Fractional Diffusion Model Research Articles

An implicit-explicit preconditioned direct method for pricing options under regime-switching tempered fractional partial differential models

Recently, fractional partial differential equations have been widely applied in option pricing problems, which better explains many important empirical facts of financial markets, but rare paper considers the multi-state options pricing problem based on fractional diffusion models. Thus, multi-state European option pricing problem under regime-switching tempered fractional partial differential equation is considered in this paper. Due to the expensive computational cost caused by the implicit finite difference scheme, a novel implicit-explicit finite difference scheme has been developed with consistency, stability, and convergence guarantee. Since the resulting coefficient matrix equals to the direct sum of several Toeplitz matrices, a preconditioned direct method has been proposed with ${\mathcal O}(\bar {S}N\log N+\bar {S}^{2} N)$ operation cost on each time level with adaptability analysis, where $\bar {S}$ is the number of states and N is the number of grid points. Related numerical experiments including an empirical example have been presented to demonstrate the effectiveness and accuracy of the proposed numerical method.

Fast calculation based on a spatial two‐grid finite element algorithm for a nonlinear space–time fractional diffusion model

AbstractIn this article, a spatial two‐grid finite element (TGFE) algorithm is used to solve a two‐dimensional nonlinear space–time fractional diffusion model and improve the computational efficiency. First, the second‐order backward difference scheme is used to formulate the time approximation, where the time‐fractional derivative is approximated by the weighted and shifted Grünwald difference operator. In order to reduce the computation time of the standard FE method, a TGFE algorithm is developed. The specific algorithm is to iteratively solve a nonlinear system on the coarse grid and then to solve a linear system on the fine grid. We prove the scheme stability of the TGFE algorithm and derive a priori error estimate with the convergence result O(Δt2 + hr + 1 − η + H2r + 2 − 2η). Finally, through a two‐dimensional numerical calculation, we improve the computational efficiency and reduce the computation time by the TGFE algorithm.

A Fractional Diffusion Model for Dye-Sensitized Solar Cells.

Dye-sensitized solar cells have continued to receive much attention since their introduction by O’Regan and Grätzel in 1991. Modelling charge transfer during the sensitization process is one of several active research areas for the development of dye-sensitized solar cells in order to control and improve their performance and efficiency. Mathematical models for transport of electron density inside nanoporous semiconductors based on diffusion equations have been shown to give good agreement with results observed experimentally. However, the process of charge transfer in dye-sensitized solar cells is complicated and many issues are in need of further investigation, such as the effect of the porous structure of the semiconductor and the recombination of electrons at the interfaces between the semiconductor and electrolyte couple. This paper proposes a new model for electron transport inside the conduction band of a dye-sensitized solar cell comprising of as its nanoporous semiconductor. This model is based on fractional diffusion equations, taking into consideration the random walk network of . Finally, the paper presents numerical solutions of the fractional diffusion model to demonstrate the effect of the fractal geometry of on the fundamental performance parameters of dye-sensitized solar cells, such as the short-circuit current density, open-circuit voltage and efficiency.

Hierarchical matrix approximations for space-fractional diffusion equations

Space fractional diffusion models generally lead to dense discrete matrix operators, which lead to substantial computational challenges when the system size becomes large. For a state of size N, full representation of a fractional diffusion matrix would require O(N2) memory storage requirement, with a similar estimate for matrix–vector products. In this work, we present H2 matrix representation and algorithms that are amenable to efficient implementation on GPUs, and that can reduce the cost of storing these operators to O(N) asymptotically. Matrix–vector multiplications can be performed in asymptotically linear time as well. Performance of the algorithms is assessed in light of 2D simulations of space fractional diffusion equation with constant diffusivity. Attention is focused on smooth particle approximation of the governing equations, which lead to discrete operators involving explicit radial kernels. The algorithms are first tested using the fundamental solution of the unforced space fractional diffusion equation in an unbounded domain, and then for the steady, forced, fractional diffusion equation in a bounded domain. Both matrix-inverse and pseudo-transient solution approaches are considered in the latter case. Our experiments show that the construction of the fractional diffusion matrix, the matrix–vector multiplication, and the generation of an approximate inverse pre-conditioner all perform very well on a single GPU on 2D problems with N in the range 105 – 106. In addition, the tests also showed that, for the entire range of parameters and fractional orders considered, results obtained using the H2 approximations were in close agreement with results obtained using dense operators, and exhibited the same spatial order of convergence. Overall, the present experiences showed that the H2 matrix framework promises to provide practical means to handle large-scale space fractional diffusion models in several space dimensions, at a computational cost that is asymptotically similar to the cost of handling classical diffusion equations.

