Abstract
In this paper, variational iteration method is presented to solve the linear and nonlinear fuzzy differential equations. This technique provides a sequence of functions which converges to the exact solution of the problem. Sufficient condition for convergence of the proposed method is given and also a maximum absolute truncation error is estimated. This method provides remarkable accuracy in comparison with the analytical solution. Several numerical examples are given to illustrate the efficiency and performance of the presented method.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
1 Introduction
Many physical problems are governed by fuzzy differential equations, and finding the solution of these equations has been the subject of many investigators in recent years [1–12]. In [13], Ji-Huan He presented a very lucid as well as an elementary discussion of the variational iteration method (VIM); the method was further developed by the originator himself [14–16]. The main property of the method is its flexibility and ability to solve nonlinear equations accurately and conveniently, the solution procedure is simple, and results are acceptable and have been applied to a wide class of nonlinear problems [17–25]. This scheme is used for solving linear system of first-order fuzzy differential equations with fuzzy constant coefficients and nth-order fuzzy differential equations in [12, 17], respectively. The aim of this paper is to extend the VIM for solving the linear and nonlinear fuzzy differential equations, whenever these equations possess unique fuzzy solutions. The VIM provides a new approach to solve the fuzzy differential equations without discretization. Numerical examples are presented to illustrate the efficiency of the VIM. The rest of paper is organized as follows. In Sect. 2, we briefly present the basic definitions. In Sect. 3, VIM for solving fuzzy differential equations is introduced. In Sect. 4, the sufficient condition is presented to guarantee the convergence of the method, and an estimation of the maximum absolute error is presented. The proposed method is illustrated by solving three examples in Sect. 5.
2 Preliminaries
The basic concepts of fuzzy numbers are given in [11]. In this section, we review some of them.
Definition 2.1
[11] A fuzzy number U is a pair of functions \(({\underline{U}}(r),\,{\overline{U}}(r)),\) for every \(0\le r \le 1,\) which satisfies the following requirements:
-
(a)
\({\underline{U}}(r)\) is a bounded, left continuous, and nondecreasing function over [0, 1].
-
(b)
\({\overline{U}}(r)\) is a bounded, left continuous, and nonincreasing function over [0, 1].
-
(c)
\({\underline{U}}(r) \le {\overline{U}}(r),\,0\le r \le 1.\)
A crisp number \(\alpha \) is simply represented by \({\underline{U}}(r) = {\overline{U}}(r) = \alpha ,\,0\le r \le 1.\) The fuzzy number space can be embedded to the Banach space \(B={\overline{C}}[0,\,1]\times {\overline{C}}[0,\,1],\) where the metric is usually defined as
for arbitrary \( (U,\,V)\in {\overline{C}}[0,\,1]\times {\overline{C}}[0,\,1].\)
A first-order fuzzy differential equation is defined by
where x is a fuzzy function of t and \(f(t,\,x)\) is a fuzzy function of the crisp variable t and the fuzzy variable \(x^{\prime }\) is the fuzzy derivative of x. If an initial value \(x(t_{0})=x_{0}\) is given, we obtain a fuzzy Cauchy problem of first order:
Sufficient conditions for the existence of a unique solution to Eq. (2) are that f is continuous and that a Lipschitz condition
is fulfilled. By Theorem 5.2 in [8] we may replace Eq. (2) by the equivalent system
which possesses a unique solution \(({\underline{x}},\,{\overline{x}})\in B\) and it is a fuzzy function, i.e., for each t, the pair \(({\underline{x}}(t,\,r),\,{\overline{x}}(t,\,r))\) is a fuzzy number. The parametric form of Eq. (4) is given by
for \(0 \le r \le 1.\) A solution of Eq. (5) must solve Eq. (4) as well by using the sup norm, an equality between two fuzzy numbers in B yields a pointwise equality.
