a new approach for the application of adomian decomposition method for the solution of fractional space diffusion equation with insulated ends abstract this paper presents the new approach for the solution of the space fractional diffusion equation defined in finite domain by adomian decomposition method adm a typical example of special interest with fractional space derivative of order 1 2 is considered in the present analysis and solved by adm after expressing the initial condition as fourier series the explicit solution of the space fractional diffusion equation has been presented in the closed form and then the numerical solution has been represented graphically the behaviour of adomian solutions and the effects of different values of are shown graphically

stability and convergence of the difference methods for the space time fractional advection diffusion equation abstract in this paper we consider a space time fractional advection dispersion equation stfade on a finite domain the stfade is obtained from the standard advection dispersion equation by replacing the first order time derivative by the caputo fractional derivative of order 0 1 and the first order and second order space derivatives by the riemman liouville fractional derivatives of order 0 1 and of order 1 2 respectively for the space fractional derivatives d x u x t and d x u x t we adopted the grunwald formula and the shift grunwald formula respectively we propose an implicit difference method idm and an explicit difference method edm to solve this equation stability and convergence of these methods are discussed using mathematical induction we prove that the idm is unconditionally stable and convergent but the edm is conditionally stable and convergent numerical results are in good agreement with theoretical analysis

fractional high order methods for the nonlinear fractional ordinary differential equation abstract in this paper we consider the nonlinear fractional order ordinary differential equation 0 d t y t f y t t 0 n 1 n y i 0 y 0 i i 0 1 2 n 1 where f y t satisfies the l condition i e f y 1 t f y 2 t l y 1 y 2 in t 0 t fractional order linear multiple step methods are introduced the high order 2 6 approximations of the fractional order ordinary differential equation with an initial value are proposed the consistence convergence and stability of the fractional high order methods are proved finally some numerical examples are provided to show that the fractional high order methods for solving the fractional order nonlinear ordinary differential equation are computationally efficient solution methods

numerical simulation of a new two dimensional variable order fractional percolation equation in non homogeneous porous media percolation flow problems are discussed in many research fields such as seepage hydraulics groundwater hydraulics groundwater dynamics and fluid dynamics in porous media many physical processes appear to exhibit fractional order behavior that may vary with time or space or space and time the theory of pseudodifferential operators and equations has been used to deal with this situation in this paper we use a fractional darcy s law with variable order riemann liouville fractional derivatives this leads to a new variable order fractional percolation equation in this paper a new two dimensional variable order fractional percolation equation is considered a new implicit numerical method and an alternating direct method for the two dimensional variable order fractional model is proposed consistency stability and convergence of the implicit finite difference method are established finally some numerical examples are given the numerical results demonstrate the effectiveness of the methods this technique can be used to simulate a three dimensional variable order fractional percolation equation

a more fundamental approach to the derivation of nonlinear acoustic wave equations with fractional loss operators l a corrected derivation of nonlinear wave propagation equations with fractional loss operators is presented the fundamental approach is based on fractional formulations of the stress strain and heat flux definitions but uses the energy equation and thermodynamic identities to link density and pressure instead of an erroneous fractional form of the entropy equation as done in prieur and holm nonlinear acoustic wave equations with fractional loss operators j acoust soc am 130 3 1125 1132 2011 the loss operator of the obtained nonlinear wave equations differs from the previous derivations as well as the dispersion equation but when approximating for low frequencies the expressions for the frequency dependent attenuation and velocity dispersion remain unchanged

periodic solutions for a neutral nonlinear dynamical equation on a time scale abstract let t be a periodic time scale we use a fixed point theorem due to krasnosel ski to show that the nonlinear neutral dynamic system with delay x t a t x t c t x t k q t x t x t k t t has a periodic solution we assume that k is a fixed constant if t r and is a multiple of the period of t if t r under a slightly more stringent inequality we show that the periodic solution is unique using the contraction mapping principle