A Finite Difference Method for Space Fractional Differential Equations with Variable Diffusivity Coefficient

Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially more difficult the mathematical analysis of these models and the establishment of suitable numerical schemes. This paper proposes and analyzes the first finite difference method for solving {\em variable-coefficient} fractional differential equations, with two-sided fractional derivatives, in one-dimensional space. The proposed scheme combines first-order forward and backward Euler methods for approximating the left-sided fractional derivative when the right-sided fractional derivative is approximated by two consecutive applications of the first-order backward Euler method. Our finite difference scheme reduces to the standard second-order central difference scheme in the absence of fractional derivatives. The existence and uniqueness of the solution for the proposed scheme are proved, and truncation errors of order $h$ are demonstrated, where $h$ denotes the maximum space step size. The numerical tests illustrate the global $O(h)$ accuracy of our scheme, except for nonsmooth cases which, as expected, have deteriorated convergence rates.

Identifying Arguments of Space-Time Fractional Diffusion: Data-Driven Approach

A plethora of dynamical complex systems, from disordered media to living systems, exhibit anomalous diffusion behavior that is characterized by a nonlinear mean-square displacement relationship. From a mathematical perspective, the anomalous behavior is described by a partial differential equation (PDE) model that often involves fractional derivatives. In general, PDEs represent various physical laws/rules governing complex dynamical systems. Since prior knowledge about the physical laws is not available, in the diverse areas of engineering involving complex systems, we aim to discover the PDE and its parameters from system's evolution data. Towards deciphering non-trivial mechanisms driving a complex system, we propose a data-driven approach that estimates the parameters of a PDE model associated with the anomalous diffusion. We study the space-time fractional diffusion model that describes a complex stochastic process, where the magnitude and the time increments are stable processes. Starting from time-series data recorded while the system is evolving, we develop a higher order moments-based approach to determine the parameters of a generalized fractional PDE. We formulate a regression problem to allow us to estimate the arguments of the fractional diffusion. Finally, using extensive simulation studies, we show that the proposed approach is effective at retrieving the relevant parameters of the space-time fractional PDE.

Pricing Path-Independent Payoffs with Exotic Features in the Fractional Diffusion Model

We provide several practical formulas for pricing path-independent exotic instruments (log options and log contracts, digital options, gap options, power options with or without capped payoffs …) in the context of the fractional diffusion model. This model combines a tail parameter governed by the space fractional derivative, and a subordination parameter governed by the time-fractional derivative. The pricing formulas we derive take the form of quickly convergent series of powers of the moneyness and of the convexity adjustment; they are obtained thanks to a factorized formula in the Mellin space valid for arbitrary payoffs, and by means of residue theory. We also discuss other aspects of option pricing such as volatility modeling, and provide comparisons of our results with other financial models.

A New Kind of Parallel Natural Difference Method for Multi-Term Time Fractional Diffusion Model

Multi-term time fractional diffusion model is not only an important physical subject, but also a practical problem commonly involved in engineering. In this paper, we apply the alternating segment technique to combine the classical explicit and implicit schemes, and propose a parallel nature difference method alternating segment pure explicit–implicit (PASE-I) and alternating segment pure implicit–explicit (PASI-E) difference schemes for multi-term time fractional order diffusion equations. The existence and uniqueness of the solutions are proved, and stability and convergence analysis of the two schemes are also given. Theoretical analyses and numerical experiments show that the PASE-I and PASI-E schemes are unconditionally stable and satisfy second-order accuracy in spatial precision and 2 − α order in time precision. When the computational accuracy is equivalent, the CPU time of the two schemes are reduced by up to 2 / 3 compared with the classical implicit difference method. It indicates that the PASE-I and PASI-E parallel difference methods are efficient and feasible for solving multi-term time fractional diffusion equations.