3 Variational Iteration Method (VIM)
In order to solve the system given in Eq. (5), by VIM, we can construct following correction functionals:
where \(\lambda _{1}(s)\) and \(\lambda _{2}(s)\) are general Lagrange multipliers and they can be identified via variational theory. Here, \({\tilde{\underline{x}}}_{n}\) and \({\tilde{\overline{x}}}_{n}\) denote restricted variations, i.e., \(\delta {\tilde{\underline{x}}}_{n}=\delta {\tilde{\overline{x}}}_{n}=0.\)
Making the above correct functionals stationary, note that
and the following stationary conditions can be obtained as,
The Lagrange multipliers can be identified as follows:
and it implies the following iteration formula,
where \({\underline{x}}_{0}(t,\,r)={\underline{x}}(0,\,r)\) and \({\overline{x}}_{0}(t,\,r)={\overline{x}}(0,\,r).\)
Now, we define the operator \(A=[A_{1},\,A_{2}],\) as [24],
and define the components \(V_{k}=({\underline{V}}_{k},\,{\overline{V}}_{k}),\,k=0,\,1,\,2,\ldots ,\) as
It implies that,
Therefore, as a result, the solution of problem (5) can be obtained from (7) and (8), in the series form,
Here, we approximate the solutions (9) by the nth-order truncated series \(\displaystyle \sum \nolimits _{k=0}^{n}{\underline{V}}_{k}(t,\,r)\) and \(\displaystyle \sum \nolimits _{k=0}^{n}{\overline{V}}_{k}(t,\,r).\)
It is easy to see that the above procedure can be easily extended to the nth-order fuzzy differential equation,
where \(c_{i},\, 0\le i\le n-1,\) are given fuzzy numbers. Using VIM to solve Eq. (10), we have,
and the following iteration formula will be derived as,
where
and
4 Convergence Analysis
In this section, we study the convergence of the VIM when applied to problem (6) [24]. The sufficient conditions for convergence of the method and the error estimate are presented. The main results are proposed in the following theorems.
Theorem 4.1
Let A, defined in (7), be an operator from a Banach space B to B. The series solutions \(\displaystyle \sum \nolimits _{k=0}^{\infty }{\underline{V}}_{k}(t,\,r),\,\displaystyle \sum \nolimits _{k=0}^{\infty }{\overline{V}}_{k}(t,\,r)\) defined in (9), converges if there exists \(0<\gamma <1\) such that \(\Vert V_{k+1}\Vert \le \gamma \Vert V_{k}\Vert \) for \(k\in \mathbb {N}\cup \{0\}.\)
The proof is straightforward.
Theorem 4.2
If the series solution \(x(t,\,r)=\displaystyle \sum \nolimits _{k=0}^{\infty }V_{k}(t,\,r),\) defined in (9), converges then it is an exact solution of the nonlinear problem (5).
Proof
Suppose that the series solution (9) converges. Set \(S(t,\,r)=\displaystyle \sum \nolimits _{k=0}^{\infty }V_{k}(t,\,r),\) then we have
and so,
By assuming that the infinite summation (11) and derivation can be replaced, we apply the derivative operator to both sides of Eq. (11) and we obtain
On the other hand, from definition (8), we have,
when \(j\ge 1\) and so, using definition (7), we get,
for \(j\ge 1.\) It implies that,
for \(j\ge 1.\)
Consequently, we have,
and
Thus,
From (12) and (13), we can observe that \(S(t,\,r)=\displaystyle \sum \nolimits _{j=0}^{\infty }V_{j}(t,\,r)\) is an exact solution of problem (5). \(\square \)
Theorem 4.3
Assume that the series solution \(\displaystyle \sum \nolimits _{k=0}^{\infty }V_{k}(t,\,r),\) defined in (9), is convergent to the solution \(x(t,\,r).\) If the truncated series \(\displaystyle \sum \nolimits _{k=0}^{m}V_{k}(t,\,r),\) is used as an approximation to the solution \(x(t,\,r)\) of problem (5), then the maximum error, \(E_{m}(t,\,r),\) is estimated as,
Proof
From Theorem 4.1, we have
for \(n\ge m.\) Now, as \(n\rightarrow \infty \) we have \(S_{n}\rightarrow x(t,\,r).\) So,
\(\square \)
In summary, Theorems 4.1 and 4.2 state that the variational iteration solution of nonlinear problem (5), obtained using the iteration formula (6) or (8), converges to exact solution under the condition that, there exists \(0<\gamma <1\) such that \(\Vert V_{k+1}\Vert \le \gamma \Vert V_{k}\Vert ,\) for every \(k\in \mathbb {N}\cup \{0\}.\) In other words, if we define, for every \(i\in \mathbb {N}\cup \{0\},\) the parameters,
then the series solution \(\displaystyle \sum \nolimits _{k=0}^{\infty }V_{k}(t,\,r)\) of problem (5) converges to exact solution, \(x(t,\,r),\) when \(0\le \beta _{i}<1,\) for \(i\in \mathbb {N}\cup \{0\}.\) Moreover, as stated in Theorem 4.3, the maximum absolute truncation error is estimated to be
where \(\beta =\max \{\beta _{i},\, i=0,\ldots ,m\}.\) The convergence discussion, which is presented in this section, can be easily extended to nth-order fuzzy differential equation (10).
