bifurcation and exact traveling wave solutions for dual power zakharov kuznetsov burgers equation with fractional temporal evolution abstract in this article we introduce the dual power zakharov kuznetsov burgers equation with fractional temporal evolution in the sense of modified riemann liouville derivative we investigate the dynamical behavior bifurcations and phase portrait analysis of the exact traveling wave solutions of the system with and without damping effect we apply the g g expansion method in context of fractional complex transformation and seek a variety of exact traveling wave solutions including solitary wave kink type wave breaking wave and periodic wave solutions of the equation furthermore the remarkable features of the traveling wave solutions and phase portraits of dynamical system are demonstrated through interesting figures

single machine scheduling to minimize total weighted earliness subject to minimal number of tardy jobs abstract motivated by just in time manufacturing we consider a single machine scheduling problem with dual criteria i e the minimization of the total weighted earliness subject to minimum number of tardy jobs we discuss several dominance properties of optimal solutions we then develop a heuristic algorithm with time complexity o n 3 and a branch and bound algorithm to solve the problem the computational experiments show that the heuristic algorithm is effective in terms of solution quality in many instances while the branch and bound algorithm is efficient for medium size problems

classical non markovian boltzmann equation the modeling of particle transport involves anomalous diffusion x2 t t with 1 with subdiffusive transport corresponding to 0 1 these anomalies give rise to fractional advection dispersion equations with memory in space and time the usual boltzmann equation with only isolated binary collisions is markovian and in particular the contributions of the three particle distribution function are neglected we show that the inclusion of higher order distribution functions give rise to an exact non markovian boltzmann equation with resulting transport equations for mass momentum and kinetic energy with memory in both time and space the two and the three particle distribution functions are considered under the assumption that the two and the three particle correlation functions are translationally invariant that allows us to obtain advection dispersion equations for modeling transport in terms of spatial and temporal fractional derivatives

a fixed interval due date scheduling problem with earliness and due date costs in this paper we consider a problem where n jobs are to be scheduled on a single machine and assigned to one of the two due dates which are given at equal intervals tardy jobs are not allowed thus the due dates are deadlines there is a linear due date penalty and linear earliness penalty and the sum of these penalties is to be minimized this problem is shown to be np hard one case of the problem the unrestricted case is shown to be easy to solve for the restricted case lower and upper bounds are developed computational experience is reported which suggests that the bounds are quite effective

on invariant analysis of some time fractional nonlinear systems of partial differential equations i an investigation of lie point symmetries for systems of time fractional partial differential equations including ito system coupled burgers equations coupled korteweg de vries equations hirota satsuma coupled kdv equations and coupled nonlinear hirota equations has been done using the obtained symmetries each one of the systems is reduced to the nonlinear system of fractional ordinary differential equations involving erdelyi kober fractional differential operator depending on a parameter

local fractional fokker planck equation we propose a new class of differential equations which we call local fractional differential equations they involve local fractional derivatives and appear to be suitable to deal with phenomena taking place in fractal space and time a local fractional analog of the fokker planck equation has been derived starting from the chapman kolmogorov condition we solve the equation with a specific choice of the transition probability and show how subdiffusive behavior can arise

thermal wave model of bioheat transfer with modified riemann liouville fractional derivative in this paper a new fractional thermal wave model of the bioheat transfer ftwmbt caused by spatial heating is built using taylor s series expansion of modified riemann liouville fractional derivatives a one dimensional analytical solution of the ftwmbt in a finite medium is obtained the ftwmbt in the case 1 interpolates the standard thermal wave model of bioheat transfer and the well known pennes bioheat equation 0 finally numerical results are presented graphically for various values of different parameters this study demonstrates that fractional models can provide a unified approach to examine the heat transfer in biological tissue

numerical solutions for fractional reaction diffusion equations fractional diffusion equations are useful for applications in which a cloud of particles spreads faster than predicted by the classical equation in a fractional diffusion equation the second derivative in the spatial variable is replaced by a fractional derivative of order less than two the resulting solutions spread faster than the classical solutions and may exhibit asymmetry depending on the fractional derivative used fractional reaction diffusion equations combine the fractional diffusion with a classical reaction term in this paper we develop a practical method for numerical solution of fractional reaction diffusion equations based on operator splitting then we present results of numerical simulations to illustrate the method and investigate properties of numerical solutions we also discuss applications to biology where the reaction term models species growth and the diffusion term accounts for movements