A fractional diffusion model for single-well simulation in geological media

In this scholarly work, a novel mathematical model for single-phase, single-well simulation in hydrocarbon reservoirs was established using the fractional calculus theory. Specifically, the existing fractional generalization of Darcy's law is reconstructed by introducing an auxiliary parameter to ensure that the definition of the conventional rock permeability with dimension L2 was preserved. Subsequently, the newly constructed fractional flux relationship was substituted as a momentum equation in the mass conservation expression in the radial-cylindrical coordinate system. The resulting nonlocal time-radial diffusivity equation was solved numerically using the block-centered finite-difference approximation and adopting the Grünwald–Letnikov formula for the definition of the fractional-order derivative. The developed numerical scheme was verified by comparing the approximate pressure solution against the semi-analytical solution of the simplified nonlocal time-radial diffusivity equation and the classic radial flow diffusivity equation. Furthermore, incremental material balance checks were performed and exhibited to ensure the conservation of mass at each time step. Finally, a detailed sensitivity analysis was performed to evaluate the impact of the order of fractional differentiation on pressure distribution in the reservoir and wellbore. The proposed nonlocal time-radial diffusivity equation proffers a useful model for single-well simulation in geological media that exhibit distorted flow paths and near power-law behaviors.

Implicit sampling for hierarchical Bayesian inversion and applications in fractional multiscale diffusion models

This paper presents an implicit sampling method for hierarchical Bayesian inverse problems. A widely used approach for sampling posterior distribution is Markov chain Monte Carlo (MCMC). However, the samples generated by MCMC are usually strongly correlated, which may lead to a small size of effective samples. The implicit sampling method proposed in Chorin and Tu (2009) can generate independent samples and capture inherent non-Gaussian features of the posterior. In implicit sampling method, the posterior samples are generated by a map and distribution around the maximum a posterior (MAP) point. For practical applications, some parameters in prior density are often unknown and the hierarchical Bayesian formulation is used to estimate the MAP point effectively. It is applied to the Bayesian inverse problems of multi-term time fractional diffusion models in heterogeneous media. To effectively capture the heterogeneity effect, we present a mixed generalized multiscale finite element method (mixed GMsFEM) to substantially speed up the Bayesian inversion. Finally, we carry out a few numerical examples to effectively identify various unknown inputs.

APPROXIMATION OF FRACTIONAL RESOLVENTS AND APPLICATIONS TO TIME OPTIMAL CONTROL PROBLEMS

We investigate the approximation of fractional resolvents, extending and improving some corresponding results on semigroups and resolvents. As applications, we utilize the approach of Meyer approximation to analyze the time optimal control problem of a Riemann-Liouville fractional system without Lipschitz continuity. A fractional diffusion model is also presented to confirm our theoretical findings.

