numerical simulation of three dimensional telegraphic equation using cubic b spline differential quadrature method abstract this paper employs a differential quadrature scheme that can be used for solving linear and nonlinear partial differential equations in higher dimensions differential quadrature method with modified cubic b spline basis functions is implemented to solve three dimensional hyperbolic equations b spline functions are employed to discretize the space variable and their derivatives the weighting coefficients are obtained by semi explicit algorithm the partial differential equation results into a system of first order ordinary differential equations odes the obtained system of odes has been solved by employing a fourth stage runge kutta method efficiency and reliability of the method has been established with five linear test problems and one nonlinear test problem obtained numerical solutions are found to be better as compared to those available in the literature simple implementation less complexity and computational inexpensiveness are some of the main advantages of the scheme further the scheme gives approximations not only at the knots but also at all the interior points in the domain under consideration the scheme is found to be providing convergent solutions and handles different cases high order time discretization using ssp rk methods guarantee stability with respect to a given norm and a proper constraint on time step matrix method has been used for stability analysis in space and it is found to be unconditionally stable the scheme can be used effectively to handle higher dimensional pdes

non linear vibration of coupled duffing oscillators by an improved incremental harmonic balance method abstract the incremental harmonic balance method is improved by i reversing the order of increment and linearization to find the bifurcated solution branches ii using a fast fourier transform to compute the solution increments and the floquet matrix and iii finding the secondary path after a bifurcation by an improved arc length method a system of two duffing oscillators linked with a linear stiffness is taken as a numerical example the qualitative change of solution due to the change of the linking stiffness is studied in detail

design and numerical analysis of a logarithmic scheme for nonlinear fractional diffusion reaction equations abstract in this work we consider a parabolic partial differential equation with fractional diffusion that generalizes the well known fisher s and hodgkin huxley equations the spatial fractional derivatives are understood in the sense of riesz and initial boundary conditions on a closed and bounded interval are considered here the mathematical model is presented in an equivalent form and a finite difference discretization based on fractional order centered differences is proposed the scheme is the first explicit logarithmic model proposed in the literature to solve fractional diffusion reaction equations we establish rigorously the capability of the technique to preserve the positivity and the boundedness of the methodology moreover we propose conditions under which the monotonicity of the numerical model is also preserved the consistency the stability and the convergence of the scheme are also proved mathematically and some a priori bounds for the numerical solutions are proposed we provide some numerical simulations in order to confirm that the method is capable of preserving the positivity and the boundedness of the approximations and a numerical study of the convergence of the technique is carried out confirming thus the analytical results

a deterministic model for the distribution of the stopping time in a stochastic equation and its numerical solution in this work we consider a stochastic differential equation that generalizes the well known paris equation from fracture of materials the model describes the propagation of cracks on solids and it includes a deterministic summand and a stochastic component in terms of a brownian motion the use of ito s stochastic integral gives an equivalent stochastic integral equation that is further generalized here we note that the probability distribution of the stopping time of the general model satisfies a deterministic diffusion advection partial differential equation for which the solution is known only in a reduced number of particular cases motivated by these analytical results we develop a fast finite difference method to approximate the distribution of the stopping time the method is an explicit exponential like technique that preserves the main features of a probability distribution namely the non negativity the boundedness from above by 1 as well as the spatial monotonicity moreover the method is a monotone technique that is also capable of preserving the temporal monotonicity of the approximations these properties of the proposed methodology are thoroughly established in the present manuscript a continuity condition of the numerical solutions in terms of the initial conditions and the temporal computational parameter is established also together with a limiting property of the methodology when the free parameter tends to infinity for comparison purposes we are providing an implicit and stable discretization of the mathematical model which has a second order of convergence but for which conditions that guarantee the positivity the boundedness and the monotonicity of approximations are not available the numerical simulations obtained with implementations of our techniques show that the explicit method is an efficient scheme that preserves the characteristics of interest non negativity boundedness from above by 1 and monotonicity and that the numerical approximations are in good agreement with the known exact solutions

free vibration analysis of circular plates with variable thickness by the generalized differential quadrature rule the generalized differential quadrature rule proposed recently by these authors is applied here to the free vibration analyses of solid circular plates with radially varying thickness and elastic restraints the thickness can vary radially in any given continuous form and it varies exponentially and linearly in this work two regularity conditions corresponding to the plate center are expressed in explicit formulae since they have been inexactly or even wrongly expressed in the literature such errors are pointed out in the application of the differential quadrature techniques and in the expression of the regularity conditions at the plate center numerical results are presented for a number of plates illustrating the versatility and accuracy of the approach

numerical solutions of coupled kleingordonzakharov equations by quintic b spline differential quadrature method numerical solutions of the coupled kleingordonzakharov equations are obtained by using quintic b spline based differential quadrature method a rungekutta fourth method is used for time integration stability of the scheme is studied using matrix stability analysis the accuracy and efficiency of the presented method is shown by conducting some numerical experiments on test problems which includes the motion of single soliton and interactions of two solitons the numerical results are found in good agreement with the exact solutions