derivation and solutions of some fractional black scholes equations in coarse grained space and time application to merton s optimal portfolio abstract by using the new fractional taylor s series of fractional order f x h e h d x f x where e denotes the mittag leffler function and d x is the so called modified riemann liouville fractional derivative which we introduced recently to remove the effects of the non zero initial value of the function under consideration one can meaningfully consider a modeling of fractional stochastic differential equations as a fractional dynamics driven by a usual gaussian white noise one can then derive two new families of fractional black scholes equations and one shows how one can obtain their solutions merton s optimal portfolio is once more considered and some new results are contributed with respect to the modeling on one hand and to the solution on the other hand finally one makes some proposals to introduce real data and virtual data in the basic equation of stock exchange dynamics

on the solutions of nonlinear fractional klein gordon equation with modified riemann liouville derivative abstract in this paper fractional variational iteration method fvim is applied for finding approximate analytical solutions of nonlinear fractional klein gordon equation time fractional derivative is described in the riemann liouville sense a new application of fractional variational iteration method fvim is extended to derive analytical solutions in the form of a series for this equation the behavior of the solutions and the effects of different values of fractional order are indicated graphically it is shown that the solutions obtained by the fvim are reliable convenient and effective for strongly nonlinear partial equations with modified riemann liouville derivative

block pulse series analysis of linear systems incorporating observers the paper considers the analysis of linear systems incorporating observers via block pulse functions and presents recursive algorithms convenient for computers compared with walsh function approaches the results are much simpler in form an example is given to illustrate the use of the method

a wavelet operational method for solving fractional partial differential equations numerically fractional calculus is an extension of derivatives and integrals to non integer orders and a partial differential equation involving the fractional calculus operators is called the fractional pde they have many applications in science and engineering however not only the analytical solution existed for a limited number of cases but also the numerical methods are very complicated and difficult in this paper we newly establish the simulation method based on the operational matrices of the orthogonal functions we formulate the operational matrix of integration in a unified framework by using the operational matrix of integration we propose a new numerical method for linear fractional partial differential equation solving in the method we 1 use the haar wavelet 2 establish a lyapunov type matrix equation and 3 obtain the algebraic equations suitable for computer programming two examples are given to demonstrate the simplicity clarity and powerfulness of the new method

fractional order legendre operational matrix of fractional integration for solving the riccati equation with fractional order this paper is devoted to both theoretical and numerical study of riccati equation with fractional order a formulation to the fractional order legendre operational matrix of fractional integration is constructed existence and uniqueness results for the considered problem are provided and proved the fractional derivative is described in the caputo sense some numerical examples are discussed to demonstrate the efficiency and the accuracy of the proposed algorithm

approximate solution of fractional vibration equation using jacobi polynomials an approximate method using jacobi polynomials is proposed for the approximate solutions of fractional vibration equation convergence and numerical stability of the proposed method are shown results are discussed for different cases of jacobi polynomials results are compared with existing analytical methods in this paper we present a method based on the jacobi polynomials for the approximate solution to fractional vibration equation fve of large membranes proposed method converts the fve into sylvester form of algebraic equations whose solution gives the approximate solution convergence analysis of the proposed method is given it is also shown that our approximate method is numerically stable numerical results are discussed for different values of wave velocities and fractional order involved in the fve these numerical results are shown through figures for particular cases of jacobi polynomials such as 1 legendre polynomial 2 chebyshev polynomial of second kind 3 chebyshev polynomial of third kind 4 chebyshev polynomial of fourth kind 5 gegenbauer polynomial the accuracy of the proposed method is proved by comparing results of our method and other exiting analytical methods comparison of results are presented in the form of tables for particular cases of fve and jacobi polynomials

laguerre operational matrices for fractional calculus and applications the laguerre operational matrices for the integration and differentiation of a laguerre vector whose elements are laguerre polynomials are generalized to fractional calculus for investigating distributed systems the generalized operational matrices corresponding to 8 1 8 8 82 1 and exp 8 8 1 are derived as examples comparison of the laguerre series approximate inversions of irrational laplace transforms with exact solutions is very satisfactory

the fractional quadratic form identity and hamiltonian structure of an integrable coupling of the fractional ablowitz kaup newell segur hierarchy starting from a general isospectral problem of fractional order we propose a fractional quadratic form identity from which the hamiltonian structure of an integrable coupling of the fractional ablowitz kaup newell segur hierarchy is derived the method can be generalized to other fractional integrable couplings

