16,923 research outputs found
A RBF-based differential quadrature method for solving two-dimensional variable-order time fractional advection-diffusion equation
The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.Numerical simulation technique of two-dimensional variable-order time fractional advection-diffusion equation is developed in this paper using radial basis function-based differential quadrature method (RBF-DQ). To the best of the authors’ knowledge, this is the first application of this method to variable-order time fractional advection-diffusion equations. For the general case of irregular geometries, the meshless local form of RBF-DQ is used and the multiquadric type of radial basis functions is selected for the computations. This approach allows one to define a reconstruction of the local radial basis functions to treat accurately both the Dirichlet and Neumann boundary conditions on the irregular boundaries. The method is validated by the well documented test examples involving variable-order fractional modelling of air pollution. The numerical results demonstrate that the proposed method provides accurate solutions fortwo-dimensional variable-order time fractional advection-diffusion equations
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 advanced meshless method for time fractional diffusion equation
Recently, because of the new developments in sustainable engineering and renewable energy, which are usually governed by a series of fractional partial differential equations (FPDEs), the numerical modelling and simulation for fractional calculus are attracting more and more attention from researchers. The current dominant numerical method for modeling FPDE is Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings including difficulty in simulation of problems with the complex problem domain and in using irregularly distributed nodes. Because of its distinguished advantages, the meshless method has good potential in simulation of FPDEs. This paper aims to develop an implicit meshless collocation technique for FPDE. The discrete system of FPDEs is obtained by using the meshless shape functions and the meshless collocation formulation. The stability and convergence of this meshless approach are investigated theoretically and numerically. The numerical examples with regular and irregular nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of fractional partial differential equations
Numerical investigation of Differential Biological-Models via GA-Kansa Method Inclusive Genetic Strategy
In this paper, we use Kansa method for solving the system of differential
equations in the area of biology. One of the challenges in Kansa method is
picking out an optimum value for Shape parameter in Radial Basis Function to
achieve the best result of the method because there are not any available
analytical approaches for obtaining optimum Shape parameter. For this reason,
we design a genetic algorithm to detect a close optimum Shape parameter. The
experimental results show that this strategy is efficient in the systems of
differential models in biology such as HIV and Influenza. Furthermore, we prove
that using Pseudo-Combination formula for crossover in genetic strategy leads
to convergence in the nearly best selection of Shape parameter.Comment: 42 figures, 23 page
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