a construction of homologically area minimizing hypersurfaces with higher dimensional singular sets we show that a large variety of singular sets can occur for homologically area minimizing codimenision one surfaces in a riemarinian manifold in particular as a result of theorem a if n is smooth compact ni 1 dimensional manifold n 7 and if s is an embedded orientable submanifold of dimension n then we cornstruct metrics on n such that the homologically area minimizing hypersurface m homologous to s has a singular set equal to a prescribed number of spheres and tori of codimension less than n 7 near each component e of the singular set m is isometric to a product c x e where c is any prescribed strictly stable strictly minimizing cone in theorem b other singular examples are constructed

what is the fractional laplacian a comparative review with new results abstract the fractional laplacian in r d which we write as 2 with 0 2 has multiple equivalent characterizations moreover in bounded domains boundary conditions must be incorporated in these characterizations in mathematically distinct ways and there is currently no consensus in the literature as to which definition of the fractional laplacian in bounded domains is most appropriate for a given application the riesz or integral definition for example admits a nonlocal boundary condition where the value of a function must be prescribed on the entire exterior of the domain in order to compute its fractional laplacian in contrast the spectral definition requires only the standard local boundary condition these differences among others lead us to ask the question what is the fractional laplacian beginning from first principles we compare several commonly used definitions of the fractional laplacian theoretically through their stochastic interpretations as well as their analytical properties then we present quantitative comparisons using a sample of state of the art methods we discuss recent advances on nonzero boundary conditions and present new methods to discretize such boundary value problems radial basis function collocation for the riesz fractional laplacian and nonharmonic lifting for the spectral fractional laplacian in our numerical studies we aim to compare different definitions on bounded domains using a collection of benchmark problems we consider the fractional poisson equation with both zero and nonzero boundary conditions where the fractional laplacian is defined according to the riesz definition the spectral definition the directional definition and the horizon based nonlocal definition we verify the accuracy of the numerical methods used in the approximations for each operator and we focus on identifying differences in the boundary behaviors of solutions to equations posed with these different definitions through our efforts we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications

emergence of fuzzy preferences for risk in a birkhoff von neumann logics environment we show that if a portfolio of a financial derivative asset and a stock is put in an environment where the value of an asset besides its price is formalized as a superposition of price states such portfolio may not be risk free and fuzzy preferences for risk premia may exist we argue for a modification of the classical brownian motion process as used in option pricing this modification on the classical brownian motion we call the email protected brownian motion and one specific format of this email protected brownian motion can be shown to have a connection with the quantum physical schrodinger equation

quantum bohmian model for financial market we apply methods of quantum mechanics for mathematical modeling of price dynamics at the financial market the hamiltonian formalism on the price price change phase space describes the classical like evolution of prices this classical dynamics of prices is determined by hard conditions natural resources industrial production services and so on these conditions are mathematically described by the classical financial potential v q where q q1 qn is the vector of prices of various shares but the information exchange and market psychology play important and sometimes determining role in price dynamics we propose to describe such behavioral financial factors by using the pilot wave bohmian model of quantum mechanics the theory of financial behavioral waves takes into account the market psychology the real trajectories of prices are determined through the financial analogue of the second newton law by two financial potentials classical like v q hard market conditions and quantum like u q behavioral market conditions

discontinuous spectral element methods for time and space fractional advection equations we develop spectral element methods for a time and space fractional advection equation of the form prescript 0 mathcal d tau t u x t theta prescript 0 mathcal d nu x u x t f x t of order tau in 0 1 nu in 0 1 subject to dirichlet initial boundary conditions we present two spectrally accurate and efficient methods for global discretization of both temporal and spatial terms instead of employing traditional low order time integration methods to this end we first develop a petrov galerkin in time and discontinuous galerkin in space pg dg method where we carry out the time integration using a single time domain spectral method sm and we perform the spatial discretization using the discontinuous spectral hp element method dsem this scheme also leads to a more efficient time integration when the time derivative is integer order i e tau 1 we develop the sm dsem scheme based on a new spectral theory for fractional sturm liouville problems

the second kind chebyshev wavelet method for solving fractional differential equations abstract in this paper the second kind chebyshev wavelet method is presented for solving linear and nonlinear fractional differential equations we first construct the second kind chebyshev wavelet and then derive the operational matrix of fractional order integration the operational matrix of fractional order integration is utilized to reduce the fractional differential equations to system of algebraic equations in addition illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method

