stability of time stepping methods for abstract time dependent parabolic problems we consider an abstract nonautonomous parabolic problem u t a t u t quad u t 0 u 0 where a t d t subset x to x t 0 le t le t 1 is a family of sectorial operators in a banach space x this problem is discretized in time by means of either an a theta stable runge kutta or an a theta stable linear multistep method we prove that the resulting discretization is stable under some natural assumptions on the relative total variation of a t with respect to t for strongly a theta stable runge kutta methods stability holds even for variable time step sizes our results are applicable to the analysis of time dependent parabolic problems in the lp norms

a meshless local collocation method for time fractional diffusion wave equation abstract in this manuscript we present a radial basis function based local collocation method for solving time fractional diffusion wave equation the advantage of the local collocation method is only the discretization points located in each sub domain surrounding the collocation point need to be considered it also avoids the ill conditioning problem arises due to the use of global collocation method the theoretical stability and convergence of the proposed time semi discrete scheme are also discussed numerical experiments for one dimension and two dimension problems are carried out it is shown that the present method is efficient and numerical results have nice agreements with theoretical result

a robust and accurate finite difference method for a generalized black scholes equation in this paper we present a numerical method for a generalized black scholes equation which is used for option pricing the method is based on a central difference spatial discretization on a piecewise uniform mesh and an implicit time stepping technique our scheme is stable for arbitrary volatility and arbitrary interest rate and is second order convergent with respect to the spatial variable furthermore the present paper efficiently treats the singularities of the non smooth payoff function numerical results support the theoretical results

a stable runge kutta methods for semilinear evolution equations we consider semilinear evolution equations for which the linear part generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of hilbert spaces in this setting we prove the existence of solutions which are temporally smooth in the norm of the lowest rung of the scale for an open set of initial data on the highest rung of the scale under the same assumptions we prove that a class of implicit a stable runge kutta semidiscretizations in time of such equations are smooth as maps from open subsets of the highest rung into the lowest rung of the scale under the additional assumption that the linear part of the evolution equation is normal or sectorial we prove full order convergence of the semidiscretization in time for initial data on open sets our results apply in particular to the semilinear wave equation and to the nonlinear schrodinger equation 2012 elsevier inc

numerical approximation of a time fractional black scholes equation abstract in this paper a time fractional black scholes equation is examined we transform the initial value problem into an equivalent integral differential equation with a weakly singular kernel and use an integral discretization scheme on an adapted mesh for the time discretization a rigorous analysis about the convergence of the time discretization scheme is given by taking account of the possibly singular behavior of the exact solution and first order convergence with respect to the time variable is proved for overcoming the possibly nonphysical oscillation in the computed solution caused by the degeneracy of the black scholes differential operator we employ a central difference scheme on a piecewise uniform mesh for the spatial discretization it is proved that the scheme is stable and second order convergent with respect to the spatial variable numerical experiments support these theoretical results

an extension of the lax richtmyer theory we give a rigorous proof of the validity of the lax equivalence theorem when the true solution at timet and its approximation lie in different spaces thus modelling the practical situation where the former is a function of the space variables and the latter is only defined at grid points

numerically pricing american options under the generalized mixed fractional brownian motion model abstract in this paper we introduce a robust numerical method based on the upwind scheme for the pricing of american puts under the generalized mixed fractional brownian motion gmfbm model by using portfolio analysis and applying the wick ito formula a partial differential equation pde governing the prices of vanilla options under the gmfbm is successfully derived for the first time based on this we formulate the pricing of american puts under the current model as a linear complementarity problem lcp unlike the classical black scholes b s model or the generalized b s model discussed in cen and le 2011 the newly obtained lcp under the gmfbm model is difficult to be solved accurately because of the numerical instability which results from the degeneration of the governing pde as time approaches zero to overcome this difficulty a numerical approach based on the upwind scheme is adopted it is shown that the coefficient matrix of the current method is an m matrix which ensures its stability in the maximum norm sense remarkably we have managed to provide a sharp theoretic error estimate for the current method which is further verified numerically the results of various numerical experiments also suggest that this new approach is quite accurate and can be easily extended to price other types of financial derivatives with an american style exercise feature under the gmfbm model

