In this paper, the reduced differential transform method (RDTM) is applied to solve fuzzy nonlinear partial differential equations (PDEs). The solutions are considered as infinite series expansions which converge rapidly to the solutions. Some examples are solved to illustrate the proposed method.
The fuzzy sets were introduced for the first time by Zadeh in [1]. Hundreds of examples have been supplied where the nature of uncertainty in the behavior of given system processes is fuzzy rather than stochastic nature. Recently, many authors showed interest in the study of the theoretical framework of fuzzy initial value problems. Chang and Zadeh [2] introduced the concept of fuzzy derivative. Dubosi and Prade [3] presented the extension principle. The differential and integral calculus for fuzzy-set-valued functions, shortly fuzzy-valued functions was developed in resent work, see [4,5,6,7,8,9].
It is known that phenomena of nature or physical systems can be modeled using partial differential equations (PDEs) such as wave equations, heat equations, Poisson’s equation and so on. Hence, studies of PDEs become one of the main topics of modern mathematical analysis and attracted much attention. Many authors developed different methods for solving different kinds of PDEs, see [10,11,12,13,14,15,16,17].
The differential transform method (DTM) was introduced by Zhou [18] and he applied it to solve initial value problems for electric circuit analysis. The DTM is based on Taylor,s series expansion and can be applied to solve both linear and nonlinear ordinary differential equations as well as PDEs.
Keskin and Oturanc [19] proposed the RDTM, defining a set of transformation rules to overcome the complicated complex calculations of traditional DTM. Recently some researchers used RDTM for solving different equations, see [20,21,22,23,24,25,26].
This paper is structured as follows: In Section 2, we call some definition on a fuzzy number, fuzzy-valued function and strongly generalized Hukuhara differentiability. In Section 3, Taylor’s formula, one-dimensional DTM, and two-and three-dimensional RDTM is introduced. In Section 4, we provide some examples to show the efficiency and simplicity of RDTM. Finally, Section 5 consists of some brief conclusions.
2. Basic concepts
The fuzzy set is called a fuzzy number if is a normal, convex fuzzy set, upper semi-continuous and supp is compact. Here denotes the closure of We use to denote the fuzzy number space [27,28].
For , , the addition and scalar multiplication are defined by
respectively, where for any .
We use the Hausdorff distance between fuzzy numbers [28] defined by
where is the Hausdorff metric. is called the distance between and .
Definition 1. [29,30] Let be a fuzzy number defined in . The -level set of , for any denoted by is a crisp set that contains all elements in , such that the membership value of is greater or equal to , that is
Whenever we represent the fuzzy number with -level set, we mean that it is closed and bounded and it is denoted by , where they represent the lower and upper bound -level set of a fuzzy number.
The researchers [31,32] defined the parametrical representation of the fuzzy numbers as in the following definition:
Definition 2. [33] A fuzzy number in parametric form is a pair of functions and for any , which satisfies the following requirements
is a bounded non-decreasing left continuous function in (0,1];
is a bounded non-increasing left continuous function in (0,1];
Some researchers classified the fuzzy numbers into several types of the fuzzy membership function and the triangular fuzzy membership function or also often referred to as triangular fuzzy number is the most widely used membership function.
In order to avoid the inconvenience, in the whole paper, the fuzzy numbers and fuzzy-valued functions are represented with a tilde sign at the top, while the real-value function and interval-valued functions are written directly.
Definition 3. [34] A fuzzy valued function of two variable is a rule that assigns to each ordered pair of real numbers, , in a set , a unique fuzzy number denoted by . The set is the domain of and its range is the set of values taken by , i.e., .
The parametric representation of the fuzzy valued function is expressed by , for all and .
Definition 4. [34,35] A fuzzy valued function is said to be fuzzy continuous at if . We say that is fuzzy continuous on if is fuzzy continuous at every point in .
Definition 5. [36,37] The generalized Hukuhara difference of two fuzzy numbers is defined as follows:
(1)
In terms of the levels, we get and if the H-difference exists, then ; the conditions for existence of are
(2)
(3)
It is easy to show that (i) and (ii) are both valid if and only if is a crisp number. In this case, it is possible that the gH-difference of two fuzzy numbers does not exist. To address this shortcoming, a new difference of fuzzy numbers was presented in [37]
Definition 6. [38] Let and . We say that is strongly generalized Hukuhara differentiable on (gH-differentiable for short) if there exists an element such that
(i)   for all sufficiently small, and the limits (in the metric D)
or
  (ii) for all sufficiently small, and the limits
or
(iii)   for all sufficiently small, and the limits
or
(iv)   for all sufficiently small, and the limits
Lemma 1. [39] Let Then the following statements hold:
(a)   If is -partial differentiable for (i.e., is partial differentiable for under the meaning of Definition 5 , similarly to ), then
(4)
(b)   If is -partial differentiable for (i.e., is partial differentiable for under the meaning of Definition 5 , similarly to ), then
(5)
Remark 1. For the following results hold:
(6)
if - exist, and
(7)
if - exist.
