Proposal for Use the Fractional Derivative of Radial Functions in Interpolation Problems

In this document we present the construction of a radial functions that have the objective of emulating the behavior of the radial basis function thin plate spline (TPS), which we will name as function TPS, we propose a way to partially and totally apply the fractional derivative to these functions to be used in interpolation problems, a proposal is presented to precondition the matrices generated in the interpolation problem using the $QR$ decomposition and finally is proposed the form of a radial interpolant to be used when solving differential equations using the asymmetric collocation method.


Polynomials similar to the function TPS
The main idea in this section is to try to emulate the behavior of the radial basis function thin plate spline (TPS) , also known as polyharmonic spline [1]: in a domain Ω of the form towards a domain of the form to do this it must be taken into account that (1) satisfy Φ(0) = 0, Φ (0) = 0, Φ(1) = 0, Φ (1) = 1, then for our purpose we look for a radial function Φ(r) such that to satisfy the conditions given in (3) is taken a polynomial of the form where the coefficients a 0 and a 1 are determined by (4), and the value of N will be given later, then in matrix form the previous system takes the form denoting by det(B) to the determinant of the matrix B from the previous system and doing a bit of algebra we obtain that det(B) = −b 2N 0 ⇔ b 0, 1 arXiv:1906.03760v2[math.NA] 26 Oct 2019 then the system (5) It always has a solution, denoting now by adj(B) the adjoint matrix of B and using that adj(B), (6) we obtain that and it is obtained as a solution to the system ( 5) with which we obtain the polynomial by construction (7) in the domain Ω 1 fulfills that In the previous construction only two coefficients are used to perform the approximation of the function TPS , to add one more coefficient we use the fact that (1) in the domain Ω 1 fulfills that then we look for a radial function Φ(r) such that Φ(0) = 0, Φ (0) = 0, Φ (0) = 0, (8) to satisfy (8) we take the polynomial on the other hand, to satisfy (9) we arrived to the matrix system (10) where to do a bit of algebra we get that det(B) = −2b 3N 0 ⇔ b 0, and using (6) we have to , with which we get the polynomial by construction (11) in the domain Ω 1 satisfy that From the systems ( 5) and (10) it can be deduced that to construct a polynomial with n coefficients that approximates the function TPS , we need to consider the (n − 1)-derivatives from both the polynomial and the function TPS , but this path would make the coefficients a i 's become more complicated in its expressions, to try to solve this problem in the next subsection we present an alternative that allows us to obtain an approximation to the function TPS where the coefficients a i s they are maintained in a simple way.

Function false TPS
With the idea of later using the fractional derivative of polynomials [2] and keep retaining the behavior of the function TPS, which is zero at the extremes of the domain Ω 1 , we start with the idea of looking for a polynomial that becomes zero in the "extremes" with respect to the derivative, as the solution of the system (5) with the above mentioned conditions it leads to the trivial solution., we used the polynomial involved in the system (10) with a vector c of the form , with c 0 > 0 and where the minus sign is included so that the solution has a convex behavior analogous to the function TPS in the domain Ω 1 .
With the above the system (10) it can be rewritten as (12) which has as solution , with which we get the polynomial although in principle c 0 can be arbitrary, later we propose a way to select it so that the coefficients of polynomial (13) are maintain in a simple way, for the particular case c 0 = 4 we get the polynomial the election of c 0 and the construction of (14) guarantees that in the domain Ω 1 fulfills that To improve the approximation we taken a small perturbation −α, with α ∈ [0, 1), in the exponent of the term of greater power associated with a negative coefficient, modifying in turn the exponent of said coefficient with a value +α, then we can define the function (15) which in the domain Ω 1 fulfills that The equation ( 16) receives the name of false TPS while the equation (15) receives the name of false TPS generalized.

Generalizing the previous construction
To generalize the idea used in the construction of the function false TPS, it seeks to generate a polynomial of the form that satisfies the conditions this generates a matrix system of the form.
, and on the other hand using (6) we have to where are the column vectors of the inverse matrix of B, with , then the matrix system has like solution , with which we get the polynomial denoting by M = mcm (2,6), the lowest common multiple of the denominators present in the coefficients of (17), it defines to (17) we can take c 0 = 18, obtaining the polynomial (18) due to the choice of c 0 and the way it is built (18) we have to in the domain Ω 1 fulfills that To improve the approximation take a small perturbation −α, with α ∈ [0, 1), in the exponent of the term of greater power associated with a negative coefficient, modifying in turn the exponent of said coefficient with a value +α, then we can define the function which in the domain Ω 1 fulfills that Of the way in which is constructed the polynomial (7) it can be generalized by changing the vector c by the vector taking the particular case c 0 = 1 and to improve the approximation take a small perturbation −α, with α ∈ [0, 1), in the exponent of the term of greater power associated with a negative coefficient, modifying in turn the exponent of said coefficient with a value +α, then we can define the function which in the domain Omega 1 fulfills that