vector bundles of rank 2 and linear systems on algebraic surfaces let s be a smooth complex algebraic surface and let l be a divisor on s it is well known that the linear system il ksi where ks is the canonical divisor plays an important role in understanding the geometry of s several authors see for example 8 11 studied various properties of this system our paper treats the same subject but the point of view is different from the works mentioned above the philosophy underlying our approach is almost classical points on s more generally effective 0 cycles in special position with respect to il ksi see definition below contain information about the geometry of s this point of view was recently revived in 5 also 10 in fact this paper contains the technique of going from the effective 0 cycle z in special position with respect to il ksi to the geometry of s for the sake of completeness we give a definition and state a theorem which can be found in 5 10

inverse optimality in robust stabilization the concept of a robust control lyapunov function bf rclf is introduced and it is shown that the existence of an bf rclf for a control affine system is equivalent to robust stabilizability via continuous state feedback this extends artstein s theorem on nonlinear stabilizability to systems with disturbances it is then shown that every bf rclf satisfies the steady state hamilton jacobi isaacs hji equation associated with a meaningful game and that every member of a class of pointwise min norm control laws is optimal for such a game these control laws have desirable properties of optimality and can be computed directly from the bf rclf without solving the hji equation for the upper value function

stabilization of nonlinear systems in the plane abstract it is shown that every small time locally controllable system in the plane can locally be asymptotically stabilized by employing locally holder continuous feedback laws as essentially was conjectured by e sontag an explicit algorithm for the construction of such feedback laws is given

application of generalized differential quadrature rule in blasius and onsager equations the generalized differential quadrature rule gdqr proposed recently by the authors is applied here to third order non linear differential equations of the blasius type and to sixth order linear onsager differential equations high 3rd order differential equations in fluid mechanics are dealt with without using point techniques the half space domain is simplified in a practical way accurate results are obtained for both kinds of problems the wide applicability of the gdqr in high order differential equations is manifested further through this work copyright 2001 john wiley sons ltd

a note on three numerical procedures to solve volterra integro differential equations in structural analysis in this paper three numerical methods to solve volterra integro differential equations containing rational functions are discussed the first one is the differential quadrature method and to the best knowledge of the author it has never been applied to this kind of problem the second one is a new version of the iterative differential quadrature method a method proposed by the author some years ago to solve problems in space time domains revised herein for the single space variable problem the third one is a numerical picard like method proposed herein to combine successive approximations with numerical integration stability and convergence of the second and the third method are discussed the three methods have been applied to solve a real world problem in the field of the structural engineering and the numerical results compared

a bounded linear integrator for some diffusive nonlinear time dependent partial differential equations we propose a numerical method to approximate the solutions of generalized forms of two bi dimensional models of mathematical physics namely the burgers fisher and the burgers huxley equations in one dimensional form the literature in the area gives account of the existence of analytical solutions for both models in the form of traveling wave fronts bounded within an interval i of the real numbers motivated by this fact we propose a finite difference methodology that guarantees that under certain analytical conditions on the model and computer parameters estimates within i will evolve discretely into new estimates which are likewise bounded within i additionally we establish the preservation in the discrete domain of the skew symmetry of the solutions of the models under study our computational implementation of the method confirms numerically that the properties of positivity and boundedness are preserved under the analytical constraints derived theoretically our simulations show a good agreement between the analytical solutions derived in the present work and the corresponding numerical approximations

stability and accuracy of the iterative differential quadrature method in this paper the stability and accuracy of an iterative method based on differential quadrature rules will be discussed the method has already been proposed by the author in a previous work where its good performance has been shown nevertheless discussion about stability and accuracy remained open an answer to this question will be provided in this paper where the conditional stability of the method will be pointed out in addition to an examination of the possible errors which arise under certain conditions the discussion will be preceded by an overview of the method and its foundations i e the differential quadrature rules and followed by a numerical case which shows how the method behaves when applied to reduce continuous systems to two degree of freedom systems in the non linear range in particular here the case of oscillators coupled in non linear terms will be treated the numerical results used to draw poincare maps will be compared with those obtained by using the runge kutta method with a high precision goal copyright 2003 john wiley sons ltd

numerical solution for differential equations of duffing type non linearity using the generalized differential quadrature rule the generalized differential quadrature rule gdqr proposed recently by the authors is applied for the first time to second and fourth order initial value differential equations with duffing type non linearity procedures are given in detail to convert these non linear differential equations into a set of linear algebraic equations in an iterative loop using the frechet derivative the effectiveness of the gdqr for obtaining the periodic solution of the duffing equation has been demonstrated through a number of examples it is also shown that the use of the frechet derivative makes it easier for the gdqr to handle non linearity the wide applicability of the gdqr is manifested further through this work