a note on optimal homotopy asymptotic method for the solutions of fractional order heat and wave like partial differential equations optimal homotopy asymptotic method oham is prolifically implemented to find the optimal solutions of fractional order heat and wave like equations we inspect the competence of the method by examining fractional order time dependent partial differential equations it is observed that oham is a prevailing and convergent method for the solutions of linear and nonlinear fractional order time dependent partial differential problems the numerical results rendering that the applied method is explicit effective and easy to use for handling more general fractional order heat and wave like models

analytical solution of the linear fractional system of commensurate order a useful representation of fractional order systems is the state space representation for the linear fractional systems of commensurate order the state space representation is defined as for regular integer state space representation with the state vector differentiated to a real order this paper presents a solution of the linear fractional order systems of commensurate order in the state space the solution is obtained using a technique based on functions of square matrices and the cayley hamilton theorem the technique developed for linear systems of integer order is extended to derive analytical solutions of linear fractional systems of commensurate order the basic ideas and the derived formulations of the technique are presented both homogeneous and inhomogeneous cases with usual input functions are solved the solution is calculated in the form of a linear combination of suitable fundamental functions the presented results are illustrated by analyzing some examples to demonstrate the effectiveness of the presented analytical approach

variational iteration method for autonomous ordinary differential systems in this paper a new iteration technique is proposed to solve autonomous ordinary differential systems in this method general lagrange multipliers are introduced to construct correction functionals for the systems the multipliers in the functionals can be identified by the variational theory the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary initial conditions some examples are given the results reveal that the method is very effective and convenient

analytical time domain green s functions for power law media frequency dependent loss and dispersion are typically modeled with a power law attenuation coefficient where the power law exponent ranges from 0 to 2 to facilitate analytical solution a fractional partial differential equation is derived that exactly describes power law attenuation and the szabo wave equation time domain wave equations for lossy media obeying a frequency power law j acoust soc am 96 491 500 1994 is an approximation to this equation this paper derives analytical time domain green s functions in power law media for exponents in this range to construct solutions stable law probability distributions are utilized for exponents equal to 0 1 3 1 2 2 3 3 2 and 2 the green s function is expressed in terms of dirac delta exponential airy hypergeometric and gaussian functions for exponents strictly less than 1 the green s functions are expressed as fox functions and are causal for exponents greater than or equal than 1 the green s functions are expressed as fox and

the general vector solutions of such linear coupled matrix fractional differential equations by using kronecker structures in this paper we demonstrate how kronecker products and vector operator can be combined beautifully to solve and fit the non homogenous linear matrix fractional differential equations and such coupled linear matrix fractional differential equations the way exists which transform the given coupled matrix fractional differential equations by using kronecker structures into forms for which solutions may be readily computed some important and interesting special cases of the general system of non homogeneous linear matrix fractional differential equations of fractional order are also considered which includes the non homogenous linear fractional dynamical system with delays in control finally a brief comparison between the fractional and integer order vector solutions and some examples are given to illustrate our new approaches

a numerical method for solving a fractional partial differential equation through converting it into an nlp problem in this paper we propose a new approach for solving fractional partial differential equations which is very easy to use and can also be applied to equations of other types the main advantage of the method lies in its flexibility for obtaining the approximate solutions of time fractional and space fractional equations using this approach we convert a fractional partial differential equation into a nonlinear programming problem several numerical examples are used to demonstrate the effectiveness and accuracy of the method

computing eigenelements of boundary value problems with fractional derivatives variational iteration method has been successfully implemented to handle linear and nonlinear differential equations the main property of the method is its flexibility and ability to solve nonlinear equations accurately and conveniently in this paper first a general framework of the variational iteration method is presented for analytic treatment of differential equations of fractional order where the fractional derivatives are described in caputo sense second the new framework is used to compute approximate eigenvalues and the corresponding eigenfunctions for boundary value problems with fractional derivatives numerical examples are tested to show the pertinent features of this method this approach provides a new way to investigate eigenvalue problems with fractional order derivatives

analytic study on linear neutral fractional differential equations this paper is devoted to investigating linear neutral fractional differential equations with constant coefficients the existence uniqueness and iterative formula of the solution are obtained meanwhile dependence of the solution on initial value and the general solution represented by a fundamental solution matrix are discussed several examples are given to illustrate the applications of our results some conclusions in the literature are extended greatly