Optimal control problem for an equation of filtration with memory

Fractional diffusion models are generalization to the diffusion models with integer derivatives. There has been great interest in the study of this models because of their appearance in modeling various applications in the physical sciences, medicine and biology. We consider a filtration model with nonclassical Darcy's constitutive equation. Resulting equation states that the flux of fluid is proportional to not only gradient pressure but it's Riemann-Liouville fractional derivative also. This model was proposed by M. Caputo and allows the permeability varies with time depending on the previous pressure gradient. These phenomena, which we will represent mathematically with memory formalisms, have often been observed qualitatively in oil extraction, in geothermal areas and in the laboratory Similar problems arises in the study of flow of generalized second grade fluid. Existence results of initial and boundary value problems for partial fractional differential equations have been studied by E. Bazhlekova, K. Diethelm, J. Janno, A.N. Kochubei, G.P. Lopushans'ka, R. Zacher and others. Fractional optimal control problems have attracted for example R.Dorville, G.M. Mophou, V.S. Valmo\-rin, Y. Zhou, L. Peng and many techniques have been developed for solving such problems. We consider the problem of minimization of the standard cost functional $J(u)$ which is determined in the terms of generalized solution of initial-boundary problem of time-fractional differential equation under considerations. We consider a control via right hand term $u$ and an observation on the whole domain in $L_2$ norm with a Tikhonov regularizer term. First we introduce functional spaces and establish some auxiliary properties of fractional integrals and fractional derivatives. Second we prove an existence and uniqueness result for the state problem. We remind that we deals with an equation of filtration with memory. Our objectives are: a) to prove that there exists a minimizer $u$ of the cost functional $J$; b) to obtain necessary and sufficient conditions for $u$ to be an extremum; c) to obtain constructive algorithm amenable to computations for approximations of the optimal control. An unique solvability of state and conjugate problem is established by the help of Galerkin method and corresponding a priori estimates. Then we prove that the cost functional is coercive, convex and weakly lower semicontinuous. We show the existence of the optimal solution by proving the existence of the weakly convergent minimization sequence satisfying the state equation. The uniqueness follows directly from the strong convexity of the cost functional. This gives us the item a). The item b) is obtained from the first order optimality condition. We justify also the conjugated gradient method to search the optimal control function. On this way we use some results of R. Winther, which allows us to use the conjugate gradient method in our situation and prove its superlinear convergence.

A spatial fractional diffusion model for predicting the characteristics of VOCs emission in porous dry building material

Volatile organic compounds (VOCs) concentrations of dry building materials can significantly affect the quality of indoor air. In this paper, a spatial fractional diffusion model is established by introducing fractional Fick’s law, based on the fact that the pore structure of porous building materials has a strongly impact on VOCs diffusion. Furthermore, the areal porosity of the material is drawn into the convection mass transfer equation. The relevant parameters are estimated by a kind of optimization algorithm. The fractional nonlinear equations are tackled by the finite difference method combined with L2-algorithm. Results indicate that spatial fractional diffusion model agrees better with the experimental data than Deng and Kim’s model. The spatial fractional diffusion model is used to describe the formaldehyde concentration in the particleboard according to the experimental data. Taking formaldehyde as a common substance, the influences of physical parameters on VOCs diffusion are predicted by this model. The results of numerical simulation demonstrate that the diffusion is consistent with the actual situation.

Numerical investigation of a fractional diffusion model on circular comb-inward structure

In this paper, the diffusion dynamics of particles on a circular comb-inward structure with different radial and tangential mobilities are studied. The Scott-Blair time-fractional memory model is exploited to incorporate the trapping process in the diffusion. A new formulation of particle flux is proposed for which the numerical discretization is straightforward. The numerical scheme is based on the second-order L2−1σ formula for time stepping and uses the finite difference method for spatial approximations. The influences of involved parameters on the distribution of particles and the mean-square displacement (MSD) are investigated in detail. According to the model prediction, the diffusion of particles on the ring decreases with the smaller values of the fractional order α. Results also indicate that the radial and tangential mobilities exhibit the opposite effects on the MSD of particles.

Trap-controlled fractal diffusion model of the Havriliak-Negami dielectric relaxation

A new model of non-Debye dielectric relaxation, which leads to the popular in broadband dielectric spectroscopy Havriliak-Negami expression for two-power-law dielectric response, is proposed. The new model is based on the multiple trapping model and the fractal diffusion model. Explicit analytical expressions for experimentally measured values such as dielectric strength, characteristic relaxation time, and power-law exponents, which include the microscopic parameters of the medium, are obtain. The comparison of the expression for the relaxation time with the experiment is demonstrated. A clear physical interpretation of the power-law exponents of the Havriliak-Negami relaxation function is found, one makes it possible to extend the area application of this function and to use it for the fitting of dielectric data, which exhibit atypical relaxation behavior.

Parameter identification for fractional fractal diffusion model based on experimental data.