functional calculus estimates for tadmor ritt operators abstract we show h functional calculus estimates for tadmor ritt operators also known as ritt operators which generalize and improve results by vitse these estimates are in conformity with the best known power bounds for tadmor ritt operators in terms of the constant dependence furthermore it is shown how discrete square function estimates influence the functional calculus estimates

anomalous transport in external fields continuous time random walks and fractional diffusion equations extended the continuous time random walk ctrw in a homogeneous velocity field and in arbitrary force fields is studied within the extended ctrw scheme anomalous transport properties due to long tailed waiting time or jump length distributions are consistently introduced the connections with generalised diffusion equations in a potential field are discussed these equations being of fractional order in particular the problems of a constant and a hookean linear force i e of a linear and a parabolic potential are worked out

a new operator splitting method for american options under fractional black scholes models abstract a new operator splitting method is proposed for american options under time fractional black scholes models the fractional linear complementarity problem is split into two easy sub problems with the leading coefficients separated from the convolution sum and matched through a general correction step the method is implementation friendly in the sense that one can easily modify a fractional european solver to obtain the proposed method since the correction step is decoupled and is trivial to solve the method is validated through numerical experiments and demonstrated to be superior to the traditional approach the paper also provides numerical studies including the effect of fractional orders and the comparison of fractional models

a stability result for sectorial operators in branch spaces in this paper the author considers approximations of holomorphic semigroups on general banach spaces based on a acceptable rational functions a general stability theorem is proved it covers rational functions r z with r infty 1 and can be applied even to variable step cases in particular the question of the text l p stability p ne 2 of crank nicolson schemes for parabolic initial value problems is settled

analytically pricing double barrier options based on a time fractional black scholes equation this paper investigates the pricing of double barrier options when the price change of the underlying is considered as a fractal transmission system in this scenario the option price is governed by a modified black scholes equation with a time fractional derivative in comparison with standard derivatives of integer order the fractional order derivatives are characterized by their globalness i e the rate of change of a function near a point is affected by the property of the function defined in the entire domain of definition rather than just near the point itself the existence of the time fractional derivative in conjunction with the presence of two barriers of the double barrier options has added an additional degree of difficulty not only when a purely numerical solution is sought but also when an analytical method is attempted albeit difficult we have managed to find an explicit closed form analytical solution for double barrier options which has been taken to price the single barrier options and european path independent options under the same framework as a special case of the current solution in addition not only have we provided a theoretical proof for the convergence of the newly found analytic solution in series form which is a vital step to show the closeness of our solution we have also proposed an efficient numerical evaluation technique to facilitate the implementation of our formula so that it can be easily used in trading practice

fractional fokker planck equation on heterogeneous fractal structures in external force fields and its solutions abstract we introduce a heterogeneous fractional fokker planck equation hffpe involving external force fields describing systems on heterogeneous fractal structure medium the hffpe is shown to obey generalized einstein relation and its stationary solution is the generalized boltzmann distribution it is proved that the asymptotic shape of its solution is a stretched gaussian and that its solution can be expressed in the form of a function of a dimensionless similarity variable for generic potentials furthermore it is shown that the solution of ffpe with parameter has the same properties as that ones of hffpe

c α preserving operators on separable banach spaces abstract let t be a bounded linear operator from a separable banach space x to a banach space y a necessary and sufficient condition on t for the existence of a subspace z of x such that z is isomorphic to c and the restriction of t to z is an isomorphism is given the special case where x is the disc algebra is then considered and results similar to those previously obtained by the author for c k spaces are obtained for the disc algebra finally some additional results of the same type are proved for subspaces of c k with small annihilator

modified spline collocation for linear fractional differential equations we propose and analyze a class of high order methods for the numerical solution of initial value problems for linear multi term fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the initial value problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established theoretical results are verified by some numerical examples