fractional sturm liouville eigen problems we first consider a regular fractional sturm liouville problem of two kinds rfslp i and rfslp ii of order 0 2 the corresponding fractional differential operators in these problems are both of riemann liouville and caputo type of the same fractional order 2 0 1 we obtain the analytical eigensolutions to rfslp i ii as non polynomial functions which we define as jacobi poly fractonomials these eigenfunctions are orthogonal with respect to the weight function associated with rfslp i ii subsequently we extend the fractional operators to a new family of singular fractional sturm liouville problems of two kinds sfslp i and sfslp ii we show that the primary regular boundary value problems rfslp i ii are indeed asymptotic cases for the singular counterparts sfslp i ii furthermore we prove that the eigenvalues of the singular problems are real valued and the corresponding eigenfunctions are orthogonal in addition we obtain the eigen solutions to sfslp i ii analytically also as non polynomial functions hence completing the whole family of the jacobi poly fractonomials in numerical examples we employ the new poly fractonomial bases to demonstrate the exponential convergence of the approximation in agreement with the theoretical results

numerical methods for the solution of partial differential equations of fractional order anomalous diffusion is a possible mechanism underlying plasma transport in magnetically confined plasmas to model this transport mechanism fractional order space derivative operators can be used here the numerical properties of partial differential equations of fractional order 1 2 are studied two numerical schemes an explicit and a semi implicit one are used in solving these equations two different discretization methods of the fractional derivative operator have also been used the accuracy and stability of these methods are investigated for several standard types of problems involving partial differential equations of fractional order

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

the short memory principle for solving abel differential equation of fractional order in this paper the short memory principle smp is applied for solving the abel differential equation with fractional order we evaluate the approximate solution at the end of required interval and construct a suitable iteration scheme employing this end point as initial value numerical experiments show that our iteration method is simple and efficient and that a proper length of memory could maintain the validity of the short memory principle

application of homotopy perturbation method to fractional ivps fractional initial value problems fivps arise from many fields of physics and play a very important role in various branches of science and engineering finding accurate and efficient methods for solving fivps has become an active research undertaking in this paper both linear and nonlinear fivps are considered exact and or approximate analytical solutions of the fivps are obtained by the analytic homotopy perturbation method hpm the results of applying this procedure to the studied cases show the high accuracy simplicity and efficiency of the approach

the variational iteration method an efficient scheme for handling fractional partial differential equations in fluid mechanics variational iteration method has been used to handle linear and nonlinear differential equations the main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently in this work a general framework of the variational iteration method is presented for analytical treatment of fractional partial differential equations in fluid mechanics the fractional derivatives are described in the caputo sense numerical illustrations that include the fractional wave equation fractional burgers equation fractional kdv equation fractional klein gordon equation and fractional boussinesq like equation are investigated to show the pertinent features of the technique comparison of the results obtained by the variational iteration method with those obtained by adomian decomposition method reveals that the first method is very effective and convenient the basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus

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

on legendre polynomial approximation with the vim or ham for numerical treatment of nonlinear fractional differential equations the variational iteration method and the homotopy analysis method as alternative methods have been widely used to handle linear and nonlinear models the main property of the methods is their flexibility and ability to solve nonlinear equations accurately and conveniently this paper deals with the numerical solutions of nonlinear fractional differential equations where the fractional derivatives are considered in caputo sense the main aim is to introduce efficient algorithms of variational iteration and homotopy analysis methods that can be simply used to deal with nonlinear fractional differential equations in these algorithms legendre polynomials are effectively implemented to achieve better approximation for the nonhomogeneous and nonlinear terms that leads to facilitate the computational work the proposed algorithms are capable of reducing the size of calculations improving the accuracy and easily overcome the difficulty arising in calculating complicated integrals numerical examples are examined to show the efficiency of the algorithms

singular homologically area minimizing surfaces of codimension one in riemannian manifolds for n 7 it is shown how to construct examples of smooth compact riemannian manifolds n n 1 g with non trivial n dimensional integer homology such that for some h n n z the hypersurface n current m which minimizes area among all hypersurfaces representing has singularities the singular set of m consists of two isolated points and the tangent cone at these points can be prescribed as any strictly stable strictly minimizing regular cone to my knowledge these are the first examples of codimension one homological minimizers with singularities

fractal ladder models and power law wave equations extensive measurement of the ultrasonic attenuation coefficient in human and mammalian tissue has revealed a power law dependence on frequency to describe this power law behavior in tissue a hierarchical fractal network model is proposed the viscoelastic and self similar properties of tissue are captured by a constitutive equation based on a lumped parameter infinite ladder topology involving alternating springs and dashpots in the low frequency limit this ladder network yields a stress strain constitutive equation with a time fractional derivative by combining this constitutive equation with linearized conservation principles and an adiabatic equation of state the fractional partial differential equations previously proposed by m caputo linear models of dissipation whose q is almost frequency independent ii geophys j r astron soc 13 529 539 1967 and mg wismer finite element analysis of broadband acoustic pulses through inhomogeneous media with power law attenuation j acoust soc am 120 3493 3502 2006 are derived the resulting attenuation coefficient is a power law with exponents ranging between 1 and 2 while the phase velocity is in agreement with the kramers kronig relations the power law exponent is interpreted in terms of the mechanical structure and related to the spectral dimension of the underlying fractal geometry

