on the solutions and conservation laws of a coupled kdv system abstract in this paper lie symmetry analysis is performed on the coupled kdv system which describes a resonant interaction of two wave modes in a shallow stratified liquid the similarity reductions and exact solutions with the aid of simplest equation method and jacobi elliptic function method are obtained based on the optimal systems of one dimensional subalgebras for the kdv system in addition the conservation laws of the coupled kdv system are also derived using the multiplier and homotopy approach and a conservation theorem

regularity of the solution of darcy forchheimer s equation on considere l equation de darcy forchheimer v v v f div v 0 v 0 on demontre un theoreme d existence et d unicite et on etudie la regularite de la solution

a novel legendre wavelet petrov galerkin method for fractional volterra integro differential equations abstract in this article the petrov galerkin method based on legendre wavelets has been developed to approximate the solution of fractional integro differential equations properties of legendre wavelets and preliminaries of fractional calculus have been presented in the article the test functions and the trial functions of petrov galerkin method have been chosen from two different spaces existence and uniqueness of the problem have been discussed error estimate of the proposed method has been presented and some illustrative examples have been demonstrated to justify the accuracy and efficiency of the present method

a finite element method for equal width equation abstract a numerical solution of the equal width ew equation is obtained using space splitting technique and quadratic b spline galerkin finite element method solitary wave motion interaction of two solitary waves wave undulation and wave generation are studied using the proposed method comparisons are made with analytical solutions and with some spline finite element method calculations at selected times accuracy and efficiency are discussed by computing the numerical conserved laws and l 2 l error norms

a mesh free numerical method for solution of the family of kuramoto sivashinsky equations in this paper radial basis function rbfs based mesh free method is implemented to find numerical solution of the kuramoto sivashinsky equations this approach has an edge over traditional methods such as finite difference and finite element methods because it does not require a mesh to discretize the problem domain and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method the accuracy of the method is assessed in terms of the error norms l 2 l number of nodes in the domain of influence free parameter dependent parameter rbfs and time step length numerical experiments demonstrate accuracy and robustness of the method for solving a class of nonlinear partial differential equations

interactions of two dimensionally localized pulses of the regularized long wave equation abstract interactions between positive and or negative cylindrical bell shaped pulses are investigated numerically for a two dimensional regularized long wave equation overtaking collisions of two same signed pulses are found to be fairly elastic head on collisions between positive and negative pulses are inelastic and their overall features are analogous to those in the one dimentional case some properties of collision processes are discussed in terms of the conversed quantities

numerical simulation of equal width wave equation by means of the variational iteration method the solution of the equal width wave equation is numerically obtained and compared with that obtained by the adomian decomposition method numerical results show that the variational iteration method can readily be implemented with excellent accuracy to nonlinear equations

closed form solutions for a generalized benjamin bona mahony burgers equation with higher order nonlinearity we obtain closed form solutions for a generalized benjamin bona mahony burgers gbbmb equation with higher order nonlinearity which belong to a class of so called nonlinear pseudo parabolic or type sobolev equations the generalized tanh coth method is used here with the aid of mathematica periodic and soliton solutions are formally derived finally some conclusions are given

application of galerkin s method to equal width wave equation the non linear equal width equation is solved by galerkin s method using linear finite elements a linear recurrence relationship for the numerical solution of the resulting system of ordinary differential equations is found employing the crank nicolson approach including a product approximation the three invariants of the motion are calculated to determine the conservation properties of the system l 2 and l error norms are used to measure differences between the exact and numerical solutions the simulations of solitary wave motion are used to determine the properties of the algorithm finally the development of an undular bore is studied

comparison of models for flow induced deformation of soft biological tissue abstract the behaviour of a deformable porous medium during the flow of fluid under a pressure difference is examined for both infinitesimal and finite deformations models for both cases are solved for the problem of steady one dimensional compression and compared with experimental data from parker et al j appl mech 54 794 800 1987 for a polyurethane sponge the purpose of this study is to identify a simple model which agrees qualitatively with these published results to relate the stress relations for biological tissues to the data for polymer sponges parker et al 1987 a translation of 1 1 kpa was introduced this allows for some structural differences between the two media it was found that the infinitesimal models were adequate up to 20 strain but significant divergence occurred for higher strains a finite deformation model with the permeability depending exponentially on the strain gave the most consistent results and required the fitting of only two parameters