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

numerical solution of the time fractional black scholes model governing european options when considering the price change of the underlying fractal transmission system a fractional black scholes b s model with an order time fractional derivative is derived in this paper we discuss the numerical simulation of this time fractional black scholes model tfbsm governing european options a discrete implicit numerical scheme with a spatially second order accuracy and a temporally 2 order accuracy is constructed then the stability and convergence of the proposed numerical scheme are analyzed using fourier analysis some numerical examples are chosen in order to demonstrate the accuracy and effectiveness of the proposed method finally as an application we use the tfbsm and the above numerical technique to price several different european options

numerically pricing double barrier options in a time fractional black scholes model abstract the numerical solution of the time fractional black scholes model tfbsm of order 0 1 governing european options is studied zhang et al 2016 derived a numerical scheme of second order in space we improve their results by constructing a scheme of fourth order in space while keeping 2 in time the solvability stability and convergence of the proposed numerical scheme are proved using a fourier analysis the results are demonstrated on two examples

preserving operators on spaces let a c k x be a bounded linear operator where if is a compact hausdorff space and x is a separable banach space sufficient conditions are given for a to be an isomorphism into when restricted to a subspace y of c k such that y is isometrically isomorphic to a space c a of continuous functions on the space of ordinal numbers less than or equal to the countable ordinal a 1 for a compact hausdorff space k c k will denote the banach space of continuous real valued functions on k with the supremum norm the greek letters a s and y will be reserved for countable ordinal numbers we consider the set 0 a y 0 x 0 for each countable ordinal a we will define a family pa x b of open subsets g and associated measures ju g g b let p0 x b g jttg g is an open set in k ixg g b and g g x if a s 1 then pa x b g nc g jiig g p0 x b and there is a disjoint sequence g of open subsets of g and associated measures xg such that g ftg g ps x b ju gn jug and ung c g if a is a limit ordinal then x b g xg g p0 x b there is a disjoint sequence of open subsets gn of g with associated measures xg such that fic nc ung c g and g jug g pa x b where an a the notation a a means that a is an increasing sequence of ordinals with a sup an a simple example illustrating the sets pa x b is contained in 4 received by the editors march 20 1979 and in revised form july 30 1981 1980 mathematics subject classification primary 46e15 research on this paper was partially supported by the national science foundation grant number mcs 77 01690 1982 american mathematical society o0o2 9939 82 oo0o o726 so5 oo

a hybrid finite difference scheme for pricing asian options in this paper we apply a hybrid finite difference scheme to evaluate the prices of asian call options with fixed strike price we use the crank nicolson method to discretize the time variable and a hybrid finite difference scheme to discretize the spatial variable the hybrid difference scheme uses the central difference approximation whenever the mesh points are sufficiently away from the left hand side of the domain to ensure the stability of the scheme otherwise a midpoint upwind difference scheme is used the matrix associated with the discrete operator is an m matrix which ensures that the spatial discretization scheme is maximum norm stable it is proved that the scheme is second order convergent with respect to both time and spatial variables numerical experiments support these theoretical results

some boundary value problems of fractional differential equations and inclusions in this paper we study the existence of solutions for nonlinear fractional differential equations and inclusions of order q 1 2 with families of mixed and closed boundary conditions in case of inclusion problems the existence results are established for convex as well as nonconvex multivalued maps our results are based on leray schauder degree theory nonlinear alternative of leray schauder type and some fixed point theorems for multivalued maps some interesting special cases are also discussed

