on mds codes via cauchy matrices a special form of cauchy matrix is used to obtain a tighter bound for the validity region of the maximum distance separable mds conjecture and a new compact characterization of generalized reed solomon codes the latter is further used to obtain constructions and some existence results for long 2k k double circulant mds codes

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

nonlinear self adjointness of a 2d generalized second order evolution equation abstract we study the nonlinear self adjointness of a general class of quasilinear 2d second order evolution equations which do not possess variational structure for this purpose we use the method of ibragimov devised and developed recently this approach enables one to establish the conservation laws for any differential equation we first obtain conditions determining the self adjoint sub class in the general case then we establish the conservation laws for important particular cases the ricci flow equation the modified ricci flow equation and the nonlinear heat equation

conservation laws for the maxwell dirac equations with dual ohm s law using a general theorem on conservation laws for arbitrary differential equations proved by ibragimov j math anal appl 333 311 320 2007 we have derived conservation laws for dirac s symmetrized maxwell lorentz equations under the assumption that both the electric and magnetic charges obey linear conductivity laws dual ohm s law we find that this linear system allows for conservation laws which are nonlocal in time

on the conservation laws and invariant solutions of the mkdv equation abstract in this paper we consider modified korteweg de vries mkdv equation by using the nonlocal conservation theorem method and the partial lagrangian approach conservation laws for the mkdv equation are presented it is observed that only nonlocal conservation theorem method lead to the nontrivial and infinite conservation laws in addition invariant solution is obtained by utilizing the relationship between conservation laws and lie point symmetries of the equation

generic formalism and discrete variational derivative method for the two dimensional vorticity equation the vorticity equation for two dimensional incompressible viscous flows is formulated within the generic formalism for non equilibrium thermodynamics the laws of conservation of energy and increasing entropy derived from the generic formulation are properly inherited by the finite difference equations obtained by invoking the discrete variational derivative method the law of increasing entropy corresponds to the dissipation of enstrophy for the vorticity equation some numerical experiments have been done to examine the usefulness of the proposed method

bracket formulations and energy and helicity preserving numerical methods for the three dimensional vorticity equation abstract the vorticity equation for three dimensional viscous incompressible fluid flows is formulated within different bracket formalisms using the poisson or nambu bracket together with a dissipative bracket the budgets of kinetic energy helicity and enstrophy derived from the bracket formulations are properly inherited by the finite difference equations obtained by invoking the discrete variational derivative method combined with the mimetic finite difference method in particular energy and helicity are conserved precisely in inviscid flow computations the energy and enstrophy dissipate properly owing to viscosity in viscous flow computations and the enstrophy is appropriately produced by the vortex stretching effect in both inviscid and viscous flow computations the relationships between the stream function velocity and vorticity as well as the solenoidal conditions on the velocity and vorticity fields are also inherited numerical experiments on a periodic array of rolls that permits analytical solutions have been done to examine the properties and usefulness of the proposed method

group properties and conservation laws for nonlocal shallow water wave equation abstract symmetry groups symmetry reductions optimal system conservation laws and invariant solutions of the shallow water wave equation with nonlocal term are studied first lie symmetries based on the invariance criterion for nonlocal equations and the solution approach for nonlocal determining equations are found and then the reduced equations and optimal system are obtained finally new conservation laws are generated and some similarity solutions for symmetry reduction forms are discussed

invariant conservation law preserving discretizations of linear and nonlinear wave equations symmetry and conservation law preserving finite difference discretizations are obtained for linear and nonlinear one dimensional wave equations on five and nine point stencils using the theory of lie point symmetries of difference equations and the discrete direct multiplier method of conservation law construction in particular for the linear wave equation an explicit five point scheme is presented that preserves the discrete analogs of its basic geometric point symmetries and six of the corresponding conservation laws for a class of nonlinear wave equations arising in hyperelasticity a nine point implicit scheme is constructed preserving four point symmetries and three local conservation laws other discretizations of the nonlinear wave equations preserving different subsets of conservation laws are discussed

