Abstract
A matrix method called the Bernoulli wavelet method is presented for numerically solving the fuzzy fractional integrodifferential equations. Using the collocation points, this method transforms the fuzzy fractional integrodifferential equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown coefficients. To illustrate the method, it is applied to certain fuzzy fractional integrodifferential equations, and the results are compared.
1. Introduction
Dynamical systems with fractional order derivatives have found many applications in various problems in science and engineering like viscoelasticity, heat conduction, electrode-electrolyte polarization, electromagnetic waves, diffusion wave, control theory, and so on. In fractional equations, the vagueness may be appearing in each part of the equation like initial condition, boundary condition, and so on. So solving fractional equations in the sense of real conditions leads to the use of interval or fuzzy calculations.
The concept of the fuzzy derivative was first introduced by Chang and Zadeh [1], followed by many authors. The starting point of the topic in the set valued differential equation and also fuzzy differential equation is Hukuhara’s paper [2]. The Hukuhara derivative was the starting point for the topic of set differential equations and later also for fuzzy fractional differential equations. By the concept of Hukuhara differentiability, the fuzzy Riemann-Liouville fractional differential equation is introduced by Agarwal et al. in [3], which was the starting point of the topic in fuzzy fractional derivative. They have considered the Riemann-Liouville differentiability concept based on the Hukuhara differentiability to solve uncertain fractional differential equations. The existence and uniqueness of solutions of Riemann-Liouville fuzzy fractional differential equations is proved in [4, 5]. Allahviranloo et al. in [6] presented the explicit solutions of uncertain fractional differential equations under Riemann-Liouville H-differentiability using Mittag-Leffler functions and in [7] introduced the fuzzy fractional differential equations under Riemann-Liouville H-differentiability and obtained the solution of this equation by fuzzy Laplace transforms. They showed two new uniqueness results for fuzzy fractional differential equations involving Riemann-Liouville generalized H-differentiability with fuzzy version of Nagumo and Krasnoselskii-Krein conditions [8]. Consequently, the Caputo generalized Hukuhara derivative is introduced in [9]; the authors introduced an ordinary fractional differential equation under the generalized Hukuhara differentiability and studied the existence and uniqueness of the solution. The nonlinear fuzzy fractional integrodifferential equation under generalized fuzzy Caputo derivative is introduced in [10, 11] and proved the existence and uniqueness of the solutions of this set of equations by considering the type of differentiability. Recently, Sahu and Saha Ray [12] applied the two-dimensional Bernoulli wavelet method to solve the fuzzy integrodifferential equations and they developed the Bernoulli wavelet method to solve the nonlinear fuzzy Hammerstein Volterra integral equations with constant delay [13].
The main aim of the presented paper is concerned with the application of the proposed approach to obtain the numerical solution of fuzzy fractional integrodifferential equations of the formwhere is the fuzzy Caputo fractional derivative of order , is a given function satisfying some assumptions that will be specified later, is an element of , and is a nonlinear integral operator given bywhere , with .
The paper is organized as follows. Section 2 collects some definitions of basic notions and notations concerning fuzzy calculus. In Section 3 we discuss the properties of Bernoulli wavelets. To determine the approximate solution for the fuzzy fractional integrodifferential equation, two-dimensional Bernoulli wavelet method has been applied in Section 4. Moreover according to the type of differentiability, solutions of a fuzzy fractional integrodifferential equations are investigated in different scenarios. In Section 6, numerical examples are given to solve the fuzzy fractional integrodifferential equation and show the accuracy of the method. Finally, in Section 7, the report ends with a brief conclusion and some remarks.
2. Preliminaries
In this section, we introduce notation, definitions, and preliminary results, which will be used throughout this paper. Let denote the set of fuzzy subsets of the real axis, if , satisfying the following properties:(i) is upper semicontinuous on .(ii) is fuzzy convex.(iii) is normal.(iv)closure of is compact.
Then is called the space of fuzzy numbers.
For , set , and . We represent , so if , the -level set is a closed interval for all . For arbitrary and , the addition and scalar multiplication are defined by , , respectively.
A triangular fuzzy number is defined as a fuzzy set in , which is specified by an ordered triple with such that and are the endpoints of -level sets for all .
