46,084 research outputs found
Novel analytical and numerical methods for solving fractional dynamical systems
During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been successfully applied to problems in system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance. In many cases these new fractional-order models are more adequate than the previously used integer-order models, because fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes that are governed by anomalous diffusion. Hence, there is a growing need to find the solution behaviour of these fractional differential equations. However, the analytic solutions of most fractional differential equations generally cannot be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behaviour of such fractional equations and exploring their applications. The main objective of this thesis is to develop new effective numerical methods and supporting analysis, based on the finite difference and finite element methods, for solving time, space and time-space fractional dynamical systems involving fractional derivatives in one and two spatial dimensions. A series of five published papers and one manuscript in preparation will be presented on the solution of the space fractional diffusion equation, space fractional advectiondispersion equation, time and space fractional diffusion equation, time and space fractional Fokker-Planck equation with a linear or non-linear source term, and fractional cable equation involving two time fractional derivatives, respectively. One important contribution of this thesis is the demonstration of how to choose different approximation techniques for different fractional derivatives. Special attention has been paid to the Riesz space fractional derivative, due to its important application in the field of groundwater flow, system biology and finance. We present three numerical methods to approximate the Riesz space fractional derivative, namely the L1/ L2-approximation method, the standard/shifted Gr¨unwald method, and the matrix transform method (MTM). The first two methods are based on the finite difference method, while the MTM allows discretisation in space using either the finite difference or finite element methods. Furthermore, we prove the equivalence of the Riesz fractional derivative and the fractional Laplacian operator under homogeneous Dirichlet boundary conditions – a result that had not previously been established. This result justifies the aforementioned use of the MTM to approximate the Riesz fractional derivative. After spatial discretisation, the time-space fractional partial differential equation is transformed into a system of fractional-in-time differential equations. We then investigate numerical methods to handle time fractional derivatives, be they Caputo type or Riemann-Liouville type. This leads to new methods utilising either finite difference strategies or the Laplace transform method for advancing the solution in time. The stability and convergence of our proposed numerical methods are also investigated. Numerical experiments are carried out in support of our theoretical analysis. We also emphasise that the numerical methods we develop are applicable for many other types of fractional partial differential equations
Numerical solution for the time-Fractional Diffusion-wave Equations by using Sinc-Legendre Collocation Method
In this paper the numerical solution of fractional diffusion wave equation is proposed. The fractional derivative will be in the Caputo sense. The proposed method will be based on shifted Legendre collocation scheme and sinc function approximation for time and space respectively. The problem is reduced to the problem into a system of algebraic equations after implementing this method. For demonstrating the validity and applicability of the proposed numerical scheme some examples are presented. Keywords: Fractional diffusion equation, Sinc functions, shifted Legendre  polynomials, Collocation method
Matrix approach to discrete fractional calculus II: partial fractional differential equations
A new method that enables easy and convenient discretization of partial
differential equations with derivatives of arbitrary real order (so-called
fractional derivatives) and delays is presented and illustrated on numerical
solution of various types of fractional diffusion equation. The suggested
method is the development of Podlubny's matrix approach (Fractional Calculus
and Applied Analysis, vol. 3, no. 4, 2000, 359--386). Four examples of
numerical solution of fractional diffusion equation with various combinations
of time/space fractional derivatives (integer/integer, fractional/integer,
integer/fractional, and fractional/fractional) with respect to time and to the
spatial variable are provided in order to illustrate how simple and general is
the suggested approach. The fifth example illustrates that the method can be
equally simply used for fractional differential equations with delays. A set of
MATLAB routines for the implementation of the method as well as sample code
used to solve the examples have been developed.Comment: 33 pages, 12 figure
Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation
Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally efficient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the finite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to efficiently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated efficiently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples, including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and efficiency of the approach
Subordinated diffusion and CTRW asymptotics
Anomalous transport is usually described either by models of continuous time
random walks (CTRW) or, otherwise by fractional Fokker-Planck equations (FFPE).
The asymptotic relation between properly scaled CTRW and fractional diffusion
process has been worked out via various approaches widely discussed in
literature. Here, we focus on a correspondence between CTRWs and time and space
fractional diffusion equation stemming from two different methods aimed to
accurately approximate anomalous diffusion processes. One of them is the Monte
Carlo simulation of uncoupled CTRW with a L\'evy -stable distribution
of jumps in space and a one-parameter Mittag-Leffler distribution of waiting
times. The other is based on a discretized form of a subordinated Langevin
equation in which the physical time defined via the number of subsequent steps
of motion is itself a random variable. Both approaches are tested for their
numerical performance and verified with known analytical solutions for the
Green function of a space-time fractional diffusion equation. The comparison
demonstrates trade off between precision of constructed solutions and
computational costs. The method based on the subordinated Langevin equation
leads to a higher accuracy of results, while the CTRW framework with a
Mittag-Leffler distribution of waiting times provides efficiently an
approximate fundamental solution to the FFPE and converges to the probability
density function of the subordinated process in a long-time limit.Comment: 10 pages, 7 figure
New High-Order Compact ADI Algorithms for 3D Nonlinear Time-Fractional Convection-Diffusion Equation
Numerical approximations of the three-dimensional (3D) nonlinear time-fractional convection-diffusion equation is studied, which is firstly transformed to a time-fractional diffusion equation and then is solved by linearization method combined with alternating direction implicit (ADI) method. By using fourth-order Padé approximation for spatial derivatives and classical backward differentiation method for time derivative, two new high-order compact ADI algorithms with orders O(τmin(1+α,2−α)+h4) and O(τ2−α+h4) are presented. The resulting schemes in each ADI solution step corresponding to a tridiagonal matrix equation can be solved by the Thomas algorithm which makes the computation cost effective. Numerical experiments are shown to demonstrate the high accuracy and robustness of two new schemes
Stability analysis of a second-order difference scheme for the time-fractional mixed sub-diffusion and diffusion-wave equation
This study investigates a class of initial-boundary value problems pertaining
to the time-fractional mixed sub-diffusion and diffusion-wave equation (SDDWE).
To facilitate the development of a numerical method and analysis, the original
problem is transformed into a new integro-differential model which includes the
Caputo derivatives and the Riemann-Liouville fractional integrals with orders
belonging to (0,1). By providing an a priori estimate of the solution, we have
established the existence and uniqueness of a numerical solution for the
problem. We propose a second-order method to approximate the fractional
Riemann-Liouville integral and employ an L2 type formula to approximate the
Caputo derivative. This results in a method with a temporal accuracy of
second-order for approximating the considered model. The proof of the
unconditional stability of the proposed difference scheme is established.
Moreover, we demonstrate the proposed method's potential to construct and
analyze a second-order L2-type numerical scheme for a broader class of the
time-fractional mixed SDDWEs with multi-term time-fractional derivatives.
Numerical results are presented to assess the accuracy of the method and
validate the theoretical findings
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