affine automata and related techniques for generation of complex images abstract in this paper we introduce probabilistic affine automata paa which are probabilistic finite generators having transitions labeled with affine transformations it is shown that paa are capable of generating highly complex images barnsley s 1988 ifs method to generate fractals is a special case of paa when the automaton happens to have only a single state a number of theoretical results on paa are shown the relationship between paa mutually recursive function systems mrfs and affine regular sets is investigated

a new framework for multivariate general conformable fractional calculus and potential applications abstract some applications of fractional partial differential equations fpdes based on conformable fractional derivative cfd are discussed recently as an extension of cfd power kernel is generalized to other normalized probability distribution kernels general conformable fractional derivative gcfd provides a framework to explain the physical meaning of cfd and describe which phenomena or processes gcfd cfd are suitable to model we will establish the multivariate theory of gcfd in this paper compared with traditional hereditary fractional calculus like riemann liouville type or caputo type the unified theory of gcfd vector calculus has well forms with simplicity and elegance as applications fractional maxwell s equations of gcfd are given to describe electromagnetic fields of media that demonstrate inhomogeneous complicated and local properties and some fractional parabolic equations are presented as natural extensions of some nonlinear parabolic equations that cover several important famous equations

new exact solutions to some variable coefficients problems abstract with the aid of computer symbolic computation system such as maple an extended tanh method is applied to determine the exact solutions for some nonlinear problems with variable coefficients several new soliton solutions and periodic solutions can be obtained if we taking paraments properly in these solutions the employed methods are straightforward and concise and it can also be applied to other nonlinear evolution equations in mathematical physics

new solutions of distinct physical structures to high dimensional nonlinear evolution equations abstract in this work we employ the tanh coth method for a reliable treatment on nonlinear evolution equations three models namely the 3 1 dimensional jimbo miwa equation the 2 1 dimensional calogero bogoyavlenskii schiff equation and the 3 1 dimensional potential ytsf equation are use as vehicles to conduct the analysis new travelling wave solutions are formally derived the work shows the power of the proposed scheme and the variety of the obtained solutions

multiple soliton solutions for the lax seventh order equation abstract in this work we study the lax seventh order equation the tanh method is used for single soliton solution of this equation the hirota s bilinear formalism combined with the simplified hereman form are applied to determine the n soliton solutions for this equation the work confirms the significant features of the employed methods

tree based picture generation the concept of tree based picture generation is introduced it is shown that there are equivalent tree based definitions of four picture generating devices known from the literature namely collage

tree based generation of languages of fractals the notion of p interpreted top down tree generators is introduced combining the nondeterministic nature of grammars as known from formal language theory with the infinite refinement of pictures studied in fractal geometry

a search for bilinear equations passing hirota s three soliton condition ii mkdv type bilinear equations in this paper the results of a search for complex bilinear equations with two soliton solutions are presented the following basic types are discussed a the nonlinear schrodinger equation b dx g f 0 a dx dt f f gg and b the benjamin ono equation p dx f f 0 it is found that the existence of two soliton solutions is not automatic but introduces conditions that are like the usual three and four soliton conditions the search was limited by the degree of a 2 and by degree of p 4 the main results are the following 1 iad3x dxdt idy b g f 0 d2xf f gg 2 d2x ad2y idt b g f 0 dxdy f f gg 3 iad3x d2x idt f f 0 and 4 dxdt i adx bdt f f 0

the extended tanh method for new solitons solutions for many forms of the fifth order kdv equations abstract the extended tanh method is used to derive new solitons solutions for several forms of the fifth order nonlinear kdv equation the forms include the lax sawada kotera sk sawada kotera parker dye skpd kaup kupershmidt kk kaup kupershmidt parker dye kkpd and the ito equations the criteria established in a m wazwaz abundant solitons solutions for several forms of the fifth order kdv equation by using the tanh method appl math comput in press doi 10 1016 j amc 2006 02 047 to build up reliable relations between the parameters of the equation are confirmed by using this new approach entirely new bell shaped solitons are determined