lie symmetry analysis conservation laws and numerical approximations of time fractional fokker planck equations for special stochastic process in foreign exchange markets abstract in this paper the transition joint probability density function of the solution of the ornstein uhlenbeck process is presented by a deterministic parabolic time fractional pde fpde named time fractional fokker planck equation the article generally is divided to two parts theoretical and approximated analysis in the theoretical sections lie group method and invariant subspace method are applied for finding exact solutions and conservation laws of the considered equation in the next part by chebyshev wavelets s method the numerical solutions are driven then the usefulness of this approximated method is comparing with the exact solutions by some plotted graphs

fractional variational calculus for nondifferentiable functions we prove necessary optimality conditions in the class of continuous functions for variational problems defined with jumarie s modified riemann liouville derivative the fractional basic problem of the calculus of variations with free boundary conditions is considered as well as problems with isoperimetric and holonomic constraints

solutions of the initial value problem for nonlinear fractional ordinary differential equations by the rach adomian meyers modified decomposition method abstract in this paper we present the generalized adomian rach theorem and the generalized rach adomian meyers modified decomposition method for solving multi order nonlinear fractional ordinary differential equations we consider different classes of initial value problems for nonlinear fractional ordinary differential equations including the case of real valued orders and another case of rational valued orders which are solved by the present method this method can treat any analytic nonlinearity the coefficients of the solution in the form of a generalized power series are determined by a convenient recurrence scheme which does not involve integration operations compared with the classic adomian decomposition method

on the design of discrete time optimal dynamic controllers using the partial state observer abstract this paper considers the problem of designing optimal dynamic controllers based on a partial state observer for discrete time linear time invariant systems with inaccessible state it is assumed that the initial system state is an unknown random vector with known mean and covariance the performance index is taken to be the expectation with respect to the initial state of the standard quadratic one for the discrete time regulator necessary and sufficient conditions for optimality are derived provided that the subspace which ensures the existence of a partial state observer is given a design algorithm is also obtained which can be directly solved for the parameters of the optimal observer and requires the solution of a matrix riccati equation of the observer order as in the continuous time case the separation property in the restricted sense is seen to hold finally an example is presented to illustrate the procedure

derivation of a new mertons optimal problem presented by fractional stochastic stock price and its applications in this article a new model of mertons optimal problem is derived this derivation is based on stock price presented by fractional order stochastic differential equation an extension of hamiltonjacobibellman is used to transfer our proposed model to a fractional partial differential equation as an application of our proposed model two optimal problems are discussed and solved analytically

a fractional calculus of variations for multiple integrals with application to vibrating string we introduce a fractional theory of the calculus of variations for multiple integrals our approach uses the recent notions of riemann liouville fractional derivatives and integrals in the sense of jumarie the main results provide fractional versions of the theorems of green and gauss fractional euler lagrange equations and fractional natural boundary conditions as an application we discuss the fractional equation of motion of a vibrating string

on the fractional solution of the equation f x y f x f y and its application to fractional laplace s transform abstract it is shown that if the problem is defined in the setting of fractional calculus via fractional difference on non differentiable functions then the solution of the functional equation f x y f x f y is exactly defined as the solution of a linear fractional differential equation the dual or counterpart problem that is the fractional solution of the equation g xy g x g y is also considered and it is shown that the corresponding solution is the logarithm of fractional order defined as the inverse of a generalized mittag leffler function which is nowhere differentiable this framework suggests a definition of fractional laplace s transform expressed in terms of generalized mittag leffler function and its main properties are outlined mainly inverse function and convolution one takes this opportunity to display a new fractional taylor s series for functions f x y of two variables x and y many open problems are stated which are directly related to the non differentiability of the functions so involved therefore the title on the fractional solution

a nonlinear fractional sharma tasso olver equation due to its importance in scientific applications the fractional nonlinear sharma tasso olver equation has been studied by several authors from point of view of exact solutions in this paper new explicit and exact solutions are obtained by using the fractional complex transform and the improved extended tanh coth method the results can be compared with other methods used to solve the same model