a novel numerical method for the time variable fractional order mobile immobile advection dispersion model evolution equations containing fractional derivatives can provide suitable mathematical models for describing anomalous diffusion and transport dynamics in complex systems that cannot be modeled accurately by normal integer order equations recently researchers have found that many physical processes exhibit fractional order behavior that varies with time or space the continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable in this paper we consider the mobile immobile advection dispersion model with the coimbra variable time fractional derivative which is preferable for modeling dynamical systems and is more efficient from the numerical standpoint a novel implicit numerical method for the equation is proposed and the stability of the approximation is investigated as for the convergence of the numerical method we only consider a special case i e the time fractional derivative is independent of the time variable t the case where the time fractional derivative depends on both the time variable t and the space variable x will be considered in a future work finally numerical examples are provided to show that the implicit difference approximation is computationally efficient

on nirenberg type conditions for higher order systems on time scales we study the existence of periodic solutions for a nonlinear system of nth order differential equations on time scales assuming a suitable nirenberg type condition we prove the existence of at least one solution of the problem using mawhin s coincidence degree

existence of periodic solutions in shifts δ for neutral nonlinear dynamic systems abstract this paper focuses on the existence of a periodic solution of the delay neutral nonlinear dynamic systems x t a t x t q t x s t g t x t x s t in our analysis we utilize a new periodicity concept in terms of shifts operators which allows us to extend the concept of periodicity to time scales where the additivity requirement t t t for all t t and for a fixed t 0 may not hold more importantly the new concept will easily handle time scales that are not periodic in the conventional way such as q z and k 1 3 k 2 3 k 0 hence we will develop the tool that enables us to investigate the existence of periodic solutions of q difference systems since we are dealing with systems in order to convert our equation to an integral systems we resort to the transition matrix of the homogeneous floquet system y t a t y t and then make use of krasnoselskii s fixed point theorem to obtain a fixed point

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

numerical methods and analysis for a class of fractional advection dispersion models in this paper a class of fractional advection dispersion models fadms is considered these models include five fractional advection dispersion models i e the time fadm the mobile immobile time fadm with a time caputo fractional derivative 0 c 1 the space fadm with two sides riemann liouville derivatives the time space fadm and the time fractional advection diffusion wave model with damping with index 1 c 2 these equations can be used to simulate the regional scale anomalous dispersion with heavy tails we propose computationally effective implicit numerical methods for these fadms the stability and convergence of the implicit numerical methods are analysed and compared systematically finally some results are given to demonstrate the effectiveness of theoretical analysis

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

the fractional order governing equation of levy motion a governing equation of stable random walks is developed in one dimension this fokker planck equation is similar to and contains as a subset the second order advection dispersion equation ade except that the order a of the highest derivative is fractional e g the 1 65th derivative fundamental solutions are levy s a stable densities that resemble the gaussian except that they spread proportional to time 1 a have heavier tails and incorporate any degree of skewness the measured variance of a plume undergoing levy motion would grow faster than fickian plume at a rate of time 2 a where 0 a 2 the equation is parsimonious since the parameters are not functions of time or distance the scaling behavior of plumes that undergo levy motion is accounted for by the fractional derivatives which are appropriate measures of fractal functions in real space the fractional derivatives are integrodifferential operators so the fractional ade describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space

a parallel cgs block centered finite difference method for a nonlinear time fractional parabolic equation abstract in this article a parallel conjugate gradient squared cgs block centered finite difference scheme is introduced and analyzed to cast about the numerical solution of a nonlinear time fractional parabolic equation with the neumann condition and a nonlinear reaction term the unconditionally stable result which just depends on initial value and source item is derived some a priori estimates of discrete l 2 norm with optimal order of convergence o t 2 h 2 k 2 with pressure and velocity are established on both uniform and non uniform rectangular grids where t is the time step h and k are maximal mesh sizes of x and y directional grids in our model because the simulation is iterated over a series of time steps it is most beneficial if all the calculation units in a time step can be run separately in cgs algorithm adding openmp instructions to the circulation calculations in an iterative step can realize parallel computing to examine the efficiency and accuracy of the proposed method numerical experiments using the schemes are studied the results clearly show the benefit of using the proposed approach in terms of execution time reduction and speedup with respect to the sequential running in a single thread