fundamental properties of infinite trees infinite trees naturally arise in the formalization and the study of the semantics of programming languages this paper investigates some of their combinatorial and algebraic properties that are especially relevant to semantics

stratified self calibration with the modulus constraint in computer vision and especially for 3d reconstruction one of the key issues is the retrieval of the calibration parameters of the camera these are needed to obtain metric information about the scene from the camera often these parameters are obtained through cumbersome calibration procedures there is a way to avoid explicit calibration of the camera self calibration is based on finding the set of calibration parameters which satisfy some constraints e g constant calibration parameters several techniques have been proposed but it often proved difficult to reach a metric calibration at once therefore in the paper a stratified approach is proposed which goes from projective through affine to metric the key concept to achieve this is the modulus constraint it allows retrieval of the affine calibration for constant intrinsic parameters it is also suited for use in conjunction with scene knowledge in addition if the affine calibration is known it can also be used to cope with a changing focal length

dynamical behavior of reaction diffusion neural networks and their synchronization arising in modeling epileptic seizure a numerical simulation study abstract excessive synchronizations of neurons in the brain networks can be a reason for some episodic disorders such as epilepsy in this paper we simulate neural dynamic models and their synchronizations numerically by a new numerical algorithm in order to overcome the complexity of the problem a suitable combination of the legendre spectral element method and operator splitting technique is proposed to convert a complex system of partial differential equations into sparse linear algebraic systems the simulations are explored in different aspects such as accuracy computing the wavefront and angular speeds computing time the condition number of obtained algebraic systems to demonstrate the proposed scheme effectiveness the numerical simulations and comparisons confirm the method is suitable in terms of accuracy and speed

application of the extended g g expansion method to solve the pochhammer chree equations abstract in the present paper the extended g g expansion method is used for a reliable treatment of the pochhammer chree equations which include variety of models travelling wave solutions of distinct structures namely bell shaped solitons kink shaped solitons and periodic solutions are formally derived the method demonstrates power reliability and efficiency

n soliton solutions for shallow water waves equations in 1 1 and 2 1 dimensions abstract a variety of shallow water waves equations in 1 1 and 2 1 dimensions are investigated we first show that these models are completely integrable we next determine multiple soliton solutions for each equation the simplified hirota s bilinear method developed by hereman will be employed to achieve this goal a comparison between dispersion relations and the phase shifts will be conducted but possess the same coefficients for the polynomials of exponentials

multiple soliton solutions for the boussinesq equation abstract in this work we use the hirota s direct method combined with the simplified version of this method to determine the n soliton solutions for the boussinesq equation the one soliton solutions will be handled by the tanh coth method the work highlights the significant features of the employed methods

a function transformation method and exact solutions to a generalized sinh gordon equation based on a transformed painleve property and the variable separated ode method a function transformation method is proposed to search exact solutions to some partial differential equations pdes with hyperbolic or exponential functions the new approach provides a more systematical and convenient handling of the solution process for the nonlinear equations its key point is to eradicate the hyperbolic or exponential terms by a transformed painleve property and reduce the given pdes to a variable coefficient ordinary differential equations then we look for solutions to the resulting equations by some methods as an application exact solutions for a generalized sinh gordon equation are formally derived

a variety of exact travelling wave solutions for the 2 1 dimensional boiti leon pempinelli equation abstract in this paper the 2 1 dimensional boiti leon pempinelli blp equation will be studied the tanh coth method will be used to obtain exact travelling wave solutions for this equation the exp function method will also be applied to the blp equation to derive a new variety of travelling wave solutions with distinct physical structures

an analytic study of fisher s equation by using adomian decomposition method in this paper we develop a framework to obtain exact solutions to fisher s equation and to a nonlinear diffusion equation of the fisher type by employing adomian decomposition method the proposed scheme is supported by examining nonlinear diffusion equations of the fisher type