a petrov galerkin method for solving the generalized equal width gew equation the generalized equal width gew equation is solved numerically by the petrov galerkin method using a linear hat function as the test function and a quadratic b spline function as the trial function product approximation has been used in this method a linear stability analysis of the scheme shows it to be conditionally stable test problems including the single soliton and the interaction of solitons are used to validate the suggested method which is found to be accurate and efficient finally the maxwellian initial condition pulse is studied

the classification of the single traveling wave solutions to the generalized equal width equation abstract the complete discrimination system for polynomial is applied to the generalized equal width ew equation to obtain the classifications of single traveling wave solutions in particular we construct the corresponding solutions for the concrete parameters to show that each solution in the classification can be realized

nonlinear equation governing flow in a saturated porous medium it is argued that the appropriate generalization of darcy s law when inertia effects are included takes the form p k v c k v v div v 0 where k is the permeability of the medium and the form drag constant c is a coefficient which is independent of the pressure p the seepage velocity v and the density and viscosity of the fluid but which is dependent on the geometry of the medium we formulate a nonlinear extension of brinkman s self consistent theory for the flow of a viscous fluid through a swarm of spherical particles we equate the drag per unit volume given by the right hand side of the first of the above equations to the total drag nd on the n particles contained within that unit volume in an infinite region where d is the drag on a single particle placed in a velocity field v subject to v v grad p 2v k v c k v v div v 0 v is a prescribed constant where is the viscosity without solving these equations we obtain an estimate for c from the known experimental drag law for a solid sphere placed in a uniform stream

rbf ps scheme for solving the equal width equation the equal width equation is solved by a rbf ps scheme this technique is meshless and there is no need to linearize the nonlinear terms the radial kernels are used to transform the pde into a system of odes a higher order ode solver is used to solve the ode system the method is tested for single solitary wave interaction of two and three solitary waves the development of an undular bore and the interaction of two solitary waves the results are compared with earlier work and other methods

solitary waves of the regularized long wave equation abstract a finite element solution of the regularised long wave equation based on galerkin s method using cubic splines as element shape functions is set up a linear stability analysis shows the scheme to be unconditionally stable test problems including the migration and interaction of solitary waves are used to validate the method which is found to be accurate and efficient the three invariants of the motion are evaluated to determine the conservation properties of the algorithm the temporal evolution of a maxwellian initial pulse is then studied

solitary waves of the equal width wave equation a numerical solution of the equal width wave equation based on galerkin s method using cubic b spline finite elements is used to simulate the migration and interaction of solitary waves the interaction of two solitary waves is seen to cause the creation of a source for solitary waves usually these are of small magnitude but when the amplitudes of the two interacting waves are equal and opposite the source produces trains of solitary waves whose amplitudes are of the same order as those of the initiating waves the three invariants of the motion are evaluated to determine the conservation properties of the system finally the temporal evolution of a maxwellian initial pulse is studied for small ut uux uxxt 0 only positive waves are formed and the behaviour mimics that of the kdv and filw equations for larger values of d both positive and negative solitary waves are generated

petrov galerkin methods for nonlinear dispersive waves abstract petrov galerkin methods based on piecewise linear interpolants for the korteweg de vries and related equations are studied both accuracy and stability are analysed for the linearised case it is shown that in the nonlinear case the order of accuracy of the standard galerkin procedure is reduced and an alternative technique is therefore proposed which retains the fourth order accuracy in space this is found to perform well when compared with finite difference or other finite element schemes

collocation method using quartic b spline for the equal width ew equation the equal width ew equation is solved numerically by giving a new algorithm based on collocation method using quartic b splines at the mid knots points as element shape also we use the fourth runge kutta method for solving the system of first order ordinary differential equations instead of finite difference method our test problems including the migration and interaction of solitary waves are used to validate the algorithm which is found to be accurate and efficient the three invariants of the motion are evaluated to determine the conservation properties of the algorithm the temporal evaluation of a maxwellian initial pulse is then studied

