Solution of fuzzy fractional order differential equations by fractional Mellin transform method

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Abstract

This paper deals with the application of fuzzy Mellin transform for fuzzy valued functions. It is an attempt to investigate fuzzy Mellin transform in fractional sense and is called fuzzy fractional Mellin transform. Some techniques are proposed for the solution of fuzzy fractional order differential equations by using left-sided Riemann–Liouville fractional derivative. Moreover, some illustrative examples are solved to show the capability and effectiveness of fuzzy Mellin transforms method by converting it into a general solution for an analogous equation with the left-sided Riemann–Liouville derivative and then presenting its explicit representation.

Introduction

Fractional Calculus has become an important part in the field of mathematical modeling due to the involvement of non-integer order of derivatives and integrals. From the last two decades, Fractional calculus (FC) is applied to model different physical and chemical process problems of science, mostly in engineering [1], [2], [3], [4], [5]. It is exceptional to discuss fractional order operators from classical calculus. The fractional calculus deals with investigation of integrals and derivatives of arbitrary real or complex order and thus an extension of integer order calculus, which is applicable in many areas of science and engineering, such as fluid flow, rheology, electric network, viscoelasticity, and​ electrochemistry of corrosion [6], [7], [8], [9]. Most of the work is done on FC for the sake of theoretical and practical solutions of fractional order differential equations, which is the generalization of ordinary differential equations to an arbitrary non-integer order. Agarwal et al. introduced and gave the concept of fractional order differential equations [1]. Kilbas and Lakshmikanthan have worked on basic theory and applications of differential equations of fractional order [8], [10]. Many integral transforms were also employed in the solution of fractional order differential equations, mainly Fourier and Laplace transform. As they show their contribution in Fractional calculus in terms of the Reisz–Feller fractional derivatives defined by Fourier and Inverse Fourier transform [11], as well as the Caputo and the Riemann–Liouville fractional derivatives defined by Laplace transform for the solution of Fractional order differential equations [12], [13].

Mellin integral transform has also become important in fractional calculus, fractional differential equation for complex valued function can be solved particularly by this technique [14]. In fractional calculus Riemann–Liouville and Caputo fractional derivatives appeared to be more common and have widespread application in solution of fractional order differential equations. Since the role of Mellin Transform in FC is considerable and extensive as we note that the FC operators applied in Laplace integral transform, the sine and cosine Fourier transforms can all be represented as Mellin convolution type integral transforms. Moreover, by means of the Mellin integral transform, it can even easily prove that some of the special functions of FC are completely monotone. The special functions of FC are all particular cases of the Fox H-function that is defined as an inverse Mellin transform of a quotient of some products of the Gamma functions [15].

In this research article we will discover some useful aspects of Mellin transform in fuzzy fractional order differential equations (FDE). Since fuzzy differential equations provide the information about the behavior of uncertain dynamical system in order to obtain a more realistic and adaptable model. Different researchers solve FDE with different approaches like with Hukuhara derivatives of a fuzzy number-valued function [16], [17]. With the same concept, to determine the existence of periodic solutions or asymptotic phenomena Buckley & Feuring solved the fuzzy differential equations by using Zadeh’s extension principle [18] and with strongly generalized differentiability concepts. The fuzzy differential equation of fractional order can be considered with the same basic concepts as classical fuzzy differential equations, both types do not have exact solutions so numerical techniques are applied to find an approximate solution with error analysis [19]. On the other hand, the complexity of many parameters in mathematical modeling of natural phenomena appears as an uncertain fractional model. Recently, Noeiaghdam & Sidorov used Caputo–Fabrizio derivative to solve fractional model of energy supply–demand system [20]. Fractional order differential equations are solvable by using fractional integral transform methods like Fourier and Laplace fractional integral transforms [21], [22]. In this paper we able to solve fractional order fuzzy differential equations by using Fuzzy Mellin transform. As per our knowledge no one has attempted to solve FDE by this method before. The main advantage of this solution is to utilize one of the characteristics of Mellin integral transform that it holds a scale invariant property is analogous to the Fourier transform’s shift invariance property which has a benefit that the original function remains unchanged when specified transforms are applied. The fractional order differential equations seem to have some drawbacks and limitations in some aspects, one of them is the initial value assigned to the model. In general, the determination of initial values is very hard due to its involvement of uncertainty quantities. This is true when dealing with real physical phenomena. To deal with uncertainty quantities, many researchers deliberated different new concepts, the concept of fuzzy set theory is very enhancing among them [23] as the theory contained ability to deal with differential equations possessing uncertainties at initial values. Agarwal et al. made their first contribution on handling with fractional order differential equations with uncertainties [1]. Later, many researchers were influenced to further explore and extend this field [24], [25], [26], [27].