This paper studies the techniques of parameter estimation and their application in determining parameters of the fractional fractal diffusion model. On account of the basic structural characteristics of the porous coal matrix, the fractional fractal diffusion model is established to express the gas transport mechanism in the heterogeneous coal matrix. A L1 finite difference method in the temporal direction while spectral collocation method in the spatial direction is proposed to solve the model numerically. Then, by means of the gas adsorption and desorption experiments in coal samples, attempts have been made by the BFGS method, nonlinear conjugate gradient method, and Bayesian method to compare and contrast to obtain the physical parameters of the model. Furthermore, advantages and limitations of different estimation methods are discussed.

Multi-dimensional spectral tau methods for distributed-order fractional diffusion equations

The distributed-order fractional diffusion equation is a generalization of the standard fractional diffusion equation that can model processes lacking power-law scaling over the whole time-domain. An important application of distributed-order diffusions is to model ultraslow diffusion where a plume of particles spreads at a logarithmic rate. To broaden the range of applicability of distributed-order fractional diffusion models, efficient numerical methods are needed to solve the model equation. In this work, we develop spectral tau schemes to discretize the fractional diffusion equation with distributed-order fractional derivative in time and Dirichlet boundary conditions. The model solution is expanded in multi-dimensions in terms of Legendre polynomials and the discrete equations are obtained with the tau method. Numerical examples are provided to highlight the convergence rate and the flexibility of this approach. The proposed spectral tau methods yield an exponential rate of convergence when the solution is smooth. Our results confirm that nonlocal numerical methods are best suited to discretize distributed-order fractional differential equations as they naturally take the global behavior of the solution into account.

Effects of fractional mass transfer and chemical reaction on MHD flow in a heterogeneous porous medium

This paper presents a study on space fractional anomalous convective-diffusion and chemical reaction in the magneto-hydrodynamic fluid over an unsteady stretching sheet. The fractional diffusion model is derived from decoupled continuous time random walks in a heterogeneous porous medium. A novel transformation which features time finite difference is introduced to reduce the governing equations into ordinary differential ones in each time level. Numerical solutions are established by an implicit finite difference scheme. The stability and convergence of the method are analyzed. Results show that increasing fractional derivative parameter enhances concentration near the surface while an opposite phenomenon occurs far away from the wall. There is a reduction of mass transfer rate on the sheet with an increase in the fractional derivative parameter. Moreover, the numerical solutions are compared with exact solutions and good agreement has been observed.

Analytical solutions and numerical schemes of certain generalized fractional diffusion models

We give the analytical solutions of the fractional diffusion equations in one-, and two-dimensional space described by the Caputo left generalized fractional derivative. We introduce the forward Euler method for fractional diffusion equations represented by the Caputo left generalized fractional derivative. The contribution of this paper is to evaluate the impact of the second parameter of the Caputo left generalized fractional derivative in the behavior of the analytical solutions of the fractional diffusion equations, and to propose a numerical method for the generalized fractional diffusion equations. We will present the difference existing between the classical diffusion equation, the fractional diffusion equation described by Caputo fractional derivative and the fractional diffusion equation expressed by the Caputo left generalized fractional derivative. The Fourier-sine-Laplace-transform method is used to determine the analytical solutions of the fractional diffusion equations described by the Caputo left generalized fractional derivative. Some particular cases of diffusion equations are discussed, and the numerical simulations of their analytical solutions are presented and analyzed.

A fractional mass transfer model for simulating VOC emissions from porous, dry building material

Dry building material is a complex porous medium with a fractal structure. A developed fractional mass transfer model can accurately simulate anomalous diffusion in complex media, such as dry building material. In this work, a new time and space fractional diffusion model is first established to study the transport and emission behaviors of volatile organic compounds (VOCs) in a dry building material in an environmental chamber. In this model, the space fractional derivative is expressed by the weight coefficients of the upward and downward transfer probabilities, and the time fractional derivative and areal porosity of the material are introduced to describe anomalous diffusion. With the newly developed finite difference and parameter estimation methods, the present model can describe the early breakthrough and heavy-tailed phenomena well. Furthermore, the results of the present model show good agreement with the experimental data, as the error value (67.44 μg/m3) is much less than that of Deng and Kim's model (190 μg/m3). The advantages of long-term prediction are shown. Moreover, the effects of key parameters on VOC transport are revealed.