approximate solutions for boundary value problems of time fractional wave equation in this article we implement relatively a new analytical technique the adomian decomposition method for solving the boundary value problems of time fractional wave equation the fractional derivative is described in the caputo sense the decomposition method is used to construct analytical approximate solutions of time fractional wave equation subject to specified boundary conditions the solutions are calculated in the form of a convergent series with easily computable components some examples are given the results reveal that the adomian method is very effective and convenient

anomalous diffusion with ballistic scaling a new fractional derivative abstract anomalous diffusion with ballistic scaling is characterized by a linear spreading rate with respect to time that scales like pure advection ballistic scaling may be modeled with a symmetric riesz derivative if the spreading is symmetric however ballistic scaling coupled with a skewness is observed in many applications including hydrology nuclear physics viscoelasticity and acoustics the goal of this paper is to find a governing equation for anomalous diffusion with ballistic scaling and arbitrary skewness to address this problem we propose a new operator called the zolotarev derivative which is valid for all orders 0 2 the fourier symbol of this operator is related to the characteristic function of a stable random variable in the zolotarev m parameterization in the symmetric case the zolotarev derivative reduces to the well known riesz derivative for 1 the zolotarev derivative is a linear combination of riemann liouville fractional derivatives and a first derivative for 1 the zolotarev derivative is a non local operator that models ballistic anomalous diffusion we prove that this operator is continuous with respect to we derive generator caputo and riemann liouville forms of this operator and provide two examples the solutions of diffusion equations utilizing the zolotarev derivative with an impulse initial condition are shifted and scaled stable densities in the m parameterization

galerkin finite element approximations for stochastic space time fractional wave equations the traditional wave equation models wave propagation in an ideal medium to characterize wave propagation in inhomogeneous media with frequency dependent power law attenuation the space time fractional wave equation is needed further incorporating the additive white gaussian noise external noise coming from many natural sources leads to the stochastic space time fractional wave equation this paper discusses the galerkin finite element approximations for the stochastic space time fractional wave equation forced by an additive space time white noise we first discretize the space time additive noise which introduces a modeling error and results in a regularized stochastic space time fractional wave equation then the regularity of the regularized equation is analyzed for the discretization in space the finite element approximation is used and the definition of the discrete fractional laplacian is introduced we derive the mean squared l 2 norm a priori estimates for the modeling error and for the

adjoint state method for fractional diffusion parameter identification fractional partial differential equations provide models for sub diffusion among which the fractal mobile immobile model fmim is often used to represent solute transport in complex media the fmim involves four parameters among which we have the order of an integro differential operator that accounts for the possibility for solutes to be sequestered during very long times to guess fmim parameters from experiments an accurate method consists in optimizing an objective function that measures how much model solutions deviate from data we show that solving an adjoint problem helps accurate computing of the gradient of such an objective function with respect to the parameters we illustrate the method by applying it on experimental data issued from tracing tests in porous media

variational iteration method new development and applications variational iteration method has been favourably applied to various kinds of nonlinear problems the main property of the method is in its flexibility and ability to solve nonlinear equations accurately and conveniently in this paper recent trends and developments in the use of the method are reviewed major applications to nonlinear wave equation nonlinear fractional differential equations nonlinear oscillations and nonlinear problems arising in various engineering applications are surveyed the confluence of modern mathematics and symbol computation has posed a challenge to developing technologies capable of handling strongly nonlinear equations which cannot be successfully dealt with by classical methods variational iteration method is uniquely qualified to address this challenge the flexibility and adaptation provided by the method have made the method a strong candidate for approximate analytical solutions this paper outlines the basic conceptual framework of variational iteration technique with application to nonlinear problems both achievements and limitations are discussed with direct reference to approximate solutions for nonlinear equations a new iteration formulation is suggested to overcome the shortcoming a very useful formulation for determining approximately the period of a nonlinear oscillator is suggested examples are given to illustrate the solution procedure

modified szabo s wave equation models for lossy media obeying frequency power law szabo s models of acoustic attenuation szabo j acoust soc am 96 1 491 500 1994 comply well with the empirical frequency power law involving noninteger and odd integer exponent coefficients while guaranteeing causality but nevertheless encounter the troublesome issues of hypersingular improper integral and obscurity in implementing initial conditions the purpose of this paper is to ease or remove these drawbacks of the szabo s models via the caputo fractional derivative concept the positive time fractional derivative is also introduced to include the positivity of the attenuation processes