Radial functions similar to the function TPS
The functions ( 22), ( 15) and ( 19) behave similarly to the TPS function in the domain Ω 1 , but our purpose is to obtain radial functions [1,3] that satisfied the previously mentioned, to solve this we impose the restrictions donde N > 0 y α ∈ [0, 1).
From now on we will take that all the functions used will have implicitly the restrictions given in (24) unless otherwise mentioned.
Imposing the restrictions (24) to the polynomials ( 22), ( 15) and ( 19) it is guaranteed that we have radial functions that behave similarly to the function TPS , to visualize this we choose the false function TPS and allow (1) take rational values obtaining the following graphs

Conditionally positive definite functions
We start the next section giving a definition and a theorem [3] that will be very useful later where r > 0, it is called a completely monotone functions in [0, ∞).
We now consider the following example Example 1.3 Suppose that φ is given by where r > 0, then

UNAM
Faculty of Sciences rewriting the last expression which implies that with which (−1) β/2 φ ( β/2 ) is completely monotone, it should be noted that m = β/2 is the smallest number for which this is fulfilled.Since β is not a natural number, φ it is not a polynomial, and therefore the powers they are strictly conditionally positive definite of order β/2 and radials in R d for all d.
A conditionally positive definite function of order m , is also conditionally positive definite function of order l ≥ m.It is also true that if a function is conditionally positive definite of order m in R d , then it is conditionally positive definite of order m in R k , for k ≤ d [1].
With the previous example we have the false function TPS is conditionally positive definite of order

Interpolation with Radial Functions
where Given a set of values (x j , u j ) , where (x j , u j ) ∈ When is used a radial function Φ conditionally positive definite, an interpolant of the form is proposed [1,3] where Solve the problem of interpolation (27) using the interpolant (28) together with the moment conditions (29) is equivalent to solving the linear system where A and P are matrices of (N p × N p ) y (N p × Q) respectively, whose components are The one that a function Φ be conditionally positive definite of order m , it can be interpreted as the matrix In this sense, A is positive definite in the vector space c that are "perpendiculars" to the polynomials.So, if in (28) the function Φ is conditionally positive definite of order m and the set of centers x j N p j=1 contains a unisolvent subset, then the interpolation problem will have a solution (the condition of unisolvency is to ensure the uniqueness) [1].

Examples with Radial Functions
Defining a domain and using the function we get that the graph of (33) is given by Then to carry out the interpolation problem a set of values is generated u(x i , y i ) N p i=1 and it is taken α ∈ [0, 1), here the option of using α for fixed values is presented, although it can also be used by looking for a value that minimizes the error.
Denoting by σ i = σ (x i , y i ) and u i = u(x i , y i ), then the error that we will use will be the root of the mean square error given by

Fractional Derivative
The perturbations −α previously used have a structure similar to the fractional derivative of Riemann-Liouville [2,4], which in its unified form with the fractional integral of Riemann-Liouville [5] is given by where n = Re (α) + 1.For a monomial given by f (x) = x s , the fractional derivative of Riemann-Liouville takes the form then to implement the fractional derivative to the radial functions (15) and ( 19) is taken getting the functions (38)

Preconditioning of system
Before continuing we must note that in the previous examples the condition number of the matrices obtained is too high , also that the linear system (30) generated to carry out the interpolation can be written compactly as where G is a matrix of (N p +Q)×(N p +Q), similarly Λ and U they are column vectors of (N p + Q) entries, then to try to solve the problem of having a condition number too high we propose to use the decomposition QR [6] of the matrix G G = QR, and change the linear system (30) by the equivalent linear system where taking the value of n in such a way that it is satisfied In the following examples will be used the linear system (39) using M = 10.

