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

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

solutions of twelfth order boundary value problems using thirteen degree spline abstract numerical solutions of the twelfth order linear special case boundary value problems are obtained using thirteen degree spline 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 j comput appl math 78 1997 371 390 presented the solutions of twelfth 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 the algorithm developed approximates not only the solution but its higher order derivatives as well numerical illustrations are tabulated to demonstrate the practical usefulness of the method

the numerical solution of third order boundary value problems using quintic splines we present a fourth order method based on quintic splines for the solution of third order linear and non linear boundary value problems bvps of the form y f x y a

series expansion of the percolation probability for the directed square lattice by extrapolation from finite lattices the authors extend the known series for the percolation probability on the directed square lattice from eight terms to 41 analysing the series they obtain the estimates qc 0 355 299 or 0 000 001 beta 0 2764 or 0 0001 for the critical probability and the critical exponent from this together with scaling relations and previous results on the moments of the pair connectedness function the authors conjecture that beta may be exactly 199 720

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

septic spline solutions of sixth order boundary value problems septic spline is used for the numerical solution of the sixth order linear special case boundary value problem end conditions for the definition of septic spline are derived consistent with the sixth order boundary value problem the algorithm developed approximates the solution and their higher order derivatives the method has also been proved to be second order convergent three examples are considered for the numerical illustrations of the method developed the method developed in this paper is also compared with that developed in m el gamel j r cannon j latour a i zayed sinc galerkin method for solving linear sixth order boundary value problems mathematics of computation 73 247 2003 1325 1343 as well and is observed to be better

nonic spline solutions of eighth order boundary value problems nonic spline is used for the numerical solutions of the eighth order linear special case boundary value problems the end conditions are derived for the definition of spline the algorithm developed not only approximates the solutions but their higher order derivatives as well the method presented in this paper has also been proved to be second order convergent two examples compared with those considered by inc et al m inc d j evans an efficient approach to approximate solutions of eighth order boundary value problems int j comput math 81 6 2004 685 692 and siddiqi et al s s siddiqi e h twizell spline solution of linear eighth order boundary value problems comput methods appl mech eng 131 1996 309 325 show that the method developed in this paper is more efficient

parametric cubic spline solution of two point boundary value problems in this paper we use parametric cubic spline function to develop a numerical method which is fourth order for a specific choice of the parameter for computing smooth approximations to the solution for second order boundary value problems some numerical evidence is also included to demonstrate the superiority of our method

estimates of the site percolation probability exponents for some directed lattices series expansion analysis has been used to obtain estimates of the percolation probability exponent beta for directed site percolation on the square triangular simple cubic and body centred cubic lattices these estimates are consistent with those obtained by please 1977 for the corresponding bond problems and the results for site and bond problems on all lattices considered are summarised by beta 0 28 or 0 01 d 2 and beta 0 59 or 0 02 d 3

novel position space renormalization group for bond directed percolation in two dimensions abstract a new position space renormalization group approach is investigated for bond directed percolation in two dimensions the threshold value for the bond occupation probabilities is found to be p c 0 6443 correlation length exponents on time parallel and space transverse directions are found to be 1 719 and 1 076 respectively which are in very good agreement with the best known series expansion results

percolation as a fractal growth problem abstract we consider percolation in two dimensions as a fractal growth problem and apply to it the theory of fractal growth based on the fixed scale transformation approach developed for diffusion limited aggregation and the dielectric breakdown model this represents an important test for this new theoretical method based on an additional invariance property with respect to the renormalization group we compute the fractal dimension of the percolating cluster including terms up to third order the result is d 1 8830 for the square lattice and d 1 8650 for the triangular lattice these values are in excellent agreement with the universal exact result d 91 48 1 8958 and also show the potential of this new method for standard problems

spectral galerkin methods for the primary two point boundary value problem in modelling viscoelastic flows the essential character of the primary elliptic hyperbolic operator encountered in the modelling of viscoelastic flows is encaptured in a non linear fifth order two point boundary value problem in one dimension expansions in terms of beam functions and chebyshev polynomials are used to compute solutions to the primary two point boundary value problem within a spectral galerkin formulation an investigation of the performance of the methods with respect to the level of non linearity is carried out accurate results are obtained with chebyshev polynomials for high levels of non linearity whereas the behaviour of beam function expansions proves far more sensitive to the level of non linearity