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

an efficient method for solving systems of fractional integro differential equations in this study a decomposition method for approximating the solution of systems of fractional integro differential equations is implemented the fractional derivative is considered in the caputo sense the approximate solutions are calculated in the form of a convergent series with easily computable components numerical results show that this approach is easy to implement and accurate when applied to integro differential equations

a constant enclosure method for validating existence and uniqueness of the solution of an initial value problem for a fractional differential equation abstract this paper presents a new method for validating existence and uniqueness of the solution of an initial value problems for fractional differential equations an algorithm selecting a stepsize and computing a priori constant enclosure of the solution is proposed several illustrative examples with linear and nonlinear fractional differential equations are given to demonstrate the effectiveness of the method

numerical steady state and hopf bifurcation analysis on the diffusive nicholson s blowflies equation1research partially supported by natural sciences and engineering research council of canada 1 for the dirichlet boundary value problem of the diffusive nicholson s blowflies equation it was shown in ref 17 that in a certain range of the parameter space there is a unique positive steady state solution in this paper we propose a scheme to compute this steady state numerically in addition we describe an iterative procedure to locate the critical values of the delay where a hopf bifurcation of time periodic solutions takes place near the steady state some numerical simulations of both schemes are given

solving differential equations of fractional order using an optimization technique based on training artificial neural network the current study aims to approximate the solution of fractional differential equations fdes by using the fundamental properties of artificial neural networks anns for function approximation in the first step we derive an approximate solution of fractional differential equation fde by using anns in the second step an optimization approach is exploited to adjust the weights of anns such that the approximated solution satisfies the fde different types of fdes including linear and nonlinear terms are solved to illustrate the ability of the method in addition the present scheme is compared with the analytical solution and a number of existing numerical techniques to show the efficiency of anns with high accuracy fast convergence and low use of memory for solving the fdes

application of the variational iteration method to the regularized long wave equation this paper applies the variational iteration method to the regularized long wave equation in this method a generalized lagrange multiplier is introduced to construct a correction functional for the discussed problem the multiplier can be identified optimally via variational theory three invariants of the motion are evaluated to determine the conservation properties of the system

solving systems of fractional differential equations using differential transform method this paper presents approximate analytical solutions for systems of fractional differential equations using the differential transform method the fractional derivatives are described in the caputo sense the application of differential transform method developed for differential equations of integer order is extended to derive approximate analytical solutions of systems of fractional differential equations the solutions of our model equations are calculated in the form of convergent series with easily computable components some examples are solved as illustrations using symbolic computation the numerical results show that the approach is easy to implement and accurate when applied to systems of fractional differential equations the method introduces a promising tool for solving many linear and nonlinear fractional differential equations

analytical solution of a time fractional navier stokes equation by adomian decomposition method abstract the aim of the present analysis is to apply adomian decomposition method for the solution of a time fractional navier stokes equation in a tube by using an initial value the explicit solution of the equation has been presented in the closed form and then its numerical solution has been represented graphically the present method performs extremely well in terms of efficiency and simplicity

new ideas for proving convergence of decomposition methods in this paper we give new formulae which calculate easily the adomian s polynomials used in decomposition methods then the proof of convergence of the adomian s technique becomes almost obvious by using a weak hypothesis on the nonlinear operator of the functional equation

wavelets method for solving fractional optimal control problems in this paper an efficient and accurate computational method based on the legendre wavelets lws is proposed for solving a class of fractional optimal control problems focps in the proposed method the focp under consideration is reduced to a system of nonlinear algebraic equations which can be simply solved to this end the fractional derivative of the state variable and the control variable are expanded by the lws with unknown coefficients then the operational matrix of the riemann liouville fractional integration with some properties of the lws are employed to achieve a nonlinear algebraic equation in place of the performance index and a linear system of algebraic equations in place of the dynamical system in terms of the unknown coefficients finally the method of constrained extrema which consists of adjoining the constraint equations derived from the given dynamical system to the performance index by a set of undetermined lagrange multipliers is applied as a result the necessary conditions of optimality are derived as a system of algebraic equations in the unknown coefficients of the state variable control variable and lagrange multipliers furthermore the efficiency of the proposed method is shown for some concrete examples the results reveal that the proposed method is very accurate and efficient

analytical approximate solutions of systems of fractional algebraic differential equations by homotopy analysis method abstract in this paper we develop a framework to obtain approximate solutions to systems of algebraic differential equations of fractional order by employing the homotopy analysis method ham the study highlights the efficiency of the method and its dependence on the auxiliary parameter numerical examples are examined to highlight the significant features of the ham method the method shows improvements over existing analytical techniques