Examples with Fractional Derivative implemented partially
Using again the equation (33), the distribution of nodes of the Figure 9 and the set of values {u i } N p i=1 , also how it is used the definition of fractional derivative given in (35) is taken α ∈ (−1, 1), here is presented the option to use alpha for fixed values although it can also be used looking for a value that minimizes the error.The following examples are presented • Using the radial function taking N = 3.22, then to use the interpolant (28) it defines obtaining the following results • Using the radial function

Examples with Fractional Derivative
Because the previous examples where partial fractional derivative is implemented did not present any problem to carry out the problem of interpolation, is proceeded to implement the fractional derivative in its entirety.To implement the fractional derivative to the radial functions ( 15) and ( 19) is taken getting the functions Using again the equation ( 33), the distribution of nodes of the Figure 9 and the set of values {u i } N p i=1 , also how it is used the definition of fractional derivative given in ( 35) is taken α ∈ (−1, 1), here is presented the option to use alpha for fixed values although it can also be used looking for a value that minimizes the error.The following examples are presented • Using the radial function taking N = 3.22, then to use the interpolant (28) it defines obtaining the following results • Using the radial function taking N = 2.55, then to use the interpolant (28) it defines obtaining the following results

A change in the interpolant
In the previous sections we use the interpolator given by (28) where Q = dim P m−1 R d , this causes the value of Q to grow considerably, take for example a polynomial in R 2 of degree 4 which makes that Q be equal to 15, considering that sometimes "less is more" is changes the polynomial present in (28) by a radial polynomial obtaining the following interpolant where now Q = dim (P m−1 (R)), with which the moment conditions takes the form With which if we take now a radial polynomial in R of degree 4 the value of Q would be equal to 5. The following examples are presented with the interpolant mentioned above • Using the radial function taking N = 3.22, then to use the interpolant (28) it defines obtaining the following results taking N = 2.55, then to use the interpolant (28) it defines obtaining the following results

Asymmetrical collocation
The interpolation technique presented above can also be applied to the solution of differential equations [1].Assuming we have a domain Ω ⊂ R d and the problem where f and g are functions given , L and B linear differential operators, and u is the solution to find.
Before continuing we will make a change in the interpolant (42) that will help us avoid discontinuities due to the application of the operators L and B. Denoting by ord(L) the order of the differential operator L, we define q = max ord(L), ord(B) , defining now o = q − 1, si q > 0 0, si q ≤ 0 , then the interpolant (42) it can be rewritten as where Q = dim (P m−1 (R)), then the moment con- ditions take the form finally to the restrictions given in (24) we must add a more restriction given by When replacing the interpolant (45) in the system (44) is obtained with which the linear system is obtained where LA, BA, LP , BP and P are matrices of (N I × N p ), ((N p − N I ) × N p ), (N I × Q), ((N p − N I ) × Q) and (N p × Q) respectively, whose components are

Examples with Fractional Derivative
The form of the interpolant (45) it will be very useful to solve differential equations in radial form.Taking the definition of fractional derivative of Caputo we can build the next differential operator taking identity as the differential operator B we can build the differential equation to use the interpolant (45) is taken and it is defined obtaining the following results then it is enough to define Although the definition of fractional derivative of Riemann-Liouville was used for the functions constructed in the previous sections, in general, any other definition of fractional derivative can be used as long as this definition is used at par with the fractional integral of Riemann-Liouville.

Figure 1 :
Figure 1: They are presented with black and red the functions r N log(r) and r N − r N +1 respectively.

Figure 2 :
Figure 2: They are presented with black and red the functions r N log(r) and − 1 2 r N +2 + 2r N +1 − 3 2 r N respectively.

Figure 3 :
Figure 3: They are presented with black and red the functions r N log(r) and −2r N +2 + 4r N +1 − 2r N respectively.

Figure 4 :
Figure 4: They are presented with black and red the functions r N log(r) and −2r N −α+2 +4r N +1 −2r N , using different values of α, respectively.

Figure 7 :
Figure 7: They are presented with black and red the functions r N log(r) and r N +1 − r N −α , using different values of α, respectively.

Figure 8 :
Figure 8: They are presented with black and red the functions r N log(r) and −2r N −α+2 +4r N +1 −2r N , using different values of α, respectively.

Figure 9 :
Figure 9: Nodes used for the interpolation problem, where N B and N I are the boundary and interior nodes respectively.

Figure 12 :
Figure 12: Graph of the numerical solution (with minimal error) to the problem raised.

Figure 13 :
Figure 13: Graph of the numerical solution (with minimal error) to the problem raised.

Figure 15 :
Figure 15: Graph of the numerical solution (with minimal error) to the problem raised.
55, then to use the interpolant (28) it defines