Space Fractional Partial Differential Equations Research Articles

Stock loan valuation based on the Finite Moment Log-Stable process

The empirical test suggests that the log-return series of stock price in US market reject the normal distribution and admit instead a subclass of the asymmetric distribution. In this paper, we investigate the stock loan problem under the assumption that the return of stock follows the finite moment log-stable process (FMLS). In this case, the pricing model of stock loan can be described by a space-fractional partial differential equation with time-varying free boundary condition. Firstly, a penalty term is introduced to change the original problem to be defined on a fixed domain, and then a fully-implicit difference scheme has been developed. Secondly, based on the fully-implicit scheme, we prove that the stock loan value generated by the penalty method cannot fall below the value obtained when the stock loan is exercised early. Thirdly, the numerical experiments are carried out to demonstrate differences of stock loan model under the FMLS and the standard normal distribution. Optimal redemption strategy of stock loan has been achieved. Furthermore the impact of key parameters in our model on the stock loan evaluation are analyzed, and some reasonable explanation are given.

A fast finite difference method for distributed-order space-fractional partial differential equations on convex domains

Fractional partial differential equations (PDEs) provide a powerful and flexible tool for modeling challenging phenomena including anomalous diffusion processes and long-range spatial interactions, which cannot be modeled accurately by classical second-order diffusion equations. However, numerical methods for space-fractional PDEs usually generate dense or full stiffness matrices, for which a direct solver requires O(N3) computations per time step and O(N2) memory, where N is the number of unknowns. The significant computational work and memory requirement of the numerical methods makes a realistic numerical modeling of three-dimensional space-fractional diffusion equations computationally intractable.Fast numerical methods were previously developed for space-fractional PDEs on multidimensional rectangular domains, without resorting to lossy compression, but rather, via the exploration of the tensor-product form of the Toeplitz-like decompositions of the stiffness matrices. In this paper we develop a fast finite difference method for distributed-order space-fractional PDEs on a general convex domain in multiple space dimensions. The fast method has an optimal order storage requirement and almost linear computational complexity, without any lossy compression. Numerical experiments show the utility of the method.

Domain decomposition methods for space fractional partial differential equations

A two-level additive Schwarz preconditioner is proposed for solving the algebraic systems resulting from the finite element approximations of space fractional partial differential equations (SFPDEs). It is shown that the condition number of the preconditioned system is bounded by C(1+H/δ), where H is the maximum diameter of subdomains and δ is the overlap size among the subdomains. Numerical results are given to support our theoretical findings.

Invariant Subspace Method: A Tool for Solving Fractional Partial Differential Equations

Abstract In this paper invariant subspace method has been employed for solving linear and non-linear time and space fractional partial differential equations involving Caputo derivative. A variety of illustrative examples are solved to demonstrate the effectiveness and applicability of the method.

Second order accuracy finite difference methods for space-fractional partial differential equations

Space-fractional partial differential equations are used for simulations of, for example, diffusion of radioactive materials, and financial and other models, which are characterized by heavy-tailed distributions. A number of first order accuracy finite difference methods have been proposed. In the present paper, we introduce second order accuracy finite difference methods with Dirichlet boundary conditions. These methods have a parameter in these schemes, and the parameter stabilizes the schemes. This means that there exist various schemes with second order accuracy, but the stability of each scheme is different. In the present paper, we introduce the most stable scheme for any fractional calculus order by choosing the optimal parameter. In addition, we describe a phenomenon whereby the expected accuracy cannot be obtained if the analytical solution can be expanded to a series having less than second order around boundaries. This also happens in both existing methods and the proposed methods. In the present paper, we develop the stability conditions for the proposed schemes, and numerical examples of second order accuracy and accuracy decay are shown.

