Two hybrid and non-hybrid methods for solving fuzzy integral equations based on Bernoulli polynomials

In this paper, our aim is to provide two hybrid and non-hybrid efficient method based on non-orthogonal Bernoulli polynomials to approximate solution of linear fuzzy Fredholm integral equations. At first, using Bernoulli basis polynomials and also combining them with known block-pulse functions, we convert the fuzzy integral equations to two algebraic systems. The convergence and error estimates of the methods is also given. Finally, we present some illustrative examples and compare the numerical computational results to confirm the theoretical topics and demonstrate the convergence rate of the methods.


Introduction
System of linear and nonlinear equations aries from various areas of science and engineering.since many realvalued systems are too complex to be defined in precise terms, imprecision is often involved.Analyzing such systems requires the use of fuzzy information.The concept of fuzzy integrals was initiated by Dubios and Prade (1982) and then investigated by Kaleva (1987) Goetschel and Voxman (1986) Nanda (1989) and others.Fuzzy integral equations are extremely important for studying and solving large properties of problems in many topics in applied mathematics, particulary in relation to physics, medicine, biology, geography, economic, social science, etc.Many researchers such as [3,4,8,14,21,24,23,25] have proposed various methods, including iterative and direct methods for approximate the numerical solution of these equations.The block-pulse functions technique is frequently applied in control and systems science to reduce the complexity of numerical problems.Since the operations of block-pulse series are much simpler than those of the original functions, see [15].The problems containing integral equations, can be solved with block-pulse functions.Many authors used the block-pulse functions to obtain numerical solution of differential equations, integral and integro-differential equations in the crisp case.In recent years these functions have also been used to solve the fuzzy integral equations.Another approach that used in finding approximate solution of integral equations is utilization of hybrid methods.For example, hybrid block-pulse functions with Legendre polynomials, Chebyshev polynomials, Taylor and Fourier functions, (see [2,16,18,19]).In this paper we proposed two non-hybrid and hybrid efficient methods based on bernoulli basis polynomials and block-pulse functions to approximate numerical solution of linear fuzzy Fredholm integral equations.The convergence and error analysis of the method are presented in two theorem.The rest of paper is organized as follows: In Section 2, we review some elementary and necessary concepts of Bernoulli polynomials, block-pulse functions and fuzzy set theory.In Section 3, we derive the proposed non-hybrid and hybrid methods to obtain numerical solution of fuzzy Fredholm integral equations.The convergence and error analysis of the methods are given in this section too.Section 4 presents some numerical examples to show the accuracy and efficiency of the introduced methods.Finally, we conclude in Section 5.

Preliminaries
In this section, we briefly review some definitions, notions and results related to fuzzy numbers, fuzzy-valued functions, Bernoulli polynomials and block-pulse functions which will be referred to throughout this paper.
The set of all fuzzy numbers denoted by R .Any real number α ∈ R can be interpreted as a fuzzy number α = χ {α} and therefore R ⊂ R .Also, the neutral element respect to ⊕ in R denoted by 0 = χ {0} .According to [11,24] for any 0 < r ≤ 1 an arbitrary fuzzy number is represented, in parametric form, by an ordered pair of functions (u, ū) r = (u r − , u r + ), which satisfies the following properties: u r − is bounded left continuous non-decreasing function over [0, 1], u r + is bounded left continuous non-increasing function over [0, 1] and u r − ≤ u r + .
Definition 2.2.[12] For any u ∈ R the r-level set of u is denoted by [u] r and defined by is the closure of the support of u and is a compact set, where A denotes the closure of A. It follows that the level sets of u are closed and bounded intervals in R. It is well-known that the addition and multiplication operations of real numbers can be extended to R .In other words, for any u, v ∈ R and λ ∈ R, we define uniquely the sum u ⊕ v and the product λ ⊙ u by r means the usual addition of two intervals (as subset of R) and λ [u] r means the usual product between a scalar and a subset of R. We use the same symbol ∑ both for the sum of real numbers and for the sum ⊕ (when the terms are fuzzy numbers).Also, according to [1,24], the following algebraic properties for any u, v, w ∈ R hold: Definition 2.3.[13,24] For arbitrary fuzzy numbers u ) is a complete metric space and following properties hold:

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According to [7] and with our notation , for all u, v with positive supports.In addition, for all strictly increasing and positive real function ψ, we have ψ(u) = ψ(u r − , u r + ) = (ψ(u r − ), ψ(u r + )).

