Granural differentiability of fuzzy valued functions and applications

HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS DUONG THU HOAN GRANULAR DIFFERENTIABLITY OF FUZZY-VALUED FUNCTIONS AND APPLICATIONS BACHELOR THESIS Speciality: Analysis Hanoi, 2019 HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS DUONG THU HOAN GRANULAR DIFFERENTIABLITY OF FUZZY-VALUED FUNCTIONS AND APPLICATIONS BACHELOR THESIS Speciality: Analysis Supervisor’s Supervisor’s Khuat Van Ninh Nguyen Phuong Dong Hanoi, 2019 Confirmation The thesis was written on the basis of my study under the guidance of Assoc Prof Khuat Van Ninh ,Nguyen Phuong Dong and my effort I have studied and presented the results from bibliographies The thesis does not coincide others The author Duong Thu Hoan Acknowledgment On this occation, firstly, I would like to thank all people who helped me in my study and preparation of this thesis I emphasize to thank Hanoi Pedagogical University where I finished this thesis with the teaching of lectures Especially, I would like to express my profound gratitude to my supervisor, Assoc Prof Khuat Van Ninh and Nguyen Phuong Dong who helped me carefully in the processing of researching and writing of this thesis, for their valuable instructive comments and their illuminating advices as well My sincere thanks are also sent to my teachers in the Department of Mathematics who educated me over around four years I take this opportunity to thank all my friends who always help and encourage me I also give special thanks and deep gratitude towards my family for their vital support and encouragement Finally, this thesis may have some errors, anyway I am very pleased and would like to receive constructive comments and suggestions to improve the quality of this thesis Ha Noi, May 2019 Duong Thu Hoan Contents Introduction 1 Preliminaries 1.1 The space of fuzzy numbers 1.2 Characterization of fuzzy numbers 1.2.1 Some types of fuzzy numbers 1.2.2 Zadeh’s Extension Principle 1.2.3 The Sum and Scalar Multiplication 1.2.4 The product of fuzzy numbers 1.2.5 The difference of fuzzy numbers 9 12 13 1.3 Fuzzy derivative and integral 14 Granular differentiability of fuzzy-valued functions 15 2.1 Granular representation 15 2.2 Granular operations 16 2.3 The granular metric space (E, Dgr ) 16 2.4 Granular derivative and intergral 21 Applications to fuzzy differential equations 27 3.1 Theoretical results 27 3.2 Numerical examples 28 Conclusions References 32 32 Introduction Rationale In many real world problems, there is often a need to interpret and solve the problems operating in the environment inherent uncertainties and vagueness When engineers want to handle these disadvantages, they may use either stochastic and statistical models or fuzzy models, but stochastic and statistical uncertainty occur due to the natural randomness in the process It is generally expressed by a probability density or frequency distribution function For the estimation of the distribution, it requires sufficient information about the variables and parameters involved in it On the other hand, fuzzy set theory refers to the uncertainty when we may have lack of knowledge or incomplete information about the variables and parameters In general, science and engineering systems are governed by ordinary and partial differential equations, but the type of differential equation (DEs) depends upon the applications, domains, complicated environments, the effect of coupling, and so on As such, the complicacy needs to be handled by recently developed differential equations