the variational iteration method for solving riesz fractional partial differential equations in this paper the variational iteration method is applied to obtain the solution for space fractional partial differential equations where the space fractional derivative is in the riesz sense on the basis of the properties and definition of the fractional derivative the iterative technique is carried out in a straightforward manner without the need for transforms or numerical approximations examples demonstrate that the series solution obtained shows agreement with the exact solutions of the problems solved

a new modification of the adomian decomposition method for linear and nonlinear operators in this paper we present an efficient modification of the adomian decomposition method that will facilitate the calculations we then conduct a comparative study between the new modification and the modified decomposition method the study is conducted through illustrative examples the new modification introduces a promising tool for many linear and nonlinear models

a generalized fractional order elastodynamic theory for non local attenuating media this study presents a generalized elastodynamic theory based on fractional order operators capable of modelling the propagation of elastic waves in non local attenuating solids and across complex

fracfit a robust parameter estimation tool for fractional calculus models anomalous transport cannot be adequately described with classical fickian advection dispersion equations ade with constant coefficients rather fractional calculus models may be used which capture salient features of anomalous transport e g skewness and power law tails fracfit is a parameter estimation tool based on space fractional and time fractional models used by the hydrology community currently four fractional models are supported 1 space fractional advection dispersion equation sfade 2 time fractional dispersion equation with drift tfde 3 fractional mobile immobile fmim equation and 4 temporally tempered levy motion ttlm model solutions using pulse initial conditions and continuous injections are evaluated using stable distributions or subordination integrals parameter estimates are extracted from measured breakthrough curves btcs using a weighted nonlinear least squares wnls algorithm optimal weights for btcs for pulse initial conditions and continuous injections are presented two sample applications are analyzed 1 pulse injection btcs in the selke river and 2 continuous injection laboratory experiments using natural organic matter model parameters are compared across models and goodness of fit metrics are presented facilitating model evaluation

acoustical finite difference time domain simulations of subwavelength geometries accurate simulation of wave propagation around subwavelength geometries using standard finite difference time domain fdtd techniques require a fine spatial and temporal sampling resulting in high computational costs in this paper an extension to this standard fdtd technique is proposed by means of quasi stationary solutions on a subwavelength scale the fdtd equations in the region near the subwavelength geometry are extended with some correction terms of which the magnitude is extracted from the quasi stationary pressure distribution around these geometries these pressure distributions can be calculated from the laplace equation using this new technique fdtd simulations can be based on a more coarse grid thus reducing computational cost considerably the accuracy of this technique is mainly determined by the accuracy of the laplace solutions this technique was tested with success on the simulation of resonators it was also shown that the new fdtd equations can be extended to include viscosity ef

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

a finite dimensional quantum model for the stock market abstract we present a finite dimensional version of the quantum model for the stock market proposed in c zhang and l huang a quantum model for the stock market physica a 389 2010 5769 our approach is an attempt to make this model consistent with the discrete nature of the stock price and is based on the mathematical formalism used in the case of the quantum systems with finite dimensional hilbert space the rate of return is a discrete variable corresponding to the coordinate in the case of quantum systems and the operator of the conjugate variable describing the trend of the stock return is defined in terms of the finite fourier transform the stock return in equilibrium is described by a finite gaussian function and the time evolution of the stock price directly related to the rate of return is obtained by numerically solving a schrodinger type equation

time domain wave equations for lossy media obeying a frequency power law for attenuation described by a slowly varying power law function of frequency 0 y classical lossy time domain wave equations exist only for the restricted cases where y 0 or y 2 for the frequently occurring practical situation in which attenuation is much smaller than the wave number a lossy dispersion characteristic is derived that has the desired attenuation general power law dependence in order to obtain the corresponding time domain lossy wave equation time domain loss operators similar in function to existing derivative operators are developed through the use of generalized functions three forms of lossy wave equations are found depending on whether y is an even or odd integer or a noninteger a time domain expression of causality analogous in function to the kramers kronig relations in the frequency domain is used to derive the causal wave equations final causal versions of the time domain wave equations are obtained even for the cases where y 1 which according to the paley wiener th

variational iteration method for solving integral equations in this paper several integral equations are solved by he s variational iteration method comparison with exact solution shows that the method is very effective and convenient for solving integral equations

the synchronization of fractional order rossler hyperchaotic systems abstract the synchronization of fractional order hyperchaotic systems is studied using the rossler system as an example based on the laplace transformation theory sufficient conditions for global synchronization of the systems are given analytically also the variational iteration method is implemented to give the approximate solution for the fractional order error system of the two identical hyperchaotic systems which is in good agreement with the approximate solution using the classical laplace transformation method numerical methods and simulations on the master slave systems are presented to verify the results obtained