Definition 2.4. A fuzzy-number-valued function
If f continuous for each s ∈ A then we say that f is continuous on A. A fuzzy number v ∈ R is an upper bound for a fuzzy-number-valued function f : Definition 2.5.In [13,14,23]

the fuzzy Riemann integral for a fuzzy-number-valued function f on
Then I is called the fuzzy Riemann integral of f on [a, b] and denoted by (FR)

and hence, f is fuzzy Riemann integrable on [a, b].
Lemma 2.2.[24,13] If f ∈ C([a, b], R ), its definite integral exists, and also for all r ∈ [0, 1], Bernoulli polynomials: In mathematics, the Bernoulli polynomials occur in the study of many special functions.Unlike orthogonal polynomials, the Bernoulli polynomials are remarkable in that the number of crossings of the xaxis in the unit interval does not go up as the degree of the polynomials goes up.In the limit of large degree, the Bernoulli polynomials, appropriately scaled, approach the sine and cosine functions.Now, we review some properties of Bernoulli polynomials.Definition 2.6.[17] The Bernoulli basis polynomials B n (t) of degree n are constructed from the following relation:

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The first five Bernoulli polynomials are: Theorem 2.2.[10] Bernoulli polynomials satisfy the following relations: Also, the Bernoulli polynomials B n (t) can be presented with these two explicit formulas: where n and m are the order of block-pulse functions and Bernoulli polynomials, respectively.
Fuzzy Fredholm integral equations: The fuzzy type of Fredholm integral equations is as where k(x,t) is an arbitrary kernel function over [a, b] 2 and u(t, r) is a fuzzy-valued function.The sufficient conditions for existence of unique solution of (2.1) are presented in [9].Throughout this paper we consider Eq. (2.1) with real kernel function k(x,t) , (a, b) = (0, 1) and λ = 1.That is, for any fixed 0 ≤ r ≤ 1, u(x, r) = (u, ū)(x, r) and f (x, r) = ( f , f )(x, r) are fuzzy functions on [0, 1].In order to provide numerical methods for solving Eq. (2.1), initially we replace it with the following system where and International Scientific Publications and Consulting Services

Main results
In this section, we present two methods for numerical solution of Eq. (2.1) based on Bernoulli polynomyals (BP) and hybrid block-pulse functions and Bernoulli polynomials (HBPBP).To this end, we apply BP and HBPBP to approximate solution of fuzzy Fredholm integral equations.Indeed, to solving this equation we shall to find two BP and HBPBP in the interval [0, 1] for the known function f (x, r) and the crisp kernel k(x,t) to approximate u(x, r).

BP method
Let g(x, r) = (g, ḡ)(x, r) and g ∈ C [0, 1] be a fuzzy Riemann integrable bounded functions on closed interval [0, 1].According to [5], we establish a fuzzy-like formula to approximate the function g(x) = (g, ḡ)(x) for fixed 0 ≤ r ≤ 1 in terms of Bernoulli basis polynomials as follows where , then the coefficients g i and ḡi , i = 0, 1, ..., M, can be calculated from the following relations: Lemma 3.2.[22] Suppose that the fuzzy-valued function g ∈ C [0, 1] is approximated on the interval [0, 1] by Bernoulli polynomials as argued in previous lemma.Then the coefficients g i and ḡi decays as follows where G i and Ḡi denotes the maximum of g (i)(x) and ḡ(i) (x) in the interval [0, 1], respectively.
where K = [k i j ] M i, j=0 is an (M + 1) × (M + 1) matrix.In addition, the previous lemmas can be used for approximating two variable real-function k in terms of truncated Bernoulli series.

be an arbitrary crisp function and is approximated by the two variable truncated
, then the coefficients k i j can be calculated from the following relation: ∂ x i ∂t j dxdt, i, j = 0, 1, ..., M.
(3.10) Also, the coefficients k i j decay as follows

.11)
where K i j denotes the maximum of ∂ i+ j k(x,t) ∂ x i ∂t j in the [0, 1] 2 .
International Scientific Publications and Consulting Services now, we consider Eq.(2.1) with fixed 0 ≤ r ≤ 1.By (2.5), (3.6) and (3.9) we get where ] T and U, Ū, F and F maybe obtained same as before respect to u, ū, f and f , respectively.In addition, from (3.9) one can approximate k(x,t) ≈ B T (x)KB(t).Considering Eq. (2.1) and using above relations we derive where D 1 is an invertible matrix, therefore we have