Discontinuous Galerkin time stepping method for solving linear space fractional partial differential equations

In this paper, we consider the discontinuous Galerkin time stepping method for solving the linear space fractional partial differential equations. The space fractional derivatives are defined by using Riesz fractional derivative. The space variable is discretized by means of a Galerkin finite element method and the time variable is discretized by the discontinuous Galerkin method. The approximate solution will be sought as a piecewise polynomial function in t of degree at most q−1,q≥1, which is not necessarily continuous at the nodes of the defining partition. The error estimates in the fully discrete case are obtained and the numerical examples are given.

A New Family of Difference Schemes for Space Fractional Advection Diffusion Equation

AbstractThe second order weighted and shifted Grünwald difference (WSGD) operators are developed in [Tian, Zhou and Deng, Math. Comput., 84 (2015), pp. 1703–1727] to solve space fractional partial differential equations. Along this direction, we further design a new family of second order WSGD operators; by properly choosing the weighted parameters, they can be effectively used to discretize space (Riemann-Liouville) fractional derivatives. Based on the new second order WSGD operators, we derive a family of difference schemes for the space fractional advection diffusion equation. By von Neumann stability analysis, it is proved that the obtained schemes are unconditionally stable. Finally, extensive numerical experiments are performed to demonstrate the performance of the schemes and confirm the convergence orders.

Fourier Spectral Methods for Some Linear Stochastic Space-Fractional Partial Differential Equations

Fourier spectral methods for solving some linear stochastic space-fractional partial differential equations perturbed by space-time white noises in the one-dimensional case are introduced and analysed. The space-fractional derivative is defined by using the eigenvalues and eigenfunctions of the Laplacian subject to some boundary conditions. We approximate the space-time white noise by using piecewise constant functions and obtain the approximated stochastic space-fractional partial differential equations. The approximated stochastic space-fractional partial differential equations are then solved by using Fourier spectral methods. Error estimates in the L 2 -norm are obtained, and numerical examples are given.

A New Fractional Calculus Model for the Two-dimensional Anomalous Diffusion and its Analysis

In this paper, a special model for the two-dimensional anomalous diffusion is first deduced from the basic continuous time random walk equations in terms of a time- and space-fractional partial differential equation with the Caputo time-fractional derivative of order α / 2 and the Riesz space-fractional derivative of order α . For α -fractional diffusion equation describes the so called Levy flights that correspond to the continuous time random walk model, where both the mean waiting time and the jump length variance of the diffusing particles are divergent. The fundamental solution to the α -fractional diffusion equation is shown to be a two-dimensional probability density function that can be expressed in explicit form in terms of the Mittag-Leffler function depending on the auxiliary variable |x |/(2√t ) as in the case of the fundamental solution to the classical isotropic diffusion equation. Moreover, we show that the entropy production rate associated with the anomalous diffusion process described by the α -fractional diffusion equation is exactly the same as in the case of the classical isotropic diffusion equation. Thus the α -fractional diffusion equation can be considered to be a natural generalization of the classical isotropic diffusion equation that exhibits some characteristics of both anomalous and classical diffusion.

An implicit numerical method of a new time distributed-order and two-sided space-fractional advection-dispersion equation

Distributed-order differential equations have recently been investigated for complex dynamical systems, which have been used to describe some important physical phenomena. In this paper, a new time distributed-order and two-sided space-fractional advection-dispersion equation is considered. Firstly, we transform the time distributed-order fractional equation into a multi-term time-space fractional partial differential equation by applying numerical integration. Then an implicit numerical method is constructed to solve the multi-term fractional equation. The uniqueness, stability and convergence of the implicit numerical method are proved. Some numerical results are presented to demonstrate the effectiveness of the method. The method and techniques can be extended to other time distributed-order and space-fractional partial differential equations.

Multigrid methods for space fractional partial differential equations

We propose some multigrid methods for solving the algebraic systems resulting from finite element approximations of space fractional partial differential equations (SFPDEs). It is shown that our multigrid methods are optimal, which means the convergence rates of the methods are independent of the mesh size and mesh level. Moreover, our theoretical analysis and convergence results do not require regularity assumptions of the model problems. Numerical results are given to support our theoretical findings.