contained uncertainty or fuzziness Since the first time introduced in 1965 by Zadeh, many extensive research have been studied on the applications of the fuzzy sets in various fields of sciences, e.g in control theory, in medicine and so forth In the recent years, the application of the fuzzy sets in differential equations has captured much attention The differential equations in which parameters and/or conditions are uncertain, and this uncertainty is expressed by a class of fuzzy sets - usually fuzzy numbers - are called fuzzy differential equations (FDEs) Since without any definition of the derivative, differential equations make no sense, in the recent years, several typical definitions of a derivative of fuzzy-valued functions have been proposed such as Hukuhara derivative (H-derivative), generalized Hukuhara derivative (gH-derivative), generalized derivative (g-derivative) and Granular derivative (gr-derivative) Among the mentioned definitions the gr-derivative is more effective and practical than the others in the viewpoint of computation and engineering The point about the horizontal membership function approach (or the use of granular derivative) is that it not only circumvents shortcomings associated to the previously mentioned approaches, but also inherits some of their benefits The most essentially important merits of gr-derivative and multidimensional fuzzy arithmetic based on relative-distance-measure in fuzzy dirrerential equations studies are outlined below: • Obtaining fuzzy function derivative and/or solving FDEs is simple; • This approach does not compel that solution support closure length of FDE be necessarily monotonic; • Solving each FDE is equivalent to solving just one individual differential equation called granular differential equation That is, it avoids the doubling property disadvantage; • An FDE has only one solution on condition that its equivalent granular differential equation has a solution That is, it avoids the multiplicity of solutions drawback; • This approach does not result in unnatural behavior in modeling phenomenon Motivated by aforesaid, in this graduation thesis, we pay more attention in studying this novel concept of differentiability of fuzzy-valued functions and extending a new concept of integrability, named granular integrability (gr-integrability), that play vital roles to investigate the class of fuzzy differential equations under granular differentiability Aim of the study Study the granular differentiability and integrability of fuzzy-valued function Task of the study • Study horizontal membership function representation of fuzzy numbers and fuzzyvalued functions; • Study the granular differentiability - integrability of fuzzy-valued function and compare with some other previous concepts; • Apply to solve some classes of fuzzy differential equations and then simulate their solutions The object and scope of the study 4.1 The object of the study: fuzzy-valued functions and fuzzy differential equations 4.2 The scope of the study: The granular differentiability and integrability Research method In the thesis, we use some techniques of set-valued functional analysis combined with fuzzy analysis to study the granular differentiability and integrability of fuzzyvalued functions Overview of the study The graduation thesis consists of chapters Chapter 1: Preliminaries, Chapter 2: Granular differentiability of fuzzy-valued functions, Chapter 3: Applications to fuzzy differential equations, Chapter 4: Conclusions The thesis is written on the basis of the