Error estimate of the MQ-RBF collocation method for fractional differential equations with Caputo–Fabrizio derivative

A collocation method based on multiquadric radial basis functions is proposed for numerical solution of fractional differential equations. The fractional derivative is sense of Caputo–Fabrizio derivative. An efficient error bound of the method is also introduced in the $$L_2$$L2 norm, using properties of native spaces. We test this approach for two examples. The obtained numerical results confirm theoretical prediction of the convergence of this method.


Introduction
Over the last several decades, radial basis functions (RBFs) were known as a powerful tool for the scattered data interpolation problems. The multiquadric (MQ) RBF interpolation method was introduced by Hardy in 1971 [11] and the method of meshless radial basis functions was introduced by Kansa for solving differential equations in 1990 [14,15]. More recently, the RBF methods have been used to obtain numerical solution of a large type of ordinary and partial differential equations [8,12,23].
In the last decades, fractional differential equations have been demonstrated applications in the several fields of mathematics, physics, and out of them [6,13,19] and the several methods have been proposed to solve them. There are various definitions of derivative with fractional order, such as Riemann-Liouville, Caputo, etc. [16,20]. Caputo and Fabrizio have suggested a new definition of fractional derivative which, contrary to the old definition, has no singularity [5].
In this paper, the collocation method based on MQ-RBFs is applied to solve the Caputo-Fabrizio fractional differential equations. The first author and Ghoreishi have applied this method for fractional differential equations with Caputo derivative [7].
Definition 1 (From [5]) Assume that u 2 H 1 ða; bÞ; a\b, the Caputo-Fabrizio derivative with fractional order 0 a 1 of the function u(x) is defined by CF D a uðxÞ ¼ MðaÞ where MðaÞ is a normalization function, such that Mð0Þ ¼ Mð1Þ ¼ 1.
Losada and Nieto proposed the fractional integral of Caputo-Fabrizio type as follows (see [18]).
Definition 2 Let 0\a\1. The fractional integral of order a of a function u is defined by CF I a uðxÞ ¼ 2ð1 À aÞ ð2 À aÞMðaÞ uðxÞ þ 2a ð2 À aÞMðaÞ Due to the above definition, the fractional integral of order a is an average between of the function and its integral of order one. Using this idea, they obtain the normalized function to be ; 0 a 1: Therefore, the Caputo-Fabrizio fractional derivative of order 0\a\1 was reformulated by Losada and Nieto as The remainder this paper is organized as follows. In Sect.

Basic definitions
Let u be a continuous function on R d which is conditionally positive definite of order m. Given a set of N distinct data points X ¼ fx j g N j¼1 corresponding data value fg j g N j¼1 , the RBF interpolant is defined by where k:k is the Euclidean norm on R d , fw i g g i¼1 is a basis of the polynomial space P mÀ1 d , that is, all polynomials of total degree less than m in d variables and The expansion coefficients fk j g N j¼1 and fk 0 i g g i¼1 are then determined from the interpolation conditions and the further conditions: which leads to the following linear system: where A NÂN and B NÂg are matrices with the elements A ji ¼ /ðx j À x i Þ; i; j ¼ 1; . . .; N and B ji ¼ w i ðx j Þ; j ¼ 1; . . .; N; i ¼ 1; . . .; g, respectively. Furthermore, k and k 0 are the vectors of coefficients of PgðxÞ and the components of g are the data gðx j Þ; j ¼ 1; . . .; N.
It is worth mentioning that system (6) has a unique solution when X contains a P mÀ1 d -unisolvent subset. We call a set of points X & R d P mÀ1 d -unisolvent if the only polynomial of total degree at most m À 1 interpolating zero data on X is the zero polynomial. A complete treatment of this topic can be seen in [3,9].
We will use multiquadric radial basis functions for the numerical scheme introduced in Sect. 3. The multiquadric RBFs can be defined as where dm À d=2e indicates the smallest integer greater than m À d=2. The MQ-RBFs in (7) were named one of the shifted surface spline functions by Yoon [22]. We need the native and Sobolev spaces to discuss about error estimates of RBF interpolation.
Each radial basis function uðxÞ induces a native space denoted by F u which is a semi-Hilbert space for m [ 0.

Definition 3 Consider the Fourier transform of
For the given basis function u, there arises a function space which is called the ''native'' space for u. This function space F u is equipped with the semi-inner product The definition and relevant properties of the native space can be found in [3,9]. Now, we recall some basic definitions about the Sobolev spaces. Assume that X is an open domain in R d ; q ¼ ðq 1 ; . . .; q d Þ is an d-tuple of non-negative integers and jqj ¼ P d k¼1 q k : We set and consider the following definition from [1].
Definition 4 Assume that 1 p 1 and k be a nonnegative integer, the Sobolev space W k p ðXÞ is defined by W k p ðXÞ ¼ ff 2 L p ðXÞ : for each non-negative multi-index q with jqj k; D q f 2 L p ðXÞg: It is equipped with the following norm: The Sobolev spaces W k p form a hierarchy of spaces for is denoted by H k ðXÞ. This space is a Hilbert space for the inner product: The kth order L p -Sobolev semi-norm jf j k;L p ðXÞ is defined by