eventual periodicity of the forced oscillations for a korteweg de vries type equation on a bounded domain using a sinc collocation method abstract we demonstrate numerically the eventual time periodicity of solutions u t to the korteweg de vries type equation with periodic forcing at one end using the sinc collocation method this method approximates the space dimension of the solution with a cardinal expansion of sinc functions thus allowing the avoidance of a costly finite difference grid for a third order boundary value problem the first order time derivative is approximated with a weighted finite difference method the sinc collocation method was found to be more robust and more efficient than other numerical schemes when applied to this problem

lie symmetries group invariant solutions and conservation laws of the vasicek pricing equation of mathematical finance abstract the one factor term structure model by vasicek is analysed from the point of view of lie symmetry analysis its one parameter lie point symmetries and corresponding group of adjoint representations are obtained an optimal system of one dimensional subalgebras is derived and is then used to obtain symmetry reductions and group invariant solutions the group invariant solutions presented here are new and have not appeared in the literature moreover we derive conservation laws for the vasicek equation by employing the theorem due to ibragimov

second noether theorem for quasi noether systems quasi noether differential systems are more general than variational systems and are quite common in mathematical physics they include practically all differential systems of interest at least those that have conservation laws in this paper we discuss quasi noether systems that possess infinite dimensional infinite symmetries involving arbitrary functions of independent variables for quasi noether systems admitting infinite symmetries with arbitrary functions of all independent variables we state and prove an extension of the second noether theorem in addition we prove that infinite sets of conservation laws involving arbitrary functions of all independent variables are trivial and that the associated differential system is under determined we discuss infinite symmetries and infinite conservation laws of two important examples of non variational quasi noether systems the incompressible euler equations and the navier stokes equations in vorticity formulation and we show that the infinite sets of conservation laws involving arbitrary functions of all independent variables are trivial we also analyze infinite symmetries involving arbitrary functions of not all independent variables prove that the fluxes of conservation laws in these cases are total divergences on solutions and demonstrate examples of this situation

conservation laws of the 2 1 dimensional kp equation and burgers equation with variable coefficients and cross terms in this paper conservation laws for the 2 1 dimensional kp equation and burgers equation with variable coefficients and cross terms are studied due to the existence of cross terms the new conservation theorem given by ibragimov cannot be applied to the two equations directly we propose two rules of modifications which ensure the theorem can be applied to nonlinear evolution equations with cross terms formulas of conservation laws for the kp equation and burgers equation are given using these formulas non trivial and time dependent conservation laws for these equations are derived according to their different lie symmetries

symmetry analysis of a model of stochastic volatility with time dependent parameters we provide the solutions for the heston model of stochastic volatility when the parameters of the model are constant and when they are functions of time in the former case the solution follows immediately from the determination of the lie point symmetries of the governing 1 1 evolution partial differential equation this is not the situation in the latter case but we are able to infer the essential structure of the required nonlocal symmetry from that of the autonomous problem and hence can present the solution to the nonautonomous problem as in the case of the standard black scholes problem the presence of time dependent parameters is not a hindrance to the demonstration of a solution

the adjoint equation method for constructing first integrals of difference equations a new method for finding first integrals of discrete equations is presented it can be used for discrete equations which do not possess a variational lagrangian or hamiltonian formulation the method is based on a newly established identity which links symmetries of the underlying discrete equations solutions of the discrete adjoint equations and first integrals the method is applied to an invariant mapping and to discretizations of second order and third order ordinary differential equations in examples the set of independent first integrals makes it possible to find the general solution of the discrete equations

an optimal system and group invariant solutions of the cox ingersoll ross pricing equation the valuation partial differential equation of standard european interest rate derivatives in the cox ingersoll ross model is analysed its one parameter lie point symmetries and corresponding group of adjoint representations are obtained an optimal system of one dimensional subalgebras is derived and used to construct distinct families of special closed form solutions of the equation

lie symmetry analysis and conservation law of variable coefficient davey stewartson equation abstract a variable coefficient davey stewartson vcds equation is investigated in this paper infinitesimal generators and symmetry groups are presented by the lie group method and the optimal system is presented with adjoint representation based on the optimal system similarity reductions to partial differential equations pdes are obtained then some pdes are reduced to ordinary differential equations odes by two dimensional subalgebras and the similarity solutions are provided including periodic solutions and elliptic function solutions with lagrangian it is shown that vcds is nonlinearly self adjoint furthermore based on nonlinear self adjointness conservation laws for vcds equation are derived

