sharp error bounds for the derivatives of lidstone spline interpolation ii abstract in this paper we shall derive explicit error estimates in l norm between a given function f x pc n a b 4 n 6 and its quintic lidstone spline interpolate the results obtained are then used to establish precise error bounds for the approximated and biquintic lidstone spline interpolates we also include applications to integral equations and boundary value problems as well as sufficient numerical examples which dwell upon the sharpness of the obtained results

on existence of positive solutions of sturm liouville boundary value problems for a nonlinear singular differential system abstract by employing the fixed point theorem of cone expansion and compression of norm type we investigate the existence of positive solutions of sturm liouville boundary value problems for a nonlinear singular differential system some well known results in the literature are generalized and improved an example is presented to illustrate the application of our main result

shift in the velocity of a front due to a cutoff we consider the effect of a small cutoff ensuremath varepsilon on the velocity of a traveling wave in one dimension simulations done over more than ten orders of magnitude as well as a simple theoretical argument indicate that the effect of the cutoff ensuremath varepsilon is to select a single velocity that converges when ensuremath varepsilon ensuremath rightarrow 0 to the one predicted by the marginal stability argument for small ensuremath varepsilon the shift in velocity has the form k mathrm ln ensuremath varepsilon ensuremath 2 and our prediction for the constant k agrees very well with the results of our simulations a very similar logarithmic shift appears in more complicated situations in particular in finite size effects of some microscopic stochastic systems our theoretical approach can also be extended to give a simple way of deriving the shift in position due to initial conditions in the fisher kolmogorov or similar equations

monotone convergence of iterative methods for right focal point boundary value problems abstract we provide sufficient conditions so that the picard iterative scheme for the right focal point boundary value problems converges monotonically an example which dwells upon the usefulness of this technique is also included

numerical methods for the solution of special and general sixth order boundary value problems with applications to benard layer eigenvalue problems a family of numerical methods is developed for the solution of special nonlinear sixth order boundary value problems methods with second fourth sixth and eighth order convergence are contained in the family the problem is also solved by writing the sixth order differential equation as a system of three second order differential equations a family of second and fourth order convergent methods is then used to obtain the solution a second order convergent method is discussed for the numerical solution of general nonlinear sixth order boundary value problems this method with modifications where necessary is applied to the sixth order eigenvalue problems associated with the onset of instability in a benard layer numerical results are compared with asymptotic estimates appearing in the literature

symmetric simple exclusion process with free boundaries we consider the one dimensional symmetric simple exclusion process with additional births and deaths restricted to a subset of configurations where there is a leftmost hole and a rightmost particle at a fixed rate birth of particles occur at the position of the leftmost hole and at the same rate independently the rightmost particle dies we prove convergence to a hydrodynamic limit and discuss its relation with a free boundary problem

speed and fluctuations of n particle branching brownian motion with spatial selection we consider branching brownian motion on the real line with the following selection mechanism every time the number of particles exceeds a large given number n only the n right most particles are kept and the others killed after rescaling time by log 3n we show that the properly recentred position of the lceil alpha n rceil th particle from the right alpha in 0 1 converges in law to an explicitly given spectrally positive levy process this behaviour has been predicted to hold for a large class of models falling into the universality class of the fkpp equation with weak multiplicative noise brunet et al in phys rev e 73 5 056126 2006 and is proven here for the first time for such a model

a fourth order compact finite difference method for nonlinear higher order multi point boundary value problems a fourth order compact finite difference method is proposed for a class of nonlinear 2nth order multi point boundary value problems the multi point boundary condition under consideration includes various commonly discussed boundary conditions such as the three or four point boundary condition n 2 point boundary condition and 2 n m point boundary condition the existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear term the convergence and the fourth order accuracy of the method are proved an efficient monotone iterative algorithm is developed for solving the resulting nonlinear finite difference systems various sufficient conditions for the construction of upper and lower solutions are obtained some applications and numerical results are given to demonstrate the high efficiency and advantages of this new approach

higher order lidstone boundary value problems for elliptic partial differential equations the aim of this paper is to show the existence and uniqueness of a solution for a class of 2nth order elliptic lidstone boundary value problems where the nonlinear functions depend on the higher order derivatives sufficient conditions are given for the existence and uniqueness of a solution it is also shown that there exist two sequences which converge monotonically from above and below respectively to the unique solution the approach to the problem is by the method of upper and lower solutions together with monotone iterative technique for nonquasimonotone functions all the results are directly applicable to 2nth order two point lidstone boundary value problems

