17 research outputs found
Differential quadrature method for space-fractional diffusion equations on 2D irregular domains
In mathematical physics, the space-fractional diffusion equations are of
particular interest in the studies of physical phenomena modelled by L\'{e}vy
processes, which are sometimes called super-diffusion equations. In this
article, we develop the differential quadrature (DQ) methods for solving the 2D
space-fractional diffusion equations on irregular domains. The methods in
presence reduce the original equation into a set of ordinary differential
equations (ODEs) by introducing valid DQ formulations to fractional directional
derivatives based on the functional values at scattered nodal points on problem
domain. The required weighted coefficients are calculated by using radial basis
functions (RBFs) as trial functions, and the resultant ODEs are discretized by
the Crank-Nicolson scheme. The main advantages of our methods lie in their
flexibility and applicability to arbitrary domains. A series of illustrated
examples are finally provided to support these points.Comment: 25 pages, 25 figures, 7 table
Alikhanov Legendre–Galerkin spectral method for the coupled nonlinear time-space fractional Ginzburg–Landau complex system
A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg-Landau system is proposed and analyzed. The Alikhanov L2-1 sigma difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efficiently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Gronwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to confirm the theoretical claims
An Efficient Hybrid Numerical Scheme for Nonlinear Multiterm Caputo Time and Riesz Space Fractional-Order Diffusion Equations with Delay
In this paper, we construct and analyze a linearized finite difference/Galerkin-Legendre spectral scheme for the nonlinear multiterm Caputo time fractional-order reaction-diffusion equation with time delay and Riesz space fractional derivatives. The temporal fractional orders in the considered model are taken as 0<β0<β1<β2<⋯<βm<1. The problem is first approximated by the L1 difference method on the temporal direction, and then, the Galerkin-Legendre spectral method is applied on the spatial discretization. Armed by an appropriate form of discrete fractional Grönwall inequalities, the stability and convergence of the fully discrete scheme are investigated by discrete energy estimates. We show that the proposed method is stable and has a convergent order of 2-βm in time and an exponential rate of convergence in space. We finally provide some numerical experiments to show the efficacy of the theoretical results. © 2021 A. K. Omran et al.A. K. Omran is funded by a scholarship under the joint executive program between the Arab Republic of Egypt and Russian Federation. M. A. Zaky wishes to acknowledge the support of the Nazarbayev University Program (091019CRP2120). M. A. Zaky wishes also to acknowledge the partial support of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant “Dynamical Analysis and Synchronization of Complex Neural Networks with Its Applications”)
High order approximation scheme for a fractional order coupled system describing the dynamics of rotating two-component Bose-Einstein condensates
A coupled system of fractional order Gross-Pitaevskii equations is under consideration in which the time-fractional derivative is given in Caputo sense and the spatial fractional order derivative is of Riesz type. This kind of model may shed light on some time-evolution properties of the rotating two-component Bose¢ Einstein condensates. An unconditional convergent high-order scheme is proposed based on L2- finite difference approximation in the time direction and Galerkin Legendre spectral approximation in the space direction. This combined scheme is designed in an easy algorithmic style. Based on ideas of discrete fractional Grönwall inequalities, we can prove the convergence theory of the scheme. Accordingly, a second order of convergence and a spectral convergence order in time and space, respectively, without any constraints on temporal meshes and the specified degree of Legendre polynomials . Some numerical experiments are proposed to support the theoretical results
Application of the B-spline Galerkin approach for approximating the time-fractional Burger's equation
This paper presents a numerical scheme based on the Galerkin finite element method and cubic B-spline base function with quadratic weight function to approximate the numerical solution of the time-fractional Burger's equation, where the fractional derivative is considered in the Caputo sense. The proposed method is applied to two examples by using the and error norms. The obtained results are compared with a previous existing method to test the accuracy of the proposed method
Fractional Calculus and the Future of Science
Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding
On the bilateral preconditioning for an L2-type all-at-once system arising from time-space fractional Bloch-Torrey equations
Time-space fractional Bloch-Torrey equations (TSFBTEs) are developed by some
researchers to investigate the relationship between diffusion and
fractional-order dynamics. In this paper, we first propose a second-order
implicit difference scheme for TSFBTEs by employing the recently proposed
L2-type formula [A. A. Alikhanov, C. Huang, Appl. Math. Comput. (2021) 126545].
Then, we prove the stability and the convergence of the proposed scheme. Based
on such a numerical scheme, an L2-type all-at-once system is derived. In order
to solve this system in a parallel-in-time pattern, a bilateral preconditioning
technique is designed to accelerate the convergence of Krylov subspace solvers
according to the special structure of the coefficient matrix of the system. We
theoretically show that the condition number of the preconditioned matrix is
uniformly bounded by a constant for the time fractional order . Numerical results are reported to show the efficiency of our
method.Comment: 24 pages, 6 tables, 4 figure
Numerical approximations of fractional differential equations: a Chebyshev pseudo-spectral approach.
Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.This study lies at the interface of fractional calculus and numerical methods. Recent
studies suggest that fractional differential and integral operators are well suited to model
physical phenomena with intrinsic memory retention and anomalous behaviour. The global
property of fractional operators presents difficulties in fnding either closed-form solutions
or accurate numerical solutions to fractional differential equations. In rare cases, when
analytical solutions are available, they often exist only in terms of complex integrals and
special functions, or as infinite series. Similarly, obtaining an accurate numerical solution
to arbitrary order differential equation is often computationally demanding. Fractional
operators are non-local, and so it is practicable that when approximating fractional
operators, non-local methods should be preferred. One such non-local method is the
spectral method. In this thesis, we solve problems that arise in the
ow of non-Newtonian
fluids modelled with fractional differential operators. The recurrent theme in this thesis
is the development, testing and presentation of tractable, accurate and computationally
efficient numerical schemes for various classes of fractional differential equations. The
numerical schemes are built around the pseudo{spectral collocation method and shifted
Chebyshev polynomials of the first kind. The literature shows that pseudo-spectral
methods converge geometrically, are accurate and computationally efficient. The objective
of this thesis is to show, among other results, that these features are true when the method
is applied to a variety of fractional differential equations. A survey of the literature
shows that many studies in which pseudo-spectral methods are used to numerically
approximate the solutions of fractional differential equations often to do this by expanding
the solution in terms of certain orthogonal polynomials and then simultaneously solving
for the coefficients of expansion. In this study, however, the orthogonality condition of
the Chebyshev polynomials of the first kind and the Chebyshev-Gauss-Lobatto quadrature
are used to numerically find the coefficients of the series expansions. This approach is
then applied to solve various fractional differential equations, which include, but are not
limited to time{space fractional differential equations, two{sided fractional differential
equations and distributed order differential equations. A theoretical framework is provided
for the convergence of the numerical schemes of each of the aforementioned classes of
fractional differential equations. The overall results, which include theoretical analysis
and numerical simulations, demonstrate that the numerical method performs well in
comparison to existing studies and is appropriate for any class of arbitrary order differential
equations. The schemes are easy to implement and computationally efficient