bmo and morrey campanato estimates for stochastic convolutions and schauder estimates for stochastic parabolic equations abstract in this paper we are aiming to prove several regularity results for the following stochastic fractional heat equations with additive noises d u t x 2 u t x d t g t x d t u 0 0 t 0 t x g for a random field u t x 0 t g u t x u t x r where 2 2 0 2 is the fractional laplacian t 0 is arbitrarily fixed g r d is a bounded domain g 0 t g r is a joint measurable coefficient and t t 0 is either a brownian motion or a levy process on a given filtered probability space f p f t t 0 t to this end we derive the bmo estimates and morrey campanato estimates respectively for stochastic singular integral operators arising from the equations concerned then by utilizing the embedding theory between the campanato space and the holder space we establish the controllability of the norm of the space c 2 d where 0 d 0 t g with all these in hand we are able to show that the q th order bmo quasi norm of the q 0 order derivative of the solution u is controlled by the norm of g under the condition that t is a levy process finally we derive the schauder estimate for the p moments of the solution of the above stochastic fractional heat equations driven by levy noise

algorithm for minimizing weighted earliness penalty in single machine problem in this paper the problem of minimizing the weighted earliness penalty in a single machine scheduling problem is addressed for this problem every job is assumed to be available at time zero and must be completed before or on its deadline no tardy job is allowed each job has its own earliness penalty and deadline the paper identifies several local optimality conditions for sequencing of adjacent jobs a heuristic algorithm is developed based on these local optimality conditions sample problems are solved and the solutions obtained from the heuristic are compared to solutions obtained from the heuristics developed by chand and schneeberger also comparisons are performed between the solutions obtained from the heuristic and the optimal solutions obtained from a mathematical modeling approach for problems involving 10 and 15 jobs the results show that the developed heuristic produces solutions close to optimal in small size problems and it also outperforms the chand and schneeberger s method

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

existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay a class of nonlinear fractional order partial differential equations with delay email protected au x t t a a t u x t f t u x t 1 t u x t l t email protected 0 t 0 be investigated in this paper where cd a is the standard caputo s fractional derivative of order email protected email protected 1 and l is a positive integer number the function f is defined as f t u 1 u l rxrx xr r and email protected email protected is a m dimension space using lebesgue dominated convergence theorem leray schauder fixed point theorem and banach contraction mapping theorem we obtain some sufficient conditions for the existence of the solutions of the above fractional order partial differential equations

minimizing the weighted number of early and tardy jobs in a stochastic single machine scheduling problem we study a static stochastic single machine scheduling problem in which jobs have random processing times with arbitrary distributions due dates are known with certainty and fixed individual penalties or weights are imposed on both early and tardy jobs the objective is to find an optimal sequence that minimizes the expected total weighted number of early and tardy jobs the general problem is np hard to solve however in this paper we develop certain conditions under which the problem is solvable exactly an efficient heuristic is also introduced to find a candidate for the optimal sequence of the general problem our illustrative examples and computational results demonstrate that the heuristic performs well in identifying either optimal sequences or good candidates with low errors furthermore we show that special cases of the problem studied here reduce to some classical stochastic single machine scheduling problems including the problem of minimizing the expected weighted number of early jobs and the problem of minimizing the expected weighted number of tardy jobs which are both solvable by the proposed exact or heuristic methods

variational problems with fractional derivatives euler lagrange equations we generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized also for the standard case when these two bounds coincide we derive a new form of euler lagrange equations we use approximations for fractional derivatives in the lagrangian and obtain the euler lagrange equations which approximate the initial euler lagrange equations in a weak sense

reduction of the characteristic initial value problem to the cauchy problem and its applications to the einstein equations a method is described by means of which the characteristic initial value problem can be reduced to the cauchy problem and examples are given of how it can be used in practice as an application it is shown that the characteristic initial value problem for the einstein equations in vacuum or with perfect fluid source is well posed when data are given on two transversely intersecting null hypersurfaces a new discussion is given of the freely specifiable data for this problem

a generalized fractional trace identity and hamiltonian structure of the fractional soliton equation hierarchy a generalized fractional trace identity and hamiltonian structure of the fractional soliton equation hierarchy are presented by using the modified riemann liouville derivative and exterior derivatives of fractional orders example of the fractional kaup newell kn soliton equation hierarchy and hamiltonian system are considered which is a new integrable hierarchy and possesses hamiltonian structure