the hirota s direct method and the tanh coth method for multiple soliton solutions of the sawada kotera ito seventh order equation abstract in this work the sawada kotera ito seventh order equation is studied the tanh coth method is applied to obtain soliton solution of this equation the hirota s direct method combined with the simplified hereman s method are applied to determine the n soliton solutions for this equation the study highlights the significant features of the employed methods and its capability of handling completely integrable equations

exact wave solutions for a 3 1 dimensional generalized b type kadomtsev petviashvili equation abstract in this work we investigate a 3 1 dimensional generalized b type kadomtsev petviashvili equation based on the simplified hirota bilinear method we first construct its soliton solutions meanwhile we correct the formula of n soliton solution for this equation on the basis of these solitons we further calculate its lump solutions periodic waves meanwhile rogue waves as well as interaction solutions of this equation are also obtained by a direct algebraic method some figures are given to display the behavior of these solutions

projective reconstruction and invariants from multiple images this correspondence investigates projective reconstruction of geometric configurations seen in two or more perspective views and the computation of projective invariants of these configurations from their images a basic tool in this investigation is the fundamental matrix that describes the epipolar correspondence between image pairs it is proven that once the epipolar geometry is known the configurations of many geometric structures for instance sets of points or lines are determined up to a collineation of projective 3 space spl pscrsup 3 by their projection in two independent images this theorem is the key to a method for the computation of invariants of the geometry invariants of six points in spl pscrsup 3 and of four lines in spl pscrsup 3 are defined and discussed an example with real images shows that they are effective in distinguishing different geometrical configurations since the fundamental matrix is a basic tool in the computation of these invariants new methods of computing the fundamental matrix from seven point correspondences in two images or six point correspondences in three images are given

multiple front solutions for the burgers kadomtsev petviashvili equation abstract in this work we study a completely integrable dissipative equation the burgers equation is extended by using the sense of the kadomtsev petviashvili kp equation the new established burgers kp equation is studied by using the tanh coth method to obtain kink solutions and periodic solutions we also apply the powerful hirota s bilinear method to establish exact n soliton solutions for the derived integrable equation

theory and practice of projective rectification this paper gives a new method for image rectification the process of resampling pairs of stereo images taken from widely differing viewpoints in order to produce a pair of matched epipolar projections these are projections in which the epipolar lines run parallel with the x axis and consequently disparities between the images are in the x direction only the method is based on an examination of the fundamental matrix of longuet higgins which describes the epipolar geometry of the image pair the approach taken is consistent with that advocated by faugeras 1992 of avoiding camera calibration the paper uses methods of projective geometry to determine a pair of 2d projective transformations to be applied to the two images in order to match the epipolar lines the advantages include the simplicity of the 2d projective transformation which allows very fast resampling as well as subsequent simplification in the identification of matched points and scene reconstruction

the extended tanh method for abundant solitary wave solutions of nonlinear wave equations the extended tanh method is used to establish abundant solitary wave solutions of nonlinear wave equations the obtained solutions include solitons kinks and plane periodic solutions the study is an extension to the remarkable development by malfliet 1 the extended tanh method presents a wider applicability for handling nonlinear wave equations

the tanh method for generalized forms of nonlinear heat conduction and burgers fisher equations generalized forms of the kdv the nonlinear heat conduction and burgers fisher equations are investigated the analysis rests mainly on the standard tanh method the focus is on the case where the parameter m is noninteger a variety of exact travelling wave solutions of distinct physical structures are formally derived