two dimensional systems that arise from the noether classification of lagrangians on the line abstract noether like operators play an essential role in writing down the first integrals for euler lagrange systems of ordinary differential equations odes the classification of such operators is carried out with the help of analytic continuation of lagrangians on the line we obtain the classification of 5 6 and 9 noether like operators for two dimensional lagrangian systems that arise from the submaximal and maximal dimensional noether point symmetry classification of lagrangians on the line cases in which the noether like operators are also noether point symmetries for the systems of two odes are mentioned in particular the 8 dimensional maximal noether algebra is remarkably obtained for the simplest system of the free particle equations in two dimensions from the 5 dimensional complex noether algebra of the standard lagrangian of the scalar free particle equation we present the effectiveness of noether like operators for the determination of first integrals of systems of two nonlinear differential equations which arise from scalar complex euler lagrange odes that admit noether symmetry

time fractional 2 1 dimensional wu zhang system dispersion analysis similarity reductions conservation laws and exact solutions abstract this paper examines the time fractional 2 1 dimensional wu zhang system describing dispersive long waves the dispersion relation of waves has been obtained from linear analysis and used to find the phase and group velocity the dispersion relation shows an anomalous dispersion of waves lie symmetries and the corresponding similarity reductions are carried out using riemann liouville fractional derivative and the order of the system is reduced to lower dimension furthermore the system is analysed for nonlinear self adjointness and the conservation laws are obtained associated with infinitesimal symmetries by applying new conservation theorem the adjoint variables are used to convert conservation laws into original dependent variables the exponential rational function method is employed to find exact solutions revealing kink and bell type profiles the effect of fractional order on the wave profile of solutions is discussed graphically

eventual periodicity for the kdv equation on a half line this paper studies the eventual periodicity of solutions to the initial and boundary value problem for the kdv equation on a half line and with periodic boundary data we derive a representation formula for solutions to the linearized kdv equation and rigorously establish the eventual periodicity of these solutions numerical experiments performed on the full kdv equation indicate that its solutions are also eventually periodic

lie theory applications to problems in mathematical finance and economics this paper is devoted to show and explain some applications of lie theory to solve some problems in economics and mathematical finance so we put forward and discuss mathematical aspects and approaches for several economic problems which have been previously considered in the literature besides we also show our advances on this topic mentioning some open problems for future research

algebraic resolution of equations of the black scholes type with arbitrary time dependent parameters we examine various forms of the black scholes equation using symmetry analysis and consider the standard problem of a terminal condition namely u t x u the models considered are the standard form of the equation to provide a benchmark the nonhomogeneous form and two variations in which the parameters of the two basic models are explicitly time dependent we find subject to some constraints upon the parameters a remarkable robustness in the results

lie symmetry analysis explicit solutions and conservation laws for the space time fractional nonlinear evolution equations abstract this paper studies the symmetry analysis explicit solutions convergence analysis and conservation laws cls for two different space time fractional nonlinear evolution equations with riemann liouville rl derivative the governing equations are reduced to nonlinear ordinary differential equation ode of fractional order using their lie point symmetries in the reduced equations the derivative is in erdelyi kober ek sense power series technique is applied to derive an explicit solutions for the reduced fractional odes the convergence of the obtained power series solutions is also presented moreover the new conservation theorem and the generalization of the noether operators are developed to construct the nonlocal cls for the equations some interesting figures for the obtained explicit solutions are presented

a new class of superregular matrices and mdp convolutional codes abstract this paper deals with the problem of constructing superregular matrices that lead to mdp convolutional codes these matrices are a type of lower block triangular toeplitz matrices with the property that all the square submatrices that can possibly be nonsingular due to the lower block triangular structure are nonsingular we present a new class of matrices that are superregular over a sufficiently large finite field f such construction works for any given choice of characteristic of the field f and code parameters n k such that n k we also discuss the size of f needed so that the proposed matrices are superregular