An Algorithm for the Numerical Solution of Two-Sided Space-Fractional Partial Differential Equations

Abstract We introduce an algorithm for solving two-sided space-fractional partial differential equations. The space-fractional derivatives we consider here are left-handed and right-handed Riemann–Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. We approximate the Hadamard finite-part integrals by using piecewise quadratic interpolation polynomials and obtain a numerical approximation of the space-fractional derivative with convergence order O ( Δ x 3 - α ) ${O(\Delta x^{3- \alpha })}$ , 1 < α < 2 ${1<\alpha <2}$ . A shifted implicit finite difference method is applied for solving the two-sided space-fractional partial differential equation and we prove that the order of convergence of the finite difference method is O ( Δ t + Δ x min ( 3 - α , β ) ) ${O (\Delta t + \Delta x^{\min (3- \alpha , \beta )})}$ , 1 < α < 2 ${1< \alpha <2}$ , β > 0 ${\beta >0}$ , where Δ t , Δ x ${\Delta t,\Delta x}$ denote the time and space stepsizes, respectively. Numerical examples where the solutions have varying degrees of smoothness are presented and compared with the exact analytical solution to compare the practical performance of the method with the theoretical order of convergence.

Optimal homotopy analysis method for nonlinear partial fractional differential equations

The main objective of this paper is to improve the optimal homotopy analysis method to find the approximate solutions for the linear and nonlinear partial fractional differential equations. The fractional derivatives are described in the Caputo sense. The optimal homotopy analysis method in applied mathematics can be used for obtaining the analytic approximate solutions for some nonlinear partial fractional differential equations such as the time and space fractional nonlinear Schrödinger partial differential equation and the time and space fractional telegraph partial differential equation. The optimal homotopy analysis method contains the h parameter which controls the convergence of the approximate solution series. Also, this method determines the optimal value of h as the best convergence of the series of solutions.

Application of Sinc-Galerkin Method for Solving Space-Fractional Boundary Value Problems

We employ the sinc-Galerkin method to obtain approximate solutions of space-fractional order partial differential equations (FPDEs) with variable coefficients. The fractional derivatives are used in the Caputo sense. The method is applied to three different problems and the obtained solutions are compared with the exact solutions of the problems. These comparisons demonstrate that the sinc-Galerkin method is a very efficient tool in solving space-fractional partial differential equations.

A method based on the Jacobi tau approximation for solving multi-term time–space fractional partial differential equations

In this paper, we propose and analyze an efficient operational formulation of spectral tau method for multi-term time–space fractional differential equation with Dirichlet boundary conditions. The shifted Jacobi operational matrices of Riemann–Liouville fractional integral, left-sided and right-sided Caputo fractional derivatives are presented. By using these operational matrices, we propose a shifted Jacobi tau method for both temporal and spatial discretizations, which allows us to present an efficient spectral method for solving such problem. Furthermore, the error is estimated and the proposed method has reasonable convergence rates in spatial and temporal discretizations. In addition, some known spectral tau approximations can be derived as special cases from our algorithm if we suitably choose the corresponding special cases of Jacobi parameters θ and ϑ. Finally, in order to demonstrate its accuracy, we compare our method with those reported in the literature.

Maximum principle and numerical method for the multi-term time-space Riesz-Caputo fractional differential equations

The maximum principle for the space and time-space fractional partial differential equations is still an open problem. In this paper, we consider a multi-term time-space Riesz-Caputo fractional dif...

Maximum principle and numerical method for the multi-term time–space Riesz–Caputo fractional differential equations

The maximum principle for the space and time–space fractional partial differential equations is still an open problem. In this paper, we consider a multi-term time–space Riesz–Caputo fractional differential equations over an open bounded domain. A maximum principle for the equation is proved. The uniqueness and continuous dependence of the solution are derived. Using a fractional predictor–corrector method combining the L1 and L2 discrete schemes, we present a numerical method for the specified equation. Two examples are given to illustrate the obtained results.