a game theoretic approach to h control for time varying systems a representation formula for all controllers that satisfy an mathcal l infty type constraint is derived for time varying systems it is now known that a formula based on two indefinite algebraic riccati equations may be found for time invariant systems over an infinite time support see j c doyle et al ieee trans automat control ac 34 1989 pp 831 847 k glover and j c doyle systems control lett 11 1988 pp 167 172 k glover et al siam j control optim 29 1991 pp 283 324 m green et al siam j control optim 28 1990 pp 1350 1371 d j n limebeer et al in proc ieee conf on decision and control austin tx 1988 g tadmor math control systems signal processing 3 1990 pp 301 324 in the time varying case two indefinite riccati differential equations are required a solution to the design problem exists if these equations have a solution on the optimization interval the derivation of the representation formula illustrated in this paper mak

the generalized differential quadrature rule for initial value differential equations abstract the generalized differential quadrature rule gdqr proposed recently by the authors is applied here to solve initial value differential equations of the 2nd to 4th order differential quadrature expressions are derived based on the gdqr for these equations the hermite interpolation functions are used as trial functions to obtain the explicit weighting coefficients for an easy and efficient implementation of the gdqr the numerical solutions for example problems demonstrate that the gdqr has high efficiency and accuracy a detailed discussion on the present method is presented by comparing with other existing methods the present method can be extended to other types of differential equation systems

differential games finite time partial state stabilisation of nonlinear dynamical systems and optimal robust control abstracttwo player zero sum differential games are addressed within the framework of state feedback finite time partial state stabilisation of nonlinear dynamical systems specifically finite time partial state stability of the closed loop system is guaranteed by means of a lyapunov function which we prove to be the value of the game this lyapunov function verifies a partial differential equation that corresponds to a steady state form of the hamilton jacobi isaacs equation and hence guarantees both finite time stability with respect to part of the system state and the existence of a saddle point for the system s performance measure connections to optimal regulation for nonlinear dynamical systems with nonlinear nonquadratic cost functionals in the presence of exogenous disturbances and parameter uncertainties are also provided furthermore we develop feedback controllers for affine nonlinear systems extending an inverse optimality framework tailored to the finite time partial state stabilisation pr

implementation of clamped and simply supported boundary conditions in the gdq free vibration analysis of beams and plates abstract in this paper a new methodology for implementing the clamped and simply supported boundary conditions is presented for the free vibration analysis of beams and plates using the generalized differential quadrature gdq method the proposed approach directly substitutes the boundary conditions into the governing equations and is referred to as sbcge approach the sbcge approach is presented to overcome the drawbacks of previous approaches in treating the boundary conditions a comparison of the sbcge approach with the method of modifying weighting coefficient matrices mwcm is made by their application to the vibration analysis of beams and plates with combinations of simply supported and clamped boundary conditions some details of the gdq method are also described in the paper

a convergent and dynamically consistent finite difference method to approximate the positive and bounded solutions of the classical burgers fisher equation in this work we investigate both analytically and numerically a spatially two dimensional advection diffusion reaction equation that generalizes the burgers and the fisher s equations the partial differential equation of interest is a nonlinear model for which the existence and the uniqueness of positive and bounded solutions are analytically established here at the same time we propose an exact finite difference discretization of the burgers fisher model of interest and show that as the continuous counterpart the method proposed is capable of preserving the positivity and the boundedness of the numerical approximations as well as the temporal and spatial monotonicity of the discrete initial boundary conditions it is shown that the method is convergent with first order in time and second order in space we provide some simulations that illustrate the fact that the proposed technique preserves the positivity the boundedness and the monotonicity

differential quadrature method application to initial boundary value problems two non linear dynamical systems have been considered in both cases the governing equation of motion is reduced to two second order non linear non autonomous ordinary differential equations using the differential quadrature method with a careful distribution of sampling points to check the numerical results a comparison with those obtained using the galerkin approach is proposed

simulating non linear coupled oscillators by an iterative differential quadrature method abstract an iterative method based on differential quadrature rules is proposed as a new unified frame of resolution for non linear two degree of freedom systems dynamical systems with duffing type non linearity have been considered differential quadrature rules have been applied with a careful distribution of sampling points to reduce the governing equation of motion to two second order non linear non autonomous ordinary differential equations and to solve the time domain problem the time domain of the problem is discretized by means of time intervals with the same distribution of sampling points used to discretize the space domain which can be seen as a single interval it will be shown that accurate solutions depend not only on the choice of the distribution of sampling points but also on the length of the time interval one refers to in the computations the numerical results utilized to draw poincare maps are successfully compared with those obtained using the runge kutta method

a generalization of the idq method and a dq based approach to approximate non smooth solutions in structural analysis abstract in this paper a new differential quadrature dq based approach in deducing the discretized equations governing the static and the dynamic problem of the euler bernoulli beam is proposed these equations here called generalized can be written easily also thanks to a lemma introduced for the first time which allow a compact writing of the solution by means of a careful distribution of the sampling points the proposed method overcomes the drawback of the element subdivision where load conditions change this is particularly useful in the analysis of some multi degree of freedom dynamic systems as it will be shown the cited distribution and the way of generating it allow to generalize the iterative differential quadrature method confined in its initial form to the two degree of freedom systems