Fuzzy reproducing kernel space method for solving fuzzy boundary value problems

In the beginning, we describe the fuzzy inner product space and the fuzzy Hilbert space. Our goal is to use the fuzzy reproducing kernel method to solve the second-order fuzzy boundary value problem. The fuzzy convergence analysis of introduced method is discussed in detail. We present some examples in the end.


Introduction
Reproducing kernel method is one of the most basic methods for approximation. In doing so, our main goal is to solve fuzzy boundary value problems using fuzzy reproducing kernel methods. We discuss the convergence of our procedure using the concept of fuzzy distance. At the end, there are some examples. We describe the fuzzy reproducing kernel method (FRKM for short) for solving the following fuzzy boundary value problem: here, f(x) is a fuzzy function. The functions of m(x) and n(x) are ordinary and continuous. The process of doing this paper is as follows: In the part of preliminaries, the fundamental concepts that have been used in later sections are presented. In "Fuzzy reproducing kernel space" section, we have defined the fuzzy reproducing kernel (FRK for short). Fuzzy inner product spaces and fuzzy reproducing kernel spaces have been introduced. Also, the solution to Eq. (1.1) is given with the initial boundary value conditions. In "Fuzzy convergence analysis" section, we bring the fuzzy convergence theorem. In the next section, we give some examples for a better understanding. At the end of this work, we bring the results. For more information on the reproducing kernel method and fuzzy convergence, see [1][2][3]11].

Preliminaries
We introduce the primary definitions of the generalized Hukuhara difference, generalized Hukuhara derivative, Hausdorff distance and fuzzy continuous function. For definition of the fuzzy number and -level set see [10]. Remark 2. 1 We assume that in the whole of this paper the generalized Hukuhara difference exists. We denote generalized Hukuhara difference by ⊖ gH . Definition 2.2 [5] Let x and y be two fuzzy numbers in ℝ F , where ℝ F is the set of all fuzzy numbers. If the exists a fuzzy number as z that satisfies in follows condition, then the generalized Hukuhara difference (gH-difference for short) is defined as where (i) and (ii) are both valid if and only if z is a crisp number.

Fuzzy reproducing kernel space
We will introduce the FRKM to solve the fuzzy boundary value problems. Initially, we create the fuzzy reproducing kernel space (FRKS for short) W m [0, 1] , while each fuzzy function satisfies in the y(0) = 0 and y(1) = 0.

Definition 3.1 (Fuzzy absolutely continuous function)
If for all , there is a , which has no relation with n, such that i s a f u z z y s p a c e , y(0) = 0, y(1) = 0}.
We define the fuzzy inner product (FIP for short) in this space as where ⟨y, z⟩ m is the fuzzy inner product in the fuzzy space W m [0, 1] . Also, fuzzy norm in this space is defined as ‖y‖ m = √ ⟨y, y⟩ m . where ⟨y, z⟩ L 2 is the fuzzy inner product in the fuzzy space .

D e f i n i t i o n 3 . 5
Using definition 3.1 in [8], we can deduce Lemmas 3.6 and 3.8 .

Lemma 3.6 Suppose that F(ℝ, F(ℝ)) is a vector space over
with this property for each k ∈ F(ℝ) and all functions f 1 , f 2 , f 3 ∈ F(ℝ, F(ℝ)) , satisfies the following conditions: Proof To prove, we need to apply the fuzzy integral properties, the concept of the Hausdorff distance and the fuzzy norm.
is called a fuzzy inner product space (FIPS for short).

Lemma 3.8 Suppose that F(ℝ, F(ℝ)) is a vector space over
with this property for each k ∈ F(ℝ) and all functions f 1 , f 2 , f 3 ∈ F(ℝ, F(ℝ)) , satisfies the following conditions: Proof To prove, we need to apply the fuzzy integral concepts, the Hausdorff distance and the fuzzy norm.
The remaining proofs follow from the previous lemma. Corollary 3. 9 The vector space F(ℝ, F(ℝ)) with a FIP in the form ⟨f 1 , In the following, assume that ⟨., .⟩ m is the fuzzy inner product in the fuzzy space W m [0, 1] . Also, ‖.‖ is the fuzzy norm.

Definition 3.17 (Fuzzy Gram-Schmidt process)
Given an arbitrary basis 1 , 2 , … , n for a FIPS, if all the gH-differences are present, the fuzzy Gram-Schmidt process constructs by the fuzzy orthogonal basis A 1 , A 2 , … , A n : Step 1 Let A 1 = 1 , Step By normalizing A 1 , … , A n vectors, we can obtain the fuzzy normal orthogonal vectors of the form as follows: The fuzzy orthonormal function system

Definition 3.18 We consider
as a fuzzy orthonormal system, we have: where 0 and 1 are fuzzy numbers.
Assume that Ly(x) = y �� (x) + m(x)y � (x) + n(x)y(x) in the equation of (1.1). In this case, L ∶ W m [0, 1] → W 1 [0, 1] is a fuzzy bounded linear operator. We take i (x) = R x i (x) and is the FRK. Also, L * is the fuzzy adjoint operator of L. Using the concepts of the FRK, for each y(x), the following equality is true: Regarding the above and using the properties of R x (t) , we get Moreover, In this case, i (x) = L t R x (t)| t=x i , where L t is the fuzzy operator L that applies to the function of t.
By the fuzzy Gram-Schmidt process, we orthonor malize the sequence { i (x)} ∞ i=1 and we get the fuzzy orthonormal system Proof Suppose that y(x) is the solution of equation of (1.1) is a fuzzy orthonormal system. In this case, the following equalities are true: where ⟨., .⟩ is the fuzzy inner product. Also, the approximate solution of (1.1) is as □ (3.12)
To solve examples of "Examples" section, by [6], we need to define the space W 3 [0, 1] and the inner product in this space as W 3   Using the method presented in this paper (i.e, reproducing kernel method), taking: we can simply discuss about |y(x) − y n (x)| , for every positive n, in the different reproducing kernel spaces. This process is similar to [12].
Example 5. 1 We consider the following boundary value problem. Then, we find its exact solution.

Conclusion
In this paper, the definitions of fuzzy Cauchy, fuzzy complete and fuzzy inner product were studied. Moreover, we presented the solution of fuzzy second-order two-point boundary value problem by the fuzzy reproducing kernel method. We had a lot of limitations compared to the real state, and all the lemmas and the theorems were not easily verifiable.
The basis for what we did was based on the existence of the gH-differences.