estimation of f matrix and image rectification by double quaternion fundamental matrix or f matrix is one of the most important and elemental tools in the field of computer vision in conventional methods for estimating the f matrix an eight point algorithm is adopted first an approximate f matrix is calculated by a linear solver using at least eight corresponding pairs since this linear optimization method excludes an essential property the rank 2 constraint a method based on a singular value decomposition svd is applied to impose the constraint this last step with svd however provides additional noise in the f matrix several methods introduce parameterizations taking into account the rank 2 constraint and optimized nonlinearly without svd in this paper we propose a novel parameterization for the nonlinear optimization which includes this constraint we adopt double quaternion dq and a scalar as the parameter set experimental results show that the nonlinear optimization with our parameterization is competitive with other parameterization methods moreover through the proposed parameterization we can obtain two transformations for the two input images these transformations lead to a novel method to estimate epipolar lines and to rectify the image pairs this rectification method can deal with any image pairs in the same manner whether the epipoles are inside or outside the images

stereo effect of image converted from planar research on planar to stereo image conversion has significant theoretical and practical implications for image processing in this paper we use a method which segments an image into sub blocks to perform this kind of conversion parameters such as random variables are used for conversion control we use two quantitative criteria cross entropy and root mean square error to evaluate the stereo effect furthermore the stereo effect that the random variables create is discussed the results of the experiment show that i when all random variables have the same distribution different values of these random variables only slightly affect the stereo effect and ii when different distributions are applied to the random variables the cross entropies or root mean square errors are slightly different which indicates different distributions have a small influence on the stereo effect generally we recommend normal distribution for better stereo effect in most cases

multiple front solutions for the burgers equation and the coupled burgers equations abstract in this work multiple front solutions for the burgers equation and the coupled burgers equations are examined the tanh coth method and the cole hopf transformation are used the work highlights the power of the proposed schemes and the structures of the obtained multiple front solutions

nonlinear variants of the improved boussinesq equation with compact and noncompact structures variants of the improved boussinesq equation with positive and negative exponents are investigated it is formally shown that these nonlinear variants give rise to compact and non compact physical structures where compactons solitons solitary patterns and periodic solutions are developed the presented sine cosine ansatz is reliable

observations on the tanh coth expansion method for finding solutions to nonlinear evolution equations abstract the tanh coth expansion method for finding solitary travelling wave solutions to nonlinear evolution equations has been used extensively in the literature it is a natural extension to the basic tanh function expansion method which was developed in the 1990s it usually delivers three types of solution namely a tanh function expansion a coth function expansion and a tanh coth expansion it is known that for every tanh function expansion solution there is a corresponding coth function expansion solution it is shown that there is a tanh coth expansion solution that is merely a disguised version of the coth solution in many papers such tanh coth solutions are erroneously claimed to be new however other tanh coth solutions may be delivered that are genuinely new in the sense that they would not be delivered via the basic tanh function method similar remarks apply to tan cot and tan cot expansion solutions

solutions of the generalized kdv equation with time dependent damping and dispersion solitary wave solutions are obtained for the generalized korteweg de vries gkdv equation with time dependent damping and dispersion by using the tanh coth method the exp function method and the modified sine cosine method these methods are useful and efficient and a variety of solitary wave solutions are obtained that possess variable coefficients

the tanh coth method for solitons and kink solutions for nonlinear parabolic equations abstract the tanh coth method is used to derive solitons and kink solutions for some of the well known nonlinear parabolic partial differential equations the equations include the fisher equation newell whithead equation allen cahn equation fitzhugh nagumo equation fisher s equation and the burgers fisher equation the new tanh coth approach provides abundant solitons and kink solutions in addition to the existing ones the power of this manageable method is confirmed

applications of a direct algorithm for the rectification of uncalibrated images an algorithm for the rectification of uncalibrated images is presented and applied to a variety of cases the algorithm generates the rectifying transformations directly from the geometrical relationship between the images using any three correspondences in the images to define a reference plane a small set of correspondences is used to calculate an initial rectification additional correspondences are introduced semi automatically by correlating regions of the rectified images since the rectified images of surfaces in the reference plane have no relative distortion features can be matched very accurately by correlation allowing small changes in disparity to be detected in the 3 d reconstruction of an architectural scene differences in depth are resolved to about 0 001 of the distance from camera to subject