description of the inelastic collision of two solitary waves for the bbm equation we prove that the collision of two solitary waves of the bbm equation is inelastic but almost elastic in the case where one solitary wave is small in the energy space we show precise estimates of the nonzero residue due to the collision moreover we give a precise description of the collision phenomenon change of size of the solitary waves and shifts in their trajectories to prove these results we extend the method introduced in martel and merle description of two soliton collision for the quartic gkdv equation submitted preprint http arxiv org abs 0709 2672 commun math phys 286 39 79 2009 for the generalized kdv equation in particular in the quartic case the main argument is the construction of an explicit approximate solution in the collision region

solitary waves induced by the boundary forced ew equation the development of a train of ew solitary waves induced by boundary forcing is studied using a petrov galerkin finite element scheme with shape functions taken as quadratic b spline functions a linear recurrence relationship for the numerical scheme is obtained via a crank nicholson approach involving a product approximation results are compared with those for the rlw and kdv equations

a note on multivariate gaussian estimates an upper bound of multivariate gaussian probability for a general convex domain d is given based on a geometric observation the bound is sharper than known ones on multivariate mills ratio in many cases

analytical and computational approaches on solitary wave solutions of the generalized equal width equation abstract in this article firstly numerical solutions of the generalized equal width gew equation have been obtained by a petrov galerkin finite element method using cubic b spline base functions as element shape functions and quadratic b spline base functions as the weight functions in order to prove the practicability and robustness of the numerical algorithm the error norms l2 l and three invariants i1 i2 and i3 are computed a linear stability analysis based on a fourier method states that the numerical scheme is unconditionally stable secondly we have proposed the modified extended tanh function method with the riccati differential equation which is a convenient and an effective method for getting the exact traveling wave solutions of the equation motion of single solitary wave is examined using the present methods the obtained results are indicated both in tabular and graphical form

a petrov galerkin method for equal width equation abstract the equal width equation is solved numerically by petrov galerkin method using linear hat function and quadratic b spline function as trial and test functions respectively product approximation has been used in this method a linear stability analysis of the scheme is shown to be conditionally stable test problems including the single soliton and the interaction of solitons are used to validate the suggested method which is found to be accurate and efficient the maxwellian initial condition pulse and the development of an undular bore are also studied

unique continuation property and control for the benjamin bona mahony equation on a periodic domain abstract we consider the benjamin bona mahony bbm equation on the one dimensional torus t r 2 z we prove a unique continuation property ucp for small data in h 1 t with nonnegative zero means next we extend the ucp to certain bbm like equations including the equal width wave equation and the kdv bbm equation applications to the stabilization of the above equations are given in particular we show that when an internal control acting on a moving interval is applied in the bbm equation then a semiglobal exponential stabilization can be derived in h s t for any s 1 furthermore we prove that the bbm equation with a moving control is also locally exactly controllable in h s t for any s 0 and globally exactly controllable in h s t for any s 1 in a sufficiently large time depending on the h s norms of the initial and terminal states

a least squares finite element scheme for the ew equation abstract the equal width ew equation is solved by a least squares technique using linear space time finite elements in simulations of the migration of a single solitary wave this algorithm is shown to have good accuracy and conservation the development of solitary waves from an arbitrary initial condition within an ew environment is examined

b spline quasi interpolation based numerical methods for some sobolev type equations in this article we intend to use quadratic and cubic b spline quasi interpolants to develop higher order numerical methods for some sobolev type equations in one space dimension our aim is also to compare the performance of the proposed methods in terms of the accuracy and the rate of convergence we also discuss another approach to the cubic b spline quasi interpolation based method where we achieve fourth order of accuracy in space we theoretically establish the order of accuracy for the three proposed methods and also establish the l 2 stability in the linear case using von neumann analysis as a particular case of the sobolev type equations we take the equal width and the benjamin bona mahony burgers equations and perform several numerical experiments to support our theoretical results