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

numerical computations of fractional nonlinear hartmann flow with revised heat flux model abstract in this communication we have examined the magneto hydrodynamic flow of viscoelastic liquid with revise version of thermal flux thermal conductivity of the fluid is variable effects of body forces such as gravity are also examined in terms of convection non integer time derivatives are used for the better understanding of viscoelastic flow characteristics in literature these derivatives have been proved suitable for the predictions of hereditary systems such as viscoelastic flows heat transfer is analyzed in the flow field via modified fractional thermal gradient temperature gradient exists between the flow boundaries and increases with the passage of time flow is generated by the variable movement of flow boundary all the segments of numerical modeling are completely addressed in this paper i e physical system mathematical model simulation prediction validation and verification unsteady motion of in compressible fluid is directed by differential equations with fractional partial derivatives for the solutions of these equations numerical techniques are necessary to use governing equations with appropriate flow conditions have been discretized by finite difference along with finite element algorithms theoretical error appraisals are hypothesized and created numerical plan has been approved by doing error investigations numerically regarding spatial directions we have shown the estimate error and plotted the error curves in logarithmic scales of l 2 and h 1 norms by taking absolute approximations of the space discretization local nusselt number and coefficients of skin friction are determined for non integer viscoelastic model flow field is simulated for various values of physical parameters the obtained results showed that fractional exponent has opposite effects on the velocity temperature profiles for time t 2 it is also noted that temperature gradient increases by the increase of fractional number and thermal relaxation parameter viscoelastic flows in polymer and plastics industries can be addressed by similar technique

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

a penalty method for a fractional order parabolic variational inequality governing american put option valuation a power penalty method is proposed for a parabolic variational inequality or linear complementarity problem lcp involving a fractional order partial derivative arising in the valuation of american options whose underlying stock prices follow a geometric levy process we first approximate the lcp with a nonlinear fractional partial differential equation fpde with a penalty term we then prove that the solution to the nonlinear fpde converges to that of the lcp in a sobolev norm at an exponential rate depending on the parameters used in the penalty term numerical results are presented to demonstrate the convergence rates and usefulness of the penalty method for pricing american put options of this type

a general equivalence theorem in the theory of discretization methods the lax richtmyer theorem is extended to work in the framework of stetter s theory of discretizations the new result applies to both initial and boundary value problems discretized by finite elements finite differences etc several examples are given together with a comparison with other available equivalence theorems the proof relies on a generalized banach steinhaus theorem 1 introduction in this paper we extend the classical lax richtmyer equivalence theorem 6 so as to cover in a simple way not only initial value problems but also boundary value problems mixed problems etc our theory relies on a generalized banach steinhaus theorem 9 and works essentially in the framework of stetter 13 this set up employs restriction operators to compare the true and discretized solutions as distinct to those theories which use prolongation operators one of the oldest prolongation theories is probably that of aubin summarized in 16 our main result is given in section 2 sections 3 and 4 are devoted to examples and counterexamples the former are meant to show the scope of our result and include the galerkin method for boundary value elliptic problems and semidiscrete and fully discrete schemes for initial value problems the counterexamples prove that the present hypotheses cannot be dispensed with in particular we show that a method which is consistent and convergent for all data in a banach space may be unstable the final section contains a comparison with other available equivalence theorems

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

on positivity shape and norm bound preservation of time stepping methods for semigroups abstract we use functional calculus methods to investigate qualitative properties of c 0 semigroups that are preserved by time discretization methods preservation of positivity concavity and other qualitative shape properties which can be described via positivity are treated in a banach lattice framework preservation of contractivity or norm bound of the semigroup is investigated in the banach space setting the use of the hille phillips h p functional calculus instead of the dunford taylor functional calculus allows us to extend fundamental qualitative results concerning time discretization methods and simplify their proofs including results on multi step schemes and variable step sizes since the h p functional calculus is used throughout the paper we present an elementary introduction to it based on the riemann stieltjes integral

piecewise polynomial collocation for linear boundary value problems of fractional differential equations we consider a class of boundary value problems for linear multi term fractional differential equations which involve 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 the numerical solution of boundary value problems by piecewise polynomial collocation methods is discussed in particular we study the attainable order of convergence of proposed algorithms and show how the convergence rate depends on the choice of the grid and collocation points theoretical results are verified by two numerical examples

convergence of a fitted finite volume method for the penalized black scholes equation governing european and american option pricing in this paper we present an analysis of a numerical method for a degenerate partial differential equation called the black scholes equation governing american and european option pricing the method is based on a fitted finite volume spatial discretization and an implicit time stepping technique the analysis is performed within the framework of the vertical method of lines where the spatial discretization is formulated as a petrov galerkin finite element method with each basis function of the trial space being determined by a set of two point boundary value problems we establish the stability and an error bound for the solutions of the fully discretized system numerical results are presented to validate the theoretical results