the interaction processes of the n soliton solutions for an extended generalization of vakhnenko equation the n soliton solutions for an extended generalization of vakhnenko equation are calculated by applying the improved hirota method and the variable transformations the n soliton solutions can be expressed explicitly furthermore the interaction processes are discussed for the n soliton solutions

resonant soliton and complexiton solutions for 3 1 dimensional boiti leon manna pempinelli equation abstract in this letter the linear superposition principle is used to discuss the 3 1 dimensional boiti leon manna pempinelli equation with bilinear derivatives as a result we obtain new resonant soliton and complexiton solutions by discussing two different cases involved the parameters these solutions are a class of n wave solutions of linear combinations of exponential traveling waves

extended transformed rational function method and applications to complexiton solutions abstract the transformed rational function method provides a systematical and convenient handling of the solution process of nonlinear equations unifying the tanh function type methods the homogeneous balance method the exp function method the mapping method and the f expansion type methods in this paper the transformed rational function method is improved and the extended method is used to obtain complexiton solutions to some nonlinear differential equations

the hirota s direct method for multiple soliton solutions for three model equations of shallow water waves abstract multiple soliton solutions for three model equations for shallow water waves are determined the three models are completely integrable the hirota bilinear method is used to determine multiple soliton solutions of sech squared type for these equations the tanh coth method is used to obtain single soliton solutions and other solutions for these three models the three models have different linear dispersion relations but possess the same coefficients for the polynomials of exponentials

multiple soliton solutions for the 2 1 dimensional asymmetric nizhnik novikov veselov equation abstract the asymmetric nizhnik novikov veselov equation will be investigated the hirota bilinear method will be used to derive n soliton solutions of this completely integrable equation multiple singular soliton solutions will be developed as well the asymmetric nizhnik novikov veselov equation does not show any resonant phenomenon

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

in defense of the eight point algorithm the fundamental matrix is a basic tool in the analysis of scenes taken with two uncalibrated cameras and the eight point algorithm is a frequently cited method for computing the fundamental matrix from a set of eight or more point matches it has the advantage of simplicity of implementation the prevailing view is however that it is extremely susceptible to noise and hence virtually useless for most purposes this paper challenges that view by showing that by preceding the algorithm with a very simple normalization translation and scaling of the coordinates of the matched points results are obtained comparable with the best iterative algorithms this improved performance is justified by theory and verified by extensive experiments on real images

multiple soliton solutions for the lax kadomtsev petviashvili lax kp equation a new completely integrable dispersive equation is derived the new equation is obtained by extending the lax fifth order equation using the sense of the kadomtsev petviashvili kp equation in extending the kdv equation the newly derived lax kadomtsev petviashvili lax kp equation is investigated by using the tanh coth method and the hirota bilinear method to derive single soliton solution and n soliton solutions respectively the study highlights the multiple soliton solutions of the derived completely integrable equation

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

variants of the two dimensional boussinesq equation with compactons solitons and periodic solutions variants of the two dimensional boussinesq equation with positive and negative exponents are studied the sine cosine ansatz is fruitfully used to carry out the analysis exact solutions of different physical structures compactons solitary patterns solitons and periodic solutions are obtained the quantitative change in the physical structure of the solutions is shown to depend mainly on the exponent of the wave function u x t and on the ratio a b of the derivatives of u x t

the tanh method exact solutions of the sine gordon and the sinh gordon equations in this work we use the tanh method for traveling wave solutions of the sine gordon and the sinh gordon equations several exact solutions of distinct physical structures are obtained the method is powerful with minimal algebra work and is demonstrated for four models

multiple soliton solutions for the generalized 1 1 dimensional and the generalized 2 1 dimensional ito equations abstract the generalized 1 1 dimensional and the generalized 2 1 dimensional ito equations obtained by introducing a new type of bilinearism are studied the tanh coth method is used to obtain single soliton solutions and periodic solutions for the ito equations the hirota bilinear method is used to determine the multiple soliton solutions of sech squared type for these equations new entirely solutions are obtained the analysis emphasizes the power of the two methods and its capability of handling completely integrable equations

