8 research outputs found
A space-time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives
We propose a direct numerical method for the solution of an optimal control
problem governed by a two-side space-fractional diffusion equation. The
presented method contains two main steps. In the first step, the space variable
is discretized by using the Jacobi-Gauss pseudospectral discretization and, in
this way, the original problem is transformed into a classical integer-order
optimal control problem. The main challenge, which we faced in this step, is to
derive the left and right fractional differentiation matrices. In this respect,
novel techniques for derivation of these matrices are presented. In the second
step, the Legendre-Gauss-Radau pseudospectral method is employed. With these
two steps, the original problem is converted into a convex quadratic
optimization problem, which can be solved efficiently by available methods. Our
approach can be easily implemented and extended to cover fractional optimal
control problems with state constraints. Five test examples are provided to
demonstrate the efficiency and validity of the presented method. The results
show that our method reaches the solutions with good accuracy and a low CPU
time.Comment: This is a preprint of a paper whose final and definite form is with
'Journal of Vibration and Control', available from
[http://journals.sagepub.com/home/jvc]. Submitted 02-June-2018; Revised
03-Sept-2018; Accepted 12-Oct-201
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
An unstructured mesh control volume method for two-dimensional space fractional diffusion equations with variable coefficients on convex domains
In this paper, we propose a novel unstructured mesh control volume method to
deal with the space fractional derivative on arbitrarily shaped convex domains,
which to the best of our knowledge is a new contribution to the literature.
Firstly, we present the finite volume scheme for the two-dimensional space
fractional diffusion equation with variable coefficients and provide the full
implementation details for the case where the background interpolation mesh is
based on triangular elements. Secondly, we explore the property of the
stiffness matrix generated by the integral of space fractional derivative. We
find that the stiffness matrix is sparse and not regular. Therefore, we choose
a suitable sparse storage format for the stiffness matrix and develop a fast
iterative method to solve the linear system, which is more efficient than using
the Gaussian elimination method. Finally, we present several examples to verify
our method, in which we make a comparison of our method with the finite element
method for solving a Riesz space fractional diffusion equation on a circular
domain. The numerical results demonstrate that our method can reduce CPU time
significantly while retaining the same accuracy and approximation property as
the finite element method. The numerical results also illustrate that our
method is effective and reliable and can be applied to problems on arbitrarily
shaped convex domains.Comment: 18 pages, 5 figures, 9 table
Finite element method for nonlinear Riesz space fractional diffusion equations on irregular domains
In this paper, we consider two-dimensional Riesz space fractional diffusion equations with nonlinear source term on convex domains. Applying Galerkin finite element method in space and backward difference method in time, we present a fully discrete scheme to solve Riesz space fractional diffusion equations. Our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes, which can help us to deal with space fractional terms on any convex domain. The stability and convergence of the scheme are also discussed. Numerical examples are given to verify accuracy and stability of our scheme