Common elections for .
In modeling using diffusion equations, the relationship between fractal geometry and fractional calculus arises by modeling the conditions of the medium as a fractal whose fractional dimension determines the order of this equation. For this reason, it is very useful to have numerical methods that solve them and discretization of the domain is not determinant for the efficiency of the algorithm. In this work, it is proposed to show that meshless methods, in particular methods with radial basis functions (RBF), are an alternative to schemes in differences or structured meshes. We show that we can obtain numerical solutions to some fractional partial differential equations using collocation and RBF, over non equally distributed data.
- fractional partial differential equations (FPDE)
- meshless methods
- radial basis functions (RBF)
- Caputo derivative
- Riemann-Liouville derivative
- Riesz derivative
1.1. Radial basis function methodology
The Hardy-based radial-based functions (RBF) methodology  arises from the need to apply multivariate interpolation to cartography problems, with randomly dispersed data (also known as collocation nodes). Micchelli , Powell et al.  gave it a great boost by proving non-singularity theorems. Later, Kansa [4, 5] proposed to consider the analytical derivatives of the FBR to develop numerical schemes that deal with partial differential equations (PDE).
Regarding PDE over spaces of dimension greater than one, we generally opt for finite element (FE) type discretizations on meshes, structured or not; also pseudo spectral (PS) methods through base functions such as Fourier or Chebyshev. The high degree of computational efficiency in the procedure raises the cost in the regularity constraints on the form of the computational domain.
The FE methods involve decomposition of the domain; for example, in two dimensions rectangles are constructed with curvilinear mappings that allow the refinement of the mesh in critical areas. However, this type of implementation is complex and very close nodes are needed, mainly at the domain boundaries, which impairs stability conditions in time.
For this reason, we look for numerical techniques that do not depend to a great extent on data distribution. RBF collocation methods belong to the “meshfree” methods; that is, they only require scattered collocation nodes on the domain and the boundary. They are also an alternative for dealing with problems in larger dimensions and irregular domains. Hence, in recent decades, such methods have attracted the attention of researchers in order to solve partial differential equations. See for example: Chen et al. .
This type of technique approximates the solution by means of a linear combination of radial basis functions, which are globally defined and of exponential convergence, but they produce dense and badly conditioned interpolation matrices. On the other hand, some RBFs contain a shape parameter, a number that influences the precision of the numerical results [7, 8].
There are algorithms that also produce better conditioned interpolants, even when the shape parameter tends to zero, with a double precision arithmetic. Some of them are: Contour-Padé  and RBF-QR . Related to the latter is the recent RBF-GA .
The RBF-QR method provides a numerically more stable alternative for converting basis that are very similar to each other, or almost linearly dependent on a set of scattered nodes, to a very well conditioned base that generates exactly the same space. This method was originally implemented for nodes on the surface of a sphere in Fornberg and Piret . And recently for an arbitrary set of nodes in one, two, and three dimensions [12, 13].
1.2. Fractional calculus and differential equations: about some applications
The growing interest in fractional calculus has been motivated by applications of fractional equations in different fields of research.
One of the applications considered in this work is that of convection-diffusion equations, which appear not only in many applications of models in Physics and Chemistry [14–17], in problems of flow or heat transfer, but also in other fields such as financial [18, 19]. The solution of a convection-diffusion problem can be interpreted as a probability distribution of one or more underlying stochastic processes . The classical diffusion equation (or heat equation) and its Gaussian solution existed long before Einstein established a connection with random walks.
The anomalous diffusion equations, on the other hand, were originally developed from stochastic walks models . The parameters of these equations are uniquely determined by the fractal dimension of the underlying object .
Recent applications of these anomalous diffusion-convection models—those related to oil extraction and hydrological models for aquifers, food production and water distribution in large cities, are quite important . Determining the behavior of the fluid inside the reservoir and the loss of permeability of the medium assists in the investigation of the mechanisms of oil migration. In this context, the geometric and physical interpretation of fractional derivatives in these differential equations is a “ultra slow” or “ultra fast” diffusion.
The relationship between fractal geometry and fractional calculus is given from the consideration that a particle moves in a porous medium with fractal structure . Thus, when considering the derivative over that variable, a fractional derivative is obtained.
Fuentes et al. [24, 25] proposed a model that represents anomalous diffusion of petroleum (very fast or very slow) in three types of medium with different porosities: fractures, vugs and matrix. This model has a dimensionless form, which is seen as
This model must be solved for p’s (pressure) at time t in the matrix of the medium pm, the fractured medium pf, and the vugular medium pv .
represents each permeability tensor, which is assumed to be constant; r is the dimensionless variable related to the ratios of the wellbore. Values λ are dimensionless values and they are defined in terms of transfer coefficients at each interface, the radius of the wellbore, the dynamic viscosity of the fluid and the permeability tensor.