piecewise complementary lidstone interpolation and error inequalities the purpose of this paper is to develop piecewise complementary lidstone interpolation in one and two variables and establish explicit error bounds for the derivatives in l and l 2 norms

weakly hyperbolic equations with non analytic coefficients and lower order terms in this paper we consider weakly hyperbolic equations of higher orders in arbitrary dimensions with time dependent coefficients and lower order terms we prove the gevrey well posedness of the cauchy problem under c k regularity of coefficients of the principal part and natural levi conditions on lower order terms which may be only continuous in the case of analytic coefficients in the principal part we establish the c infty well posedness the proofs are based on using the quasi symmetriser for the corresponding companion system and inductions on the order of equation and on the frequency regions the main novelty compared to the existing literature is the possibility to include lower order terms to the equation which have been untreatable until now in these problems as well as considering any space dimensions we also give results on the ultradistributional and distributional well posedness of the problem and we look at new effects for equations with discontinuous lower order terms

well posedness of hyperbolic systems with multiplicities and smooth coefficients we study hyperbolic systems with multiplicities and smooth coefficients in the case of non analytic smooth coefficients we prove well posedness in any gevrey class and when the coefficients are analytic we prove c infty well posedness the proof is based on a transformation to block sylvester form introduced by d ancona and spagnolo boll umi 8 1b 169 185 1998 which increases the system size but does not change the eigenvalues this reduction introduces lower order terms for which appropriate levi type conditions are found these translate then into conditions on the original coefficient matrix this paper can be considered as a generalisation of garetto and ruzhansky math ann 357 2 401 440 2013 where weakly hyperbolic higher order equations with lower order terms were considered

triple positive solutions and dependence on higher order derivatives abstract in this paper we consider the lidstone boundary value problem y 2 m t f y t y 2 j t y 2 m 1 t 0 t 1 y 2 i 0 0 y 2 i 1 0 i m 1 where 1 m f 0 growth conditions are imposed on f and inequalities involving an associated green s function are employed which enable us to apply the leggett williams fixed point theorem to cones in ordered banach spaces this in turn yields the existence of at least three positive symmetric concave solutions the emphasis here is that f depends on higher order derivatives

hydrodynamic limits for one dimensional particle systems with moving boundaries we analyze a new class of one dimensional interacting particle systems featuring random boundaries with a random motion that is coupled to the local particle density we show that the hydrodynamic limiting behavior in these systems corresponds to the solution of an appropriate

explicit error bounds for the derivatives of piecewise lidstone interpolation abstract the purpose of this paper is to provide explicit error bounds for the derivatives of piecewise lidstone interpolation in l and l2 norms the results obtained are best possible sharp and supplement those established in earlier papers

effect of selection on ancestry an exactly soluble case and its phenomenological generalization we consider a family of models describing the evolution under selection of a population whose dynamics can be related to the propagation of noisy traveling waves for one particular model that we shall call the exponential model the properties of the traveling wave front can be calculated exactly as well as the statistics of the genealogy of the population one striking result is that for this particular model the genealogical trees have the same statistics as the trees of replicas in the parisi mean field theory of spin glasses we also find that in the exponential model the coalescence times along these trees grow like the logarithm of the population size a phenomenological picture of the propagation of wave fronts that we introduced in a previous work as well as our numerical data suggest that these statistics remain valid for a larger class of models while the coalescence times grow like the cube of the logarithm of the population size

polynomial approximation of cm functions by means of boundary values and applications a survey we collect classical and more recent results on polynomial approximation of sufficiently regular real functions defined in bounded closed intervals by means of boundary values only the problem is considered from the point of view of the existence of explicit formulas interpolation to boundary data bounds for the remainder and convergence of the polynomial series applications to some problems of numerical analysis are pointed out such as nonlinear equations numerical differentiation and integration formulas special associated differential boundary value problems some polynomial expansions for smooth enough functions defined in rectangles or in triangles of r 2 as well as in cuboids or in tetrahedrons in r 3 and their applications are also discussed