paper: “Granular Differentiability of Fuzzy-Numbers Valued Functions”, IEEE Tran Fuzzy Syst., 26(1)(2018), 310-323 of M Mazandarani, N Pariz and A.V Kamyad and the manuscript “The solvability of Cauchy problem to fuzzy fractional evolution equations under Caputo generalized Hukuhara-differentiability, J Science, HPU2, (2019)” of N.P Dong and D.T Hoan Chapter Preliminaries 1.1 The space of fuzzy numbers A fuzzy number is a generalization of regular real number in the sense that it does not refer to one single value but rather to a contected set of possible values, where each possible value has its own weight between and The fuzzy number concept is fundamental for fuzzy analysis and fuzzy differential equations and a very useful tool in several applications of fuzzy sets and fuzzy logic Definition 1.1.1 [1] Consider a fuzzy set u: R → [0, 1] of the real line u: R → [0, 1] Then u is said to be a fuzzy number if it satisfies following properties: (i) u is normal, i.e., ∃x0 ∈ R such that u(x0 ) = 1; (ii) u is fuzzy convex, i.e., u(tx + (1 − t)y) ≥ min{u(x), (y)}, for all t ∈ [0, 1], x, y ∈ R; (iii) u is upper semicontinuous on R, i.e., for all ϵ > 0, ∃δ > such that |x − x0 | < δ u(x) − u(x0 ) < ϵ; (iv) u is compactly supported, i.e., cl{x ∈ R; u(x) > 0} is compact, where cl(A) denotes the closure of the set A Let us denote by E the space of fuzzy numbers on the real line Example 1.1.1 The fuzzy set u: R → [0, 1],      x3 u(x) = (2 − x)3    given by if x 0} Then, [u]α is called the α-level sets of the fuzzy number u The 1-level set is called the core of the fuzzy number, while the 0-level set is called the support of the fuzzy number Theorem 1.2.1 (Stacking theorem, [1]) Let u ∈ E be a fuzzy number whose α - level sets is given by [u]α for all α ∈ [0, 1] Then + − + (i) [u]α is a closed interval of the form [u]α = [u− α , uα ], where uα = inf uα , uα = sup uα for each α ∈ [0, 1]; (ii) If ≤ α1 ≤ α2 ≤ then uα2 ⊆ uα1 ; (iii) For any sequence {αn } which converges from below to r ∈ [0, 1], we have ∞ ∩ uαn = uα ; n=1 Figure 2.3: The gr-distance between two fuzzy numbers u = (0, 1, 2) and v = (3, 6, 7, 9) Example 2.3.1 a Let u = (0, 1, 2) and v = (3, 6, 7, 9) be triangular and trapezoidal fuzzy numbers, respectively (see Figure 2.3) Then, from the formula (2.2), it follows that the distance between two fuzzy numbers u = (0, 1, 2) and v = (3, 6, 7, 9) is given by Dgr (u, v) = sup max |ugr (α, αu ) − v gr (α, αv )| α αu ,αv = sup max |3 + 2α + (6 − 5α)αv − (2 − 2α)αu | α αu ,αv = sup |9 − 3α| = α where ugr (α, αu ) = α + (2 − 2α)αu , v gr (α, αv ) = + 3α + (6 − 5α)αv for α, αu , αv ∈ [0, 1] b Let u = (4, 8, 10) and v = (1, 5, 7) be triangular fuzzy numbers with HMFs ugr (α, αu ) = 4(1 + α) + 6(1 − α)αu and v gr (α, αv ) = + 4α + 6(1 − α)αv , for α, αu , αv ∈ [0, 1] Then, as a result of formula (2.2), the distance between u and v is given by Dgr (u, v) = sup max |ugr (α, αu ) − v gr (α, αv )| α αu ,αv = sup max |3 + 6(1 − α)(αu − αv )| α αu ,αv = sup |9 − 6α| = α c Let u = (1, 2, 4) and v = (0, 1, 4, 5) be triangular and trapezoidal fuzzy numbers with respective HMFs ugr (α, αu ) = + α + 3(1 − α)αu and v gr (α, αv ) = α + (5 − 2α)αv , for α, αu , αv ∈ [0, 1] (see Figure 2.4) Then, the gr-distance of u and v can be computed as follows Dgr (u, v) = sup max |ugr (α, αu ) − v gr (α, αv )| α αu ,αv = sup max |1 + 3(1 − α)αu − (5 − 2α)αv )| α αu ,αv 19 Figure 2.4: The gr-distance between two fuzzy numbers u = (1, 2, 4) and v = (0, 1, 4, 5) = sup |4 − 3α| = α Definition 2.3.2 Let f : [a, b] ⊂ R → E and t0 ∈ [a, b] Then, we say that