analysis of lie symmetries with conservation laws and solutions for the generalized 3 1 dimensional time fractional camassa holm kadomtsev petviashvili equation abstract in this paper under investigated is a generalized 3 1 dimensional camassa holm kadomtsev petviashvili gch kp equation which describes the role of dispersion in the formation of patterns in liquid drops with the help of the semi inverse method the euler lagrange equation and agrawal s method the time fractional gch kp equation is derived in the sense of riemann liouville fractional derivatives further the symmetry of the 3 1 dimensional time fractional gch kp equation is studied by fractional order symmetry meanwhile based on the new conservation theorem the conservation laws of 3 1 dimensional time fractional gch kp equation are constructed finally the solutions to the equation are given via a bilinear method and the radial basis functions rbfs meshless approach

first integrals for a general linear system of two second order odes via a partial lagrangian the partial noether operators and first integrals of a general system of two linear second order ordinary differential equations odes with variable coefficients are studied by means of a partial lagrangian the canonical form for the general system of two second order ordinary differential equations is invoked and all cases of this system are discussed with respect to partial noether operators we also tabulate the results for the special case b x c x of the system which was considered elsewhere using a lagrangian and a partial lagrangian the first integrals are obtained explicitly by exploiting a noether like theorem with the help of partial noether operators this study gives a new way to construct first integrals for systems without a variational principle as not all linear equations have a lagrangian physical applications to conservative and oscillator mechanical systems are given

lie symmetry analysis conservation laws and exact solutions of the time fractional generalized hirota satsuma coupled kdv system abstract in the present paper lie point symmetries of the time fractional generalized hirota satsuma coupled kdv hs ckdv system based on the riemann liouville derivative are obtained using the derived lie point symmetries we obtain similarity reductions and conservation laws of the considered system finally some analytic solutions are furnished by means of the invariant subspace method in the caputo sense

symmetries and exact solutions via conservation laws for some partial differential equations of mathematical physics abstract exact solutions of some classical pdes with two independent variables are achieved by exploiting a double reduction method which is applied after deducing conserved vectors associated with lie point symmetries these are generally obtained through the application of noether s theorem after introducing the notions of adjoint equation and extended lagrangian alternatively associated conserved vectors can be obtained by a direct method

on the nonlinear self adjointness of a class of fourth order evolution equations in this paper we study the property of nonlinear self adjointness for a class of nonlinear fourth order equation it is shown that if an equation itself is a conservation law then it possesses the property of nonlinear self adjointness and hence it can be rewritten in an equivalent strictly self adjoint form the property of nonlinear self adjointness is used to obtain all nontrivial conservation laws for the class under consideration

a family of wave equations with some remarkable properties we consider a family of homogeneous nonlinear dispersive equations with two arbitrary parameters conservation laws are established from the point symmetries and imply that the whole family admits square integrable solutions recursion operators are found for two members of the family investigated for one of them a lax pair is also obtained proving its complete integrability from the lax pair we construct a miura type transformation relating the original equation to the korteweg de vries kdv equation this transformation on the other hand enables us to obtain solutions of the equation from the kernel of a schrodinger operator with potential parametrized by the solutions of the kdv equation in particular this allows us to exhibit a kink solution to the completely integrable equation from the 1 soliton solution of the kdv equation finally peakon type solutions are also found for a certain choice of the parameters although for this particular case the equation is reduced to a homogeneous second order nonlinear evolution equation

invariant solutions and conservation laws to nonconservative fp equation we generate conservation laws for the one dimensional nonconservative fokker planck fp equation also known as the kolmogorov forward equation which describes the time evolution of the probability density function of position and velocity of a particle and associate these where possible with lie symmetry group generators we determine the conserved vectors by a composite variational principle and then check if the condition for which symmetries associate with the conservation law is satisfied as the fokker planck equation is evolution type no recourse to a lagrangian formulation is made moreover we obtain invariant solutions for the fp equation via potential symmetries

classification reduction group invariant solutions and conservation laws of the gardner kp equation in this paper symmetries and group invariant solutions to the gardner kp equation are obtained by using the direct symmetry method at the same time we find the corresponding lie algebra optimal system classification and the similarity reductions to the equation respectively our exact solutions generalize the corresponding results obtained by wazwaz in addition the conservation laws of gardner kp equation are also given