numerical approach to differential equations of fractional order in this paper the variational iteration method and the adomian decomposition method are implemented to give approximate solutions for linear and nonlinear systems of differential equations of fractional order the two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of differential equations in these schemes the solution takes the form of a convergent series with easily computable components this paper presents a numerical comparison between the two methods for solving systems of fractional differential equations numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order

application of legendre wavelets for solving fractional differential equations in this paper we develop a framework to obtain approximate numerical solutions to ordinary differential equations odes involving fractional order derivatives using legendre wavelet approximations the properties of legendre wavelets are first presented these properties are then utilized to reduce the fractional ordinary differential equations fodes to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique results show that this technique can solve the linear and nonlinear fractional ordinary differential equations with negligible error compared to the exact solution

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

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

the black scholes pricing formula in the quantum context a natural explanation for extreme irregularities in the evolution of prices in financial markets is provided by quantum effects the lack of simultaneous observability of relevant variables and the interference of attempted observation with the values of these variables represent such effects these characteristics have been noted by traders and economists and appear intrinsic to market dynamics this explanation is explored here in terms of a corresponding generalization of the wiener process and its role in the black scholes merton theory the differentiability of the wiener process as a sesquilinear form on a dense domain in the hilbert space of square integrable functions over wiener space is shown and is extended to the quantum context this provides a basis for a corresponding generalization of the ito theory of stochastic integration an extension of the black scholes option pricing formula to the quantum context is deduced

variational iteration method for elastodynamic green s functions abstract in this study a new application of the variational iteration method is presented we use this method to obtain approximations to 3d green s function for the dynamic system of anisotropic elasticity the numerical results obtained from convolution of green s function and data of the cauchy problem are compared with the exact solution for cubic media

the time domain solution of the wide angle parabolic equation including the effects of sediment dispersion the wide angle time domain parabolic equation tdpe which is the inverse fourier transform of the wide angle parabolic equation pe is derived a numerical solution for the model is described and a benchmark calculation is presented the narrow angle tdpe is also considered and its error is analyzed and compared with the error of the narrow angle pe the tdpe is compared with the progressive wave equation which is shown to be restricted to narrow angle propagation for practical purposes in the sediment attenuation is assumed to depend linearly on frequency and the corresponding causal dispersion law is assumed the model is used to show that the effect of sediment dispersion on pulse propagation in the ocean can be significant

spline collocation for nonlinear fractional boundary value problems abstract we consider a class of boundary value problems for nonlinear fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the boundary value problem some regularity properties of the exact solution are derived based on these properties and spline collocation techniques the numerical solution of boundary value problems by suitable non polynomial approximations is discussed optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given theoretical results are tested by two numerical examples

quantum probability and financial market can the mathematical formalism of quantum mechanics and in particular quantum probability be applied outside of physics the answer is positive in this paper we apply methods of quantum mechanics for mathematical modelling of price dynamics of the financial market we propose to describe behavioral financial factors e g expectations of traders by using the pilot wave bohmian model of quantum mechanics on the one hand our bohmian model is a quantum like model for the financial market cf with works of w segal i e segal e haven e w piotrowski j sladkowski on the other hand since bohmian mechanics provides for the possibility to describe individual price trajectories it belongs to the domain of extended research on deterministic dynamics for financial assets our model emphasizes the complexity of the financial market the traditional description of price dynamics is completed by schrodinger s dynamics for the pilot wave of expectations of traders this is a kind of socio economic model for the financial market

an iterative method for solving nonlinear functional equations an iterative method for solving nonlinear functional equations viz nonlinear volterra integral equations algebraic equations and systems of ordinary differential equation nonlinear algebraic equations and fractional differential equations has been discussed

a new algorithm for calculating adomian polynomials for nonlinear operators in this paper a reliable technique for calculating adomian polynomials for nonlinear operators will be developed the new algorithm offers a promising approach for calculating adomian polynomials for all forms of nonlinearity the algorithm will be illustrated by studying suitable forms of nonlinearity a nonlinear evolution model will be investigated

quantum model for the price dynamics the problem of smoothness of trajectories we apply methods of quantum mechanics to mathematical modelling of price dynamics in a financial market we propose to describe behavioral financial factors e g expectations of traders by using the pilot wave bohmian model of quantum mechanics our model is a quantum like model of the financial market cf with works of w segal i e segal e haven in this paper we study the problem of smoothness of price trajectories in the bohmian financial model we show that even the smooth evolution of the financial pilot wave t x representing expectations of traders can induce jumps of prices of shares

