# Computational Optimization of Residual Power Series Algorithm for Certain Classes of Fuzzy Fractional Differential Equations

Authors: Mohammad Alaroud; Mohammed Al-Smadi; Rokiah Rozita Ahmad; Ummul Khair Salma Din
Journal: International Journal of Differential Equations
 (2018)

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## Abstract

This paper aims to present a novel optimization technique, the residual power series (RPS), for handling certain classes of fuzzy fractional differential equations of order1<γ≤2 under strongly generalized differentiability. The proposed technique relies on generalized Taylor formula under Caputo sense aiming at extracting a supportive analytical solution in convergent series form. The RPS algorithm is significant and straightforward tool for creating a fractional power series solution without linearization, limitation on the problem’s nature, sort of classification, or perturbation. Some illustrative examples are provided to demonstrate the feasibility of the RPS scheme. The results obtained show that the scheme is simple and reliable and there is good agreement with exact solution.


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## Body

## 1. Introduction

Fuzzy fractional differential equation is hot and important branch of mathematics. It has attracted much attention recently due to potential applications in artificial intelligence, industrial engineering, physics, chemistry, and other fields of science. Parameters and variables in many of the nature studies and technological processes that were designed utilizing the fractional differential equation (FDE) are specific and completely defined. Indeed, such information may be vague and uncertain because of experimentation and measurement errors that then lead to uncertain models, which cannot handle these studies. The process of analyzing the relative influence of uncertainty in inputs information to outputs led us to study solutions to the qualitative behavior of equations. Therefore, it is necessary to obtain some mathematical tools to understand the complex structure of uncertainty models [1–5]. On the other hand, the theory of fractional calculus, which is a generalization of classical calculus, deals with the discussion of the integrals and derivatives of noninteger order, has a long history, and dates back to the seventeenth century [6–10]. Different forms of fractional operators are introduced to study FDEs such as Riemann–Liouville, Grunwald-Letnikov, and Caputo. Out of these forms, the Caputo concept is an appropriate tool for modeling practical situations due to its countless benefits as it allows the process to be performed based on initial and boundary conditions as is traditional and its derivative is zero for constant [11–17].The residual power series (RPS) method developed in [18] is considered as an effective optimization technique to determine and define the power series solution’s values of coefficients of first- and second-order fuzzy differential equations [19–22]. Furthermore, the RPS is characterized as an applicable and easy technique to create power series solutions for strongly linear and nonlinear equations without being linearized, discretized, or exposed to perturbation [23–27]. Unlike the classical power series method, the RPS neither requires comparing the corresponding coefficients nor is a recursion relation needed as well. Besides that, it calculates the power series coefficients through chain of equations of one or more variables and offers convergence of a series solution whose terms approach quickly, especially when the exact solution is polynomial.The remainder of this paper is organized as follows. In Section2, essential facts and results related to the fuzzy fractional calculus will be shown. In Section 3, the concept of Caputo’s H-differentiability will be presented together with some closely related results. In Section 4, basic idea of the RPS method will be presented to solve the fuzzy FDEs of order 1<γ≤2. In Section 5, numerical application will be performed to show capability, potentiality, and simplicity of the method. Conclusions will be given in Section 6.