group classification and conservation laws for a two dimensional generalized kuramoto sivashinsky equation abstract the two dimensional anisotropic kuramoto sivashinsky equation is a fourth order nonlinear evolution equation in two spatial dimensions that arises in sputter erosion and epitaxial growth on vicinal surfaces a generalization of this equation is proposed and studied via group analysis methods the complete group classification of this generalized kuramoto sivashinsky equation is carried out it is classified according to the property of the self adjointness and the corresponding conservation laws are established

strongly mds convolutional codes maximum distance separable mds convolutional codes have the property that their free distance is maximal among all codes of the same rate and the same degree in this paper a class of mds convolutional codes is introduced whose column distances reach the generalized singleton bound at the earliest possible instant such codes are called strongly mds convolutional codes they also have a maximum or near maximum distance profile the extended row distances of these codes will also be discussed briefly

time fractional cahn allen and time fractional klein gordon equations lie symmetry analysis explicit solutions and convergence analysis abstract this research analyzes the symmetry analysis explicit solutions and convergence analysis to the time fractional cahn allen ca and time fractional klein gordon kg equations with riemann liouville rl derivative the time fractional ca and time fractional kg are reduced to respective nonlinear ordinary differential equation of fractional order we solve the reduced fractional odes using an explicit power series method the convergence analysis for the obtained explicit solutions are investigated some figures for the obtained explicit solutions are also presented

special conformal groups of a riemannian manifold and lie point symmetries of the nonlinear poisson equation abstract we obtain a complete group classification of the lie point symmetries of nonlinear poisson equations on generic pseudo riemannian manifolds m using this result we study their noether symmetries and establish the respective conservation laws it is shown that the projection of the lie point symmetries on m are special subgroups of the conformal group of m in particular if the scalar curvature of m vanishes the projection on m of the lie point symmetry group of the poisson equation with critical nonlinearity is the conformal group of the manifold we illustrate our results by applying them to the thurston geometries

nonlinear self adjointness of a generalized fifth order kdv equation the new concepts of self adjoint equations formulated by ibragimov and gandarias are applied to a class of fifth order evolution equations then from ibragimov s theorem on conservation laws conservation laws for the generalized kawahara equation simplified kahawara equation and modified simplified kawahara equation are established

integrating factors adjoint equations and lagrangians integrating factors and adjoint equations are determined for linear and non linear differential equations of an arbitrary order the new concept of an adjoint equation is used for construction of a lagrangian for an arbitrary differential equation and for any system of differential equations where the number of equations is equal to the number of dependent variables the method is illustrated by considering several equations traditionally regarded as equations without lagrangians noether s theorem is applied to the maxwell equations

conservation laws of 3 α dimensional time fractional diffusion equation abstract the concept of lie backlund symmetry plays a fundamental role in applied mathematics it is clear that in order to find conservation laws for a given partial differential equations pdes or fractional differential equations fdes by using lagrangian function firstly we need to obtain the symmetries of the considered equation fractional derivation is an efficient tool for interpretation of mathematical methods many applications of fractional calculus can be found in various fields of sciences as physics classic quantum mechanics and thermodynamics biology economics engineering and etc so in this paper we present some effective application of fractional derivatives such as fractional symmetries and fractional conservation laws by fractional calculations in the sequel we obtain our results in order to find conservation laws of the time fractional equation in some special cases

symmetry reduction exact solutions and conservation laws of the sawada kotera kadomtsev petviashvili equation in this paper by applying a direct symmetry method we obtain the symmetry reduction group invariant solution and many new exact solutions of sk kp equation which include jacobi elliptic function solutions hyperbolic function solutions trigonometric function solutions and so on at last we also give the conservation laws of sk kp equation

comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics abstract the conservation laws for second order scalar partial differential equations and systems of partial differential equations which occur in fluid mechanics are constructed using different approaches the direct method noether s theorem the characteristic method the variational approach multiplier approach for arbitrary functions as well as on the solution space symmetry conditions on the conserved quantities the direct construction formula approach the partial noether approach and the noether approach for the equation and its adjoint are discussed and explained with the help of an illustrative example on a non linear field equation describing the relaxation to a maxwellian distribution the conservation laws for the non linear diffusion equation for the spreading of an axisymmetric thin liquid drop the system of two partial differential equations governing flow in a laminar two dimensional jet and the system of two partial differential equations governing flow in a laminar radial jet are discussed via these approaches