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

wavelet based quasilinearization method for semi linear parabolic initial boundary value problems finite difference free wavelets based quasilinearization methods developed for spibvp fisher huxley and newell whitehead segel equations solved very efficiently comparison study provided for quasilinearization based on haar and legendre wavelets proposed approach performs better than the methods in the recent literature in this paper numerical methods based on quasilinearization and haar and legendre wavelets to solve a class of semi linear parabolic initial boundary value problem spibvp have been presented the haar and legendre wavelet methods have been successfully combined with quasilinearization to solve spibvp efficiently the presented numerical scheme has been illustrated using appropriate examples including fisher equation and the obtained results show that the proposed numerical scheme is robust and easy to apply

modified riemann liouville derivative and fractional taylor series of nondifferentiable functions further results the paper gives some results and improves the derivation of the fractional taylor s series of nondifferentiable functions obtained recently in the form f g h e a h ad g a f g 0 a 1 where e a is the mittag leffier function here one defines fractional derivative as the limit of fractional difference and by this way one can circumvent the problem which arises with the definition of the fractional derivative of constant using riemann liouville definition as a result a modified riemann liouville definition is proposed which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered one s in order to support this f taylor series one shows how its first term can be obtained directly in the form of a mean value formula the fractional derivative of the dirac delta function is obtained together with the fractional taylor s series of multivariate functions the relation with irreversibility of time and symmetry breaking is exhibited and to some extent this f taylor s series generalizes the fractional mean value formula obtained a few years ago by kolwantar

block pulse operational matrix method for solving fractional partial differential equation in this paper we first introduce two dimensional block pulse functions and the block pulse operational matrices of the fractional order integration also the block pulse operational matrices of the fractional order differentiation are obtained then we present a computational method based on the above results for solving a class of fractional partial differential equations transforming the initial equation into a sylvester equation the error analysis of the method is given the method is computationally attractive and applications are demonstrated by some numerical examples moreover comparing the methodology with the known technique shows that our approach is more efficient and more convenient

a reliable numerical algorithm for fractional advection dispersion equation arising in contaminant transport through porous media abstract this article presents a reliable numerical approach for the fractional advection dispersion equation by making use of legendre scaling functions as a basis the fractional advection dispersion equation describes the anomalous transport in surface and subsurface water using two dimensional basis formed by legendre scaling functions we get operational matrix for fractional integrations and differentiations substituting these operational matrices in the equation leads linear algebraic equations whose solutions can be derived with the aid of sylvester s approach this in turn yields approximate solutions for advection dispersion equation convergence analysis of the proposed scheme is presented the potency and accuracy of the proposed numerical algorithm are shown by plotting error figures

analytical approximate solutions for nonlinear fractional differential equations we consider a class of nonlinear fractional differential equations fdes based on the caputo fractional derivative and by extending the application of the adomian decomposition method we derive an analytical solution in the form of a series with easily computable terms for linear equations the method gives exact solution and for non linear equations it provides an approximate solution with good accuracy several examples are discussed

the operational matrices of integration and differentiation for the fourier sine cosine and exponential series for the fourier sine cosine series basis vector varphi t and the fourier exponential series basis vector psi t a linear nonsingular transfor marion t is determined such that psi t t varphi t this result is then used to show that the operational matrices of integration p and q for varphi t and psi t respectively are related by the expression tp qt analogous results are derived for the corresponding operational matrices of differentiation d and r general expressions are derived for t p q d and r

on the smoothness of the critical sets of the cylinder at spatial infinity in vacuum spacetimes we analyze the appearance of logarithmic terms at the critical sets of friedrich s cylinder representation of spatial infinity it is shown that if the radiation field vanishes at all orders at the critical sets no logarithmic terms are produced in the formal expansions conversely it is proved that under the additional hypothesis that the spacetime has constant adm mass aspect and vanishing dual adm mass aspect by which we mean the limit of the bondi dual mass aspect on to the critical set this condition is also necessary for a spacetime to admit a smooth conformal representation at the critical sets

derivation of an amplitude of information in the setting of a new family of fractional entropies by generalizing the basic functional equation f xy f x f y in the form f b xy f b x f b y b 1 one can derive a family of solutions which are exactly the inverse of the mittag leffler function referred to as mittag leffler logarithm or logarithm of fractional order this result provides a new family of generalized informational entropies which are indexed by a parameter clearly related to fractals via fractional calculus and which is quite relevant in the presence in defect of observation the relation with shannon s entropy renyi s entropy and tsallis entropy is clarified and it is shown that tsallis generalized logarithm has a significance in terms of fractional calculus the case b 2 looks like directly relevant to amplitude of probability in quantum mechanics and provides an approach to the definition of amplitude of informational entropy one examines the kind of result one can so obtain in applying the maximum entropy principle in the presence of uncertain definition or fuzzy definition the mittag leffler function would be more relevant than the gaussian normal law to some extent this new formulation could be fully supported by the derivation of a new family of fractional fisher information