The Hausdorff distance between fuzzy numbers is given by as in [15] where is the Hausdorff metric. The metric space is complete, separable, and locally compact and the following properties from [15] for metric are valid: (1);(2);(3);(4), as long as and exist, where .
Here, is the Hukuhara difference (H-difference); it means that if and only if .
Definition 1 (see [16]). The generalized Hukuhara difference of two fuzzy numbers is defined as follows: In terms of -levels one has and if the H-difference exists, then ; the conditions for the existence of are as follows.
Case (i)Case (ii)It is easy to show that and are both valid if and only if is a crisp number.
Remark 2. Throughout the rest of this paper, we assume that .
Definition 3 (see [17]). A fuzzy-valued function is said to be continuous at if for each there is such that , whenever and . One says that is fuzzy continuous on if is continuous at each .
Definition 4 (see [16]). The generalized Hukuhara derivative of a fuzzy-valued function at is defined asIf satisfying (7) exists, one says that is generalized Hukuhara differentiable (-differentiable for short) at .
Definition 5 (see [18]). Let . One says that is fuzzy Riemann integrable in if, for any , there exists such that for any division with the norms one has where denotes the fuzzy summation. One chooses to write Note that if the fuzzy-valued function is continuous in the metric , the Lebesgue integral and the Riemann integral yield the same value, and also
Throughout this paper, we consider the notations for the space of the fuzzy-valued functions from into that are absolutely continuous on . Also, denote the set of the fuzzy-valued functions which are fuzzy continuous on all of such that the continuity is one-sided at endpoints . Also, we denote the space of all Lebesgue integrable fuzzy-valued functions on the bounded interval by .
Definition 6 (see [7]). Let . The fuzzy Riemann-Liouville integral of a fuzzy-valued function is defined as follows:for , and .
Theorem 7 (see [7]). Let be a fuzzy-valued function. The fuzzy Riemann-Liouville integral of a fuzzy-valued function can be expressed as follows: where
Definition 8 (see [9]). Let . The fuzzy -fractional Caputo differentiability of the fuzzy-valued function is defined as follows:where .
Definition 9 (see [9]). Let be -differentiable at . One says that is -differentiable at ifand that is -differentiable at if
Definition 10 (see [9]). Let be a fuzzy function. A point is said to be a switching point for the -differentiability of , if in any neighborhood of there exist points such that one has the following.
Type(I). At (15) holds while (16) does not hold and at (16) holds and (15) does not hold.
Type(II). At (16) holds while (15) does not hold and at (15) holds and (16) does not hold.
Lemma 11 (see [9]). Let be a fuzzy-valued function such that ; then
Lemma 12 (see [14]). Fuzzy initial value problem (1) is equivalent to one of the following integral equations.
Case 1. If is -differentiable, then Case 2. If is -differentiable, hence
Theorem 13 (see [14]). Assume that the following conditions hold: (H1) is fuzzy continuous.(H2)There exists a constant and real-valued positive functions such that for each , and all .Ifthen (1) has a unique solution on .
3. Bernoulli Wavelets
In this section, first we recall the definitions of wavelets and Bernoulli wavelets. Our aim is to approximate the solution by the truncated Bernoulli series.
3.1. Wavelets and Bernoulli Wavelets
The Bernoulli polynomials play an important role in different areas of mathematics, including number theory and the theory of finite differences. The classical Bernoulli polynomials are usually defined by means of following relations: Also the Bernoulli polynomials can be represented in the formwhere are the Bernoulli numbers. Thus, the first four such polynomials, respectively, are Also, these polynomials satisfy the following formula:
The properties of Bernoulli polynomials and the sequence of Bernoulli numbers are(1).(2).(3).(4).(5).(6)(7).(8).(9).
Wavelets constitute a family of functions constructed from dilation and translation of single function called the mother wavelet . They are defined by where is dilation parameter and is a translation parameter. Bernoulli wavelets have four arguments, defined on interval by with where and is the order of the Bernoulli polynomials and is a fixed positive integer. The coefficient is for orthonormality, the dilation parameter is , and translation parameter is .
The two-dimensional Bernoulli wavelets are defined as where and and and are any positive integers.