a meshless numerical procedure for solving fractional reaction subdiffusion model via a new combination of alternating direction implicit adi approach and interpolating element free galerkin efg method a new numerical algorithm has been investigated for solving time fractional reaction subdiffusion equation the fractional derivative of the considered equation is described in the riemann liouville sense firstly we discrete the temporal dimension of the considered model using a finite difference scheme a central difference scheme has been applied to discrete the first time derivative and then for discretizing the fractional integral term a difference scheme has been employed with convergence order o 1 moreover to achieve a full discretization scheme a type of meshless method has been improved that is known as element free galerkin efg method the efg method for integration uses a background mesh this method is based on the galerkin weak form in which the test and trial functions are shape functions of moving least squares mls approximation since the shape functions of traditional mls lack the kronecker property essential boundary conditions of a boundary value problem can not be directly computed and other methods must be employed for this issue to this end a new class of mls shape functions has been applied that is called shape functions of interpolating mls the new shape function has the mentioned property in the efg method calculating the appeared two dimensional integrals is a basic issue in this research work the alternating direction implicit approach is combined with the element free galerkin method then using the new proposed method the two dimensional integrals on rectangular domain will be changed to simple one dimensional integrals we prove that the new numerical algorithm is unconditionally stable and also we obtain an error bound for the new procedure using the energy method numerical examples are reported which demonstrate the theoretical results and the efficiency of proposed scheme

a bounded and efficient scheme for multidimensional problems with anomalous convection and diffusion abstract this work is motivated by a generalization of the well known burgers fisher and burgers huxley equations in multiple dimensions considering riesz fractional diffusion and convection initial boundary conditions which are positive and bounded are imposed on a closed and bounded rectangular domain in this manuscript we propose a finite difference method to approximate the positive and bounded solutions of the fractional model the methodology is a linear three steps crank nicolson technique which is based on the use of fractional centered differences the properties of fractional centered differences are employed to establish the existence and the uniqueness of solutions of the finite difference method as well as the capability of the technique to preserve the positivity and the boundedness of the approximations we show in this work that the method is capable of preserving some of the constant solutions of the continuous model additionally we prove that our technique is a second order consistent stable and quadratically convergent scheme suitable bounds for the numerical solutions are also derived in this work finally some illustrative simulations show that the method is able to preserve the positivity and the boundedness of the numerical approximations in agreement with the analytic results proved in this work numerical comparisons provided in this work confirm the rate of convergence of the numerical technique

a unified floquet theory for discrete continuous and hybrid periodic linear systems abstract in this paper we study periodic linear systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts e g hybrid dynamical systems we develop a comprehensive floquet theory including lyapunov transformations and their various stability preserving properties a unified floquet theorem which establishes a canonical floquet decomposition on time scales in terms of the generalized exponential function and use these results to study homogeneous as well as nonhomogeneous periodic problems furthermore we explore the connection between floquet multipliers and floquet exponents via monodromy operators in this general setting and establish a spectral mapping theorem on time scales finally we show this unified floquet theory has the desirable property that stability characteristics of the original system can be determined via placement of an associated but time varying system s poles in the complex plane we include several examples to show the utility of this theory

new exact solutions of nonlinear fractional acoustic wave equations in ultrasound in this paper new exact solutions of fractional nonlinear acoustic wave equations have been devised the travelling periodic wave solutions of fractional burgers hopf equation and khokhlov zabolotskaya kuznetsov kzk equation have obtained by first integral method nonlinear ultrasound modelling is found to have an increasing number of applications in both medical and industrial areas where due to high pressure amplitudes the effects of nonlinear propagation are no longer negligible taking nonlinear effects into account the ultrasound beam analysis makes more accurate in these applications the burgers hopf equation is one of the extensively studied models in mathematical physics in addition the kzk parabolic nonlinear wave equation is one of the most widely employed nonlinear models for propagation of 3d diffraction sound beams in dissipative media in the present analysis these nonlinear equations have solved by first integral method as a result new exact analytical solutions have been obtained first time ever for these fractional order acoustic wave equations the obtained results are presented graphically to demonstrate the efficiency of this proposed method

matrix approach to discrete fractional calculus ii partial fractional differential equations a new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order so called fractional derivatives and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation the suggested method is the development of podlubny s matrix approach i podlubny matrix approach to discrete fractional calculus fractional calculus and applied analysis 3 4 2000 359 386 four examples of numerical solution of fractional diffusion equation with various combinations of time space fractional derivatives integer integer fractional integer integer fractional and fractional fractional with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach the fifth example illustrates that the method can be equally simply used for fractional differential equations with delays a set of matlab routines for the implementation of the method as well as sample code used to solve the examples have been developed