In this research paper, first we discuss some preliminaries of fuzzy sets, fuzzy operators, and fractional calculus on fuzzy numbers with relevant operations. The second section includes definition and some properties of fuzzy Mellin transform with two derived propositions with respect to fuzzy valued function. In Section 3, the general procedure for solving fuzzy differential equations of fractional order is established and explained with two examples of applications, by using fuzzy Mellin transform method.

A fuzzy set as an uncertain set, is formed by ordered pairs such that the second component is the degree of membership of the first component. For any fuzzy set AinXR belongs to the family of fuzzy numbers is defined as A={x,M(x):xX}, where M(x) is the membership of its elements and has a membership degree which maps xX on its real interval [0,1]. If the degree is 0, the member does not belong to the set and if the degree is 1, the member belongs to the set completely [28].

A fuzzy membership function M(x)[0,1], is called a fuzzy number if it has the following conditions:

  • i.

    M is normal, such that M(x0)=1 for x0R

  • ii.

    M, is fuzzy convex. For any two arbitrary real points x1;x2 and μ[0,1] we have Mμx1+1μx2minMx1,Mx2

  • iii.

    M, is upper semi-continuous.

  • iv.

    The closure of the set is compact i.e. SuppM=xR|Mx>0

Different forms of fuzzy numbers like (triangular, trapezoidal and parametric form) can be operated​ with fuzzy operations, for any two arbitrary fuzzy numbers P and Q, and a continuous measurable function like f{,,}, the fuzziness of P and fuzziness of Q is less than or equal to the fuzziness of f(P,Q). Also the diameter of an interval in a fuzzy number can be called as fuzziness of P in any level of α. If f is continuous and measurable then for 0<α<1

ifPQ=R then P(α)+Q(α)=R(α), (addition)

ifPQ=R then P(α)Q(α)=R(α), (multiplication)

ifλQ=R then λQ(α)=R(α), (Scalar multiplication)

In addition, Hukuhara and generalized Hukuhara differences are defined for fuzzy numbers which further defined H-differentiability for fuzzy valued function [29].

Let f(x):(a,b)M and x0(a,b), then f(x) is strongly differentiable at x0 if there exist an element f/(x0)M such that in the first form, if for h>0 sufficiently near to zero there exist the H-differences f(x0+h)f(x0) and f(x0) f(x0h). f(x)=limh0f(x0+h)f(x0)h=limh0f(x0)f(x0h)h,or also f(x) is differentiable at x0, in the second form, if for h>0, sufficiently near 0, there exist the H-differences, f(x0)f(x0+h) and f(x0h) f(x0). f(x)=limh0f(x0)f(x0+h)h=limh0f(x0h)f(x0)h,Now we can define improper fuzzy Riemann-integrable function for every baon[a,b] [30]. Let f(x,r)=(f(x,r),f¯(x,r)) be a fuzzy valued function on [a,], for every ba there exist two positive numbers; N(r) and N¯(r) such that ab|f(x,r)|dxN(r) and ab|f¯(x,r)|dxN¯(r), then f(x) is improper fuzzy Riemann-integrable on [a,] and the improper integral for this is expressed as Iu=au(x,r)dx=au̲(x,r)dx,au¯(x,r)dx,

Fractional calculus is a powerful tool which provides the idea on how to deal with non-local operators to real world problems. If the data and information are uncertain then fuzzy fractional operators are better contenders to model the problems. There are several types of fractional differentiability on fuzzy number valued functions, some are discussed below.