time fractional generalized boussinesq equation for rossby solitary waves with dissipation effect in stratified fluid and conservation laws as well as exact solutions abstract construct fractional order model to describe rossby solitary waves can provide more pronounced effects and deeper insight for comprehending generalization and evolution of rossby solitary waves in stratified fluid in the paper from the quasi geostrophic vorticity equation with dissipation effect and complete coriolis force based on the multi scale analysis and perturbation method a classical generalized boussinesq equation is derived to describe the rossby solitary waves in stratified fluid further by employing the reduction perturbation method the semi inverse method the agrawal method we derive the euler lagrangian equation of classical generalized boussinesq equation and obtain the time fractional generalized boussinesq equation without dissipation effect by using lie group analysis method the conservation laws of time fractional boussinesq equation are given finally with the help of the improved g g expansion method the exact solutions of the above equation are generated meanwhile in order to consider the dissipation effect we have to derive the approximate solutions by adopting the new iterative method we remark that the fractional order model can open up a new window for better understanding the waves in fluid

comment on the paper on conservation laws by lie symmetry analysis for 2 1 dimensional bogoyavlensky konopelchenko equation in wave propagation by s saha ray abstract in a recent paper referred to in the title the author used the concept of quasi self adjointness to obtain conservation laws for a system of the 2 1 dimensional bogoyavlensky konopelchenko equation apart from the adjoint system of equations all the results on the quasi self adjointness and the subsequent conservation laws obtained are inaccurate in this comment we clarify these inaccuracies and also generate conservation laws for a potential form of the underlying equation through noether theorem

a new conservation theorem a general theorem on conservation laws for arbitrary differential equations is proved the theorem is valid also for any system of differential equations where the number of equations is equal to the number of dependent variables the new theorem does not require existence of a lagrangian and is based on a concept of an adjoint equation for non linear equations suggested recently by the author it is proved that the adjoint equation inherits all symmetries of the original equation accordingly one can associate a conservation law with any group of lie lie backlund or non local symmetries and find conservation laws for differential equations without classical lagrangians

nonlinear self adjointness a criterion for linearization of pdes in this communication we show that the property of nonlinear self adjointness is conserved by an invertible point transformation we use this property to give necessary conditions for the existence of an invertible point transformation mapping of a given partial differential equation pde to a linear pde

superregular lower triangular toeplitz matrices for low delay wireless streaming a matrix is termed superregular if all of its possible submatrices are non singular superregular lower triangular toeplitz matrices are useful for mds convolutional codes and sequential network codes in this paper we present the explicit matrix constructions for superregular lower triangular toeplitz matrices in mathrm gf 2 p k times k k leq 5 for k 5 we provide a greedy algorithm which over sufficiently large fields is guaranteed to find a superregular lower triangular toeplitz matrix we introduce product preserving joint superregularity and extend our explicit matrix constructions to these cases we provide methods for deriving the exact symbol loss probability and delay for any deterministic block code we derive the exact symbol loss probability and delay for codes using a superregular lower triangular matrix and for codes using two product preserving jointly superregular lower triangular matrices we then compare these results with those obtained from both simulations and our practical implementation and for each case we also compare with random based codes furthermore our experiments show a gain in coding throughput above 40 for superregular lower triangular toeplitz matrices over random matrices

symmetry properties and explicit solutions of some nonlinear differential and fractional equations abstract one generalized burgers hierarchy is derived by applying the cole hopf transforation whose dark equation hierarchy is also generated by the dark equation method from which a generalized burgers equation and a generalized kupershmidt equation respectively are followed to obtain through lie group analysis method we produce similarity reductions exact solutions of the generalized burgers and the kupershmidt equations specially we investigate the similarity reductions of the fractional kupershmidt equation and its exact solutions in addition we obtain the conservation laws of the kupershmidt equation and its adjoint equation finally we give rise to symmetries primary branch solutions as well various recursion operators of degenerated equations from the kupershmidt equation

