In this work, the collocation method to solve fuzzy Volterra integral equations is introduced. Then, solving these systems by collocation method with Bernoulli polynomials is presented. Also, we will discuss the convergence of this method. In addition, the superiority of this method compared to other methods is proved, by solving of some systems of fuzzy Volterra integral equations.
The differential equations and the integral equations are very important today due to their natural structure and compatibility. With a collocation method, many problems of applied mathematics, such as differential equations and integral equations, can be solved.
Brunner and Kauthen in [15, 24] have presented the collocation method to solve Fredholm-Volterra integral equations. In [5, 37] an application of Walsh functions for Fredholm-Hammerstein integro-differential equations was introduced. Fuzzy integral equations arise in many fields of electrical engineering, medical, biology, geographic, physics, social sciences, etc. Many practical problems in science and engineering can be transformed into fuzzy Volterra integral equations; thus, their solution is one of the main goals in various areas of applied sciences and engineering. The basic arithmetic structure for fuzzy numbers was later developed by Dubois and Prade [17, 18], Mizumoto and Tanaka [32, 33], and Nahmias [35]. The study of these equations began with the investigations of Kaleva [23] and then continued by Goetschel and Voxman [22], Nanda [36] and Seikkala [47] for the fuzzy Volterra integral equations that is equivalent to the initial value problem for the first order fuzzy differential equations. The collocation method based Bernoulli operational [45, 46] matrix for numerical solution of generalized pantograph equation is realized in [6]. The concept of fuzzy sets, originally introduced by Zadeh [54], led to the definition of the fuzzy number and implementation in approximate reasoning problems [53] and others.
In this work, we study the collocation methods using Bernoulli polynomials. In order to find the numerical solution of the systems of liner fuzzy Volterra integral equations, we used the method of approximation of the unknown function u (x) and v (x) in our collocation point.
This paper is organized as follows. In Section 2, introduce Bernoulli polynomials. In the third section, we present a brief summary of the main concepts of fuzzy. Section 4 is devoted to the study of fuzzy systems of Volterra integral equations. Section 5, explains the collocation method described above. In Section 6, we prove the existence of the solution for systems of linear fuzzy Volterra integral equations. In the seventh section, convergence analysis is proved. The proposed method is applied to several numerical examples in Section 8. Also, a conclusion is given in Section 9.
Bernoulli polynomials and some their properties
Bernoulli polynomials play an important role in various expansions and approximation formulas which are useful both in analytic theory of numbers and in classical and numerical analysis. The Bernoulli numbers have been defined here by the exponential generating function [49]:
The question considered here is how to introduce a parameter into these numbers. One option is to multiply the left-hand side of by a simple function and to see what happens. Naturally, from the view point of generating functions, the simplest possible function is an exponential one. Therefore, consider the expansion
where the coefficients on the right-hand side depend on parameter x.
It follows that
is a polynomial in x, called the Bernoulli polynomial. The first few Bernoulli polynomials are:
If, we define vector Ai+1, as
then Bi (z) = Ai+1XT (z), for i = 0, 1, ⋯, p, where
Now, we can expand the matrix
as
where
Preliminaries in fuzzy calculus
The set of all fuzzy numbers is denoted by E1 and is a convex cone. An equivalent parametric definition of fuzzy numbers is given in [13, 24–26]].
Definition 3.1. [1, 38] An arbitrary fuzzy number in the parametric form is represented by an ordered pair of functions performing the following requirements
is a bounded left-continuous non-decreasing function over [0, 1].
is a bounded left-continuous non-increasing function over [0, 1].
for all 0 ⩽ r ⩽ 1.
For arbitrary fuzzy numbers and , we define addition, subtraction, scalar multiplication and multiplication as follows
Addition: and .
Subtraction: and .
Scalar product:
Multiplication:
Definition 3.2. [2, 34] For arbitrary fuzzy numbers and , the quantity
is the distance between and .
This metric is equivalent to the one used by Puri and Ralescu [43] and Kaleva [23]. It is shown that (E1, D) is a complete metric space [42, 44]. Now we follow Goestschel and Voxman [22] and define the integral of a fuzzy function using the Riemann integral concept.
Definition 3.3. [48]. A fuzzy function is said to be continuous if for arbitrary fixed x0 ∈ [a, b] and γ > 0 there exists σ > 0 such that if |x - x0| < σ, then .
Definition 3.4. Take .
Definite integral of h (x) over [a, b] is for each partition q = {x0, x1, ⋯, xn} of [a, b] and arbitrary ξi : xi-1 ⩽ ξi ⩽ xi, i = 1, 2, ⋯, n and is continuous in metric D, its definite integral exists [11, 16] and also,
It should be noted that the fuzzy integral also can be defined by using the Lebesgue-type approach [10].
Definition 3.5. [3, 52]. is fuzzy-Riemann integrable to if for any δ > 0, there exists σ > 0 such that for any division W = {[f, g]; η} of [a, b] with the norms Δ (W) < σ, we have
where denotes the fuzzy summation.