3. Analysis of the method
In this section, we shall give some definitions and theorems of the Taylor series, one-dimensional DTM, and two-and three-dimensional RDTM.
Definition 7. [40] A Taylor series for the polynomial of degree is defined as
(8)
Theorem 1. If the function has derivatives on an interval for some , and , for all , where is the error between and the polynomial function then the Taylor series expanded about converges to Thus
(9)
3.1. Differential transform method
We consider the following one-dimensional DTM:
Definition 8. [26] The differential transform of the function for order derivative is defined as
(10)
Definition 9. [26] The inverse differential transform of is defined as
(11)
The Equation (9) is the Taylor series expansion of at . From Equations (10) and (11), the following basic operations of DTM can be deduced
  If
  If
  If .
  If .
  If
  If
  If
  If .
  If
  If
  If
3.2. Reduced differential transform method
We consider the following two-dimensional RDTM:
Definition 10. [41] If the function is analytical and differentiable continuously with respect to time and space in the domain of interest, then we get
(12)
where the t-dimensional spectrum function is the transformed function of . Here the lower case function represents the original function while the upper case stands for the transformed function.
Definition 11. [41] The inverse differential transform of is defined as
(13)
Thus combining (12) and (13), we can express the solution as
(14)
The basic concept of RDTM mainly comes from the power series expansion.
For two-dimensional function
.
  where a is constant.
Misplaced &
.
.
.
For three-dimensional function
.
.
.
.
.
.
.
.
.
4. Examples
In this section, we demonstrate how RDTM can be easily applied to obtain the exact solutions of the fuzzy partial differential equations.
Example 1. Consider the following one-dimensional initial value problem describing fuzzy heat-like equations
(15)
subject to the initial condition
(16)
where Now
(17)
and
The parametric form of (15) is
(18)
(19)
for where stands for and stands for
Applying the RDTM on Equations (18) and (19), we get the recurrence relation as
(20)
where is the transform function. From the initial condition (16), we get
(21)
(22)
Substituting into the recurrence relation (20), we get the following values successively
and
The inverse differential transform of is obtained from the relations
and the exact solution is
Example 2. Consider the following two-dimensional initial value problem describing fuzzy heat-like equations
(23)
subject to the initial condition
(24)
where Now
(25)
and
The parametric form of (23) is
(26)
(27)
for where stands for and stands for
Applying the RDTM on Equations (26) and (27), we get the recurrence relation as
(28)
where is the transform function. From the initial condition (24), we get
Substituting into the recurrence relation (28), we get the following values successively
and
The solution for is
and the exact solution is
Example 3. We consider following two-dimensional initial value problem describing heat-like equations
(29)
subject to the initial condition
(30)
where
The parametric form of (29) is
Applying the RDTM, we get the recurrence relation as
(31)
where when , and when . Moreover is the transform function. From the initial conditions, we obtain
(32)
(33)
Substituting into the recurrence relation (31), we get the following values successively
and
The solution for is
and the exact solution is
Example 4. We consider the following fuzzy partial differential equation
(34)
subject to the initial condition
(35)
where (n = 1,2,3,…). Now
(36)
and
The parametric form of (34) is
(37)
(38)
for where stands for and stands for .
Applying the RDTM on Equations (37) and (38), we get the recurrence relation as
(39)
The transformed initial condition (35) becomes
(40)
(41)
For different values of , we get the following results
and
The solution for is
and the exact solution is
Figure 1 illustrate that the left-hand functions of the r-level set of (u lower) are always increasing functions of and the right-hand functions of the r-level set of (u upper) are always decreasing functions of in the above examples.
5. Conclusion
In this paper, the reduced differential transform method (RDTM) has been successfully applied for solving fuzzy nonlinear partial differential equations under gH-differentiability. The solutions are considered as infinite series expansions that converge rapidly to the exact solutions. We solved some examples to illustrate the proposed method. The results reveal that the proposed method is a powerful and efficient technique for solving fuzzy nonlinear partial differential equations.
Author Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Conflicts of Interest
The authors declare no conflict of interest.
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