fourth order boundary value problems are solved by means of exponentially fitted methods of different orders these methods which depend on a parameter can be constructed following a six step flow chart of ixaru and vanden berghe special attention is paid to the expression for the error term and to the choice of the parameter in order to make the error as small as possible some numerical examples are given to illustrate the practical implementation issues of these methods

a smooth approximation based on exponential spline solutions for nonlinear fourth order two point boundary value problems methods of order two four and six based on exponential spline functions consisting of a polynomial part of degree three and an exponential part is developed to find approximation of linear and nonlinear fourth order two point boundary value problems it is shown that the free parameter k of the exponential part can be used to raise the order of accuracy of the new scheme convergence analysis of these methods is shown along with numerical examples each for linear and nonlinear are included to illustrate the practical usefulness of our methods

polynomial and nonpolynomial spline approaches to the numerical solution of second order boundary value problems in this paper quadratic and cubic polynomial and nonpolynomial spline functions based methods are presented to find approximate solutions to second order boundary value problems using these spline functions we drive a few consistency relations which to be used for computing approximations to the solution for second order boundary value problems the present approaches have less computational cost convergence analysis of these methods is discussed two numerical examples are included to illustrate the practical usefulness of the proposed methods

truncation errors in exponential fitting for oscillatory problems a generalization of peano s kernel theorem due to ghizzetti and ossicini quadrature formulae birkhauser basel switzerland 1970 provides expressions in the form of integrals for the truncation errors in a variety of exponential fitting formulae for oscillatory problems in some circumstances this leads to an expression analogous to the lagrange form of remainder more generally the error can be expressed as a sum of two terms of lagrange type our examples include formulae for quadrature and numerical differentiation and linear multistep methods for ordinary differential equations two families of exponential fitting quadrature formulae are investigated one with evenly spaced abscissas and the other based on the philosophy of gaussian quadrature in particular the integral representation can be used to determine the asymptotic rate of decay of the error with increasing frequency for a class of oscillatory integrands

a class of methods based on non polynomial spline functions for the solution of a special fourth order boundary value problems with engineering applications we use a quintic non polynomial spline functions to develop a numerical method for computing approximations to the solution of a system of fourth order boundary value problems associated with plate deflection theory we show that the present family of methods gives better approximations and generalize all the existing finite difference and spline functions based methods up to order six convergence of the methods is shown through standard convergence analysis numerical exampls are given to illustrate the applicability and efficiency of the new method

stability analysis of the non linear mathieu equation abstract the non linear mathieu equation is analyzed within the framework of the method of normal forms analytical conditions for explosive instability are obtained and expressions for the period as well as the amplitude of the stable response are derived

convergence of numerical solution of a fourth order boundary value problem quintic spline procedure for the solution of a fourth order boundary value problem is developed spline relations are derived boundary condition formula of orders o h 6 and o h 8 are formulated the convergence of prescribed method is proved the matrix associated with the system of linear equations that arises is not required to be monotone the only requirement is that f x be positive numerical illustration are given

numerical integration based on mixed type interpolation for a fourth order boundary value problem abstract modified finite difference methods based on mixed type interpolation are derived and applied for fourth order linear ordinary differential equations subject to a special kind of boundary conditions in this paper we present and analyse methods of orders 2 4 and 6 several numerical experiments are performed to sustain our technique

the use of quartic splines in the numerical solution of a fourth order boundary value problem abstract this paper derives via uniform quartic polynomial splines a few new consistency relations connecting the quartic spline function values at midknots and the corresponding values of the second and fourth derivatives respectively it is shown how these relations may be used in an algorithm for computing approximations to the solution and its higher derivatives for a linear fourth order boundary value problem two numerical illustrations are also included to demonstrate the practical usefulness of our algorithm

effect of ridge ridge interactions in crumpled thin sheets we study whether and how the energy scaling based on the single ridge approximation is revised in an actual crumpled sheet namely in the presence of ridge ridge interactions molecular dynamics simulation is employed for this purpose in order to improve the data quality modifications are introduced to the common protocol as crumpling proceeds we find that the average storing energy changes from being proportional to one third of the ridge length to a linear relation while the ratio of bending and stretching energies decreases from 5 to 2 the discrepancy between previous simulations and experiments on the material dependence for the power law exponent is resolved we further determine the average ridge length to scale as 1 d 1 3 the ridge number as d 2 3 and the average storing energy per unit ridge length as d 0 881 where d denotes the volume density of the crumpled ball these results are accompanied by experimental proofs and are consistent with mean field predictions finally we extend the existent simulations to the high pressure region and verify the existence of a scaling relation that is more general than the familiar power law at covering the whole density range

b spline interpolation compared with finite difference finite element and finite volume methods which applied to two point boundary value problems abstract this paper considers the b spline interpolation and compares this method with finite difference finite element and finite volume methods which applied to the two point boundary value problem d d x p x d u d x f x a x b u a u b 0