traveling wave solutions and conservation laws for nonlinear evolution equation in this work the riccati bernoulli sub ordinary differential equation and modified tanh coth methods are used to reach soliton solutions of the nonlinear evolution equation we acquire new types of traveling wave solutions for the governing equation we show that the equation is nonlinear self adjoint by obtaining suitable substitution therefore we construct conservation laws for the equation using new conservation theorem the obtained solutions in this work may be used to explain and understand the physical nature of the wave spreads in the most dispersive medium the constraint condition for the existence of solitons is stated some three dimensional figures for some of the acquired results are illustrated

self adjoint sub classes of generalized thin film equations in this work we consider a class of fourth order nonlinear partial differential equation containing several un specified coefficient functions of the dependent variable which encapsulates various mathematical models used e g for describing the dynamics of thin liquid films we determine the subclasses of these equations which are self adjoint by using a general theorem on conservation laws proved by one of the authors nhi we find conservation laws for some of these partial differential equations without classical lagrangians

a group theoretic analysis of the generalised gardner equation with arbitrary order nonlinear terms abstract in this paper we analyse a gardner equation with dual power law nonlinearity of any order which has been widely investigated due to its applications in quantum field theory solid state plasma and fluid physics a group theoretic approach is used to perform a comprehensive and detailed analysis of the equation we derive the symmetry generators of the equation in terms of its arbitrary parameters and used them to obtain symmetry reductions and exact solutions furthermore the conservation laws of the equation are derived via the noether approach after increasing the order and by the use of the multiplier method the importance of these conservation laws in finding exact solutions is proved via double reduction theory most of these solutions are new and contain many known solutions as special cases these solutions include important soliton solutions and nontrivial solutions in terms of special functions which are meromorphic in the entire complex plane since the gardner equation can model a variety of wave phenomena in plasma solid state and fluid physics these solutions possess significant features in the non linear mechanics aspects of the work they can also be used as a basis for solving other related model problems and assessing numerical and approximate analytical methods for nonlinear equations describing solitons in wave mechanics

a new dual petrov galerkin method for third and higher odd order differential equations application to the kdv equation a new dual petrov galerkin method is proposed analyzed and implemented for third and higher odd order equations using a spectral discretization the key idea is to use trial functions satisfying the underlying boundary conditions of the differential equations and test functions satisfying the dual boundary conditions the method leads to linear systems which are sparse for problems with constant coefficients and well conditioned for problems with variable coefficients our theoretical analysis and numerical results indicate that the proposed method is extremely accurate and efficient and most suitable for the study of complex dynamics of higher odd order equations

weak self adjoint differential equations the concepts of self adjoint and quasi self adjoint equations were introduced by ibragimov 2006 j math anal appl 318 742 57 2007 arch alga 4 55 60 in ibragimov 2007 j math anal appl 333 311 28 a general theorem on conservation laws was proved in this paper we generalize the concept of self adjoint and quasi self adjoint equations by introducing the definition of weak self adjoint equations we find a class of weak self adjoint quasi linear parabolic equations the property of a differential equation to be weak self adjoint is important for constructing conservation laws associated with symmetries of the differential equation

lump solutions of a generalized calogero bogoyavlenskii schiff equation abstract a generalized calogero bogoyavlenskii schiff equation is considered and based on its hirota bilinear form a class of lump solutions is explicitly generated via symbolic computations with maple together with plots of a specific lump solution the result enlarges the category of nonlinear partial differential equations which possess lump solutions

self adjointness and conservation laws of a generalized burgers equation a 2 1 dimensional generalized burgers equation is considered having written this equation as a system of two dependent variables we show that it is quasi self adjoint and find a nontrivial additional conservation law

nonlinear self adjointness and conservation laws the general concept of nonlinear self adjointness of differential equations is introduced it includes the linear self adjointness as a particular case moreover it embraces the strict self adjointness definition 1 and quasi self adjointness introduced earlier by the author it is shown that the equations possessing nonlinear self adjointness can be written equivalently in a strictly self adjoint form by using appropriate multipliers all linear equations possess the property of nonlinear self adjointness and hence can be rewritten in a nonlinear strictly self adjoint form for example the heat equation ut u 0 becomes strictly self adjoint after multiplying by u 1 conservation laws associated with symmetries are given in an explicit form for all nonlinearly self adjoint partial differential equations and systems