an implicit difference scheme with the kps preconditioner for two dimensional time space fractional convection diffusion equations abstract this paper deals with the numerical computation and analysis for a class of two dimensional time space fractional convection diffusion equations an implicit difference scheme is derived for solving this class of equations it is proved under some suitable conditions that the derived difference scheme is stable and convergent moreover the convergence orders of the scheme in time and space are also given in order to accelerate the convergence rate by combining the kronecker product splitting kps preconditioner with the generalized minimal residual gmres method a preconditioning strategy for implementing the difference scheme is introduced finally several numerical examples are presented to illustrate the computational accuracy and efficiency of the methods

numerical techniques for the variable order time fractional diffusion equation abstract in this paper we consider the variable order time fractional diffusion equation we adopt the coimbra variable order vo time fractional operator which defines a consistent method for vo differentiation of physical variables the coimbra variable order fractional operator also can be viewed as a caputo type definition although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling numerical methods for fractional partial differential equations using this definition have not yet appeared in the literature here an approximate scheme is first proposed the stability convergence and solvability of this numerical scheme are discussed via the technique of fourier analysis numerical examples are provided to show that the numerical method is computationally efficient

a novel equivalent definition of caputo fractional derivative without singular kernel and superconvergent analysis in this article we present a new second order finite difference discrete scheme for a fractal mobile immobile transport model based on equivalent transformative caputo formulation the new transformative formulation takes the singular kernel away to make the integral calculation more efficient furthermore this definition is also effective where is a positive integer besides the t caputo derivative also helps us to increase the convergence rate of the discretization of the order 0 1 caputo derivative from o 2 to o 3 where is the time step for numerical analysis a crank nicolson finite difference scheme to solve the fractal mobile immobile transport model is introduced and analyzed the unconditional stability and a priori estimates of the scheme are given rigorously moreover the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis

preconditioned iterative methods for space time fractional advection diffusion equations in this paper we propose practical numerical methods for solving a class of initial boundary value problems of space time fractional advection diffusion equations first we propose an implicit method based on two sided grunwald formulae and discuss its stability and consistency then we develop the preconditioned generalized minimal residual preconditioned gmres method and preconditioned conjugate gradient normal residual preconditioned cgnr method with easily constructed preconditioners importantly because resulting systems are toeplitz like fast fourier transform can be applied to significantly reduce the computational cost we perform numerical experiments to demonstrate the efficiency of our preconditioners even in cases with variable coefficients

error analysis of a finite difference method on graded meshes for a time fractional diffusion equation a reaction diffusion problem with a caputo time derivative of order alpha in 0 1 is considered the solution of such a problem is shown in general to have a weak singularity near the initial time t 0 and sharp pointwise bounds on certain derivatives of this solution are derived a new analysis of a standard finite difference method for the problem is given taking into account this initial singularity this analysis encompasses both uniform meshes and meshes that are graded in time and includes new stability and consistency bounds the final convergence result shows clearly how the regularity of the solution and the grading of the mesh affect the order of convergence of the difference scheme so one can choose an optimal mesh grading numerical results are presented that confirm the sharpness of the error analysis

almost automorphic solutions of discrete delayed neutral system abstract we study almost automorphic solutions of the discrete delayed neutral dynamic system x t 1 a t x t q t x t g t g t x t x t g t by means of a fixed point theorem due to krasnoselskii using discrete variant of exponential dichotomy and proving uniqueness of projector of discrete exponential dichotomy we invert the equation and obtain some limit results leading to sufficient conditions for the existence of almost automorphic solutions of the neutral system unlike the existing literature we prove our existence results without assuming boundedness of inverse matrix a t 1 hence we significantly improve the results in the existing literature we provide two examples to illustrate effectiveness of our results finally we also provide an existence result for almost periodic solutions of the system