the tanh method a tool for solving certain classes of nonlinear evolution and wave equations the tanh or hyperbolic tangent method is a powerful technique to search for travelling waves coming out from one dimensional nonlinear wave and evolution equations in particular in those problems where dispersive effects reaction diffusion and or convection play an important role to show the strength of the method an overview is given to find out which kind of problems are solved with this technique and how in some nontrivial cases this method adapted to the problem at hand still can be applied single as well as coupled equations arising from wave phenomena which appear in different scientific domains such as physics chemical kinetics geochemistry and mathematical biology will be treated next attention is focussed towards approximate solutions as a result solitary and shock wave profiles are derived together with the associated width and velocity the same method can be easily extended so that difference differential equations can be similarly solved finally some extensions of the method are discussed

the improved g g expansion method and its applications to the broer kaup equations and approximate long water wave equations by introducing a new general ansatze the improved g g expansion method is proposed to construct exact solutions of both broer kaup equations and approximate long water wave equations as a result some new travelling wave solutions involving parameters expressed by three types of functions which are the hyperbolic functions the trigonometric functions and the rational functions are obtained when the parameters are taken as special values the solitary wave solutions are derived from the hyperbolic function solutions the proposed method is straightforward concise and effective and can be applied to other nonlinear evolution equations in mathematical physics

on classes of integrable systems and the painleve property the caudrey dodd gibbon equation is found to possess the painleve property investigation of the backlund transformations for this equation obtains the kuperschmidt equation a certain transformation between the kuperschmidt and caudrey dodd gibbon equation is obtained this transformation is employed to define a class of p d e s that identically possesses the painleve property for equations within this class backlund transformations and rational solutions are investigated in particular the sequences of higher order kdv caudrey dobb gibbon and kuperschmidt equations are shown to possess the painleve property

complexiton solutions for 3 1 dimensional kdv type equation abstract 3 1 dimensional nonlinear kdv type equation is solved by wazwaz and zhaqilao s method which is arisen by employing complex parameters instead of real parameters and considered generalization of simplified hirota method complexiton solutions which include both trigonometric and exponential functions are obtained for referred equation also some special conditions to specify the non singularity and type of solutions are derived

the hirota s bilinear method and the tanh coth method for multiple soliton solutions of the sawada kotera kadomtsev petviashvili equation abstract in this work we derive a new completely integrable dispersive equation the equation is obtained by combining the sawada kotera sk equation with the sense of the kadomtsev petviashvili kp equation the newly derived sawada kotera kadomtsev petviashvili sk kp equation is studied by using the tanh coth method to obtain single soliton solution and by the hirota bilinear method to determine the n soliton solutions the study highlights the significant features of the employed methods and its capability of handling completely integrable equations

multiple soliton solutions of two extended model equations for shallow water waves abstract multiple soliton solutions for two extensions of model equations for shallow water waves presented in m j ablowitz d j kaup a c newell h segur the inverse scattering transform fourier analysis for nonlinear problems stud appl math 53 1974 249 315 r hirota j satsuma n soliton solutions of model equations for shallow water waves j phys soc jpn 40 2 1976 611 612 are obtained the two extensions possess identical dispersion relations but with different structures for the n soliton solutions for n 1 the hirota bilinear method is used to determine multiple soliton solutions of sech squared type for these extended equations the tanh coth method is used to obtain single soliton solutions

multiple soliton solutions for the fifth order caudrey dodd gibbon cdg equation abstract in this work we use the hirota s direct method combined with the simplified hereman method to determine the n soliton solutions for the fifth order caudrey dodd gibbon cdg equation the one soliton solution will be handled by the tanh coth method the work highlights the significant features of the employed methods