the function f has the finite limit as t tends to t0 if and only if there exists τ ∈ E such that lim Dgr (f (t), τ ) = 0, that is ∀ ϵ > 0, ∃ δ(t0 , ϵ) > such that ∀t ∈ [a, b] : < |t−t0 | < δ, t→t0 it implies Dgr (f (t), τ ) < ϵ Definition 2.3.3 (The continuity) A fuzzy-valued function f : (a, b) ⊂ R → E is said to be continuous on (a, b) if for each t0 ∈ (a, b), for all ϵ > 0, there exists δ > such that ∀t ∈ (a, b) : |t − t0 | < δ, Dgr (f (t), f (t0 )) < ϵ, i.e., lim Dgr (f (t), f (t0 )) = t→t0 Lemma 2.3.2 Let u, v, e ∈ E and λ ∈ R Then, the following statements hold (i) Dgr (u + e, v + e) = Dgr (u, v) (ii) Dgr (λu, λv) = |λ|Dgr (u, v) Proof (i) By the formula (2.2), we have Dgr (u + e, v + e) = sup max |[ugr (α, αu ) + egr (α, αe )] − [v gr (α, αv ) + egr (α, αe )]| α αu ,αv ,αe = sup max |ugr (α, αu ) − v gr (α, αv )| α αu ,αv gr = D (u, v) (ii) Similarly, we also have Dgr (λu, λv) = sup max |λugr (α, αu ) − λv gr (α, αv )| α αu ,αv = |λ| sup max |ugr (α, αu ) − v gr (α, αv )| α αu ,αv gr = |λ|D (u, v) Hence, the proof is complete 20 2.4 Granular derivative and intergral Definition 2.4.1 [4] The fuzzy-valued function f : [a, b] ⊆ R → E is said to be granular differentiable (gr-differentiable) at a point t0 ∈ [a, b] if there exists an element dgr f (t0 ) ∈ E such that the following limit dt dgr f (t0 ) f (t0 + h) ⊖gr f (t0 ) = , h→0 h dt lim dgr f (t0 ) is the granular derivadt tive (gr-derivative) of fuzzy-valued function f at the point t0 We say that f is grdgr f (t0 ) differentiable on [a, b] if the gr-derivative exists at each point t0 ∈ [a, b] The dt gr d f : [a, b] → E is then called gr-derivative of f on [a, b] fuzzy-valued function dt Denote C ([a, b], E) by the space of all continuously gr-differentiable fuzzyvalued functions on [a, b] exists for h sufficiently near In this case, we call Proposition 2.4.1 A necessary and sufficient condition for function f : [a, b] ⊂ R → E is gr-differentiable at a point t0 ∈ [a, b] is the differentiability of its HMF with respect ( gr ) ∂f gr (t , α, α ) f f (t0 ) to t at that point Moreover, we have H d dt = ∂t Proof Under the assumption that the fuzzy number function f is gr-differentiable at the point t0 ∈ (a, b), we obtain that for all ϵ > 0, there exists δ > such that for all h > satisfying < h < δ then ( ) f (t0 + h) ⊖gr f (t0 ) dgr f (t0 ) gr D , < ϵ h dt Next, by using the definition of granular metric, the above inequality becomes ( gr )gr gr d f gr (f (t0 + h, α, αf ) − f (t0 , α, αf )) − (t0 , α, αf ′ ) < ϵ, sup max α ,α dt f f′ h α for all αf , αf ′ , α ∈ [0, 1], that is equivalent to gr (f (t0 + h, α, αf ) − f gr (t0 , α, αf )) − h ( dgr f dt )gr (t0 , α, αf ′ ) < ϵ, for all ϵ > and h ∈ (0, δ) Then, by letting h → 0, we obtain that ( gr )gr f gr (t0 + h, α, αf ) − f gr (t0 , α, αf ) d f lim = (t0 , α, αf ′ ), h→0 h dt for all αf , αf ′ , α ∈ [0, 1], which leads to the differentiability of the horizontal membership function of the function f The converse statement can be obtained by using similar arguments The proof is complete Definition 2.4.2 [4] Suppose that f : [a, b] → E is a continuous fuzzy-valued function such that the HMF H (f (t)) = f gr (t, α, αf ) is integrable on [a, b] If there exists a fuzzy 21 ∫ b number m such that H(m) = H (f (t)) dt Then, m is said to be the granular integral ∫ b of f on [a, b] and denoted by m = f (t)dt a a Theorem 2.4.2 Let f be a continuous fuzzy-valued function defined∫ on [a, b] Then, t for each t ∈ [a, b], the fuzzy-valued function Λ, given by Λ(t) = f (s)ds, is an a anti-derivative of the function f Proof