comparison of fickian and temporally nonlocal transport theories over many scales in an exhaustively sampled sandstone slab 1 it is not a compelling argument solely on the basis of a better fit to solute breakthrough curve btc data that a temporally nonlocal model is necessary to simulate transport in an advection dominated system one may counter that the classical advection dispersion equation ade is a valid model at some small scale and that the detailed hydraulic conductivity k data must be well represented then the nonlocality is only a result of upscaling and loss of information but is the nonlocal model demonstrably necessary at all scales we examine the experiment conducted by klise et al 2008 in which a 30 5 30 5 cm slab of relatively homogeneous cross bedded sandstone was exhaustively sampled for k the slab was sealed saturated with potassium iodide and x rayed 10 times while being flushed with fresh water the 8 649 air permeameter measurements were down and upscaled to make finer and coarser grids on which the velocity field was solved and the ade applied the optimized parameters in the ade were found to scale predictably most notably the longitudinal dispersivity which grew linearly with upscaling but at all levels of up and downscaling including the original k measurement scale of 0 33 cm the ade did not adequately represent the late time tails the temporally nonlocal time fractional ade t fade was applied and the optimized parameters and the immobile capacity did not depend on scale the better fit provided by the t fade in the late btc tails did not bring about a sacrificed fit elsewhere in the btc furthermore the optimized ade and t fade solutions do not converge at the smallest scale directly implying that the temporal nonlocality is a necessary model component we conclude that the logical inference if the ade is valid in heterogeneous material then there is tailing in the btc is not a proof that the reverse is true we provide a clear counterexample a corollary is that a mismatch between data and a discretized solution to the ade does not imply that more data will improve fits or predictive ability

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

periodic solutions of abstract neutral functional differential equations abstract we characterize the existence of periodic solutions of some abstract neutral functional differential equations with finite and infinite delay when the underlying space is a umd space

on haar wavelet operational matrix of general order and its application for the numerical solution of fractional bagley torvik equation abstract in the present analysis the motion of an immersed plate in a newtonian fluid described by torvik and bagley s fractional differential equation 1 has been considered this bagley torvik equation has been solved by operational matrix of haar wavelet method the obtained result is compared with analytical solution suggested by podlubny 2 haar wavelet method is used because its computation is simple as it converts the problem into algebraic matrix equation

boundary layer flows of viscoelastic fluids over a non uniform permeable surface abstract this study investigates viscoelastic fluid which possesses both viscous and elastics properties by employing a fractional derivative model to reveal the stress relaxation phenomenon with distance the spatial fractional derivative used in the momentum conservation equation is the riemann liouville type derivative further we use non uniform boundary conditions subject to the boundary layer equations of the fluid flowing through a semi infinite permeable flat surface owing to the fractional derivative model and non uniform boundary conditions this problem is complex thus a finite difference scheme is applied after the coupled continuity equation and momentum equation are decoupled and linearized the accuracy convergence and stability of the numerical method are presented it is shown that non uniform mass transfer through a permeable surface considerably affects the velocity boundary layer by illustrating the physical interactions between the velocity fields in the boundary layer and the permeation mode in the surface this paper predicts the velocity distributions with varying permeable surface and also provides the possibility of changing the velocity fields by altering the permeable sheet the results and numerical technique used in this study will help in the understanding of fractional calculus investigation in engineering

a two grid mixed finite element method for a nonlinear fourth order reaction diffusion problem with time fractional derivative in this article we develop a two grid algorithm based on the mixed finite element mfe method for a nonlinear fourth order reaction diffusion equation with the time fractional derivative of caputo type we formulate the problem as a nonlinear fully discrete mfe system where the time integer and fractional derivatives are approximated by finite difference methods and the spatial derivatives are approximated by the mfe method to solve the nonlinear mfe system more efficiently we propose a two grid algorithm which is composed of two steps we first solve a nonlinear mfe system on a coarse grid by nonlinear iterations then solve the linearized mfe system on the fine grid by newton iteration numerical stability and optimal error estimate o k 2 h r 1 h 2 r 2 in l 2 norm are proved for our two grid scheme where k h and h are the time step size coarse grid mesh size and fine grid mesh size respectively we implement the two grid algorithm and present the numerical results justifying our theoretical error estimate the numerical tests also show that the two grid method is much more efficient than solving the nonlinear mfe system directly

multi point boundary value problems on time scales at resonance abstract we study the nonlinear second order differential equation on a time scale u t f t u t u t e t t 0 1 t subject to the multi point boundary conditions u 0 0 u 1 i 1 m i u i and u 0 0 u 1 i 1 m i u i where t is a time scale such that 0 1 t i 0 1 t i 1 m and f is a continuous function satisfying a caratheodory type growth condition and e is a lebesgue integrable function our existence results are obtained by applying a coincidence degree theorem due to mawhin

