Numerical results and absolute errors of
Fractional calculus and fuzzy calculus theory, mutually, are highly applicable for showing different aspects of dynamics appearing in science. This chapter provides comprehensive discussion of system of fractional differential models in imprecise environment. In addition, presenting a new vast area to investigate numerical solutions of fuzzy fractional differential equations, numerical results of proposed system are carried out by the Grünwald‐Letnikov's fractional derivative. The stability along with truncation error of the Grünwald‐Letnikov’s fractional approach is also proved. Moreover, some numerical experiments are performed and effective remarks are concluded on the basis of efficient convergence of the approximated results towards the exact solutions and on the depictions of error bar plots.
- fuzzy‐valued functions
- fuzzy differential equations
- fractional differential equations
- Grünwald‐Letnikov’s derivative
It is worthwhile mentioning, since last few decades, the theory of fractional calculus has gained significant importance in almost every branch of science, for having the capability to consider integrals and derivatives of any arbitrary order. The characteristic feature of generalizing the classic integer‐order differentiation and n‐fold integration to arbitrary fractional order have broadened its application in modeling several phenomena of physics, mathematics, and engineering. The differential models of fractional order, due to the nonlocal properties of fractional operator, are excellent instruments for providing information about the current as well as the historical state of the system. For these reasons, it is intensively developed and advanced, and existence of its solution is studied by well‐known authors, Euler, Laplace, Liouville, Riemann, Fourier, Abel, Caputo, etc., to further widen its scope in describing various real‐world problems of science, for instance see [1–6]. Another wide‐spreading exploration of mathematics is theory of fuzzy calculus, which has a lot of interesting applications in physics, engineering, mechanics, and many others. It is the theory of a particular type of interval‐valued functions, in which mapping is made in such a way that it takes all the possible values in and not only the crisp values as found in usual interval‐valued functions. After the inception of fuzzy set theory by Zadeh , its attributes have been extended and established to overcome impreciseness of parameters and structures in mathematical modeling, reasoning, and computing [8–12].
Advanced development of mathematical theories and techniques has gained very high standard. On the basis of classical theories, new theories are pioneered by undergoing its inadequacies and widening its scope in many disciplines. In a similar manner, the aforementioned theories have been brought together in modeling different aspects of applied sciences, to analyze the change in the respective system at each fractional step with the uncertain parameters. Agarwal et al.  initiatively incorporated uncertainty into dynamical system, modeled fractional differential equations with uncertainty, and studied its possible solutions. Ahmad et al.  described the situation of impreciseness of initial values of fractional differential equations and discussed its solutions by utilizing Zadeh’s extension principle. In [15, 16], authors considered the concept of Caputo and Riemann fractional derivative, respectively, together with the Hukuhara differentiability and demonstrated the fuzzy fractional differential equations and a lot of others [17–23].
In light of noteworthy applications of above‐mentioned theories, in this chapter, we demonstrate fractional order dynamical models in fuzzy environment to depict unequivocal fractional differential equations of dynamical system. Moreover, we investigate its numerical solutions using the well‐known Grünwald‐Letnikov's fractional definition. This definition is widely applicable as a numerical scheme to solve linear and nonlinear differential equations of fractional order [24–26]. It is considered as an extended form of the classical Euler method. Here it will be utilized, for the first time, to solve fractional differential equations of imprecise functions. Sequentially, this chapter features description of fuzzy theory and fuzzy‐valued functions for the explanation of impreciseness, modeling of system of nonlinear fractional order differential equations with imprecise functions, deliberation of Grünwald‐Letnikov’s fractional approach in conjunction with its truncation error for the proposed system, tabulated and pictorial investigations of some examples, and conclusive remarks of the undergone experiments and findings of the whole manuscript.
2. Basic descriptions
Fuzzy calculus theory is the branch of mathematical analysis that deals with the interval analysis of imprecise functions. This section comprises some rudiments of fuzzy calculus theory and acquaints the necessary notations that are prerequisite for the whole paper. All the below‐mentioned descriptions are widely elaborated and used in literature, for instance [13–23].