characteristic initial data and smoothness of scri ii asymptotic expansions and construction of conformally smooth data sets we derive necessary and sufficient conditions on characteristic initial data for friedrich s conformal field equations in 3 1 dimensions to have no logarithmic terms in an asymptotic expansion at null infinity

stochastic calculus for fractional brownian motion i theory in this paper a stochastic calculus is given for the fractional brownian motions that have the hurst parameter in 1 2 1 a stochastic integral of ito type is defined for a family of integrands so that the integral has zero mean and an explicit expression for the second moment this integral uses the wick product and a derivative in the path space some ito formulae or change of variables formulae are given for smooth functions of a fractional brownian motion or some processes related to a fractional brownian motion a stochastic integral of stratonovich type is defined and the two types of stochastic integrals are explicitly related a square integrable functional of a fractional brownian motion is expressed as an infinite series of orthogonal multiple integrals

an efficient computational method for solving system of nonlinear generalized abel integral equations arising in astrophysics abstract in this paper we aim to solve nonlinear system of generalized abel integral equations arising in astrophysics the suggested approach is operational matrix technique by using legendre scaling functions as a basis convergence analysis of the suggested technique is provided numerical experiments are performed to show the effectiveness of the proposed scheme the results are shown through figures and tables

a fractional variational iteration method for solving fractional nonlinear differential equations recently fractional differential equations have been investigated by employing the famous variational iteration method however all the previous works avoid the fractional order term and only handle it as a restricted variation a fractional variational iteration method was first proposed in g c wu e w m lee fractional variational iteration method and its application phys lett a 374 2010 2506 2509 and gave a generalized lagrange multiplier in this paper two fractional differential equations are approximately solved with the fractional variational iteration method

fractional fokker planck equation solution and application recently metzler et al phys rev lett 82 3563 1999 introduced a fractional fokker planck equation ffpe describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a boltzmann thermal heat bath in this paper we present the solution of the ffpe in terms of an integral transformation the transformation maps the solution of ordinary fokker planck equation onto the solution of the ffpe and is based on levy s generalized central limit theorem the meaning of the transformation is explained based on the known asymptotic solution of the continuous time random walk ctrw we investigate in detail i a force free particle ii a particle in a uniform field and iii a particle in a harmonic field we also find an exact solution of the ctrw and compare the ctrw result with the corresponding solution of the ffpe the relation between the fractional first passage time problem in an external nonlinear field and the corresponding integer first passage time is given an example of the one dimensional fractional first passage time in an external linear field is investigated in detail the ffpe is shown to be compatible with the scher montroll approach for dispersive transport and thus is applicable in a large variety of disordered systems the simple ffpe approach can be used as a practical tool for a phenomenological description of certain types of complicated transport phenomena

haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations haar wavelet operational matrix has been widely applied in system analysis system identification optimal control and numerical solution of integral and differential equations in the present paper we derive the haar wavelet operational matrix of the fractional order integration and use it to solve the fractional order differential equations including the bagley torvik ricatti and composite fractional oscillation equations the results obtained are in good agreement with the existing ones in open literatures and it is shown that the technique introduced here is robust and easy to apply

quadrature rules for numerical integration based on haar wavelets and hybrid functions in this paper haar wavelets and hybrid functions have been applied for numerical solution of double and triple integrals with variable limits of integration this approach is the generalization and improvement of the methods siraj ul islam et al 2010 9 where the numerical methods are only applicable to the integrals with constant limits apart from generalization of the methods 9 the new approach has two major advantages over the classical methods based on quadrature rule i no need of finding optimum weights as the wavelet and hybrid coefficients serve the purpose of optimal weights automatically ii mesh points of the wavelets algorithm are used as nodal values instead of considering the n nodes as unknown roots of polynomial of degree n the new methods are more efficient the novel methods are compared with existing methods and applied to a number of benchmark problems accuracy of the methods are measured in terms of absolute errors