5 Numerical Examples
In this section, some interesting problems are solved by proposed method. It should be noted that by VIM a continuous and smooth approximation of exact solution can be obtained, whereas based on finite difference methods, just a discrete approximate solution can be achieved. Furthermore, it can be seen that the results obtained by proposed method have high accuracy.
Example 5.1
Consider the fuzzy initial value problem [6],
for \(r\in (0,\,1].\) The exact solution is given by
According to Eq. (5), we have,
Applying VIM, defined in (6), we get,
where \({\underline{x}}_{0}(t,\,r)=0.75+0.25r\) and \({\overline{x}}_{0}(t,\,r)=1.125-0.125r.\)
Approximate solution \(({\underline{x}}_{40}(t,\,r),\,{\overline{x}}_{40}(t,\,r)),\) at \(t=1,\) and the obtained absolute errors, \(|{\underline{x}}(t,\,r)-{\underline{x}}_{40}(t,\,r)|\) and \(|{\overline{x}}(t,\,r)-{\overline{x}}_{40}(t,\,r)|,\) for \((t,\,r)\in [0,\,1]\times [0,\,1].\) are given in Figs. 1, 2, and 3, respectively.
Example 5.2
Consider the fuzzy initial value problem [6],
where \(C=(1+r,\,3-r)\) and the exact solution is
for \(0\le r\le 1.\)
Using the VIM, we obtain,
where \({\underline{x}}_{0}(t,\,r)=8+0.5r\) and \({\overline{x}}_{0}(t,\,r)=9-0.5r.\)
Approximate solution \(({\underline{x}}_{40}(t,\,r),\,{\overline{x}}_{40}(t,\,r)),\) at \(t=1\) and the obtained absolute errors \(|{\underline{x}}(t,\,r)-{\underline{x}}_{40}(t,\,r)|\) and \(|{\overline{x}}(t,\,r)-{\overline{x}}_{40}(t,\,r)|,\) for \((t,\,r)\in [0,\,1]\times [0,\,1],\) are plotted in Figs. 4, 5, and 6, respectively.
Example 5.3
Consider the following fuzzy differential equation[7],
where \(0\le r,\,t \le 1.\) The exact solution is
Using the VIM, we obtain,
where \({\underline{x}}_{0}(t,\,r)=3+r-(1+r)t+(8+r)\frac{t^{2}}{2}\) and \({\overline{x}}_{0}(t,\,r)=5-r+(r-3)t+(10-r)\frac{t^{2}}{2}.\) The results are shown in Figs. 7, 8, and 9, respectively.
Example 5.4
Consider the nonlinear fuzzy initial value problem [3],
with the exact solution,
for \(0\le r\le 1.\)
According to VIM, we have,
where \({\underline{x}}_{0}(t,\,r)=0.2+0.2r\) and \({\overline{x}}_{0}(t,\,r)=0.9-0.3r.\)
Table 1 shows the comparison between the exact solution and approximate solution \(({\underline{x}}_{6},\,{\overline{x}}_{6}),\) at \(t=1.\) Also, Fig. 10 shows the approximate solution by VIM, at \(t=1.\)
6 Conclusion
In this paper, an efficient iterative method has been presented to solve fuzzy differential equations. The theorems of convergence and error estimation have been discussed. Furthermore, several numerical examples of the proposed method have been presented, and the comparisons with the exact solutions confirm that the method is capable of generating accurate solutions. The proposed method can be easily implemented for a system of fuzzy differential equations. To solve boundary value, fuzzy problems can be studied using VIM in future work.