If time orders α’s are less than one we say that the process is subdiffusive and if it is greater than one it is called superdiffusive. Spatial orders β’s result from generalizing the classic Darcy law, which corresponds to β = 1.
The above equations, like the flow equation that considers the medium as a whole, are modified Bessel’s equations.
Different porosities imply different fractal dimensions and therefore, fractional derivatives with different orders.
In Fractional Calculus there is not a single definition of derivative and between these definitions, in general, not equivalent to each other. Definitions such as Riemann-Liouville, Riesz, Caputo, Weyl or Grünwald-Letnikov are the most used to model anomalous diffusion, from a porous medium with fractal dimension.
But, fractional equations present serious numerical and mathematical difficulties in the context of diffusion equations. From the computational point of view, the challenge to overcome is to attenuate the numerical cost of the matrices that result in discretizing the problem. The proposal is to consider radial basis interpolators , taking into account that the domain geometry does not determine the efficiency of the algorithm and sounds like an immediate alternative for generalizing larger dimensions.
The objective in this work is to deal with fractional differential operators, using radial basis functions (RBF) and optimizing discretization processes of such fractional operators, through QR matrix decomposition and to attenuate the bad condition due to the shape parameter. With RBF-QR, a very high precision and convergence are obtained without the need to increase a polynomial term to the interpolator . As the computational cost increases, so does the range of problems to which such a technique can be applied.
3. Radial basis function method for interpolation
Definition 1. Let s be a positive integer, a function ϕ:Rs→R is called radial whenever there is a function in one variable ϕ:[0,∞)→R such that
and ||⋅|| is a norm in Rs (usually the Euclidean standard norm). Table (1) shows some examples, about the more used real functions ϕ(r).
|Piecewiese smooth, global|
|Compact support (’Wendland’)||, p polynomial.|
|Inverse quadric (IQ)|
|Inverse multiquadric (IMQ)|
|Bessel (BE) ()|
A standard interpolator in terms of radial basis functions, given the data , of some real function , in the corresponding collocation nodes , , has the form 
where is one real variable function and the constant value is called shape parameter.
Note 1. The radial basis function (FBR) we will take for the approach and examples is the Gaussian function:
Because it is globally smooth and for which the RBF method can be applied together with QR matrix decomposition .
Notation 1. Considering one dimensional case, we abbreviate ϕk(x):=ϕ(ε||x−xk||). Now the interpolator (5) is written as
The unknown coefficients can be determined by interpolation conditions
producing the linear equations system
which can be described in matrix form
is called Gram matrix.
4. RBF methodology for the discretization of differential operators
Suppose we want to apply a differential operator to the function in every evaluation node from the set , given the values of the function at collocation nodes
where and .
Assuming that is linear, we apply it to the RBF interpolator, for each of the evaluation nodes, as an approximation to the values of in :
which in the matrix form is
Thus, the matrix BA−1 is that matrix D which is mentioned in Eq. (9) and that approximates the values of the differential operator applied to the function.
5. Fractional calculus definitions
Generalizing the formula of Cauchy, we obtain
Definition 2. The left-sided Riemann-Liouville fractional integral of order α of function f(x) is defined as
Definition 3. The right-sided Riemann-Liouville fractional integral of order α of function f(x) is defined as
Based on the definition of fractional integral, the following fractional derivatives are constructed, which we will use in the numerical examples
Definition 4. The left-sided Riemann-Liouville fractional derivative of order α of function f(x) is defined as
Definition 5. The right-sided Riemann-Liouville fractional derivative of order α of function f(x) is defined as
Definition 7. The left-sided Caputo fractional derivative of order α of function f(x) is defined as
The fact that the derivative of integer order appears within the integral in Definition 7, makes Caputo derivative the most suitable for dealing with initial conditions of the FPDEs.
Definition 8. The right-sided Caputo fractional derivative of order α of function f(x) is defined as
6. Numerical examples
6.1. Fractional partial differential equation with Riesz space fractional derivatives
where u can be, for example, the concentration of a dissolute substance, Kα the dispersion coefficient and the fractional derivative of Riesz is given with fractional order .