a jacobi gauss lobatto collocation method for solving generalized fitzhugh nagumo equation with time dependent coefficients in this paper we propose a new jacobi gauss lobatto collocation method for solving the generalized fitzhugh nagumo equation the jacobi gauss lobatto points are used as collocation nodes for spatial derivatives the proposed method has the advantage of obtaining the solution in terms of the jacobi parameters a and b in addition the problem is reduced to a system of ordinary differential equations in time this system can be solved by any standard numerical techniques numerical solutions obtained by this method when compared with the exact solution reveal that the obtained solution produces high accurate results numerical results show that the proposed method is of high accuracy and is efficient for solving the generalized fitzhugh nagumo equation also the results demonstrate that the proposed method is powerful algorithm for solving the nonlinear partial differential equations

multiple soliton solutions for the kp equation by hirota s bilinear method and by the tanh coth method abstract in this work we use the hirota s direct method combined with the simplified version of this method to determine the n soliton solutions n 1 2 3 4 5 for the kadomtsev petviashvili kp equation the one soliton solutions will be handled by the tanh coth method the work emphasizes the power of the two methods that will be used

solitary wave solutions of the generalized shallow water wave gsww equation by hirota s method tanh coth method and exp function method abstract the generalized short wave equation gsww is studied by three distinct methods the hirota s bilinear method is used to derive multiple soliton solutions for the completely integrable forms of the gsww equation the tanh coth method is used to obtain single soliton solutions for this equation the exp function method is also applied to derive a variety of travelling wave solutions with distinct physical structures for this non linear evolution equation

travelling wave solutions of nonlinear evolution equations using the simplest equation method in this paper the simplest equation method has been used for finding the exact solutions of three nonlinear evolution equations namely the vakhnenko parkes equation the generalized regularized long wave equation and the symmetric regularized long wave equation all three of these equations arise in fluids science so finding their exact solutions is of great importance

an explicit hybrid finite difference scheme for the allen cahn equation abstract in this paper we propose an explicit hybrid numerical method for solving the allen cahn equation which models antiphase domain coarsening process in a binary mixture the proposed method is based on an operator splitting method first we solve the linear diffusion part using the explicit euler method second we solve the nonlinear term using the closed form analytical solution we show the stability condition of the proposed numerical scheme we also show the pointwise boundedness of the numerical solution for the allen cahn equation under a solvability condition numerical experiments such as linear stability analysis traveling wave motion by mean curvature image segmentation and crystal growth are presented to demonstrate the performance of the proposed method

the tanh coth and the sine cosine methods for kinks solitons and periodic solutions for the pochhammer chree equations abstract in this paper the tanh coth method and the sine cosine method are used for a reliable treatment of the pochhammer chree equations which include a variety of models travelling wave solutions of distinct structures namely bell shaped solitons kink shaped solitons and periodic solutions are formally derived the two methods demonstrate power reliability and efficiency

multiple soliton solutions for the ninth order kdv equation and sixth order boussinesq equation abstract two nonlinear dispersive equations namely the ninth order kdv equation and the sixth order boussinesq equation are formally derived by generalizing the bilinear forms of the kdv and boussinesq equations respectively the two equations are approached by using the tanh coth method to obtain single soliton solutions and by the hirota bilinear method to determine the multiple soliton solutions the study highlights the fact that both equations are completely integrable and admits n soliton solutions

using string languages to describe picture languages a picture is a set of unit lines from the cartesian plane considered as a square grid a word over the alphabet l r u d is a picture description in the sense that it represents a traversal of a picture where the interpretation of the symbols l r u d is l go one unit line to the left of the current point r go one unit line to the right of the current point u go one unit line up from the current point and d go one unit line down from the current point a set of picture descriptions forms a picture description language this paper investigates the basic properties of pictures and picture description languages from the formal language theory point of view

the tanh method for traveling wave solutions of nonlinear equations in this work we employ the tanh method for traveling wave solutions of nonlinear equations the study is extended to equations that do not have tanh polynomial solutions the efficiency of the method is demonstrated by applying it for a variety of selected equations