Error estimation of BP method
Here, in order to show the convergence of the BP method we present a theorem that demonstrate the accuracy and efficiency of the BP method for solving fuzzy Fredholm integral equations.Theorem 3.1.Let k(x,t) be continuous and positive for 0 ≤ x, s ≤ 1, and f : [0, 1] → R be continuous fuzzy-valued functions.Then the approximate solution of Eq. (2.1) using BP method converges to the exact the solution.
Proof.Let u M be approximate solution of Eq. (2.1) using BP method.By (3.6) and (3.9) we have Using Lemmas 3. International Scientific Publications and Consulting Services

HBPBP method
Now, using hybrid block-pulse functions and Bernoulli polynomials we introduce a numerical method for solving Eq.(2.1).Suppose that X = L 2 [0, 1] and {b 10 (x), b 20 (x), ..., b NM (x)} ⊂ X is the set of HBPBP, and Therefore, there exist unique coefficients c 10 , c 20 , ..., c NM such that where Using (3.14) assume that where D 2 is an M(N + 1) spars invertible matrix [20].Therefore, for any k(x,t) ∈ L 2 [0, 1] 2 , we can approximate k(x,t) as Suppose that M(N + 1) matrix Φ defined by and k i j (x) can be expanded as k i j (x) = ∑ M q=0 ∑ N p=1 φ i j pq b pq (x), where φ i j pq can be obtained similar to (3.17).Considering Eq. (2.1) and (3.16) we observe and k(x,t) = ϒ T (x)Kϒ(t), where V , V , F, F and K are defined similar to Φ in (3.16).According to these relations and (2.5) we obtain hence, we conclude that V = (I − KD 2 ) −1 F and V = (I − KD 2 ) −1 F, where I is an NM × NM identity matrix.By using these systems, we maybe find unknown NM × NM matrix V and V .

Error estimation of HBPBP method
Now, we obtain an error estimate between the approximate and the axact solution of Eq.(2.1) using HBPBP method.
Proof.Let u NM be approximate solution of Eq. (2.1), for fixed 0 ≤ r ≤ 1 we have

Numerical experiments
The mentioned numerical methods in Subsections 3.1 and 3.2, were tested on two illustrative examples providing the accuracy and the convergence of the BP and HBPBP methods.Comparing the numerical results of applying these methods show that the hybrid method has better convergence rate and accuracy than the non-hybrid method.The methods were implemented using Mathematica.
the exact solution in this case is given by Applying BP and HBPBP methods for various M, N, r and obtaining the computational right and left errors E r − , E r + at point x = 0.5 we infer that for each 0 ≤ r ≤ 1, the norm of the errors tend to zero as M, N → ∞.The results are presented in Tables 1 and 2. The numerical results obtained by implementation of the BP and HBPBP methods are in Tables 3 and 4.  In this work, to approximate the solution of fuzzy Fredholm integral equations we used known non-orthogonal Bernoulli basis polynomials.Using these polynomials we have established two numerical methods, namely BP and HBPBP methods.In HBPBP technique, we have combined the Bernoulli polynomials with piecewise constant blockpulse functions to creating an efficient and simple method to approximate solution of fuzzy Fredholm integral equations, such that by using this hybrid functions, the Eq.(2.1) has been reduced to two algebraic systems.Also, in Theorems 3.1.and 3.2.the analysis of error estimation of the BP and HBPBP methods are given.The numerical experiment show that the methods performs well for fuzzy Fredholm integral equations and comparing between the computational results obtained from these two method demonstrates that the HBPBP method is more better than BP method.

Definition 2 . 1 .
[14] A fuzzy number is a function u : R → [0, 1] satisfying the following properties: u is upper semicontinuous on R, u(x) = 0 outside of someinterval [c, d], there are the real numbers a and b with c ≤ a ≤ b ≤ d, such that u is increasing on [c, a], decreasing on [b, d] and u

Table 1 :
Left numerical errors on the level sets for Example 4.1.at x = 0.5

Table 2 :
Right numerical errors on the level sets for Example 4.1.at x = 0.5 International Scientific Publications and Consulting Services

Table 3 :
Left numerical errors on the level sets for Example 4.2.at x = 0.25

Table 4 :
Right numerical errors on the level sets for Example 4.2.at x = 0.25