Taking advantage of the fact that for RBFs we can consider non-equispaced collocation nodes, we divide the interval into N nodes, using Chebyshev distribution (see Figure 1):
The FPDE is solved by the method of lines based on the spatial trial spaces spanned by the Lagrange basis associated to RBFs. The Lagrange basis is generated by radial functions , , taking into account the collocation nodes. This is done by solving the system
and the Gram matrix
If is a differential operator and RBF φ is sufficiently smooth, then the application of such operator to the Lagrange base is calculated through the relation
Because of Lagrange’s standard conditions, the zero boundary conditions in and are de facto fulfilled if we use an approximation generated by the functions . This approximation is then represented as
with unknown vector
Evaluating the interpolator at the PDE in each node xi, we obtain
and initial conditions
From the latter two equations we obtain the following system of ordinary differential equations
According to the equation data in articles [30, 6], the problem (23) is taken with parameters , and . For the numerical solution, Gaussian RBF is used with a shape parameter , applying the scheme shown in Section 6.1. Graphs are shown in Figure 2.
6.2. Riemann-Liouville space-fractional diffusion equation
with Riemannn-Liouville derivative, ,
Initial conditions are considered
and Dirichlet boundary conditions
The exact solution to Eq. (27) is
We apply RBF and the method of lines with Lagrange polynomials, in a similar way as it was applied in Section 6.1. With Chebyshev collocation nodes and evaluation nodes, various values for the order of the derivative and for shape parameters.
The results obtained by Sousa  were obtained by applying implicit Crank-Nicholson schemes for time. The discretization of the fractional derivative was done using splines, with second-order precision. The results of both Sousa and RBF are shown in Tables 2 and 3, taking into account the error
|Global ℓ∞ error (30) of time converged solution for three mesh resolutions at t=1 for α=1.2, α=1.4 and Δt=Δx.|
|Global error (30) of time converged solution for three mesh resolutions at for , and .|
|Global ℓ∞ error (30) of time converged solutions at t=1 for α=1.2,1.4,1.5,1.8 and shape parameters ε=0.6,1.2.|
|(r)1-2 Order||Maximal error|
where is the norm.
6.3. Caputo time fractional partial differential equations
In this part we consider examples of fractional partial differential equations (see ) of the type
where , and are real parameters, bounded initial condition u(x,0) and Dirichlet boundary conditions and , . Caputo Derivative will be considered and rbf-qr routines from www.it.uu.se/research/scientific_computing/software/rbf_qr
6.3.1. Example 1
Putting and in Eq. (31), we obtain a non-homogeneous, fractional and linear Burger equation
Using initial condition
Dirichlet boundary conditions
The exact solution (see ) is
This problem is solved using the method described in section 6.1 along with QR decomposition for spatial part and an implicit scheme for time, on domain , fractional derivative of order ; several shape parameters are considered and the one that gives the best approximation is taken. Figure 3 shows a 3D image of the solution and maximal error for the solution at time .
In Figure 4 we compare errors that result when choosing uniform collocation nodes (with a fixed step size) and Chebyshev, on definition interval . The election of Chebyshev nodes is due to attenuating instability, which manifests as oscillations in the graph, called the Gibbs phenomenon [34, 35].
[41 uniform collocation nodes and 100 evaluation nodes, maximal error for , , is .] [41 Chebyshev collocation nodes and 100 evaluation nodes, maximal error for , , is .]
6.3.2. Example 2
Putting , and in Eq. (31), we obtain
Next function is the exact solution (see ), which is used to set boundary conditions.
The problem is solved by Lagrange (like Section 6.1) and RBF-QR, for , collocation nodes on interval . In Figure 5 we notice that when the number of uniform collocation nodes grows, there are large oscillations which cause unstability. For this reason, Chebyshev nodes are also preferred, as they attenuate this misbehavior.
6.3.3. Example 3
Putting and in Eq. (31), we obtain
using initial condition
The exact solution of this problem is not known, but is shown as an example of subdiffusive phenomenon which was discussed in the introduction. The problem is solved using the method described in section 6.1 along with QR decomposition for spatial part and an implicit scheme for time, on domain , fractional derivative of order and , according to the results shown in Refs. [36, 32]. Figures 6 and 7 show these results. Figure 8 shows a what if situation when (superdiffusive phenomenon), mentioned in Section 1.2.
The idea was to show that radial basis schemes are efficient and are on par with schemes like Finite Differences. They are an option to deal with multidimensional and irregular domain problems. The challenge is to adapt them to deal with diffusive problems, particularly with multidimensional systems of equations that consider that the medium does not have a single characteristic.
Fuentes et al. show in the work “The fractal models of saturated and unsaturated flows (capillary, diffusion and matrix) at micro, macro and mega scales”, for the PEMEX oil company, how to model the pressure which the oil must leave in a wellbore, from a system of triple porosity and permeability of the fractured medium.
By converting the model into a dimensionless system, in terms of radial distances, non-mesh schemes such as RBFs sound like a viable option and it is a work that is still being addressed.