## 2. Preliminaries

In this section, necessary definitions and results relating to fuzzy fractional calculus are presented. For the fuzzy derivative concept, the strongly generalized differentiability will be adopted, which is considered H-differentiability modification.A fuzzy setv in a nonempty set U is described by its membership function v:U→0,1. So, for each η∈U the degree of membership of η in v is defined by vη.Definition 1 ([28]).
Suppose thatv is a fuzzy subset of ℝ. Then, v is called a fuzzy number such that v is upper semicontinuous membership function of bounded support, normal, and convex.Ifv is a fuzzy number, then vσ=v1σ,v2σ, where v1σ=min⁡η∣η∈vσ and v2σ=max⁡η∣η∈vσ for each σ∈0,1. The symbol vσ is called the σ-level representation or the parametric form of a fuzzy number v.Theorem 2 ([29]).
Suppose thatv1,v2:0,1→R satisfy the following conditions:(1)
v1 is a bounded nondecreasing function.(2)
v2 is a bounded nonincreasing function.(3)
v11≤v21.(4)
for eachk∈0,1, limσ→k-v1σ=v1k and limσ→k-v2σ=v2k.(5)
limσ→0+v1σ=v10 and limσ→0+v2σ=v20.
Thenv:R→0,1 given by vx=sup⁡σ∣v1σ≤x≤vσ is a fuzzy number with parameterization v1σ,v2σ.Definition 3 ([29]).
Letv,w∈RF. If there exists an element P∈RF such that v=w+P, then we say that P is the Hukuhara difference (H-difference) of v and w, denoted by v⊖w.The sign⊖ stands always for Hukuhara difference. Thus, it should be noted that v⊖w≠v+-1w. Normally, v+-1w is denoted by v-w. If the H-difference v⊖w exists, then [v⊖w]σ=v1σ-w1σ,v2σ-w2σ.Definition 4 ([30]).
The complete metric structure onRF is given by the Hausdorff distance mapping DH:RF×RF→R+∪0 such that(1)DHv,w=sup0≤σ≤1⁡max⁡v1σ-w1σ,v2σ-w2σ,for arbitrary fuzzy numbers v= (v1,v2) and w=(w1,w2).Definition 5 ([30]).
Letφ:[a,b]→RF. Then the function φ is continuous at x0∈[a,b] if for every ϵ>0, ∃δ=δ(x0,ϵ)>0 such that DHφx,φx0<ϵ, for each x∈a,b, whenever x-x0<δ.Remark 6.
If the functionφ(x) is continuous for each x∈[a,b], where the continuity is one-sided at endpoints of [a,b], then φ(x) is continuous function on [a,b]. This means that φ(x) is continuous on [a,b] if and only if φ1σ and φ2σ are continuous on [a,b].Definition 7 ([28]).
For fixedx0∈a,b and φ:a,b→RF, the function φ is called a strongly generalized differentiable at x0, if there is an element φ′x0∈RF such that either(i)
the H-differencesφx0+ξ⊖φx0,φx0⊖φx0-ξ exist, for each ξ>0 sufficiently tends to 0 and limξ→0+⁡((φx0+ξ⊖φx0)/ξ⁡)=φ′x0=limξ→0+⁡((φx0⊖φx0-ξ)/ξ), or(ii)
the H-differencesφx0⊖φx0+ξ,φx0-ξ⊖φx0 exist, for each ξ>0 sufficiently tends to 0 and limξ→0+⁡((φx0⊖φx0+ξ)/-ξ)=φ′x0=limξ→0+⁡((φx0-ξ⊖φx0)/-ξ),where the limit here is taken in the complete metric spaceRF,DH.Theorem 8 ([31]).
Suppose thatφ:a,b→RF, where [φ(x)]σ=φ1σx,φ2σx,∀σ∈0,1, then(1)