a fractional alternating direction implicit method for a multi term time space fractional bloch torrey equations in three dimensions abstract the time space fractional bloch torrey equation ts fbte has been proposed to simulate anomalous diffusion in the human brain however the development of numerical methods and theoretical analysis for multi term time space fractional bloch torrey equations in three dimensions is still limited in this paper we consider a class of 3 d multi term time space fractional bloch torrey equations 3d mtts fbtes firstly we adopt a fractional centred difference scheme to discretize the riesz fractional derivative and propose a fractional alternating direction implicit method fadim for the model secondly the solvability stability and convergence of the method are investigated finally numerical results are presented to support our theoretical analysis in addition to demonstrate the applicability of our method we consider a coupled 3 d fbte as an example to solve and exhibit the effects on the behaviour of the transverse magnetization

existence of periodic solutions for abstract neutral non autonomous equations with infinite delay in this paper by using sadovskii fixed point theorem we study the existence of solutions and periodic solutions for a class of abstract neutral functional evolution equations with infinite delay an example is presented in the end to show the applications of the obtained results

compact difference scheme for distributed order time fractional diffusion wave equation on bounded domains in this paper we derive and analyse a compact difference scheme for a distributed order time fractional diffusion wave equation this equation is approximated by a multi term fractional diffusion wave equation which is then solved by a compact difference scheme the unique solvability of the difference solution is discussed using the discrete energy method we prove the compact difference scheme is unconditionally stable and convergent finally numerical results are presented to support our theoretical analysis

integration on time scales in this paper we study the process of riemann and lebesgue integration on time scales the relationship of the riemann and lebesgue integrals is considered and a criterion for riemann integrability is established

fitted reproducing kernel hilbert space method for the solutions of some certain classes of time fractional partial differential equations subject to initial and neumann boundary conditions latterly many problems arising in different fields of science and engineering can be reduced by applying some appropriate discretization to a series of time fractional partial differential equations unlike the normal case derivative the differential order in such equations is with a fractional order which will lead to new challenges for numerical simulation the purpose of this analysis is to introduce the reproducing kernel hilbert space method for treating classes of time fractional partial differential equations subject to neumann boundary conditions with parameters derivative arising in fluid mechanics chemical reactions elasticity anomalous diffusion and population growth models the method provides appropriate representation of the solutions in convergent series formula with accurately computable components numerical experiments with different order derivatives degree are performed to support the theoretical analyses which are acquired by interrupting the n term of the exact solutions finally the obtained outcomes showed that the proposed method is competitive in terms of the quality of the solutions found and is very valid for solving such time fractional neumann problems

periodicity of scalar dynamic equations and applications to population models easily verifiable sufficient criteria are established for the existence of periodic solutions of a class of nonautonomous scalar dynamic equations on time scales which incorporate as special cases many single species models governed by ordinary differential and difference equations when the time scale is the set of all real and all integer numbers respectively

simplest equation method for some time fractional partial differential equations with conformable derivative abstract the conformable fractional derivative was proposed by r khalil et al in 2014 which is natural and obeys the leibniz rule and chain rule based on the properties a class of time fractional partial differential equations can be reduced into odes using traveling wave transformation then the simplest equation method is applied to find exact solutions of some time fractional partial differential equations the exact solutions solitary wave solutions periodic function solutions rational function solutions of time fractional generalized burgers equation time fractional generalized kdv equation time fractional generalized sharma tasso olver fsto equation and time fractional fifth order kdv equation 3 1 dimensional time fractional kdv zakharov kuznetsov kdv zk equation are constructed this method presents a wide applicability to solve some nonlinear time fractional differential equations with conformable derivative