a wavelet based approach for vibration analysis of framed structures in this paper an explicit time integration scheme is proposed for structural vibration analysis by using wavelet functions initially the differential equation of vibration governing sdof single degree of freedom systems has been solved by wavelet operators and later the proposed approach has been generalized for mdof multi degrees of freedom systems for this purpose two different types of wavelet functions have been exemplified including complex chebyshev wavelet functions and simple haar wavelet functions in the proposed approach a straightforward formulation has been derived from the numerical approximation of response through the wavelet definition emphasizing on frequency domain approximation a simple step by step algorithm has been implemented and improved to calculate the response of mdof systems moreover stability and accuracy of results have been evaluated the effectiveness of the proposed approach is demonstrated using three examples compared with some of the existing numerical integration schemes such as family of newmark b wilson q and central difference method in all the procedures computation time involved has also been considered finally it is concluded that the vibration analysis of structures is improved by lesser computation time and high accuracy of proposed approach particularly in large scaled systems

legendre polynomials approximation to dynamic linear state equations with initial or boundary value conditions the state equations of the linear dynamic system with initial or boundary value conditions are solved by an effective shifted legendre polynomials approximation a powerful calculation algorithm is proposed to integrate the state equation with very accurate results as time tends to infinity a recursive formula is developed to calculate the expansion coefficients of the state functions for saving computer time and minimizing the computational error the proposed method is very suitable to solve a class of problems with peculiar control functions such as periodic functions or piecewise discontinuous functions several illustrative examples are given satisfactory computational results are obtained

analytical treatment of differential equations with fractional coordinate derivatives this paper outlines a reliable strategy to use the homotopy perturbation method based on jumarie s derivative for solving fractional differential equations in this framework compact structures of fourth order fractional diffusion wave equations are considered as prototype examples moreover convergence of the proposed approach for these types of equations is investigated results show that the response expressions are mittag leffler stable

idea of invariant subspace combined with elementary integral method for investigating exact solutions of time fractional npdes abstract in this paper inspired by the idea of invariant subspace method and combined with elementary integral method we introduced a novel approach for investigating exact solutions of a time fractional nonlinear partial differential equation npde based on hypothetical structure of solution of separated variable a time fractional npde defined by time and space variables can be reduced to a nonlinear ordinary differential equation node or nodes defined by space variable alone and then using the elementary integral method to solve the node or nodes different kinds of exact solutions of a time fractional npde are obtained finally as examples the time fractional hunter saxton equation and time fractional li olver equation were studied different kinds of exact solutions of these equations were obtained and their dynamical properties were illustrated

a review of definitions of fractional derivatives and other operators abstract given the increasing number of proposals and definitions of operators in the scope of fractional calculus it is important to introduce a systematic classification nonetheless many of the definitions that emerged in the literature can not be considered as fractional derivatives we analyze a list of expressions to have a general overview of the concept of fractional integrals derivatives moreover some formulae that do not involve the term fractional are also included due to their particular interest in the area

haar wavelet collocation method for three dimensional elliptic partial differential equations a new collocation method based on haar wavelet is presented for numerical solution of three dimensional elliptic partial differential equations with dirichlet boundary conditions an important advantage of the method is that it can be applied to both linear as well as nonlinear problems the algorithm based on this new method is simple and can be easily implemented in any programming language experimental rates of convergence of the proposed method are calculated which are in agreement with theoretical results the proposed method is applied to several benchmark problems from the literature including linear and nonlinear elliptic problems as well as systems of elliptic partial differential equations the numerical experiments confirm the accuracy and diverse applicability of the method

new algorithms for the numerical solution of nonlinear fredholm and volterra integral equations using haar wavelets two new algorithms based on haar wavelets are proposed the first algorithm is proposed for the numerical solution of nonlinear fredholm integral equations of the second kind and the second for the numerical solution of nonlinear volterra integral equations of the second kind these methods are designed to exploit the special characteristics of haar wavelets in both one and two dimensions formulae for calculating haar coefficients without solving the system of equations have been derived these formulae are then used in the proposed numerical methods in contrast to other numerical methods the advantage of our method is that it does not involve any intermediate numerical technique for evaluation of the integral present in integral equations the methods are validated on test problems and numerical results are compared with those from existing methods in the literature

different methods for 3 1 dimensional space time fractional modified kdv zakharov kuznetsov equation in this paper the fractional derivatives in the sense of modified riemann liouville derivative and the exp function method the g g expansion method and the generalized kudryashov method are used to construct exact solutions for 3 1 dimensional space time fractional modified kdv zakharov kuznetsov equation this fractional equation can be turned into another nonlinear ordinary differential equation by fractional complex transformation and then these three methods are applied to solve it as a result some new exact solutions are obtained the three methods demonstrate power reliability and efficiency