the functionsφ1σ and φ2σ are two differentiable functions and [D11φ(x)]σ=φ1σ′x,φ2σ′x, when φ is (1)-differentiable;(2)
the functionsφ1σ and φ2σ are two differentiable functions and [D21φ(x)]σ=φ2σ′x,φ1σ′x, when φ is (2)-differentiable.Definition 9 ([31]).
Suppose thatφ:a,b→RF. One can say that φ is (n,m)-differentiable at x0∈a,b, if Dn1φ exists on a neighborhood of x0 as a fuzzy function and it is (m)-differentiable at x0. The second-order derivatives of φ at x are indicated by φ′′x=Dn,m2φx for n,m={1,2}.Theorem 10 ([32]).
LetD11φ:a,b→RF and D21φ:a,b→RF, where [φ(x)]σ=φ1σx,φ2σx for each σ∈0,1:(1)
IfD11φ is (1)-differentiable, then φ1σ′ and φ2σ′ are differentiable functions and D1,12φxσ=φ1σ′′x,φ2σ′′x,(2)
IfD11φ is (2)-differentiable, then φ1σ′ and φ2σ′ are differentiable functions and D1,22φxσ=φ2σ′′x,φ1σ′′x,(3)
IfD21φ is (1)-differentiable, then φ1σ′ and φ2σ′ are differentiable functions and D2,12φxσ=φ2σ′′x,φ1σ′′x,(4)
IfD21φ is (2)-differentiable, then φ1σ′ and φ2σ′ are differentiable functions and D2,22φxσ=φ1σ′′x,φ2σ′′x.Definition 11 ([32]).
Letφ:[a,b]→RF and φ∈CFa,b∩LFa,b. One can say that φ is Caputo fuzzy H-differentiable at x when (Da+γCφ)(x)=(1/Γ(1-γ))∫axφ′τ/x-τγdτ exists, where 0<γ≤1. Also, we say that φ is Caputo [(1)-γ]-differentiable if φ is (1)-differentiable and φ is Caputo [(2)-γ] differentiable if φ is (2)-differentiable, where CFa,b and LFa,b stand for the space of all continuous and Lebesque integrable fuzzy-valued functions on [a,b], respectively.Theorem 12 ([33]).
Let0<γ≤1 and φ∈CFa,b. Then, for each σ∈0,1, the Caputo fuzzy fractional derivative exists on (a,b) such that(2)Da+γCφxσ=1Γ1-γ∫axφ1σ′τx-τγdτ,1Γ1-γ∫axφ2σ′τx-τγdτfor (1)-differentiable and(3)Da+γCφxσ=1Γ1-γ∫axφ2σ′τx-τγdτ,1Γ1-γ∫axφ1σ′τx-τγdτfor (2)-differentiable.The next characterization theorem shows a way to convert the FFDEs into a system of ordinary fractional differential equations (OFDEs), ignoring the fuzzy setting approach.Theorem 13 ([34]).
Consider the below fuzzy fractional IVPs(4)Dt0+γCφt=ft,φt,t>t0,subject to(5)φt0=φ0,where f:a,b×RF→RF such that
(i)[ft,φ(t)]σ=[f1σt,φ1σt,φ2σt,f2σ(t,φ1σt,φ2σt)].
(ii) for anyϵ>0 there exist δ>0 such that f1σt,s,u-f1σt1,s1,u1<ϵ and f2σt,s,u-f2σt1,s1,u1<ϵ,∀σ∈0,1, whenever t,s,u and t1,s1,u1∈a,b×R2,t,s,u-t1,s1,u1R3<δ and f1σ,f2σ are uniformly bounded on any bounded set.
(iii) there is a constant (say)l>0 such that(6)f1σt2,s2,u2-f1σt1,s1,u1≤l.max⁡s2-s1,u2-u1,∀σ∈0,1and(7)f2σt2,s2,u2-f2σt1,s1,u1≤l.max⁡s2-s1,u2-u1,∀σ∈0,1.Therefore, there are two systems of OFDEs that are equivalent to FFDEs (4) and (5) as follows:
Case 1. When φt is Caputo [(1)-γ]-differentiable(8)Dt0+γCφ1σt=f1σt,φ1σt,φ2σt,Dt0+γCφ2σt=f2σt,φ1σt,φ2σt,with φ1σ(t0)=φ01σ, φ2σ(t0)=φ02σ.
Case 2. When φt is Caputo [(2)-γ]-differentiable(9)Dt0+γCφ1σt=f2σt,φ1σt,φ2σt,Dt0+γCφ2σt=f1σt,φ1σt,φ2σt,with φ1σ(t0)=φ01σ, φ2σ(t0)=φ02σ.