runge kutta convolution quadrature methods for well posed equations with memory runge kutta based convolution quadrature methods for abstract well posed linear and homogeneous volterra equations non necessarily of sectorial type are developed a general representation of the numerical solution in terms of the continuous one is given the error and stability analysis is based on this representation which for the particular case of the backward euler method also shows that the numerical solution inherits some interesting qualitative properties such as positivity of the exact solution numerical illustrations are provided

answer to an open problem proposed by r metzler and j klafter in a positive answer to the open problem proposed by r metzler and j klafter it is proved that the asymptotic shape of the solution for a wide class of fractional fokker planck equations is a stretched gaussian for the initial condition being a pulse function in the homogeneous and heterogeneous fractal structures whose mean square displacement behaves like x 2 t t and x 2 t x t 0 1 respectively

stability of rational multistep approximations of holomorphic semigroups in this paper we prove the stability of semidiscretizations in time of holomorphic semigroups in banach spaces by means of a stable rational multistep methods no assumptions on the method other than a stability are required our result is applicable in the maximum norm analysis of parabolic problems

runge kutta methods without order reduction for linear initial boundary value problems it is well known the loss of accuracy when a runge kutta method is used together with the method of lines for the full discretization of an initial boundary value problem we show that this phenomenon called order reduction is caused by wrong boundary values in intermediate stages with a right choice the order reduction can be avoided and the optimal order of convergence in time is achieved we prove this fact for time discretizations of abstract initial boundary value problems based on implicit runge kutta methods moreover we apply these results to the full discretization of parabolic problems by means of galerkin finite element techniques we present some numerical examples in order to confirm that the optimal order is actually achieved

operators on c k spaces preserving copies of schreier spaces it is proved that an operator t c k x k compact metrizable x a separable banach space for which the e szlenk index of t bx is greater than or equal to 1 is an isomorphism on a subspace of c k isomorphic to x the schreier space of order as a corollary one obtains that a complemented subspace of c k with szlenk index equal to 1 contains a subspace isomorphic to x

fast numerical simulation of a new time space fractional option pricing model governing european call option abstract when the fluctuation of option price is regarded as a fractal transmission system and the stock price follows a levy distribution a time space fractional option pricing model tsfopm is obtained then we discuss the numerical simulation of the tsfopm a discrete implicit numerical scheme with a second order accuracy in space and a 2 order accuracy in time is constructed where is a transmission exponent the stability and convergence of the obtained numerical scheme are analyzed moreover a fast bi conjugate gradient stabilized method is proposed to solve the numerical scheme in order to reduce the storage space and computational cost then a numerical example with exact solution is presented to demonstrate the accuracy and effectiveness of the proposed numerical method finally the tsfopm and the above numerical technique are applied to price european call option the characteristics of the fractional option pricing model are analyzed in comparison with the classical black scholes b s model

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

a new numerical method for backward parabolic problems in the maximum norm setting a new method for solving numerically backward parabolic problems is proposed as usual for this kind of ill posed problems it is assumed that an a priori bound for the solution is available the algorithm consists of two basic steps first a standard forward integration is performed in order to approximate the solution at suitable future time levels second a holomorphic recovery procedure is carried out providing the required approximations for the preceding times the analysis is valid in the maximum norm setting and rigorous estimates are derived among other advantages the method can also be applied to nonlinear problems and it produces a continuous output some numerical illustrations are presented

on the stability of variable stepsize rational approximations of holomorphic semigroups we consider variable stepsize time approximations of holomorphic semigroups on general banach spaces for strongly a o acceptable rational functions a general stability theorem is proved which does not impose any constraint on the ratios between stepsizes