3.2. Function Approximation
A function defined over can be expanded in terms of Bernoulli wavelets asIf the infinite series in (30) is truncated, then it can be written aswhere is matrix, given by Also, is matrix and
3.3. The Fractional Order Integration of the Bernoulli Wavelet
The fractional order integration of the Bernoulli wavelets is as follows: where for , and is the order of the Bernoulli polynomial and is a fixed positive integer.
4. The Numerical Method
In this paper, we focus on the fuzzy fractional integrodifferential equation:with initial conditionwhere and are fuzzy functions and . Applying on both sides of (36), using Lemmas 11 and 12, if is -differentiable, thenand is -differentiable:
Consider is the solution of (36) and we approximate the unknown function as given by (30). Assume that is -differentiable, so by Theorem 7 and (38) we have the following fractional integrodifferential equations system:We see that Hence we obtainIn order to apply the Bernoulli wavelets in (42), we first approximate the unknown function as
Putting (44) in (42) we obtain Now we collocate (45) at collocation points by , for , yielding
Now consider (43); we approximate the unknown function by Bernoulli wavelet asWorking the same way we find
Equations (46) and (48) yield equations in unknowns in and . By solving this system of equations using mathematical software, the Bernoulli wavelet coefficients and can be obtained and, hence, substituting them in (44) and (47), the approximate solutions can be obtained.
Now, consider is -differentiable; we have the following fractional integrodifferential equations system:Using (39) and definition of Hukuhara difference, this system can be written in the form Then in similar way to previous case, we get the values of unknown vectors and and then obtain the solutions and from (44) and (47), respectively.
5. Error Estimation Algorithm and Convergence Analysis
In this section, we will show an efficient estimation for the Bernoulli wavelets approximation and also a technique to obtain the corrected solution of problem (40) by using the residual method and to describe the convergence behavior of the proposed numerical method.
Let be the truncated series which approximate the unknown function of (40), so we observe that where is the residual function. Now, let us considerHence, we havewith initial condition for all .
Furthermore, the error function can be defined aswhere is the exact solution of problem (40).
By using (53) and (54), we have the error equationwith initial condition Subsequently, the error problem by using (52) and (55) can be written asSolving (57) in the way as in Section 4, we get the approximation which is the error function based on residual function. We note that if the exact solution of problem (40) is unknown, then the error function can be estimated by which is found without the exact solution and also clearly seen from given error estimation algorithm.
Let us consider can be expanded in terms of Bernoulli wavelets as And can be the truncated series: Hence, the truncated error term can be calculated as Now, we have the following Theorems based on [13].
Theorem 14 (see [13]). If is defined on and , then the Bernoulli wavelets expansion of defined in (31) converges uniformly and also where and .
Theorem 15 (see [13]). If a continuous function defined on is bounded, namely, , then where and .
6. Numerical Examples
In order to illustrate the effectiveness of the proposed method, we consider numerical examples of fuzzy fractional differential equation.
Example 1. Consider the following fuzzy fractional integrodifferential equation:where denotes the crisp set and the exact solution of (63) is given by . The exact solution is plotted in Figure 1(a) and its is plotted in Figure 1(b). As you see, is -differentiable. So, by applying the method which is discussed in detail, we presented numerical solution of this example for , . Also, we calculated the absolute errors as and . To compare the absolute errors of presented method and Legendre method proposed in [14], see Table 1.
It is evident from Table 1 that the numerical solutions converge to the exact solution. It is also concluded that the proposed method is very efficient for numerical solutions of these problems.
(a)
(b)
Example 2. Consider the following fuzzy fractional integrodifferential equation: where denotes the crisp set . The exact solution of this equation is given by . By applying the proposed method to obtain -differentiable solution, we solve this problem numerically and calculate the absolute errors by using . To see these absolute errors, one can refer to Table 2. Also, in Figure 2, we present the graph of the Bernoulli wavelet approximation error for , and , .
7. Conclusion
In the present paper, the two-dimensional Bernoulli wavelet method was applied to approximate the solution of fuzzy fractional integrodifferential equation. We transformed our problem to a system of algebraic equations so that by solving this system we obtained the solution of this kind of equation by considering the type of differentiability. Finally, the solution obtained using the suggested method shows that this approach can solve the problem effectively.
Conflicts of Interest
The authors declare that they have no conflicts of interest.