on two dimensional directed percolation extended series expansions for the mean size and the first and second moments of the pair connectedness for both bond and site percolation on the directed square and triangular lattices have been obtained analysis based on differential approximants allows the critical percolation probabilities and exponents to be estimated and as a result the critical exponents are conjectured to be gamma 41 18 nu perpendicular to 79 72 and nu sub 26 15 scaling then gives beta 199 720 alpha 299 360 and delta 1839 199

real space renormalisation group for directed systems in arbitrary dimensions a real space renormalisation group approach to intrinsically anisotropic systems i e systems having two different correlation lengths is defined the problems of directed self avoiding walks dsaw and directed percolation in d dimensions are discussed in detail for dsaw one obtains the exact critical fugacity in all dimensions for directed percolation the percolation threshold pc obtained is very good in two dimensions asymptotically exact for large dimensions and quite good in between the parallel and perpendicular correlation length exponents nu sub and nu perpendicular to are computed for the two problems

real space renormalisation group for directed systems a new real space renormalisation group method is developed to treat systems with directionally dependent critical behaviour here two effective lengths s sub b and sperpendicular to b are defined for a given cell of linear size b the renormalised lattice is constructed from these effective lengths and is deformed from the original lattice it is the anisotropic rescaling of these effective lengths which give the anisotropic exponents nu sub and nu perpendicular to this method gives the exact result for the case of fully directed saws on a square lattice in the b to infinity limit for directed bond and directed site lattice animals the results obtained here are also in good agreement with the other known values

self organized critical forest fire model a forest fire model is introduced which contains a lightning probability f this leads to a self organized critical state in the limit f ensuremath rightarrow 0 provided that the time scales of tree growth and burning down of forest clusters are separated we derive scaling laws and calculate all critical exponents the values of the critical exponents are confirmed by computer simulations for a two dimensional system we show that the forest density in the critical state assumes its minimum possible value i e that energy dissipation is maximum

fixed scale transformation for ising and potts clusters the fractal dimension of ising and potts clusters are determined via the fixed scale transformation approach which exploits both the self similarity and the dynamical invariance of these systems at criticality the results are easily extended to droplets a discussion of the interrelationships between the present approach and renormalization group methods as well as glauber type dynamics is provided

solutions of tenth order boundary value problems using eleventh degree spline abstract numerical solutions of the tenth order linear special case boundary value problems are obtained using eleventh degree spline the end conditions consistent with the bvp are also derived siddiqi and twizell s s siddiqi e h twizell spline solutions of linear tenth order boundary value problems int j comput math 68 1998 345 362 presented the solutions of tenth order boundary value problems using tenth degree spline where some unexpected results for the solution and higher order derivatives were obtained near the boundaries of the interval no such unexpected situation is observed in this method near the boundaries of the interval and the results are better in the whole interval the algorithm developed approximates the solutions and their higher order derivatives numerical illustrations are tabulated to compare the errors with those considered by siddiqi and twizell s s siddiqi e h twizell spline solutions of linear tenth order boundary value problems int j comput math 68 1998 345 362 and the method is observed to be better

theory of laplacian fractals diffusion limited aggregation and dielectric breakdown model abstract we describe a new theoretical approach that clarifies the origin of fractal structures in irreversible growth models based on the laplace equation and a stochastic field this new theory provides a systematic method for the calculation of the fractal dimension d and of the multifractal spectrum of the growth probability a detailed application to the dielectric breakdown model and diffusion limited aggregation in two dimensions is presented our approach exploits the scale invariance of the laplace equation that implies that the structure is self similar both under growth and scale transformation this allows one to introduce a fixed scale transformation instead of coarse graining as in the renormalization group theory that defines a functional equation for the fixed point of the distribution of basic diagrams used in the coarse graining process for the calculation of the matrix elements of this transformation one has to consider an infinite but rapidly convergent number of processes that occurs outside a considered diagram

avalanches and correlations in driven interface depinning we study the critical behavior of a driven interface in a medium with random pinning forces by analyzing spatial and temporal correlations in a lattice model recently proposed by sneppen phys rev lett bf 69 3539 1992 the static and dynamic behavior of the model is related to the properties of directed percolation we show that due to the interplay of local and global growth rules the usual method of dynamical scaling has to be modified we separate the local from the global part of the dynamics by defining a train of causal growth events or avalanche which can be ascribed a well defined dynamical exponent z loc 1 zeta c simeq 1 63 where zeta c is the roughness exponent of the interface we observe that the avalanche size distribution obeys a power law decay with an exponent kappa simeq 1 25