Series expansion solutions for the multi-term time and space fractional partial differential equations in two- and three-dimensions

Fractional partial differential equations with more than one fractional derivative in time describe some important physical phenomena, such as the telegraph equation, the power law wave equation, or the Szabo wave equation. In this paper, we consider two- and three-dimensional multi-term time and space fractional partial differential equations. The multi-term time-fractional derivative is defined in the Caputo sense, whose order belongs to the interval (1,2],(2,3],(3,4] or (0,m], and the space-fractional derivative is referred to as the fractional Laplacian form. We derive series expansion solutions based on a spectral representation of the Laplacian operator on a bounded region. Some applications are given for the two- and three-dimensional telegraph equation, power law wave equation and Szabo wave equation.

Numerical Solutions for the Time and Space Fractional Nonlinear Partial Differential Equations

We implement relatively analytical techniques, the homotopy perturbation method, and variational iteration method to find the approximate solutions for time and space fractional Benjamin-Bona Mahony equation. The fractional derivatives are described in the Caputo sense. These methods are used in applied mathematics to obtain the analytic approximate solutions for the nonlinear Bejamin-Bona Mahoney (BBM) partial fractional differential equation. We compare between the approximate solutions obtained by these methods. Also, we present the figures to compare between the approximate solutions. Also, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. We use the improved <svg style="vertical-align:-2.3205pt;width:47.275002px;" id="M1" height="23.612499" version="1.1" viewBox="0 0 47.275002 23.612499" width="47.275002" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,20.662)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,5.944,20.662)"><path id="x1D43A" d="M713 296l-5 -25q-47 -7 -59 -20t-23 -72l-15 -79q-9 -48 -3 -74q-15 -3 -55 -13t-63 -15t-59.5 -10t-70.5 -5q-149 0 -243 80t-94 220q0 169 127.5 276.5t336.5 107.5q91 0 206 -36l-10 -165l-29 -1q1 85 -47.5 126t-146.5 41q-153 0 -243.5 -97t-90.5 -242&#xA;q0 -122 68.5 -198.5t188.5 -76.5q121 0 139 75l20 86q13 58 -1.5 70.5t-99.5 20.5l5 26h267z" /></g> <g transform="matrix(.008,-0,0,-.008,18.25,6.8)"><path id="x2032" d="M227 744l-123 -338l-31 15l73 368q12 3 41.5 -8t36.5 -20z" /></g> <g transform="matrix(.017,-0,0,-.017,22.025,20.662)"><path id="x2F" d="M368 703l-264 -866h-60l265 866h59z" /></g><g transform="matrix(.017,-0,0,-.017,29.028,20.662)"><use xlink:href="#x1D43A"/></g><g transform="matrix(.017,-0,0,-.017,41.336,20.662)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g> </svg>-expansion function method to find exact solutions of nonlinear fractional BBM equation.

Analytical solutions for the multi-term time–space Caputo–Riesz fractional advection–diffusion equations on a finite domain

Generalized fractional partial differential equations have now found wide application for describing important physical phenomena, such as subdiffusive and superdiffusive processes. However, studies of generalized multi-term time and space fractional partial differential equations are still under development. In this paper, the multi-term time–space Caputo–Riesz fractional advection diffusion equations (MT-TSCR-FADE) with Dirichlet nonhomogeneous boundary conditions are considered. The multi-term time-fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2] and [0,2], respectively. These are called respectively the multi-term time-fractional diffusion terms, the multi-term time-fractional wave terms and the multi-term time-fractional mixed diffusion-wave terms. The space fractional derivatives are defined as Riesz fractional derivatives. Analytical solutions of three types of the MT-TSCR-FADE are derived with Dirichlet boundary conditions. By using Luchkoʼs Theorem (Acta Math. Vietnam., 1999), we proposed some new techniques, such as a spectral representation of the fractional Laplacian operator and the equivalent relationship between fractional Laplacian operator and Riesz fractional derivative, that enabled the derivation of the analytical solutions for the multi-term time–space Caputo–Riesz fractional advection–diffusion equations.