last passage percolation and traveling fronts we consider a system of n particles with a stochastic dynamics introduced by brunet and derrida phys rev e 70 016106 2004 the particles can be interpreted as last passage times in directed percolation on 1 n of mean field type the particles remain grouped and move like a traveling front subject to discretization and driven by a random noise as n increases we obtain estimates for the speed of the front and its profile for different laws of the driving noise as shown in brunet and derrida phys rev e 70 016106 2004 the model with gumbel distributed jumps has a simple structure we establish that the scaling limit is a levy process in this case we study other jump distributions we prove a result showing that the limit for large n is stable under small perturbations of the gumbel in the opposite case of bounded jumps a completely different behavior is found where finite size corrections are extremely small

exact solution and precise asymptotics of a fisher kpp type front the present work concerns a version of the fisher kpp equation where the nonlinear term is replaced by a saturation mechanism yielding a free boundary problem with mixed conditions following an idea proposed in brunetderrida 2015 we show that the laplace transform of the initial condition is directly related to some functional of the front position mu t we then obtain precise asymptotics of the front position by means of singularity analysis in particular we recover the so called ebert and van saarloos correction ebertvansaarloos 2000 we obtain an additional term of order log t t in this expansion and we give precise conditions on the initial condition for those terms to be present

explicit polynomial expansions of regular real functions by means of even order bernoulli polynomials and boundary values for a function f c2n 1 a b an explicit polynomial interpolant in a and in the even derivatives up to the order 2n 1 at the end points of the interval is derived explicit cauchy and peano representations and bounds for the error are given and the analysis of the remainder term allows to find sufficient conditions on f so that the polynomial approximant converges to f these results are applied to derive a new summation formula with application to rectangular quadrature rule the polynomial interpolant is related to a fairly interesting boundary value problem for odes we will exhibit solutions for this problem in some special situations

lidstone polynomials and boundary value problems abstract we define lidstone polynomials provide their explicit representations give their relations with bernoulli polynomials obtain best possible error estimates in lidstone interpolation and establish several best possible sharp inequalities which compare favourably with the known results next we use these results to study even order differential equations together with lidstone boundary conditions

third order homogeneous weakly hyperbolic equations with nonanalytic coefficients abstract we consider the cauchy problem for homogeneous linear third order weakly hyperbolic equations with time depending coefficients we study the relation between the regularity of the coefficients and the gevrey class in which the cauchy problem is well posed

upper and lower solution methods for fully nonlinear boundary value problems abstract sufficient conditions are given for the existence of a solution of a fourth order nonlinear boundary value problem with nonlinear boundary conditions the conditions assume the existence of a strong upper solution lower solution pair a concept that is defined in the paper the differential equation has nonlinear dependence on all lower order derivatives of the unknown in particular appropriate nagumo conditions are obtained and employed

eigenvalues and the one dimensional p laplacian we consider the boundary value problem p u f t u 0 with p 1 t 0 1 u 0 u 1 0 and with 0 the value of is chosen so that the boundary value problem has a positive solution in addition we derive an explicit interval for such that for any in this interval the existence of a positive solution to the boundary value problem is guaranteed in addition the existence of two positive solutions for in an appropriate interval is also discussed

branching brownian motion seen from its tip it has been conjectured since the work of lalley and sellke ann probab 15 1052 1061 1987 that branching brownian motion seen from its tip e g from its rightmost particle converges to an invariant point process very recently it emerged that this can be proved in several different ways see e g brunet and derrida a branching random walk seen from the tip 2010 poissonian statistics in the extremal process of branching brownian motion 2010 arguin et al the extremal process of branching brownian motion 2011 the structure of this extremal point process turns out to be a poisson point process with exponential intensity in which each atom has been decorated by an independent copy of an auxiliary point process the main goal of the present work is to give a complete description of the limit object via an explicit construction of this decoration point process another proof and description has been obtained independently by arguin et al the extremal process of branching brownian motion 2011

monotone iterative technique and positive solutions of lidstone boundary value problems the purpose of this paper is to investigate the existence and iteration of n symmetric positive solutions for lidstone boundary value problem 1 nw 2 n t f t w t 0

a new representation for the error function for the hermite interpolation and sharper pointwise and uniform error bounds for the derivatives let a a a r 2 2 be the given points and x e c a a n 2 2 be a given function satisfying the conditions a a j l j i 0 1 k j 1 2 r 1 1 where j kj r n let p r be the hermite interpolation polynomial of degree n 1 satisfying 1 1 for the error function e r x f p t everyone knows the integral representation in terms of peano s kernel green s function however the explicit representation of the green s function is unwieldy l 21 and hence except in some simple cases it cannot be used to obtain the bounds for the derivatives of e t the situation further worsens because the derivatives of the green s function change sign in the interval at a in fact even for the case n 6 r 2 k k 2 the computation is quite lengthy 3 the purpose of this paper is to give a new representation of the error function e t and use it to obtain pointwise as well as uniform error bounds for je k t 0 5 k 5 n 1 in terms of max ix t in