## 3. Formulation of Fuzzy Fractional IVPs of Order1<γ≤2

Consider the below fuzzy fractional differential equation(10)Da+γCφt=gtφ′t+ft,φt,a≤t≤b,1<γ≤2,subject to fuzzy initial conditions(11)φa=α,φ′a=β.where α,β∈RF,f:a,b×RF→RF is a linear or nonlinear continuous fuzzy-valued function, gt is a continuous real valued function with nonnegative values on [a,b], and φ(t) is unknown analytical fuzzy function to be determined. We assume that the fuzzy fractional IVPs (10) and (11) have unique smooth solution on the domain of interest.Next, some theorems and definitions which are used later in this paper are presented.Definition 14.
Letφ:[a,b]→RF be fuzzy function such that φ,φ′∈CFa,b∩LFa,b. Then, for 1<γ≤2, Caputo’s H-derivative of φ at x∈(a,b) is defined as(12)Da+γCφx=1Γ2-γ∫axφ′′τx-τ1-γdτ.
Also, we say thatφ is Caputo [(n,m)-γ]-differentiable for n,m∈1,2, when (Da+γCφ)x exists, and φ is (n,m)-differentiable.Theorem 15.
Letφ,φ′∈CFa,b, such that [φx]σ=φ1σx,φ2σx,∀σ∈0,1. Caputo’s H-derivative of order 1<γ≤2 exists on (a,b) such that
(i) Ifφ is (1,1)-differentiable, then [(Da+γCφ)x]σ=M∫axφ1σ′′τx-τ1-γdτ,M∫axφ2σ′′τx-τ1-γdτ=[(Da+γCφ1σ)x, (Da+γCφ2σ)x].
(ii) Ifφ is (1,2)-differentiable, then [(Da+γCφ)x]σ=M∫axφ2σ′′τx-τ1-γdτ,M∫axφ1σ′′τx-τ1-γdτ=[(Da+γCφ2σ)x,(Da+γCφ1σ)x].
(iii) Ifφ is (2,1)-differentiable, then [(Da+γCφ)x]σ=M∫axφ2σ′′τx-τ1-γdτ,M∫axφ1σ′′τx-τ1-γdτ=[(Da+γCφ2σ)x,(Da+γCφ1σ)x].
(iv) Ifφ is (2,2)-differentiable, then [(Da+γCφ)x]σ=M∫axφ1σ′′τx-τ1-γdτ,M∫axφ2σ′′τx-τ1-γdτ=[(Da+γCφ1σ)x,Da+γCφ2σ)x], where M=1/Γ(2-γ).The(n,m)-solution of fuzzy fractional IVPs (10) and (11) is a function φ:[a,b]→RF that has Caputo [n,m-γ]-differentiable and satisfies the FFIVPs (10) and (11). To compute it, we firstly convert the fuzzy problem into equivalent system of second OFDEs, called correspondence (n,m)-system, based upon the type of derivative chosen. Then, by utilizing the σ-cut representation of φt, ft,φt, and the initial data in (11) such that φtσ=φ1σt,φ2σt, ft,φtσ=f1σt,φ1σt,φ2σt,f2σt,φ1σt,φ2σt, φaσ=φ1σa,φ2σa=α1σ,α2σ, and φ′aσ=φ1σ′a,φ2σ′a=β1σ,β2σ, the following corresponding (n,m)-systems will be hold:(i) (1,1)-system such that(13)Da+γCφ1σt=gtφ1σ′t+f1σt,φ1σt,φ2σt,Da+γCφ2σt=gtφ2σ′t+f2σt,φ1σt,φ2σt,(ii) the (1,2)-system such that(14)Da+γCφ2σt=gtφ1σ′t+f1σt,φ1σt,φ2σt,Da+γCφ1σt=gtφ2σ′t+f2σt,φ1σt,φ2σt,(iii) the (2,1)-system such that(15)Da+γCφ2σt=gtφ2σ′t+f1σt,φ1σt,φ2σt,Da+γCφ1σt=gtφ1σ′t+f2σt,φ1σt,φ2σt,(iv) the (2,2)-system such that(16)Da+γCφ1σt=gtφ2σ′t+f1σt,φ1σt,φ2σt,Da+γCφ2σt=gtφ1σ′t+f2σt,φ1σt,φ2σt,subject to initial conditions(17)φ1σa=α1σ,φ1σ′a=β1σ,φ2σa=α2σ,φ2σ′a=β2σ.Theorem 16 ([33]).
Letn,m∈{1,2} and let [φt]σ=[φ1σt,φ2σt] be an (n,m)-solution of FFIVPs (10) and (11) on a,b. Then, φ1σt and φ2σt will be a solution to the associated (n,m)-system.Theorem 17 ([33]).
Letn,m∈{1,2} and let φ1σt and φ2σt be the solution of (n,m)-system for each σ∈0,1. If [φt]σ=[φ1σt,φ2σt] has valid level sets and φ(t) is Caputo [n,m-γ]-differentiable, then φt is an (n,m)-solution of FFIVPs (10) and (11) on a,b.The aim of the next algorithm is to perform a strategy to solve the FFIVPs (10) and (11) in terms of its σ-cut representation form. Indeed, there are four cases that depend on type of differentiability.Algorithm 18.
To determine the solutions of FFIVPs (10) and (11), do the following:
Case  (I). If φt is Caputo [(1,1)-γ]-differentiable and the FFIVPs (10) and (11) will be converted to crisp system described in (13) and (17), then do the following steps:Step 1: Solve the required system.Step 2: Ensure that φ1σt,φ2σt,φ1σ′t,φ2σ′t and φ1σ′′t,φ2σ′′t are valid level sets for each σ∈0,1.Step 3: Construct (1,1)-solution φt whose σ-cut representation is φ1σt,φ2σt.
Case  (II). If φt is Caputo [(1,2)-γ]-differentiable and the FFIVPs (10) and (11) will be converted to crisp system described in (14) and (17), then do the following steps:Step 1: Solve the required system.Step 2: Ensure that φ1σt,φ2σt,φ1σ′t,φ2σ′t and φ2σ′′t,φ1σ′′t are valid level sets for each σ∈0,1.Step 3: Construct (1,2)-solution φt whose σ-cut representation is φ1σt,φ2σt.
Case  (III). If φt is Caputo [(2,1)-γ]-differentiable and the FFIVPs (10) and (11) will be converted to crisp system described in (15) and (17), then do the following steps:Step 1: Solve the required system.Step 2: Ensure that φ1σt,φ2σt,φ2σ′t,φ1σ′t and φ2σ′′t,φ1σ′′t are valid level sets for each σ∈0,1.Step 3: Construct (2,1)-solution φt whose σ-cut representation is φ1σt,φ2σt.
Case  (IV). If φt is Caputo [(2,2)-γ]-differentiable and the FFIVPs (10) and (11) will be converted to crisp system described in (16) and (17), then do the following steps:Step 1: Solve the required system.Step 2: Ensure that φ1σt,φ2σt,φ2σ′t,φ1σ′t and φ1σ′′t,φ2σ′′t are valid level sets for each σ∈0,1.Step 3: Construct (2,2)-solution φt whose σ-cut representation is φ1σt,φ2σt.