approximate analytical solution for seepage flow with fractional derivatives in porous media abstract in this paper a new and more exact model for seepage flow in porous media with fractional derivatives has been proposed which has modified the well known darcy law and overcome the continuity assumption of seepage flow a new kind of analytical method of nonlinear problem called the variational iteration method is described and used to give approximate solutions of the problem the results show that the proposed iteration method requiring no linearization or small perturbation is very effective and convenient

convergence of the variational iteration method for solving multi order fractional differential equations in this paper the variational iteration method vim is applied to obtain approximate solutions of multi order fractional differential equations m fdes we can easily obtain the satisfying solution just by using a few simple transformations and applying the vim a theorem for convergence and error estimates of the vim for solving m fdes is given moreover numerical results show that our theoretical analysis are accurate and the vim is a powerful method for solving m fdes

application of he s methods to nonlinear chemistry problems in this paper he s variational iteration method vim and homotopy perturbation method hpm are implemented for solving analytically two systems of ordinary differential equations that often appear in chemical applications in both methods the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions the solution procedure is of utter simplicity and the obtained solution is of high accuracy

a quantum model of option pricing when black scholes meets schrodinger and its semi classical limit abstract the black scholes equation can be interpreted from the point of view of quantum mechanics as the imaginary time schrodinger equation of a free particle when deviations of this state of equilibrium are considered as a product of some market imperfection such as transaction cost asymmetric information issues short term volatility extreme discontinuities or serial correlations the classical non arbitrage assumption of the black scholes model is violated implying a non risk free portfolio from haven 2002 1 we know that an arbitrage environment is a necessary condition to embedding the black scholes option pricing model in a more general quantum physics setting the aim of this paper is to propose a new black scholes schrodinger model based on the endogenous arbitrage option pricing formulation introduced by contreras et al 2010 2 hence we derive a more general quantum model of option pricing that incorporates arbitrage as an external time dependent force which has an associated potential related to the random dynamic of the underlying asset price this new resultant model can be interpreted as a schrodinger equation in imaginary time for a particle of mass 1 2 with a wave function in an external field force generated by the arbitrage potential as pointed out above this new model can be seen as a more general formulation where the perfect market equilibrium state postulated by the black scholes model represent a particular case finally since the schrodinger equation is in place we can apply semiclassical methods of common use in theoretical physics to find an approximate analytical solution of the black scholes equation in the presence of market imperfections as it is the case of an arbitrage bubble here as a numerical illustration of the potential of this schrodinger equation analogy the semiclassical approximation is performed for different arbitrage bubble forms step linear and parabolic and compare with the exact solution of our general quantum model of option pricing

variational iteration method and homotopy perturbation method for fourth order fractional integro differential equations in this article linear and nonlinear boundary value problems for fourth order fractional integro differential equations are solved by variational iteration method and homotopy perturbation method the fractional derivatives are described in the caputo sense the solutions of both problems are derived by infinite convergent series which are easily computable and then graphical representation shows that both methods are most effective and convenient one to solve linear and nonlinear boundary value problems for fourth order fractional integro differential equations

backward fractional advection dispersion model for contaminant source prediction the forward fractional advection dispersion equation fade provides a useful model for non fickian transport in heterogeneous porous media the space fade captures the long leading tail skewness and fast spreading typically seen in concentration profiles from field data this paper develops the corresponding backward fade model to identify source location and release time the backward method is developed from the theory of inverse problems and then explained from a stochastic point of view the resultant backward fade differs significantly from the traditional backward advection dispersion equation ade because the fractional derivative is not self adjoint and the probability density function for backward locations is highly skewed finally the method is validated using tracer data from a well known field experiment where the peak of the backward fade curve predicts source release time while the median or a range of percentiles can be used to determine the most likely source location for the observed plume the backward ade cannot reliably identify the source in this application since the forward ade does not provide an adequate fit to the concentration data

simulation of the approximate solutions of the time fractional multi term wave equations abstract in this paper simulations of the approximation solutions of time fractional wave forced wave shear wave and damped wave equations are given the common finite difference rules besides the backward grunwald letnikov scheme are used to find the approximation solution of these models the paper discusses also the effects of the memory the internal force resistance and the external force on the travelling wave the von neumann stability conditions are also considered and discussed for these models besides the simulations of the time evolutions of the approximation solutions the stationary solutions are also simulated the numerical results are obtained by the mathematica software

approximate solution of nonlinear differential equations with convolution product nonlinearities in this paper a new iteration method is proposed to solve nonlinear problems special attention is paid to nonlinear differential equations with convolution product nonlinearities the results reveal the approximations obtained by the proposed method are uniformly valid for both small and large parameters in nonlinear problems furthermore the first order of approximations are more accurate than perturbation solutions at high order of approximation