Definition 1.1 [31], [32]

Riemann–Liouville fuzzy fractional order derivative operator.

Let us consider f(x,r) be a fuzzy number valued function, for (0<r<1) Da+α[f(x)]=1Γ(mα)(ddx)max(xt)mα1f(t)dt,(x>a;R(α)+1>0) Dbα[f(x)]=1Γ(mα)(ddx)mxb(tx)mα1f(t)dt,(x<b;R(α)+1>0)These are left sided and right sided fractional derivatives of order α and R(α)>0

Definition 1.2 [33], [34]

Riemann–Liouville fuzzy fractional order integral operator.

The Riemann–Liouville fractional integral operator of order α > 0 of a fuzzy valued function f(x,r) is defined as Da+α[f(x)]=Ja+αf(x)=1Γ(α)ax(xt)α1f(t)dt,(x>a;R(α)>0)and Dbα[f(x)]=Jbαf(x)=1Γ(α)xb(tx)α1f(t)dt,(x<b;R(α)>0) Da+α and Dbα are the left sided and right sided fractional integrals of order α.

Particularly, J0f(x)=f(x).

for this, J0 is the integral operator and used for computing DRα at the point t0.

Definition 1.3

Fuzzy Caputo Fractional Derivative operator.

In the definition of Riemann–Liouville fractional derivative, suppose the integer order of the derivative is an operator inside of the integral and operating on operand function x(t)FR(fuzzy number valued function),tε[t0,T] DCαfx=JmαDnfx,and DCαfx=1Γ(mα)t0x(xt)mα1fgHm(t)dt,m1<α<m(ddx)m1f(x),α=m1In the Caputo differential fractional operator, there is a derivative of order m on fuzzy number valued function, fm(t). To existence of this derivative for any m, it should be a fuzzy number at any point t. The gH-difference in the definition of the derivative should be defined as fm(t)=limh0fm1(t+h)gHfm1(t)h, which are defined in the following level wise form for xε[t0,T],(m1<α<m) DCαflx,r=1Γmαt0xxtmα1flmt,rdt, DCαfux,r=1Γmαt0xxtmα1fumt,rdt,Particularly, for m=1 Eq. (7) becomes DCαfx=1Γ(1α)t0x(xt)αfgH/(t)dt,(0<α<1)

Section snippets

Fuzzy fractional Mellin transform

In this section, we are going to define fuzzy Mellin transform for fuzzy-valued function. Here we will consider the properties of the fuzzy Mellin transforms, then a derivative theorem is given to connect between Mellin integral transform of fractional derivative and corresponding fuzzy-valued function.

From [35] Sun & Yang proposed the Mellin transform for fuzzy-valued functions and investigated some well-known properties of it.

Definition 2.1

[35]

Let f:(a,b)A be continuous complex fuzzy-valued function,

Suppose

Fuzzy fractional differential equation

The use of fractional operators on fuzzy differential equations (FDE) makes it fuzzy fractional order differential equation FFDEs. It is helpful for the solution of uncertain fractional model in various physical or natural problems and provides an approximate solution.

Here we consider fuzzy fractional differential equation with fuzzy initial value problem, Dxαf(x)=f(x,y(x)),y(x0)=y0x[x0,X](0<α<1)here f:[x0,X]×RR is a continuous mapping, y0,M is a fuzzy number with r-level intervals [y0]r=[y0̲

CRediT authorship contribution statement

Noreen Azhar: Writing of the paper. Saleem Iqbal: Writing of the paper.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

The authors are grateful to the learned referee for a very careful reading of the manuscript with many useful suggestions for improvement. The authors read and approved the manuscript.

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