## 4. Description of Fractional RPS Method

In this section, the RPS scheme is presented for constructing an analytical solution of FFIVPs (10) and (11) through substituting the expansion of fractional power series (FPS) among the truncated residual functions. In view of that, the resultant equation helps us to derive a recursion formula for the coefficients’ computation, where the coefficients can be computed recursively through the recurrent fractional differentiating of the truncated residual function.Definition 19 ([35]).
A fractional power series (FPS) representation att0 has the following form:(18)∑n=0∞cnt-t0nγ=c0+c1t-t0γ+c2t-t02γ+…,where 0≤m-1<γ≤m, t≥t0, and cn’s are the coefficients of the series.Theorem 20 ([35]).
Suppose thatf has the following FPS representation at t0:(19)ft=∑n=0∞cnt-t0nγ,where f(t)∈C[t0,t0+R) and DnγCft∈C(t0,t0+R) for n=0,1,2,…; then the coefficients cn will be in the form cn=DnγCft0/Γ1+nγ such that DnγC=DγC·DγC·…·DγC (n-times).Conveniently, for obtaining(n,m)-solution of FFIVPs (10) and (11) utilizing the solution of the corresponding (n,m)-system, we will explain the fashion to determine (1,1)-solution equivalent to the solution for the system of OFDEs (13) and (17). Further, same manner can be applied to construct other type of (n,m)-solutions. To achieve our goal, assume that the solution of OFDEs (13) and (17) at t0=0 has the following form:(20)φ1σt=∑n=0∞cntnγΓ1+nγ,φ2σt=∑n=0∞dntnγΓ1+nγ.Sinceφ1σt and φ2σt satisfy the initial conditions in (17), then the following polynomials φ1σt=α1σ+β1σt and φ2σt=α2σ+β2σt will be the initial guesses for the system and the solutions can also be represented by(21)φ1σt=α1σ+β1σt+∑n=1∞cntnγΓ1+nγ,φ2σt=α2σ+β2σt+∑n=1∞dntnγΓ1+nγ.Consequently, thekth-truncated series solutions can be given by(22)φk,1σt=α1σ+β1σt+∑n=1kcntnγΓ1+nγ,φk,2σt=α2σ+β2σt+∑n=1kdntnγΓ1+nγ.The residual functionsRes1σ(t) and Res2σ(t) are defined as follows:(23)Res1σt=D0+γCφ1σt-gtφ1σ′t-f1σt,φ1σt,φ2σt,Res2σt=D0+γCφ2σt-gtφ2σ′t-f2σt,φ1σt,φ2σt,and the kth-residual functions Resk,1σ(t) and Resk,2σ(t) for k=1,2,3,…n are defined as follows:(24)Resk,1σt=D0+γCφk,1σt-gtφk,1σ′t-f1σt,φk,1σt,φk,2σt,Resk,2σt=D0+γCφk,2σt-gtφk,2σ′t-f2σt,φk,1σt,φk,2σt.From (23), we have Resnσt=0 and limk→∞⁡Resk,nσ(t)=Resnσ≡0 for n=1,2 and each t≥0, which leads to DtmγCResnσ(t)=0. Also, the fractional derivatives DtmγCResnσ(t) and DtmγCResk,nσ(t) are equivalent at t=0 for each m=0,1,2,…,k, that is, DtmγCResnσ(0)=DtmγCResk,nσ(0)=0. However, Dtk-1γCResk,nσ0=0 holds for n=1,2.Regarding employing the RPS algorithm to obtain the1st unknown coefficients, c1and d1, substitute the 1st approximations φ1,1σt=α1σ+β1σt+c1(tγ/Γ1+γ) and φ1,2σt=α2σ+β2σt+d1(tγ/Γ1+γ) into the 1st residual functions Res1,1σ(t) and Res1,2σ(t) of (24) such that(25)Res1,1σt=D0+γCφ1,1σt-gtφ1,1σ′t-f1σt,φ1,1σt,φ1,2σt,Res1,2σt=D0+γCφ1,2σt-gtφ1,2σ′t-f2σt,φ1,1σt,φ1,2σt,and based upon the facts Res1,1σ0=Res1,2σ0=0, we have c1=g0φ1,1σ′0-f1σ0,α1σ,α2σ and d1=g0φ1,2σ′0-f2σ0,α1σ,α2σ. Therefore, the 1st RPS approximate solutions can be written as(26)φ1,1σt=α1σ+β1σt+g0φ1,1σ′0-f1σ0,α1σ,α2σtγΓ1+γ,φ1,2σt=α2σ+β2σt+g0φ1,2σ′0-f2σ0,α1σ,α2σtγΓ1+γ.Currently, for the2nd unknown coefficients, c2 and d2 substitute φ2,1σt=α1σ+β1σt+∑n=12cn(tnγ/Γ1+nγ) and φ2,2σt=α2σ+β2σt+∑n=12dn(tnγ/Γ1+nγ) into the 2nd residual functions, Res2,1σt and Res2,2σt of (24) such that(27)Res2,1σt=D0+γCφ2,1σt-gtφ2,1σ′t-f1σt,φ2,1σt,φ2,2σt,Res2,2σt=D0+γCφ2,2σt-gtφ2,2σ′t-f2σt,φ2,1σt,φ2,2σt.Then, by applying the fractional derivativeDtγC on both sides of Res2,1σ(t) and Res2,2σt, using the facts DtγCRes2,1σ0=DtγCRes2,2σ0=0 as well, the values of c2 and d2 will be given by(28)c2=Γ2γΓγΓγβ1σDtγCgtt=0+c1DtγCgt.tγ-1t=0+f1σ0,c1,d1Γ2γ-DtγCgt.t2γ-1t=0,d2=Γ2γΓγΓγβ2σDtγCgtt=0+d1DtγCgt.tγ-1t=0+f2σ0,c1,d1Γ2γ-DtγCgt.t2γ-1t=0.For the3rd unknown coefficients, c3 and d3 substitute φ3,1σt=α1σ+β1σt+∑n=13cn(tnγ/Γ1+nγ) and φ3,2σt=α2σ+β2σt+∑n=13dn(tnγ/Γ1+nγ) into the 3rd residual functions, Res3,1σt and Res3,2σt of (24), and then by computing Dt2γCRes3,1σ(t) and Dt2γCRes3,2σ(t) and using the facts Dt2γCRes3,1σ0=Dt2γCRes3,2σ0=0, the coefficients, c3 and d3, will be given such that(29)c3=Γ3γΓγΓ2γΓγΓ2γβ1σDt2γCgtt=0+c1Dt2γCgt.tγ-1t=0+c2Dt2γCgt.t2γ-1t=0+f1σ0,c2,d2Γ3γ-Dt2γCgt.t3γ-1t=0,d3=Γ3γΓγΓ2γΓγΓ2γβ2σDt2γCgtt=0+d1Dt2γCgt.tγ-1t=0+d2Dt2γCgt.t2γ-1t=0+f2σ0,c2,d2Γ3γ-Dt2γCgt.t3γ-1t=0.Using similar argument, the4th unknown coefficients, c4 and d4, will be given utilizing the facts Dt3γCRes4,1σ0=Dt3γCRes3,1σ0=0. The same manner can be repeated until we obtain on the coefficients’ arbitrary order of the FPS solution for the OFDE (13).