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

periodic solutions of neutral nonlinear system of differential equations with functional delay abstract we study the existence of periodic solutions of the nonlinear neutral system of differential equations of the form d d t x t a t x t d d t q t x t g t g t x t x t g t in the process we use the fundamental matrix solution of y a t y and convert the given neutral differential equation into an equivalent integral equation then we construct appropriate mappings and employ krasnoselskii s fixed point theorem to show the existence of a periodic solution of this neutral differential equation we also use the contraction mapping principle to show the existence of a unique periodic solution of the equation

fractal mobile immobile solute transport 1 a fractal mobile immobile model for solute transport assumes power law waiting times in the immobile zone leading to a fractional time derivative in the model equations the equations are equivalent to previous models of mobile immobile transport with power law memory functions and are the limiting equations that govern continuous time random walks with heavy tailed random waiting times the solution is gained by performing an integral transform on the solution of any boundary value problem for transport in the absence of an immobile phase in this regard the output from a multidimensional numerical model can be transformed to include the effect of a fractal immobile phase the solutions capture the anomalous behavior of tracer plumes in heterogeneous aquifers including power law breakthrough curves at late time and power law decline in the measured mobile mass the made site mobile tritium mass decline is consistent with a fractional time derivative of order 0 33 while haggerty et al s 2002 stream tracer test is well modeled by a fractional time derivative of order 0 3

improved fractional sub equation method for 3 1 dimensional generalized fractional kdv zakharov kuznetsov equations in the present paper we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics namely the time fractional kdv zakharov kuznetsov kdv zk and space time fractional modified kdv zakharov kuznetsov mkdv zk equations by using improved fractional sub equation method as a result new types of exact analytical solutions are obtained the obtained results are shown graphically here the fractional derivative is described in the jumarie s modified riemann liouville sense

the use of finite difference element approaches for solving the time fractional subdiffusion equation in this paper two finite difference element approaches for the time fractional subdiffusion equation with dirichlet boundary conditions are developed in which the time direction is approximated by the fractional linear multistep method and the space direction is approximated by the finite element method the two methods are unconditionally stable and convergent of order o tau q h r 1 in the l 2 norm where q 2 beta or 2 when the analytical solution to the subdiffusion equation is sufficiently smooth beta 0 beta 1 is the order of the fractional derivative tau and h are the step sizes in time and space respectively and r is the degree of the polynomial space the corresponding schemes for the subdiffusion equation with neumann boundary conditions are presented as well where the stability and convergence are shown numerical examples are provided to verify the theoretical analysis comparisons between the algorithms derived in this paper and the existing algorithms are given

the first integral method to study the burgers korteweg de vries equation in this paper applying the theory of commutative algebra we propose a new approach which we currently call the first integral method to study the burgers korteweg de vries equation

a fast second order implicit scheme for non linear time space fractional diffusion equation with time delay and drift term abstract in this paper a second order accurate implicit scheme based on the l2 1 formula for temporal variable and the fractional centered difference formula for spatial discretization is established to solve a class of time space fractional diffusion equations with time drift term and non linear delayed source function the stability of this scheme is proved rigorously by the discrete energy method under several auxiliary assumptions then we theoretically and numerically show that the proposed scheme converges in the l2 norm with the order o t 2 h 2 with time step t and mesh size h moreover it finds that the discreted linear systems are symmetric toeplitz systems in order to solve these systems efficiently the conjugate gradient method with suitable circulant preconditioners is designed in each iterative step the storage requirements and the computational complexity of the resulting equations are o n and o n log n respectively where n is the number of grid nodes numerical experiments are carried out to demonstrate the effectiveness of our proposed circulant preconditioners

high order algorithms for riesz derivative and their applications ii in this paper we firstly develop two high order approximate formulas for the riesz fractional derivative secondly we propose a temporal second order numerical method for a fractional reaction dispersion equation where we discretize the riesz fractional derivative by using two numerical schemes we prove that the numerical methods for a spatial riesz fractional reaction dispersion equation are both unconditionally stable and convergent and the orders of convergence are o 2 h 6 and o 2 h 8 in which and h are spatial and temporal step sizes respectively finally we test our numerical schemes and observe that the numerical results are in good agreement with the theoretical analysis

computational algorithm for solving fredholm time fractional partial integrodifferential equations of dirichlet functions type with error estimates abstract numerical modeling of partial integrodifferential equations of fractional order shows interesting properties in various aspects of science which means increased attention to fractional calculus this paper is concerned with a feasible and accurate technique for obtaining numerical solutions for a class of partial integrodifferential equations of fractional order in hilbert space within appropriate kernel functions the algorithm relies on the reproducing kernel hilbert space method that provides the solutions in rapidly convergent series representations for the reproducing kernel based upon the fourier coefficients of orthogonalization process the caputo fractional derivatives are introduced to address these issues moreover the error estimate of the generated solutions is established as well as the convergence of the iterative method is investigated under some theoretical assumptions the superiority and applicability of the present technique is illustrated by handling linear and nonlinear numerical examples the outcomes obtained are compared with exact solutions and existing methods to confirm the effectiveness of the reproducing kernel method