b spline finite element studies of the non linear schrodinger equation abstract the non linear schrodinger equation is solved numerically by a b spline finite element method the approach used is based on collocation of cubic b splines over spatial finite elements so that we have continuity of the dependent variable and its first two derivatives throughout the solution range time integration of the resulting system of ordinary differential equations is effected using a crank nicolson approximation standard problems are used to validate the algorithm comparisons are made with published numerical and analytical solutions the proposed method exhibits good conservation properties and performs well the development of temporally recurrent states from spatially periodic initial conditions is examined through computer experiments the birth of mobile as well as static solitons out of maxwellian initial data is observed

scattering of regularized long wave solitary waves abstract the lagrangian density for the regularized long wave equation also known as the bbm equation is presented using the trial function technique ordinary differential equations that describe the time dependence of the position of the peaks amplitudes and widths for the collision of two solitary waves are obtained these equations are analyzed in the born and equal width approximations and compared with numerical results obtained by direct integration utilizing the split step fast fourier transform method the computations show that collisions are inelastic and that production of solitary waves may occur

a space time least square finite element scheme for advection diffusion equations abstract a space time least square finite element scheme is presented for the advection diffusion problems at moderate to high peclet numbers this scheme is designed to eliminate spurious oscillations and can be used to define the steady state solution as the asymptotic transient solution for large time numerical results using linear elements in a 1d space and bilinear elements in a 2d space demonstrate the accuracy and the stability of the new scheme

least squares finite element method for the advection diffusion equation the space time least squares finite element methods are constructed for the advection diffusion equation by using both linear shape functions and quadratic b spline shape functions two test problems are studied to demonstrate the accuracy of the present methods results of the two schemes have been compared

solitary wave solutions for the general kdv equation by adomian decomposition method in recent publications chaos solitons fractals 12 2001 2283 int j appl math 3 4 2000 361 we have dealt with the numerical solutions of the korteweg de vries kdv and modified korteweg de vries mkdv equations we extend this study to a more general nonlinear equation which is the general korteweg de vries gkdv equation in which the previous studies is a special case of it the method applied here is adomian decomposition method which has been developed by george adomian solving frontier problems of physics the decomposition method kluwer academic publishers boston ma 1994 numerical examples are tested to illustrate the pertinent feature of the proposed algorithm

numerical schemes for a model for nonlinear dispersive waves abstract a description is given of a number of numerical schemes to solve an evolution equation that arises when modelling the propagation of water waves in a channel the discussion also includes the results of numerical experiments made with each of the schemes it is suggested on the basis of these experiments that one of the schemes may have discrete solitary wave solutions

a numerical solution of the equal width wave equation by a lumped galerkin method in this paper the equal width wave ew equation is solved by a numerical technique based on a lumped galerkin method using quadratic b spline finite elements to investigate the motion of a single solitary wave development of two solitary waves interaction and an undular bore the obtained results are compared with published numerical solutions a linear stability analysis of the method is also investigated

painleve analysis and lie group symmetries of the regularized long wave equation the painleve formulation is given for the regularized long wave equation which has been used to describe shallow water waves and drift waves in a plasma the lie group symmetries of the equation is then studied and it is shown that the only symmetries are due to the three conserved quantities

certain approximations to achieve sharp lower and upper bounds for the mills ratio of the inverse gaussian distribution abstract in this note we prove new complete monotonicity properties of some functions associated with the inverse gaussian distribution based on these properties we present some lower and upper bounds with explicit expressions for the mills ratio of the inverse gaussian distribution finally for demonstrating the efficiency of our estimates some numerical computations are provided

asymptotic stability of solitary wave solutions to the regularized long wave equation abstract we study the asymptotic stability of solitary wave solutions to the regularized long wave equation rlw in h 1 r rlw is an equation which describes the long waves in water to prove the result we make use of the monotonicity of the local h1 norm and apply the liouville property of rlw as in merle and martel j math pures appl 79 2000 339 arch rational mech anal 157 2001 219

a new general asymptotic formula for the mills ratio of the skew generalized normal distribution abstract in this paper we establish some lower and upper bounds with explicit expressions for the mills ratio of the skew generalized normal distribution and for demonstrating the efficiency of our estimates some numerical computations are provided