## 5. Numerical Simulation and Discussion

This section aims to verify the efficiency and applicability of the proposed algorithm by applying the RPS method to a numerical example. Here, all necessary calculations and analysis are done using Mathematica 10.For this purpose, let us consider the fuzzy fractional differential equation(30)D0+γCφt=μ,0≤t≤1,with the fuzzy initial conditions(31)φ0=α,φ′0=β,where γ∈1,2 and μ,α,β are the fuzzy numbers whose σ-cut representation is [σ-1,1-σ].Based on the type of differentiability, the FFIVPs (30) and (31) can be converted into one of the following systems.Case 1.
Ifφt is (1,1)-solution, then the corresponding (1,1)-system will be(32)D0+γCφ1σt=σ-1,D0+γCφ2σt=1-σ,φ1σ0=φ1σ′0=σ-1,φ2σ0=φ2σ′0=1-σ.If γ=2, then the exact solution of (32) is [φt]σ=σ-1,1-σ1+t+tγ/Γ(γ+1), t∈0,1. In finding the fuzzy (1,1)-solution of FFDEs (30), let φ(t) be Caputo [(1,1)-γ]-differentiable. Sequentially, after selecting the initial guesses as φ0,1σt=σ-1+σ-1t and φ0,2σt=1-σ+1-σt, the FPS expansion of solutions for OFDEs (32) can be represented as follows:(33)φ1σt=σ-1+σ-1t+∑n=1∞cntnγΓ1+nγ,φ2σt=1-σ+1-σt+∑n=1∞dntnγΓ1+nγ.
To determine the1st RPS approximate solution for OFDEs (32), substitute the 1st-truncated series φ1,1σt=σ-1+σ-1t+c1(tγ/Γ1+γ) and φ1,2σt=1-σ+1-σt+d1(tγ/Γ1+γ) into the 1st-residual functions Res1,1σt and Res1,2σt such that Res1,1σt=1-σ+c1 and Res1,2σt=σ-1+d1. Thus, based upon the facts Res1,1σ0=0 and Res1,2σ0=0, we have c1=σ-1 and d1=1-σ. Hence, the 1st RPS approximate solution for OFDEs (32) can be written in the form of (34)φ1,1σt=σ-1+σ-1t+σ-1tγΓ1+γ,φ1,2σt=1-σ+1-σt+1-σtγΓ1+γ.
Similarly, to find out the2nd RPS approximate solution for OFDEs (32), substitute the 2nd truncated series φ2,1σt=σ-1+σ-1t+c1(tγ/Γ1+γ)+c2(t2γ/Γ1+2γ) and φ2,2σt=1-σ+1-σt+d1(tγ/Γ1+γ)+d2(t2γ/Γ1+2γ) into the 2nd residual functions Res2,1σt and Res2,2σt such that Res2,1σt=(CD0+γφ2,1σ)t-σ-1=1-σ+c1+c2(tγ/Γ1+γ) and Res2,2σt=(CD0+γφ2,2σ)t-(1-σ)=-1+σ+d1+d2(tγ/Γ1+γ). Now, applying the fractional derivative DtγC on both sides of Res2,1σ(t) and Res2,2σt yields the following: DtγCRes2,1σt=c2 and DtγCRes2,2σt=d2. So, the 2nd unknown coefficients are c2=0 and d2=0 through using the facts DtγCRes2,1σ0=DtγCRes2,2σ0=0. Therefore, the 2nd RPS approximate solution for OFDEs (32) is given by(35)φ2,1σt=σ-1+σ-1t+σ-1tγΓ1+γ,φ2,2σt=1-σ+1-σt+1-σtγΓ1+γ.
Accordingly, the unknown coefficientscn and dn will be vanished for n≥3 by continuing in the similar approach, that is, ∑n=3∞cntnγ/Γ1+nγ=0 and ∑n=3∞dn(tnγ/Γ1+nγ)=0. Hence, the RPS approximate solutions corresponding to (1,1)-system are coinciding well with the exact solutions φ1σt=1+t+(tγ/Γ(1+γ))σ-1 and φ2σt=1+t+tγ/Γ(1+γ)1-σ. Here, φ1σt,φ2σt, φ1σ′t,φ2σ′t, and φ1σ′′t,φ2σ′′t are valid level sets for σ∈0,1 and t∈[0,1]. Moreover,φt=μ1+t+tγ/Γ(1+γ) is a (1,1)-solution for FFIVPs (30) and (31) on [0,1].Case 2.
Ifφt is (1,2)-solution, then the corresponding (1,2)-system will be(36)D0+γCφ1σt=1-σ,D0+γCφ2σt=σ-1,φ1σ0=φ1σ′0=σ-1,φ2σ0=φ2σ′0=1-σ,If γ=2, then the exact solution of (36) is [φt]σ=σ-1,1-σ1+t-tγ/Γ(γ+1), t∈0,1. In finding the fuzzy (1,2)-solution of FFDEs (30), let φt be Caputo [(1,2)-γ]-differentiable. Sequentially, after selecting the initial guesses as in case 1, the FPS expansion of solutions for OFDEs (36) can be represented by(37)φ1σt=σ-1+σ-1t+∑n=1∞cntnγΓ1+nγ,φ2σt=1-σ+1-σt+∑n=1∞dntnγΓ1+nγ.