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

analysis of a new finite difference local discontinuous galerkin method for the fractional diffusion wave equation in this paper a finite difference local discontinuous galerkin method for the fractional diffusion wave equation is presented and analyzed we first propose a new finite difference method to approximate the time fractional derivatives and give a semidiscrete scheme in time further we develop a fully discrete scheme for the fractional diffusion wave equation and prove that the method is unconditionally stable and convergent with order o hk 1 t 3 where k is the degree of piecewise polynomial extensive numerical examples are carried out to confirm the theoretical convergence rates

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

on using random walks to solve the space fractional advection dispersion equations the solution of space fractional advection dispersion equations fade by random walks depends on the analogy between the fade and the forward equation for the associated markov process the forward equation which provides a lagrangian description of particles moving under specific markov processes is derived here by the adjoint method the fade however provides an eulerian description of solute fluxes there are two forms of the fade based on fractional flux ff ade and fractional divergence fd ade the ff ade is derived by taking the integer order mass conservation of non local diffusive flux while the fd ade is derived by taking the fractional order mass conservation of local diffusive flux the analogy between the fade and the forward equation depends on which form of the fade is used and on the spatial variability of the dispersion coefficient d in the fade if d does not vary in space then the fades can be solved by tracking particles following a markov process with a simple drift and an stable levy noise with index that corresponds to the fractional order of the fade if d varies smoothly in space and the solute concentration at the upstream boundary remains zero the fd ade can be solved by simulating a markov process with a simple drift an stable levy noise and an additional term with the dispersion gradient and an additional levy noise of order 1 however a non markov process might be needed to solve the ff ade with a space dependent d except for specific d such as a linear function of space

a generalization of trigonometric transform splitting methods for spatial fractional diffusion equations abstract the spatial fractional diffusion equation can be discretized by employing the implicit finite difference scheme with the shifted grunwald formula and the given discretized linear systems have a diagonal plus toeplitz structure in this paper based on trigonometric transformation splitting tts we study efficient iterative method called gtts has two parameters as a focus we give the simple and effective optimal forms of these two parameters respectively we carry out some numerical experiments to illustrate the effectiveness and accuracy of this new algorithm

a fast method for a generalized nonlocal elastic model we develop a numerical method for a generalized nonlocal elastic model which is expressed as a composition of a riesz potential operator with a fractional differential operator by composing a collocation method with a finite difference discretization by carefully exploring the structure of the coefficient matrix of the numerical method we develop a preconditioned fast krylov subspace method which reduces the computations to n log n per iteration and the memory to o n the use of the preconditioner significantly reduces the number of iterations and the preconditioner can be inverted in o n log n computations numerical results show the utility of the method

traveling waves to a burgers korteweg de vries type equation with higher order nonlinearities abstract in this paper first we survey some recent advances in the study of traveling wave solutions to the burgers korteweg de vries equation and some comments are given then we study a burgers korteweg de vries type equation with higher order nonlinearities a qualitative analysis to a two dimensional autonomous system which is equivalent to the burgers kdv type equation is presented and indicates that under certain conditions the burgers korteweg de vries type equation has neither nontrivial bell profile solitary waves nor periodic waves finally a solitary wave solution is obtained by means of the first integral method which is based on the ring theory of commutative algebra

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

high order unconditionally stable difference schemes for the riesz space fractional telegraph equation in this paper a class of unconditionally stable difference schemes based on the pade approximation is presented for the riesz space fractional telegraph equation firstly we introduce a new variable to transform the original differential equation to an equivalent differential equation system then we apply a second order fractional central difference scheme to discretize the riesz space fractional operator finally we use 1 1 2 2 and 3 3 pade approximations to give a fully discrete difference scheme for the resulting linear system of ordinary differential equations matrix analysis is used to show the unconditional stability of the proposed algorithms two examples with known exact solutions are chosen to assess the proposed difference schemes numerical results demonstrate that these schemes provide accurate and efficient methods for solving a space fractional hyperbolic equation