References
Abbasbandy, S., Allahviranloo, T.: Numerical solutions of fuzzy differential equations by Taylor method. Comput. Methods Appl. Math. 2, 113–124 (2002)
Bede, B., Stefanini, L.: Generalized differentiability of fuzzy-valued functions. Fuzzy Sets Syst. 230, 119–141 (2013)
Chalco-Cano, Y., Roman-Flores, H.: Comparation between some approaches to solve fuzzy differential equations. Fuzzy Sets Syst. 160, 1517–1527 (2009)
Fard, O.S., Kamyad, A.V.: Modified k-step method for solving fuzzy initial value problems. Iran. J. Fuzzy Syst. 8(3), 49–63 (2011)
Georgiou, D.N., Nieto, J.J., Rodriguez-Lopez, R.: Initial value problems for higher-order fuzzy differential equations. Nonlinear Anal. 63, 587–600 (2005)
Ghazanfari, B., Shakerami, A.: Numerical solutions of fuzzy differential equations by extended Runge–Kutta-like formulae of order four. Fuzzy Sets Syst. 189, 74–91 (2011)
Jayakumar, T., Kanagarajan, K., Indrakumar, S.: Numerical solution of \(n\)th-order fuzzy differential equation by Runge–Kutta method of order five. Int. J. Math. Anal. 6(58), 2885–2896 (2012)
Kaleva, O.: Fuzzy differential equations. Fuzzy Sets Syst. 24, 301–317 (1987)
Khastan, A., Ivaz, K.: Numerical solution of fuzzy differential equations by Nystrom method. Chaos Solitons Fractals 41, 859–868 (2009)
Khastana, A., Nietoa, J.J., Rodriguez-Lopez, R.: Variation of constant formula for first order fuzzy differential equations. Fuzzy Sets Syst. 177, 20–33 (2011)
Ma, M., Friedman, M., Kandel, A.: Numerical solutions of fuzzy differential equations. Fuzzy Sets Syst. 105, 133–138 (1999)
Fard, O.S., Ghal-Eh, N.: Numerical solutions for linear system of first-order fuzzy differential equations with fuzzy constant coefficients. Inf. Sci. 181, 4765–4779 (2011)
He, J.H.: Variational iteration method—a kind of non-linear analytical technique: some examples. Int. J. Nonlinear Mech. 34(4), 699–708 (1999)
He, J.H.: Variational iteration method—some recent results and new interpretations. J. Comput. Appl. Math. 207(1), 3–17 (2007)
He, J.H., Wu, X.H.: Variational iteration method: new development and applications. Comput. Math. Appl. 54, 881–894 (2007)
He, J.H., Wu, G.C., Austin, F.: The variational iteration method which should be followed. Nonlinear Sci. Lett. A 1, 1–30 (2010)
Abbasbandy, S., Allahviranloo, T., Darabi, P., Sedaghatfar, O.: Variational iteration method for solving \(n\)th order fuzzy differential equations. Math. Comput. Appl. 16(4), 819–829 (2011)
Ates, I., Yildirim, A.: Application of variational iteration method to fractional initial-value problems. Int. J. Nonlinear Sci. Numer. Simul. 10, 877–883 (2009)
Biazar, J., Ghazvini, H.: He’s variational iteration method for solving linear and non-linear systems of ordinary differential equations. Appl. Math. Comput. 191, 287–297 (2007)
Ghaneai, H., Hosseini, M.M.: Variational iteration method with auxiliary parameter for solving wave-like and heat-like equations in large domains. Comput. Math. Appl. 69(5), 363–373 (2015)
Hesameddini, E., Latifizadeh, H.: Reconstruction of variational iteration algorithms using the Laplace transform. Int. J. Nonlinear Sci. Numer. Simul. 10, 1377–1382 (2009)
Hosseini, M.M., Mohyud-Din, S.T., Ghaneai, H.: Variational iteration method for Nonlinear Age-Structured population models using auxiliary parameter. Z. Nat. 65A, 1137–1142 (2010)
Jafari, M., Hosseini, M.M., Mohyud-Din, S.T., Ghovatmand, M.: Solution of the nonlinear PDAEs by variational iteration method and its applications in nanoelectronics. Int. J. Phys. Sci. 6, 1535–1539 (2011)
Odibat, Z.M.: A study on the convergence of variational iteration method. Math. Comput. Model. 51, 1181–1192 (2010)
Tatari, M., Dehghan, M.: On the convergence of He’s variational iteration method. J. Comput. Appl. Math. 207, 121–128 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hosseini, M.M., Saberirad, F. & Davvaz, B. Numerical Solution of Fuzzy Differential Equations by Variational Iteration Method. Int. J. Fuzzy Syst. 18, 875–882 (2016). https://doi.org/10.1007/s40815-016-0156-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40815-016-0156-2