To determine the1st RPS approximate solution for OFDEs (36), substitute the 1st truncated series φ1,1σt=σ-1+σ-1t+c1(tγ/Γ1+γ) and φ1,2σt=1-σ+1-σt+d1(tγ/Γ1+γ) into the 1st residual functions Res1,1σt and Res1,2σt such that Res1,1σt=σ-1+c1 and Res1,2σt=1-σ+d1. Thus, based upon the facts Res1,1σ0=Res1,2σ0=0, we have c1=1-σ and d1=σ-1. Hence, the 1st RPS approximate solution for OFDEs (36) can be written in the form of (38)φ1,1σt=σ-1+σ-1t+1-σtγΓ1+γ,φ1,2σt=1-σ+1-σt+σ-1tγΓ1+γ.
Similarly, to find out the2nd RPS approximate solution for OFDEs (36), substitute the 2nd truncated series φ2,1σt=σ-1+σ-1t+c1(tγ/Γ1+γ)+c2(t2γ/Γ1+2γ) and φ2,2σt=1-σ+1-σt+d1(tγ/Γ1+γ)+d2(t2γ/Γ1+2γ) into the 2nd residual functions Res2,1σt and Res2,2σt such that Res2,1σt=1-σ+c1+c2(tγ/Γ1+γ) and Res2,2σt=-1+σ+d1+d2(tγ/Γ1+γ). Then, applying the fractional derivative DtγC on both sides of Res2,1σ(t) and Res2,2σtyields the following: DtγCRes2,1σt=c2 and DtγCRes2,2σt=d2. So, the 2nd unknown coefficients are c2=0 and d2=0 through using the facts DtγCRes2,1σ0=DtγCRes2,2σ0=0. Therefore, the 2nd RPS approximate solution for OFDEs (36) is given by(39)φ2,1σt=σ-1+σ-1t+1-σtγΓ1+γ,φ2,2σt=1-σ+1-σt+σ-1tγΓ1+γ.
By continuing in the similar manner, the unknown coefficientscn and dn will be vanished for n≥3, that is, ∑n=3∞cn(tnγ/Γ1+nγ)=0 and ∑n=3∞dn(tnγ/Γ1+nγ)=0. Hence, the RPS approximate solutions corresponding to (1,2)-system are coinciding well with the exact solutions φ1σt=1+t-tγ/Γ(1+γ)σ-1 and φ2σt=1+t-tγ/Γ(1+γ)1-σ. Here, φ1σt,φ2σt, φ1σ′t,φ2σ′t, and φ1σ′′t,φ2σ′′t are valid level sets for σ∈0,1 and t∈[0,1]. On the other hand, φt=μ1+t-tγ/Γ(1+γ) is a (1,2)-solution for FFIVPs (30) and (31) on [0,1].Case 3.
Ifφt is (2,1)-solution, then the corresponding (2,1)-system will be(40)D0+γCφ1σt=1-σ,D0+γCφ2σt=σ-1,φ1σ0=φ2σ′0=σ-1,φ2σ0=φ1σ′0=1-σ,If γ=2, then the exact solution of (40) is [φt]σ=σ-1,1-σ1-t-tγ/Γ(1+γ), t∈0,3-1. To obtain the fuzzy (2,1)-solution of FFDEs (30), let φ(t) is Caputo [(2,1)-γ]-differentiable. By using the same manner in previous cases, the solutions for (2,1)-system can be obtained such as φ1σt=1-t-tγ/Γ(1+γ)σ-1 and φ2σt=1-t-tγ/Γ(1+γ)1-σ. It is easy to check that φ1σt,φ2σt,φ2σ′t,φ1σ′t and φ2σ′′t,φ1σ′′t are also valid level sets for σ∈0,1 and t∈[0,3-1]. Thus, φt=μ1-t-tγ/Γ(1+γ) is a (2,1)-solution for FFIVPs (30) and (31) on (0,3-1].Case 4.
Ifφt is (2,2)-solution, then the corresponding (2,2)-system will be(41)D0+γCφ1σt=σ-1,D0+γCφ2σt=1-σ,φ1σ0=φ2σ′0=σ-1,φ2σ0=φ1σ′0=1-σ,If γ=2, then the exact solution of OFDEs (41) is [φt]σ=σ-1,1-σ1-t+tγ/Γ(γ+1), t∈0,1. Finally, to determine the fuzzy (2,2)-solution of FFDEs (30), let φ(t) be Caputo [(2,2)-γ]-differentiable. By using the same manner in previous cases, the solutions for (2,2)-system can be obtained such as φ1σt=σ-11-t+tγ/Γ(1+γ) and φ2σt=1-σ1-t+tγ/Γ(1+γ). Here, φ1σt,φ2σt,φ2σ′t,φ1σ′t and φ1σ′′t,φ2σ′′t are also valid level sets for σ∈0,1 and t∈[0,1]. However, φt=μ1-t+tγ/Γ(γ+1) defines as a (2,2)-solution for FFIVPs (30) and (31) on (0,1].
