A conformable calculus of radial basis functions and its applications

Article History: Received 03 October 2017 Accepted 06 April 2018 Available 22 April 2018 In this paper we introduced the conformable derivatives and integrals of radial basis functions (RBF) to solve conformable fractional differential equations via RBF collocation method. For that, firstly, we found the conformable derivatives and integrals of power, Gaussian and multiquadric basis functions utilizing the rule of conformable fractional calculus. Then by using these derivatives and integrals we provide a numerical scheme to solve conformable fractional differential equations. Finally we presents some numerical results to confirmed our method.


Introduction
Recently, the question of how to take non-integer order of derivative or integration was phenomenon among the scientists.However together with the development of mathematics knowledge, this question was answered via Fractional Calculus which is a generalization of ordinary differentiation and integration to arbitrary (non-integer) order.Then In conjunction with the development of theoretical progress of fractional calculus, a number of mathematicians have started to applied the obtained results to real world problems consist of fractional derivatives and integrals [1,2].An significant point is that the fractional derivative at a point x is a local property only when a is an integer; in non-integer cases we cannot say that the fractional derivative at x of a function f depends only on values of f very near x, in the way that integer-power derivatives certainly do.Therefore it is expected that the theory involves some sort of boundary conditions, involving information on the function further out.To use a metaphor, the fractional derivative requires some peripheral vision.As far as the existence of such a theory is concerned, the foundations of the subject were laid by Liouville in a paper from 1832.The fractional derivative of a function to order a is often now defined by means of the Fourier or Mellin integral transforms.Various types of fractional derivatives were introduced: Riemann-Liouville, Caputo, Hadamard, Erdelyi-Kober, Grunwald-Letnikov, Marchaud and Riesz are just a few to name [3,4].Now, all these definitions satisfy the property that the fractional derivative is linear.This is the only property inherited from the first derivative by all of the definitions.However, all definitions do not provide some properties such as Product Rule (Leibniz Rule), Quotient Rule, Chain Rule, Rolls Theorem and Mean Value Theorem.In addition most of the fractional derivatives except Caputo-type derivatives, do not satisfy D α (f ) (1) = 0 if α is not a natural number.
Note that if f is fully differentiable at t; then the derivative is D α (f ) (t) = t 1−α f ′ (t).(Here, operators of a very similar form, t α D 1 , have been applied in combinatorial theory [18]).Of course, for t = 0 this is not valid and it would be useful to deal with equations and solutions with singularities.Additionally it must be noted that conformable derivative is conformable at α = 1, as lim On the other hand radial basis functions method is one of the more practical ways of solving fractional order of models.The most significant property of an RBF technique is that there is no need to generate any mesh so it called meshfree method.One only requires the pairwise distance between points for an RBF approximation.Therefore it can be easily applied to high dimensional problems since the computation of distance in any dimensions is straightforward.On the other hand in order to solve partial differential equations (PDEs) in [19,20] Kansa proposed RBF collocation method which is meshfree and easy-to-handle in comparison with the other methods.Not only integer order PDEs [21] but also Kansa's approach has been used fractional order of PDEs [22].
In this paper we find the conformable derivatives and integrals of needed function of RBF interpolation such as powers, Gaussians and multiquadric.This derivatives play a significant role in the numerical solution of conformable differential equations by the help of RBF method.The remainder of this work is organized as follows: In Section 2, the related definitions and theorems are summarised.In Section 3, the conformable derivative and integrals have been obtained for the radial basis functions which will use in the RBF computations.Numerical experiments are given in Section 4, while some conclusions and further directions of research are discussed in Section 5.

Review of fractional derivatives and integrals
Here we review the Riemann-Liouville fractional derivatives and integrals introduced in [3,4,23].
Definition 1.The left-sided Riemann-Liouville fractional derivative of order α of function u(t) is described as Definition 2. The right-sided Riemann-Liouville fractional derivative of order α of function u(t) is described as where τ = ⌈α⌉.Definition 3. The left-sided Riemann-Liouville fractional integral of order α of function u(t) is described as Then Khalil et.al.[6] have introduced the conformable fractional derivative and integrals by following definition.
η where α ∈ (0, 1) and for all t > 0. In other words where prime denotes the classical derivative operator.
Similarly, one can define the conformable fractional integral operator.
The left sided conformable integral of u(t) of order α described by where α ∈ (0, 1) and the integral is classical integral operator.
The right sided conformable integral of u(t) of order α described by where α ∈ (0, 1) and the integral is classical integral operator.