a note on weakly hyperbolic equations with analytic principal part in this note we show how to include low order terms in the c well posedness results for weakly hyperbolic equations with analytic time dependent coefficients this is achieved by doing a different reduction to a system from the previously used one we find the levi conditions such that the c well posedness continues to hold

discrete polynomial interpolation green s functions maximum principles error bounds and boundary value problems abstract we construct discrete interpolating polynomials provide explicit representations of discrete green s functions give several identities and inequalities for these green s functions use the explicit forms of the interpolating polynomials and that of green s functions to establish several maximum principles further we obtain error bounds in discrete polynomial interpolation and use them to study existence and uniqueness of the discrete boundary value problems these bounds are also used to provide sufficient conditions for the convergence of the picard s method the approximate picard s method quasilinearization and the approximate quasilinearization the monotone convergence of the picard s iterative method is also analysed

numerical methods for unilateral problems abstract the theory of variational inequalities besides being elegant and synthetic also provides a unified general framework for applying the numerical methods for solving many unrelated free boundary value problems in this paper we describe numerical experience on the use of variational inequalities and pade approximants to obtain approximate solutions to a class of unilateral boundary value problems of elasticity like those describing the equilibrium configuration of an elastic beam stretched over an elastic obstacle in the case of a known obstacle these problems can be alternatively formulated as nonlinear boundary value problems without constraints for which the technique of pade approximants can be successfully employed the variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution

eigenvalues of a general class of boundary value problem with derivative dependent nonlinearity we consider a general class of boundary value problem bvp comprising the differential equation y m t f t y t y t y t y q t t 0 1 where 1 q m 1 and 0 together with multi point boundary conditions y 0 y 0 y 0 y q 1 0 0 a i y q t j y q 1 t j y m 1 t j 0 j r 0 1 i m q where 0 t 0 t 1 t r 1 particular cases of this general bvp include the well known abel gontscharoff focal m p sturm liouville and complementary lidstone bvps it is noted that bvps with derivative dependent nonlinear terms are less investigated in the literature due to technical difficulty in this paper a new technique is developed to characterize the eigenvalues so that the bvp has a positive solution explicit eigenvalue intervals are also established we include several well known examples in the literature to illustrate the results obtained

internal dla and the stefan problem generalized internal diffusion limited aggregation is a stochastic growth model on the lattice in which a finite number of sites act as poisson sources of particles which then perform symmetric random walks with an attractive zero range interaction until they reach the first site which has been visited by fewer than alpha particles at which point they stop sites on which particles are frozen constitute the occupied set we prove that in appropriate regimes the particle density has a hydrodynamic limit which is the one phase stefan problem this is then used to study the asymptotic behavior of the occupied set in two dimensions when the walks are independent with one source at the origin and alpha 1 we obtain in particular that the occupied set is asymptotically a disc of radius k sqrt t where k is the solution of exp k 2 4 pi k 2 settling a conjecture of lawler bramson and griffeath

on the bivariate shepard lidstone operators we propose a new combination of the bivariate shepard operators coman and trimbitas 2001 2 by the three point lidstone polynomials introduced in costabile and dell accio 2005 7 the new combination inherits both degree of exactness and lidstone interpolation conditions at each node which characterize the interpolation polynomial these new operators find application to the scattered data interpolation problem when supplementary second order derivative data are given kraaijpoel and van leeuwen 2010 13 numerical comparison with other well known combinations is presented

hyperbolic equations with non analytic coefficients we prove that the cauchy problem for a hyperbolic homogeneous equation with mathcal c infty coefficients depending on time is well posed in every gevrey class although in general it is not well posed in mathcal c infty provided the characteristic roots satisfy the condition lambda i t 2 lambda j t 2 leq m lambda i t lambda j t 2 quad i neq j