MQ-RBF collocation method
Consider the fractional differential equation: According to the RBF collocation method, the approximation of the unknown function u(x) may be written as a linear combination of MQ-RBF radial basis functions and the monomial basis for the vector space P mÀ1 1 . Thuŝ Using (9) and collocation points fx k g N k¼0 & ð0; 1Þ, Eq. (3) takes the following form: To obtain the integral term in (10), we will transfer the integral interval ½0; x k to a fixed interval ½À1; 1 and then make use of some appropriate quadrature rules. Therefore, we make a simple linear transformation: Then Therefore, (10) becomes Using a ðM þ 1Þ-point Gauss quadrature formula relative to the Legendre weights fw ' g M '¼1 gives where the set fn ' g M '¼0 coincides with the Gauss-Legendre nodal points on ½À1; 1.
To obtain approximate solution of (8) using the MQ-RBF collocation method, it is necessary that (8) be satisfied exactly by (9) at a set of collocation points fx k g N k¼1 : By inserting (16) into the above equation, the numerical scheme (17) leads to a system of N linear equations in N þ m unknowns. Therefore To determine the unknown coefficients fk j g N j¼1 and fk 0 i g mÀ1 i¼0 to the N equations resulting from (18), an extra m equations are required. This is ensured by the m conditions (5) as

Error estimate
The aim of this section is to provide an error estimate of the numerical scheme (18), where the MQ-RBF is conditionally positive definite of order m ¼ 1, in the other words: First, we state some lemmas and theorems, and then, we prove our main theorem. In this position for subsequent error analysis, we have to assume that X R d has Lipschitz boundary and the interior cone property. For simplicity, in the sequel, we will assume that X ¼ ðÀ1; 1Þ. To prove the error estimate for the proposed method, we need the generalized Hardy's inequality and estimates for the interpolation error (see [4,10]). holds if and only if for the case 1\p q\1. Here, T is an operator of the form with k(x, t) a given kernel, u, v weight functions, and À1 a\b 1.
Lemma 2 (Estimates for the interpolation error) Let f 2 H q ðXÞ with q ! 1, x j ; 0 j M, be the Gauss, or the Gauss-Radau, or the Gauss-Lobatto points relative to the Legendre weight wðxÞ 1 and I M f denote the polynomial of degree M that interpolates f at one of these sets of points, namely where L j is the jth Lagrange basis function. Then In all the estimates contained in this chapter, C will denote a positive constant that depends upon the type of norm involved in the estimate, but which is independent of the function u, the integer N, M, and the diameter of the domain.
Yoon proposed L p error estimates for the so-called ''shifted surface splines'' for functions f is the standard Sobolev spaces.
Lemma 3 (From [22]) Let Pf ðxÞ in (4) be an interpolant to f on X using the basis function u in (7) and f be a function in the native space F u . Then, for every function f 2 W m 2 , we have jf À Pf j m;L 2 ðXÞ jf À Pf j u jf j u ; where as before stated, u is strictly conditionally positive definite of order m.
Theorem 1 (From [22]) Let Pf ðxÞ in (4) be the interpolant to f using the ''shifted'' surface spline u. Assume that the parameter a in the basis function u in (7) is chosen to be proportional to h. Then, there is a positive constant C, independent of X , such that for every f 2 W m 2 ðXÞ \W m 1 ðXÞ, we have an error bound of the form: kf À Pf k L p ðXÞ Ch c p jf j m;L 2 ðR d Þ ; 1 p 1; Corollary 1 A function that is (strictly) conditionally positive definite of order m on R d is also (strictly) conditionally positive definite of any higher order; therefore, Lemma 3 and Theorem 1 are satisfied for any l ! m.

Numerical experiments
In this section, we will use the MQ-RBF collocation method to solve Caputo-Fabrizio FDEs. These examples are considered, because exact solutions are available for them. All computations were performed on a running Mathematica software. We consider the MQ-RBF as (19) with shape parameter a ¼ 1 for a good balance between accuracy and stability. In these experiments, we have used zeros of the shifted Legendre polynomials on (0, 1) as the centers of MQ-RBFs and the Gauss-Legendre quadrature method with M ¼ 2N. We tested the algorithm using various differential equations. All tests were performed with three different values for a, namely a ¼ 0:1; 0:5, and 0.9. In the following, we show a representative selection of our results.
Example 1 For our first example, we consider the fractional differential equation: CF D a uðxÞ À f ðxÞ ¼ 2uðxÞ; uð0Þ ¼ 1; where The exact solution is uðxÞ ¼ x þ cos x: Numerical results of MQ-RBF collocation method with several values of N and 40 digits precision are displayed in Fig. 1 and Table 1.
As we know the accuracy of the approximation solution depends on the value of the number of centers N, the distances between them [2,21]. Increasing the number of data points has a severe effect on the stability of the linear system. For a fixed a, the condition number of the matrix in the linear system grows exponentially as the number of data points is increased [21]. Numerical experiments indicate that we can obtain good numerical results with small N in a short time and the value of a influences the error rate. According to concept of ''effective condition number'', we can achieve good results with huge condition number using the arbitrary precision ability of the Mathematica software [17].

Conclusion
In the present paper, MQ-RBF method for a Caputo-Fabrizio fractional-order ordinary differential equation was described. The error estimate was proved and numerical examples were presented. These results indicate that the desired accuracy is obtained and MQ-RBF collocation method is an effective method for solving the fractionalorder ordinary differential equations. Radial basis functions in their usual form lead to the solution of an ill-conditioned system of equations. To avoid such limitation of radial basis function collocation schemes, we can explained by the concept of ''effective condition number''.