Lemma 3.6.[4, 41] If is fuzzy continuous functions, then the function H : [a, b] → R+ by
The pair (E1, ⊕) is a commutative semigroup with zero element.
For fuzzy numbers which are not crisp, there is no opposite element (that is, (E1, ⊕) cannot be a group).
The function of ||.||F : E1 → R by has the usual properties of norm, that is, ||u||F = 0 if and only if and
and for any .
Definition 3.8. [28]. The fuzzy linear system
where A = (aij), B = (bij), 1 ⩽ i, j ⩽ n are crisp coefficient matrices and a fuzzy number vector, is called dual fuzzy linear system.
Definition 3.9. [19, 20]. A fuzzy number vector given by i = 1, 2, ⋯, n, 0 ⩽ r ⩽ 1, is called a solution of (3.5) if for all i = 1, 2, ⋯, n, we have
Fuzzy systems of Volterra integral equations
The general fuzzy Volterra integral equation systems of the second kind are fuzzy integral equations for x ∈ [a, b] of the from [39, 40]:
where
are unknown functions. Also,
and
and
Collocation method based on Bernoulli polynomials
In this section, we use the collocation method based on Bernoulli polynomials (BP) to solve Fuzzy Systems of Volterra Integral Equations (FSVIE).
Let P (x) = [p1 (x), p2 (x), ⋯, pn (x)] be a BP on x ∈ D. Then, we have
Let us consider
We can write the system 6 in the matrix form
and
where for all i = 1, 2, ⋯, n, we have
Existence of the solution
In this section, we will study the solvability of the linear fuzzy Volterra integral equation systems 6. First, we introduce some simplifying notations. Given an n × m matrix (or, in particular, a vector) C with (i, j) entry ci,j, we define |C|, the absolute value of C, to be the n × m matrix whose (i, j) entry is |cij|. Also, given two n × m matrices C and F, we will write C ⩽ F if and only if cij ⩽ fij for all i and j. With these definitions, we can show [[7–9, 90]
If |C| ⩽ |F| (elementwise), then ||C||∞ ⩽ ||F||∞.
||C||∞ = || |C| ||∞.
Theorem 6.1.Let and be the fuzzy continuous functions and ki,j (x, t), i, j = 1, 2, ⋯, n be continuous for a ⩽ x, t ⩽ b, and .
Let , i, j = 1, 2, ⋯, n} and be a space of fuzzy continuous functions with the metric (that is called the uniform distance between fuzzy number value functions). If then the fuzzy system 6 has an unique solution in Yn which can be obtained by the following successive approximations method
Moreover, the sequence of successive approximations, and converges to the solution and . Furthermore, the following error bound holds
where
and
Proof. First, note that since ki,j (x, t), i, j = 1, 2, ⋯, n, x ∈ [a, b] is continuous for a ⩽ x, t ⩽ b and uniformly continuous with respect to t, there exists kij > 0 such that . To prove this theorem, we investigate the conditions of the Banach fixed point principle. First, we define the operator Q : Yn → Yn by
for each . We show that Q maps Yn into Yn (i.e.Q (Yn) ⊂ Yn). To the end, we show that the operator Q is uniformly continuous. Since is continuous on the compact set of [a, b], we deduce that it is uniformly continuous and hence for δi > 0, i = 1, 2, ⋯, n.
There exists σi > 0 such that
x1, x2 ∈ [a, b].As described above, ki,j (x, t), i, j = 1, 2, ⋯, n, also is uniformly continuous, thus for γij > 0 there exists σij > 0 such that |ki,j (x1, t) - ki,j (x2, t) | < γij, whenever |x1 - x2| < σi,j, x1, x2 ∈ [a, b]. Take
and
Therefore
and
So, we have
Take
Therefore,
By choosing and , we derive
This shows that Q is uniformly continuous for any , and continuous on [a, b], and hence Q (Yn) ⊂ Yn. Now, we prove that the operator Q is a continuous map. So, for and x ∈ [a, b], we have
where
and thus,
Since Θ < 1, the operator Q is a contraction on the Banach space (Yn, D*). Consequently, the Banach fixed point principle implies that 6 has a unique solution in Yn and the following inequality holds
since
We have
Therefore, from 11 we have
□
Convergence analysis
In this section, we prove that the present numerical method converges to the exact solution.
Theorem 7.1.Suppose that and are the approximate and exact solution of system 6, respectivly. In linear system 6, if ki,j (x, t), i, j = 1, 2, ⋯, n and a ⩽ x, t ⩽ b are bounded and continuous, then
as P → ∞.
The approximate solution and absolute error of and of Example 8.1 for N = 4
r
Approximate Solution
Approximate Solution
Absolute Error
Absolute Error
0.