the first integral method for some time fractional differential equations abstract in this paper the fractional derivatives in the sense of modified riemann liouville derivative and the first integral method are employed for constructing the exact solutions of nonlinear time fractional partial differential equations the power of this manageable method is presented by applying it to several examples this approach can also be applied to other nonlinear fractional differential equations

lie symmetry analysis of the time fractional kdv type equation abstract the lie symmetry analysis method is extended to deal with the time fractional kdv type equation it is shown that this equation can be reduced to an equation with the erdelyi kober fractional derivative this method can also be applied to symmetry classification of fractional equations with some arbitrary functions

laguerre series approach to the analysis of a linear control system incorporating observers application of the laguerre polynomial expansion to the analysis of a linear optimal control system incorporating observers is studied the method simplifies the system of equations into the successive solution of a set of linear algebraic recursive equations compared with other orthogonal polynomials the present method is simpler in form and more convenient for use with a digital computer a numerical example is given to illustrate the use of this method

on group invariant solutions of fractional order burgers poisson equation abstract in this paper symmetry property of fractional order space time burgers poisson fbp equation is investigated the equation is obtained by replacing first order time and space derivatives to the corresponding fractional derivatives of order and respectively in the classical burgers poisson equation we present lie symmetries and corresponding infinitesimal generators for the fbp equation by using fractional lie group method with the help of these infinitesimal generators some group invariant solutions are sought by reducing the order of the equation

holder continuous of the solutions to stochastic nonlocal heat equations abstract in this paper we aim to obtain the holder continuous of solutions to stochastic nonlocal heat equations by using campanato estimates and sobolev embedding theorem we first prove the holder continuous of the mild solution of stochastic nonlocal diffusion equations locally in r d in the sense that the solution u belongs to the space c d t l p then by using tail estimates we obtain the estimates of the mild solution in l p c d t lastly we prove the holder continuous of the mild solution on bounded domain

a fokker planck equation of fractional order with respect to time by combining the maximum entropy principle with some considerations related to derivatives of fractional order one is led to suggest a fokker planck of fractional order with respect to time which could be related to dynamical systems subject to fractional brownian motion the relation with the process associated with the equation p t 1 n 1 2np x2n is exhibited

jacobi collocation method for the approximate solution of some fractional order riccati differential equations with variable coefficients abstract this paper presents a computational method for the approximate solution of arbitrary order non linear fractional riccati differential equations with variable coefficients proposed computational method is a combination of the operational matrix of integration method and the collocation method associated with the jacobi polynomials convergence analysis of the proposed method is provided numerical results for different fractional orders of the riccati differential equations are discussed figures and tables are used to show the numerical results derived from the proposed computational method for particular cases of jacobi polynomials such as the legendre polynomials the chebyshev polynomials of the second kind the chebyshev polynomials of the third kind the chebyshev polynomial of the fourth kind and the gegenbauer or ultraspherical polynomials numerical results from the proposed methods are compared from those derived by using the existing analytical and numerical methods it is observed that the results from the proposed method are more accurate maximum absolute error and the root mean square error tables are given for all five kinds of polynomials for comparison purposes

haar wavelet approximation for the solution of cubic nonlinear schrodinger equations abstract in this study haar wavelet collocation method is used for the numerical solution of 1d and 2d cubic nonlinear schrodinger equations with initial and dirichlet boundary conditions the space derivatives are estimated through haar wavelet collocation method whereas for time derivative we have used crank nicolson scheme the proposed method is implemented upon several test problems and the numerical results of these test problems establish that the proposed method is accurate

new approach to find exact solutions of time fractional kuramoto sivashinsky equation abstract in the present paper we construct the analytical exact solutions of a nonlinear evolution equation in mathematical physics viz time fractional kuramoto sivashinsky equation by a new proposed method via fractional complex transform as a result new types of exact analytical solutions are obtained here the fractional derivative is described in jumarie s modified riemann liouville sense

generalized einstein relation and the metzler klafter conjecture in a composite subdiffusive regime abstract in this paper a generalized diffusion model driven by the composite subdiffusive fractional brownian motion fbm is employed based on this stochastic process we derive a fractional fokker planck equation ffpe and obtain its solution it is proved that the generalized einstein relation ger and the metzler and klafter conjecture on the asymptotic behavior of stretched gaussian hold the ffpe in a composite subdiffusive regime