To demonstrate the agreement between the exact and approximate solution, Table1 shows the absolute error of the 10th PRS approximate solution for FFIVPs (30) and (31) obtained for different values of σ-cut representations and nodes with fractional order γ=1.9. Some graphical results are also presented in Figures 1 and 2. The numerical results obtained indicate that the RPS approximate solutions are in good agreement with each other and with the exact solutions for all cases of differentiability.Table 1
The absolute error of10th approximation of FFIVPs (30) and (31).
( n , m )-solution t σ = 0 σ = 0.2 5 σ = 0.5 σ = 0.75 1,1-system 0.2 2.0578567 × 10 - 4 1.5433927 × 10 - 4 1.0289283 × 10 - 4 5.1446419 × 10 - 5 0.4 3.8465419 × 10 - 4 2.8849064 × 10 - 4 1.9232709 × 10 - 4 9.6163547 × 10 - 5 0.6 5.5428820 × 10 - 4 4.1571615 × 10 - 4 2.7714410 × 10 - 4 1.3857205 × 10 - 4 0.8 7.1821803 × 10 - 4 5.3866352 × 10 - 4 3.5910901 × 10 - 4 1.7955450 × 10 - 4  ( 1,2 ) -system 0.2 2.9676224 × 10 - 4 2.2257168 × 10 - 4 1.4838112 × 10 - 4 5.1446419 × 10 - 5 0.4 4.7033136 × 10 - 4 3.5274852 × 10 - 4 2.3516568 × 10 - 4 9.6163547 × 10 - 5 0.6 6.3684312 × 10 - 4 4.7763234 × 10 - 4 2.7714410 × 10 - 4 1.3857205 × 10 - 4 0.8 7.9857642 × 10 - 4 5.3866352 × 10 - 4 3.5910901 × 10 - 4 1.7955450 × 10 - 4  ( 2,1 ) -system 0.2 2.0578567 × 10 - 4 1.5433925 × 10 - 4 1.0289283 × 10 - 4 5.1446419 × 10 - 5 0.4 3.8465419 × 10 - 4 2.8849064 × 10 - 4 1.9232709 × 10 - 4 9.6163547 × 10 - 5 0.6 5.5428820 × 10 - 4 4.1571615 × 10 - 4 2.7714410 × 10 - 4 1.3857205 × 10 - 4 0.8 6.3684312 × 10 - 4 5.3746353 × 10 - 4 3.5910901 × 10 - 4 1.7955450 × 10 - 4  ( 2,2 ) -system 0.2 2.0578567 × 10 - 4 1.5433925 × 10 - 4 1.0289283 × 10 - 4 5.1446419 × 10 - 5 0.4 3.8465419 × 10 - 4 2.8849064 × 10 - 4 1.9232709 × 10 - 4 9.6163547 × 10 - 5 0.6 5.5428820 × 10 - 4 4.1571615 × 10 - 4 2.7714410 × 10 - 4 1.3857205 × 10 - 4 0.8 7.1821803 × 10 - 4 5.3866352 × 10 - 4 3.5910901 × 10 - 4 1.7955450 × 10 - 4Figure 1
Plots ofα-cut representations of φk,1σt,φk,2σt with k=10, σ=0.25, and different values of γ∈{2,1.9,1.8,1.7} (--- Exact, ⋯ RPS-approximation).
(a)
RPS solution of1,1-system (b)
RPS solution of1,2-system (c)
RPS solution of2,1-system (d)
RPS solution of2,2-systemFigure 2
Plots of exact and RPS-approximation atγ=2 with different values of σ-levels, σ∈{0,0.25,0.5,0.75,1} (--- Exact, ⋯ RPS-approximation).
(a)
σ-levels of 1,1-system (b)
σ-levels of 1,2-system (c)
σ-levels of 2,1-system (d)
σ-levels of 2,2-system


## 6. Conclusion

In this paper, the RPS algorithm is successfully developed, investigated, and applied to solve the fuzzy differential equation of fractional order1<γ≤2 with fuzzy initial constraints under the fuzzy concept of Caputo H-differentiability. The fuzziness is represented using upper semicontinuous membership function of bounded support, convex, and normalized fuzzy numbers based on its single parametric form. The behavior of approximate solution for different values of fractional order γ is discussed quantitatively as well as graphically. The numerical results in this paper demonstrate the efficiency of the algorithm. We conclude that the proposed scheme is highly accurate in solving widely array of fuzzy fractional issues.



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