2.1. Fuzzy numbers
is bounded non‐decreasing lower function, left continuous on and right continuous at is bounded non‐increasing upper function, left continuous on and right continuous at
The sum and scalar product of any fuzzy number is the consequence of Zadeh’s extension principal. Let , and be the symbols of addition, multiplication and subtraction, accordingly, for fuzzy numbers, which will be greatly used throughout the paper, then, for
The distance between any two fuzzy numbers and is given by the Hausdorff metric
2.2. Fuzzy-valued Function and its fractional derivative
Any interval‐valued function
where is taken in a way that . For
In a similar manner, fractional order differential of
2.3. System of fractional order fuzzy differential equations
In particular, modeling of differential equations of fractional order in imprecise characteristics is obtained by encompassing fuzzy‐valued functions. Let
where the unknown fuzzy‐valued function
Here, we consider the system of fractional order fuzzy differential equations of the following form:
with the initial conditions,
where are the fuzzy numbers that can be written as, for all
Therefore, Eq. (8) can be remodeled as:
And as mentioned earlier,
3. Grünwald‐Letnikov’s fractional derivative
This section comprises the description of Grünwald‐Letnikov’s fractional derivative in conjunction with the algorithm to solve the system of Eq. (11) and undergoes some requisite theorem and lemma of the governing approach.
Consider a function
where are the binomial coefficients that are obtained by the formula:
and represents the integral part.
This definition is considered to be equivalent to the definition of Riemann‐Liouville fractional derivative and for equivalence to Caputo’s fractional definition the following term of initial value is added to the right hand side of Eq. (15), i.e.
That becomes zero if initial values of Caputo‐type differential equations are homogeneous and again reduces to that of Riemann‐Liouville definition. Since here the fuzzy Caputo‐type fractional differential equations are considered with inhomogeneous initial values, the definition in Eq. (16) will be used for the approximation of Eq. (11).
Next consider the fractional system in Eq. (11), for the cases of inhomogeneous initial values. Assume the uniform grids , where , such that
Solving above system fuzzy-valued functions of respective fuzzy functions are generated at different grid points.
3.2. Theorem: truncation error
Let fuzzy‐valued functions
Assume the nth equation of the system (19) and on applying Grünwald‐Letnikov’s fractional derivative we have,
for and from Lemma 3.1 we can attain,
Let, for ,
or it can be rearranged as:
4. Numerical illustrations
Subsequent to the algorithm demonstrated in Section 3, here numerical experiments of some system of fuzzy fractional differential equations are presented. Results for fuzzy‐valued functions are depicted in tabular form in the finite interval at different values of . In addition, error bar pictorials are given for each respective example. All the exact values and calculations are carried out through Mathematica 10.
4.1. Example 1
Following nonlinear fractional system is solved in  using homotopy analysis method, here the system is restructured with imprecise functions
with and subjected to initial conditions
On applying Grünwald‐Letnikov’s fractional definition on left hand side of Eq. (27) and following the algorithm, the differential equations are reduced to nonlinear algebraic equations as:
which on expanding to
Solving this system, numerical approximations of Eq. (27) are obtained. Tables 1 and 2 represent absolute error of
4.2. Example 2
Consider the following nonlinear fractional system  with imprecise functions
with and subjected to initial conditions
On employing Grünwald‐Letnikov’s approach, the differential equations are converted into nonlinear algebraic equations as:
Thus, numerical results of Eq. (31) are obtained from the above system. Tables 4–6 present absolute error of
In this chapter, system of fractional differential equations with fuzzy‐valued functions was constructed to study the system in imprecise environment. We assessed numerical interpretations of the system using Grünwald‐Letnikov’s fractional derivative scheme, which has not been considered for fuzzy differential equations in literature hitherto. In addition, we illustrated the stability of the scheme for the system of fuzzy fractional differential equations. Furthermore, we conducted experiment on some nonlinear fuzzy fractional systems and successfully attained the approximated solutions. From the entire discussion and analysis, collectively, we come up with the following remarks:
Scrutinizing differential models with arbitrary fractional order in combination with fuzzy theory is effectively advantageous to analyze the change in the system at each fractional step with imprecise parameters rather than crisp values.
Grünwald‐Letnikov’s fractional definition is equivalent to either Riemann‐Liouville fractional definition or Caputo-type fractional definition in case of homogeneous and inhomogeneous initial values, respectively. Since Riemann‐Liouville fractional definition and Caputo‐type fractional definition are greatly applicable for defining fractional derivative of fuzzy‐valued functions, so is Grünwald‐Letnikov’s fractional definition found to be.
Approximations of examples attained by undertaking Grünwald‐Letnikov’s fractional derivative approach are efficaciously convergent towards the exact solutions that prove the method to be appropriate for the solutions of fuzzy differential equations of fractional order to a great extent.
Pointwise explanation of errors through bar graph is conspicuously helpful in locating the error between exact and calculated solutions at each time by simply measuring the length of the bar at the respective point.