singular lidstone boundary value problem with given maximal values for solutions abstract the singular problem 1 nx 2n f t x x 2n 2 x 2j 0 x 2j t 0 0 j n 1 max x t 0 t t a depending on the parameter is considered here the positive caratheodory function f may be singular at the zero value of all its phase variables the paper presents conditions which guarantee that for any a 0 there exists a 0 such that the above problem with a has a solution x ac2n 1 0 t which is positive on 0 t the proofs are based on the regularization and sequential techniques and use the leray schauder degree and vitali s convergence theorem

driven tracer particle in one dimensional symmetric simple exclusion consider an infinite system of particles evolving in a one dimensional lattice according to symmetric random walks with hard core interaction we investigate the behavior of a tagged particle under the action of an external constant driving force we prove that the diffusively rescaled position of the test particle ex e 2t t 0 converges in probability as e 0 to a deterministic function v t the function v depends on the initial distribution of the random environment through a non linear parabolic equation this law of large numbers for the position of the tracer particle is deduced from the hydrodynamical limit of an inhomogeneous one dimensional symmetric zero range process with an asymmetry at the origin an einstein relation is satisfied asymptotically when the external force is small

monotone convergence of iterative methods for n p and p n boundary value problems abstract we provide sufficient conditions so that the picard iterative scheme for n p and p n boundary value problems converges monotonically an example which illustrates the usefulness of this technique is also included

polynomial approximation to a non local boundary value problem in this paper we construct a special class of polynomials which converge uniformly to the solution of a non local boundary value problem nbvp the use of this special class is justified by the physics of the model which is described by this nbvp this nbvp has been studied by palamides et al 2009 in 2 where the existence of solutions is established

brunet derrida particle systems free boundary problems and wiener hopf equations we consider a branching selection system in r with n particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased keeping the number of particles constant we show that as n the empirical measure process associated to the system converges in distribution to a deterministic measure valued process whose densities solve a free boundary integrodifferential equation we also show that this equation has a unique traveling wave solution traveling at speed c or no such solution depending on whether c a or c a where a is the asymptotic speed of the branching random walk obtained by ignoring the removal of the leftmost particles in our process the traveling wave solutions correspond to solutions of wiener hopf equations

existence of positive solutions of lidstone boundary value problems abstract we show that there exists at least one positive solution for lidstone boundary value problem 1 n u 2 n t f t u t u t u 2 n 1 t 0 t 1 u 2 i 0 u 2 i 1 0 i 0 1 n 1 under some suitable conditions our proof is based upon global bifurcation techniques

brunet derrida behavior of branching selection particle systems on the line we consider a class of branching selection particle systems on mathbb r similar to the one considered by e brunet and b derrida in their 1997 paper shift in the velocity of a front due to a cutoff based on numerical simulations and heuristic arguments brunet and derrida showed that as the population size n of the particle system goes to infinity the asymptotic velocity of the system converges to a limiting value at the unexpectedly slow rate log n 2 in this paper we give a rigorous mathematical proof of this fact for the class of particle systems we consider the proof makes use of ideas and results by r pemantle and by n gantert y hu and z shi and relies on a comparison of the particle system with a family of n independent branching random walks killed below a linear space time barrier

large deviations for the branching brownian motion in presence of selection or coalescence the large deviation function has been known for a long time in the literature for the displacement of the rightmost particle in a branching random walk brw or in a branching brownian motion bbm more recently a number of generalizations of the bbm and of the brw have been considered where selection or coalescence mechanisms tend to limit the exponential growth of the number of particles here we try to estimate the large deviation function of the position of the rightmost particle for several such generalizations the l bbm the n bbm and the coalescing branching random walk cbrw which is closely related to the noisy fkpp equation our approach allows us to obtain only upper bounds on these large deviation functions one noticeable feature of our results is their non analytic dependence on the parameters such as the coalescence rate in the cbrw

homogeneous weakly hyperbolic equations with time dependent analytic coefficients abstract this paper concerns the cauchy problem for homogeneous weakly hyperbolic equations with time depending analytic coefficients we give a sufficient condition for the c well posedness which is also necessary if the space dimension is equal to one the main point of the paper consists in expressing our condition only in terms of the coefficients of the operator without needing to know the behavior of the characteristic roots this is made possible by using the so called standard symmetrizer of a companion hyperbolic matrix