Radial basis function method
One of the properly approach to solving PDE is radial basis functions (RBFs).The main idea of the RBFs is to calculate distance to any fixed center points x i with the form ϕ( x − x i 2 ).Additionally RBF may also have scaling parameter called shape parameter ε.This can be done in the manner that ϕ(r) is replaced by ϕ(εr).Generally shape parameter have been chosen arbitrarily because there are no exact consequence about how to choose best shape parameter.Some of the RBFs are listed in Table 1.
RBFs ϕ(r) Inverse Quadratic (IQ) The main advantageous of RBF technique is that it does not require any mesh hence it called meshfree method.Therefore the RBF interpolation can be represent as a linear combination of RBFs as follows: where the a i 's the coefficients which are usually calculated by collocation technique.Some of the greatest advantages of RBF interpolation method lies in its practicality in almost any dimension and their fast convergence to the approximated target function.

Conformable derivatives of RBFs in one dimension
In order to construct conformable derivatives and integrals we will make use of the fractional calculus.Namely the relationship between Riemann-Liouville fractional integral and conformable fractional integral can be given as follows: 8. Let α ∈ (ǫ, ǫ + 1], then the left sided relationship between Riemann-Liouville fractional integral and conformable fractional integral is Proof.The proof is similar to Theorem 1.
For instance if we take a = 0 and b = 0 for the above results, we obtain (−t) α+γ respectively.Now, similarly, we can get the conformable derivative of function (t − a) γ .Namely, the derivative of and again if we choose a = 0, we get α D(t) γ = γt γ−α .Now, by using the above results, one can find the conformable derivatives and integration of radial basis functions.Additionally throughout this and next sections n C k denotes the combina- Proof.In order to prove the above theorem we use the Taylor expansion of t m about the point t = a.Namely, If we substitute the equation ( 1) into conformable integration definition, we have Proof.The proof is similar to Theorem 3.
Theorem 5.For a = 0, t > a and m ∈ N Proof.In order to prove the above theorem we use the Taylor expansion of t m about the point t = a again.In other words if we substitute the equation ( 1) into conformable integration definition, we have 3.2.For ϕ(t) = e (−t 2 /2) (Gaussian basis function) Now we can make use of the conformable derivatives and integration of power basis function, we are able to find out the Gaussian basis function derivatives and integrations.
Theorem 6.For a = 0, t > a and m ∈ N Proof.In order to prove the above theorem we use the Taylor expansion of e −t 2 /2 about the point t = 0. Namely, If we substitute the equation ( 3) into conformable integration definition, we have Proof.The proof is similar to Theorem 7.
Theorem 8.For a = 0, t > a and m ∈ N Proof.Similarly by using the Taylor expansion of Gaussian function about t = 0 we can calculate the conformable derivative of it.That is, Similarly one can compute the conformable derivatives and integrations.
Theorem 9.For a = 0, t > a and m ∈ N Proof.In order to prove the above theorem we use the Taylor expansion of √ 1 + t 2 about the point t = 0. Namely, If we substitute the equation ( 4) into conformable integration definition, we have Theorem 10.For a = 0, b > t and m ∈ N Proof.The proof is similar to Theorem 9.
Theorem 11.For a = 0, t > a and m ∈ N Proof.Similarly by using the Taylor expansion of multiquadric basis function about t = 0 we can calculate the conformable derivative of it.That is,

Numerical example
In this section we will give some results of numerical solution of conformable differential equations to validate our numerical scheme.For that we will use RBF interpolation method by the help of collocation technique.Consider the general form of following conformable differential equation: α Dy(t) + p(t)y(t) = q(t), y 0 (t) = y(t 0 ).( 5) Let t j be equally spaced grid points in the interval 0 ≤ t j ≤ K such that 1 ≤ j ≤ L, t 1 = 0 and t L = K.Additionally, because collocation approach has been used we not only require an expression for the value of the function but also for the conformal derivative given in (5).Thus, by conformal differentiating (6), we get where α D denotes the conformable derivative the with respect to t.In order to compute conformable derivative of radial basis functions we take the advantage of formulas which are derived in the previous section.Then using the RBF collocation method, one can compute the unknown coefficients a k 's by solving following matrix system: with boundary condition.In order to illustrate this scheme by numerically we take the following conformable differential equations: (1) α Dy(t) + y(t) = 0  Here we use the multiquadric basis function with ε = 10 −4 .In Figures 1, 2 and 3, we present the numerical solutions of given conformable differential equations with different α values.
These results are in accord with the exact solutions of them.

Conclusion
In this paper we gave the derivatives and integrals of three kinds of radial basis functions such as powers, Gaussians and multiquadric by using the conformable derivatives and integrals which are new type of fractional calculus.These findings allow to solve conformable differential equations by the RBF's.Then we gave three differential equations to show that this technique is applicable.These differential equations are solved by the help of RBF collocation method.