a high order uniformly convergent alternating direction scheme for time dependent reaction diffusion singularly perturbed problems in this work we design and analyze an efficient numerical method to solve two dimensional initial boundary value reaction diffusion problems for which the diffusion parameter can be very small with respect to the reaction term the method is defined by combining the peaceman and rachford alternating direction method to discretize in time together with a hodie finite difference scheme constructed on a tailored mesh we prove that the resulting scheme is e uniformly convergent of second order in time and of third order in spatial variables some numerical examples illustrate the efficiency of the method and the orders of uniform convergence proved theoretically we also show that it is easy to avoid the well known order reduction phenomenon which is usually produced in the time integration process when the boundary conditions are time dependent

determination of diffusion kernel on fractals when the diffusion set is a fractal whether the generating mappings of the diffusion set are linear or nonlinear the concrete expressions for the diffusion kernel are obtained moreover the fractional diffusion exponent of the diffusion kernel must satisfy 0 1 in addition the inverse problem of the above case is also discussed

fractional diffusion equation on fractals three dimensional case and scattering function a fractional equation for diffusion in isotropic and homogeneous fractal structures is discussed it generalizes the fractional diffusion equation valid for d dimensional euclidean systems the asymptotic behaviour of the probability density function is obtained exactly analytical expressions are derived for the scattering and relaxation functions which can be studied by x ray and neutron scattering experiments on fractals

avoiding the order reduction of runge kutta methods for linear initial boundary value problems a new strategy to avoid the order reduction of runge kutta methods when integrating linear autonomous nonhomogeneous initial boundary value problems is presented the solution is decomposed into two parts one of them can be computed directly in terms of the data and the other satisfies an initial value problem without any order reduction a numerical illustration is given this idea applies to practical problems where spatial discretization is also required leading to the full order both in space and time

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

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

on some explicit adams multistep methods for fractional differential equations in this paper we present a family of explicit formulas for the numerical solution of differential equations of fractional order the proposed methods are obtained by modifying in a suitable way fractional adams moulton methods and they represent a way for extending classical adams bashforth multistep methods to the fractional case the attention is hence focused on the investigation of stability properties intervals of stability for k step methods k 1 5 are computed and plots of stability regions in the complex plane are presented

high order methods for elliptic and time dependent reaction diffusion singularly perturbed problems the objective of this paper is to construct some high order uniform numerical methods to solve linear reaction diffusion singularly perturbed problems first for 1d elliptic problems based on the central finite difference scheme a new hodie method is defined on a piecewise uniform shishkin mesh using this hodie scheme jointly with a two stage sdirk method we solve a 1d parabolic singularly perturbed problem in both cases we prove that the methods are third order uniform convergent in the maximum norm finally for a 2d parabolic problem of the same type we show numerically that the combination of the hodie scheme with a fractional step rk method gives again a third order uniform convergent scheme

on a connection between a class of q deformed algebras and the hausdorff derivative in a medium with fractal metric abstract over the recent decades diverse formalisms have emerged that are adopted to approach complex systems amongst those we may quote the q calculus in tsallis version of non extensive statistics with its undeniable success whenever applied to a wide class of different systems kaniadakis approach based on the compatibility between relativity and thermodynamics fractional calculus fc that deals with the dynamics of anomalous transport and other natural phenomena and also some local versions of fc that claim to be able to study fractal and multifractal spaces and to describe dynamics in these spaces by means of fractional differential equations the question we might ask is whether or not there are common aspects that connect these alternative approaches in this short communication we discuss a possible relationship between q deformed algebras in two different contexts of statistical mechanics namely tsallis framework and kaniadakis scenario with local form of fractional derivative operators defined in fractal media the so called hausdorff derivatives mapped into a continuous medium with a fractal measure this connection opens up new perspectives for theories that satisfactorily describe the dynamics for the transport in media with fractal metrics such as porous or granular media possible connections with other alternative definitions of fc are also contemplated insights on complexity connected to concepts like coarse grained space time and physics in general are pointed out