on hyperbolic equations and systems with non regular time dependent coefficients in this paper we study higher order weakly hyperbolic equations with time dependent non regular coefficients the non regularity here means less than holder namely bounded coefficients as for second order equations in 14 we prove that such equations admit a very weak solution adapted to the type of solutions that exist for regular coefficients the main idea in the construction of a very weak solution is the regularisation of the coefficients via convolution with a mollifier and a qualitative analysis of the corresponding family of classical solutions depending on the regularising parameter classical solutions are recovered as limit of very weak solutions finally by using a reduction to block sylvester form we conclude that any first order hyperbolic system with non regular coefficients is solvable in the very weak sense

boundary value problems occurring in plate deflection theory abstract the paper deals with several iterative methods for solving some boundary value problems occurring in plate deflection theory making it possible to avoid the lengthy problem preparation which is required to solve these problems by finite difference methods

general lidstone problems multiplicity and symmetry of solutions abstract for the 2mth order lidstone boundary value problem y 2 m t f y t y t y 2 i t y 2 m 1 t t 0 1 y 2 i 0 y 2 i 1 0 0 i m 1 where 1 mf r m 0 thinsp is continuous growth conditions are imposed on f which yield the existence of at least three symmetric positive solutions this generalizes earlier papers which have applied avery s generalization of the leggett williams theorem to lidstone problems we then prove the analogous result for difference equations

finite difference methods for twelfth order boundary value problems the effect of rotation acting on a layer of fluid heated from below is similar to the effect of a magnetic field acting under the same conditions they both inhibit the onset of instability and they both elongate the cells which appear at marginal stability however it must not be assumed that acting together rotation and magnetic field reinforce each other in fact they have conflicting tendencies when acting together the effect of rotation and magnetic field together leads to a twelfh order eigenvalue problem chandrasekhar 1961 which will be addressed in a future paper experience in solving high order boundary value problems has shown that considerable insight may be obtained by solving the special problem first moreover beside the mathematical interest the computational aspects of the special problem need to be considered to this end finite difference methods of order two and global extrapolation on two grids are proposed to solve the following special nonlinear twelfth order boundary value problem w xii x f x w a x b w 2i a a2i w 2i b b2i i 0 1 5 a simple example is carried out to illustrate the results given by the methods

the genealogy of branching brownian motion with absorption we consider a system of particles which perform branching brownian motion with negative drift and are killed upon reaching zero in the near critical regime where the total population stays roughly constant with approximately n particles we show that the characteristic time scale for the evolution of this population is of order logn 3 in the sense that when time is measured in these units the scaled number of particles converges to a variant of neveu s continuous state branching process furthermore the genealogy of the particles is then governed by a coalescent process known as the bolthausen sznitman coalescent this validates the non rigorous predictions by brunet derrida muller and munier for a closely related model

existence and uniqueness of positive solutions for a higher order boundary value problem the goal of this paper is to study the existence and uniqueness of positive solutions for the 2nth order boundary value problem 1 nu 2 n f t u u 1 n 1u 2 n 2 a 0u 2 i 0 email protected 0u 2 i 1 0 0 i 0 1 n 1 a 1u 2 i 1 email protected 1u 2 i 1 1 0 i 0 1 n 1 the krein rutman theorem and the krasnoselskii zabreiko fixed point theorem are the main tools that have been used to develop our work

wavelet like bases for thin wire integral equations in electromagnetics in this paper wavelets are used in solving by the method of moments a modified version of the thin wire electric field integral equation in frequency domain the time domain electromagnetic quantities are obtained by using the inverse discrete fast fourier transform the retarded scalar electric and vector magnetic potentials are employed in order to obtain the integral formulation the discretized model generated by applying the direct method of moments via point matching procedure results in a linear system with a dense matrix which have to be solved for each frequency of the fourier spectrum of the time domain impressed source therefore orthogonal wavelet like basis transform is used to sparsify the moment matrix in particular dyadic and m band wavelet transforms have been adopted so generating different sparse matrix structures this leads to an efficient solution in solving the resulting sparse matrix equation moreover a wavelet preconditioner is used to accelerate the convergence rate of the iterative solver employed these numerical features are used in analyzing the transient behavior of a lightning protection system in particular the transient performance of the earth termination system of a lightning protection system or of the earth electrode of an electric power substation during its operation is focused the numerical results obtained by running a complex structure are discussed and the features of the used method are underlined

upper and lower estimates for positive solutions of the higher order lidstone boundary value problem we consider the higher order lidstone boundary value problem new upper and lower estimates for positive solutions of the problem are obtained a discussion of the sharpness of the estimates is included