(-0.877584, 1.755168)
(0.479424, 1.438272)
(1.850488E-6, 3.700978E-6)
(1.403340E-6, 4.209990E-6)
0.1
(-0.702068, 1.667410)
(0.431961, 1.428683)
(1.480403E-6, 3.515930E-6)
(1.264414E-6, 4.181920E-6)
0.2
(-0.526551, 1.579652)
(0.387375, 1.399920)
(1.110320E-6, 3.330881E-6)
(1.133897E-6, 4.097721E-6)
0.3
(-0.351034, 1.491893)
(0.348541, 1.351980)
(7.402350E-7, 3.145834E-6)
(1.020224E-6, 3.957383E-6)
0.4
(-0.175517, 1.404135)
(0.318338, 1.284860)
(3.701506E-6, 2.960790E-6)
(9.318133E-7, 3.760913E-6)
0.5
(-6.61E-11, 1.31638)
(0.299640, 1.198560)
(6.612821E-11, 2.775742E-6)
(8.770820E-7, 3.508310E-6)
0.6
(0.175517, 1.228618)
(0.295325, 1.093090)
(3.700183E-7, 2.590690E-6)
(8.644510E-7, 3.199573E-6)
0.7
(0.351034, 1.140860)
(0.308270, 0.968437)
(7.401027E-7, 2.405642E-6)
(9.023404E-7, 2.834701E-6)
0.8
(0.526551, 1.053101)
(0.341350, 0.824610)
(1.110187E-6, 2.220601E-6)
(9.991703E-7, 2.413700E-6)
0.9
(0.702068, 0.965343)
(0.397443, 0.661605)
(1.480272E-6, 2.035575E-6)
(1.163360E-6, 1.936560E-6)
1.
(0.877584, 0.877584)
(0.479424, 0.479424)
(1.850356E-6, 1.85057E-6)
(1.403331E-6, 1.403282E-6)
The approximate solution and absolute error of and of Example 8.2 for N = 8
r
Approximate Solution
Approximate Solution
Absolute Error
Absolute Error
0.
(0.000000, 6.594890)
(-0.606531, 1.819591)
(0.000000, 1.932765E-11)
(1.950110E-12, 5.851100E-12)
0.1
(0.131898, 6.589940)
(0.539812, 1.758332)
(0.263795, 1.931344E-11)
(1.735612E-12, 5.654144E-12)
0.2
(0.197847, 6.555320)
(0.460963, 1.693434)
(0.395693, 1.921219E-11)
(1.482092E-12, 5.445640E-12)
0.3
(0.197847, 6.461340)
(-0.369984, 1.621260)
(0.395693, 1.893596E-11)
(1.189604E-12, 5.213390E-12)
0.4
(0.131898, 6.278330)
(-0.266873, 1.538162)
(0.263795, 1.839950E-11)
(8.850360E-13, 4.946270E-12)
0.5
(0.000000, 5.976614)
(-0.151633, 1.440510)
(0.000000, 1.751577E-11)
(4.875544E-13, 4.632294E-12)
0.6
(-0.197847, 5.526513)
(-0.024261, 1.324663)
(0.395693, 1.619682E-11)
(7.796541E-14, 4.259930E-12)
0.7
(-0.461642, 4.898350)
(0.115241, 1.186980)
(0.923284, 1.435563E-11)
(3.705092E-13, 3.817170E-12)
0.8
(-0.791386, 4.062450)
(0.266873, 1.023824)
(1.582772, 1.190514E-11)
(8.580360E-13, 3.292700E-12)
0.9
(-1.187079, 2.989131)
(0.430637, 0.831554)
(2.374160, 8.760104E-12)
(1.384615E-12, 2.674310E-12)
1.
(1.648721, 1.648721)
(0.606531, 0.606531)
(3.297442, 4.831690E-12)
(1.950110E-12, 1.950884E-12)
Proof. We have
where
Therefore
So
thus, we have
By using 4, we have
for i = 1, 2, ⋯, n. Finally, since α is bounded, we conclude that
and the proof is completed. □
Exact and Approximate Solution and Absolute Error functions obtained by the present method for N = 4 of Example 8.1.
Exact and Approximate Solution and Absolute Error functions obtained by the present method for N = 8 of Example 8.2.
Numerical examples
In order to illustrate the accuracy of the method we present below some numerical examples and the accuracy of the suggested method. As can be viewed in Tables 1 and 2, for both these cases, the absolute error of and by the presented method for N = 4 and N = 8, respectively. Figures 1 and 2 display exact and approximate solution and absolute error functions obtained by the present method for N = 4 and N = 8. All the numerical computations have been done using mathematic.
Example 8.1. Consider the system of fuzzy Volterra integral equations with
and kernel functions
The exact solution in this case is given by
Example 8.2. Consider the system of fuzzy Volterra integral equations with
and kernel functions K (x, t) = (x - t). We get
that is the exact solution.
Conclusions
In this paper, we used the collocation method based on Bernoulli polynomials to approximate the solution of the systems of linear fuzzy Volterra integral equations. Collocation method is easily implemented and is easy and efficient to approximate the solution of systems of integral Equation 6. In the above presented numerical examples, we see that the proposed method well is carried out for the system of linear fuzzy integral equations and the existence and convergence results are approved. In addition, the presented method is simple and quick to compute.
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