maximum norm analysis of completely discrete finite element methods for parabolic problems we consider full discretizations of linear parabolic initial value problems in any space dimension finite elements are used for the discretization in space the resulting semidiscrete problem is integrated in time by means of a strongly a theta acceptable rational method stability estimates are derived and error bounds are established in the maximum norm for both smooth and rough initial data

transfer processes in fractal media an irreversible process in fractal media involves coupling relation between the space and the time the present note displays how the fractional derivation has to be introduced to describe this effect as a result the law of the chemical diffusion to a fractal is given

electrified fractional nanofluid flow with suspended carbon nanotubes abstract improvement in thermal and electric conductivity is the basic requirement of many nano electrochemical systems these systems in material science energy management and chemical processing can be efficiently handled with the presence of carbon nanotubes in the base fluid basically nanofluid is used to act as coolant with enhanced thermal conductivity in equipment such as radiators heat exchangers and electronic systems transfer of heat in nanofluids has been discussed by many researchers with ordinary derivative but here we have considered fractional derivatives to achieve more control on heat transfer here in this communication we have discussed two directional viscoelastic fluid flow with suspended carbon nanotubes swcnts as well as mwcnts are considered as nanoparticles in the base fluid in order to achieve control of the flow and heat transfer flow problem is modeled with caputo fractional derivatives along with fractional momentum and thermal relaxation times and 1 in the electrically conducting flow regime electric and magnetic fields are perpendicular to each other and ohmic heating is appropriately handled in energy equation via ohm s law governing fractional pdes are nonlinear in nature and solved numerically with finite difference discretization along with l 1 algorithm simulated results of velocity and temperature are noted for both the swcnts and mwcnts velocity and temperature profiles for mwcnts remained at a higher level when compared to swcnts for all the values of physical parameters the presence of fractional derivatives swcnts and mwcnts made the results valuable that can be used to efficiently handle similar thermal management problems in engineering such as to achieve thermal control in the vehicle engines refrigerators grinders boilers chillers and heat exchangers

central difference approximation of convection in caputo fractional derivative two point boundary value problems a rigorous numerical analysis is given for a fractional derivative two point boundary value problem the highest order term of the differential operator is a caputo fractional derivative of order 1 2 the second and higher order derivatives of the solution of the problem are in general unbounded at one end of the interval which creates difficulties for the analysis the problem is discretized on an equidistant mesh of diameter h using a standard central difference approximation of the convection term and nodal convergence of order o h 1 is proved provided that h satisfies a stability condition that depends on and the convective coefficient this condition may be restrictive when is near 1 when it is violated the solution may exhibit small oscillations but these can be removed by a simple and inexpensive postprocessing technique numerical results are given to display the performance of the method

a closed form approximation for the fractional black scholes model with transaction costs abstract in this paper we investigate option valuation problems under the fractional black scholes model the aim is to propose a pricing formula for the european option with transaction costs where the costs structure contains fixed costs a cost propositional to the volume traded and a cost proportional to the value traded precisely we provide an approximate solution of the nonlinear hoggard whalley wilmott equation the comparison results reveal that our approximate solutions are close to the numerical computations moreover the comparison results demonstrate that the price of the european option decreases as the hurst exponent increases

maximum principle for the generalized time fractional diffusion equation abstract in the paper a maximum principle for the generalized time fractional diffusion equation over an open bounded domain g 0 t g r n is formulated and proved the proof of the maximum principle is based on an extremum principle for the caputo dzherbashyan fractional derivative that is given in the paper too the maximum principle is then applied to show that the initial boundary value problem for the generalized time fractional diffusion equation possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions

option pricing beyond black scholes based on double fractional diffusion abstract we show how the prices of options can be determined with the help of double fractional differential equation in such a way that their inclusion in a portfolio of stocks provides a more reliable hedge against dramatic price drops than the use of options whose prices were fixed by the black scholes formula

from finite differences to finite elements a short history of numerical analysis of partial differential equations abstract this is an account of the history of numerical analysis of partial differential equations starting with the 1928 paper of courant friedrichs and lewy and proceeding with the development of first finite difference and then finite element methods the emphasis is on mathematical aspects such as stability and convergence analysis