Let t0 ∈ [a, b] be fixed By the assumption that the function f is continuous at t0 , we deduce that ∀ ϵ > 0, ∃ δ > such that ∀t ∈ [a, b] : |t − t0 | < δ then Dgr (f (t), f (t0 )) < ϵ Next, for h sufficiently near 0, we consider following quotient ∆Λ ∆t [Λ(t0 + h) ⊖gr Λ(t0 )] h ] [∫ t0 +h ∫ t0 gr f (s)ds , f (s)ds ⊖ = h a a = t=t0 Using horizontal membership function approach, we obtain ( ] ) [∫ t0 +h ∫ t0 ∆Λ H H(f (s))ds − H(f (s))ds = ∆t t=t0 h a a ] [∫ t0 +h ∫ t0 gr gr = f (s0 + h, α, αf )ds − f (s0 , α, αf )ds h a a ∫ t0 +h gr = f (s0 + h, α, αf )ds h t0 Next, by the use of Mean Value Theorem for integral, we get ) ( ∫ t0 +h gr ∆Λ = f (s0 + h, α, αf )ds = f gr (t0 + τ h, α, αf ), H ∆t t=t0 h t0 where τ ∈ (0, 1) Here, note that t0 + τ h tends to t0 as h → Thus, it implies that ( gr ) ( ) d Λ(t0 ) ∆Λ H = lim H = lim f gr (t0 + τ h, α, αf ) = f gr (t0 , α, αf ), h→0 h→0 dt ∆t t=t0 dgr Λ(t0 ) = H−1 (f gr (t0 , α, αf )) = f (t0 ) Additionally, since t0 ∈ dt [a, b] is chosen arbitrarily, the proof is complete which follows that Corollary 2.4.3 As a consequence of Theorem 2.4.2, we have that ) (∫ t dgr f (s)ds = f (t) dt a Theorem 2.4.4 [4] Asume that fuzzy-valued function F : [a, b] → E is gr-differentiable dgr F (t) is continuous on [a, b] Then, and the function f (t) = dt ∫ b f (t)dt = F (b) ⊖gr F (a) a 22 ∫ Proof As a result of Theorem 2.4.2, function Λ(t) = t f (s)ds is an anti-derivative of a the function f on [a, b] and its granular representation is as ∫ t gr Λ (t, αΛ ) = f gr (s, α, αf )ds a gr This expression means that Λ (t, α, αΛ ) is an anti-derivative of the function f gr (t, α, αf ) on [a, b] Thus, if f gr (t, αf ) is another anti-derivative of f gr (t, α, αf ) on [a, b] then ∫ t gr gr f (t, α, αf ) = Λ (t, α, αΛ ) + C = f gr (s, α, αf )ds + C, (2.3) a where C is a real constant Next, by substituting t = a into the equality (2.3), we obtain that C = f gr (a, α, αf ) Then, the integral equality (2.3) becomes ∫ t gr f (t, α, αf ) = f gr (s, α, αf )ds + f gr (a, α, αf ), ∫ a t f gr (s, α, αf )ds = f gr (t, α, αf ) − f gr (a, α, αf ) or equivalently, a Next, if we let t = b then we immediately get following formula ∫ b f gr (s, α, αf )ds = f gr (b, α, αf ) − f gr (a, α, αf ), a Finally, by using the transformation (2.1), we can see that ∫ b H−1 (f gr (s, α, αf )) ds = H−1 (f gr (b, α, αf ) − f gr (a, α, αf )) , a ∫ b f (t)dt = F (b) ⊖gr F (a) holds that means the integral equality a Example 2.4.1 Let f (t) = e2t u + tv be a fuzzy-valued function with u = (1, 2, 3) and v = (−5, −3, −1) We have f gr (t, α, αu , αv ) = 2(1 + α)e2 t + (2α − 5)t2(1 − α)αu e2 t + 4(1 − α)tαv and its derivative with respect to t is ∂f gr (t, α, αu , αv ) = 2(1 + α)e2 t + (2α − 5) + 4(1 − α)αu e2 t + 4(1 − α)αv , ∂t for each α, αu , αv ∈ [0, 1] Then f is gr-differentiable with its horizontal membership function can be calculated by ( gr ) ∂f (t, α, αu , αv ) H = [2 + 2α, − 2α] e2t + [−5 + 2α, −1 − 2α] ∂t By using α−level sets representation theorem, we have ∪{ } dgr (f (t)) = [2 + 2α, − 2α] e2t + [−5 + 2α, −1 − 2α] dt α = (2, 4, 6)e2t + (−5, −3, −1) 23 Theorem 2.4.5 Let F, G : [a, b] ⊂ R → E be integrable fuzzy-valued functions and λ, γ ∈ R Then, the following statements hold ∫ b ∫ b ∫ b (i) [λF (t) + γG(t)]dt = λ F (t)dt + γ G(t)dt a a a (ii) Dgr (F, G) is integrable ) ∫ b (∫ b ∫ b gr (iii) D F (t)dt, G(t)dt ≤ Dgr (F (t), G(t))dt ∫ (iv) a a ∫ b ∫ c F (t)dt = a b F (t)dt, for each c ∈ (a, b) F (t)dt + a a c Proof (i) From Definition 2.4.2, we have (∫ b ) ∫ b H [f (t) + g(t)] dt = H (f (t) + g(t)) dt a a ∫ b = [H(f (t)) + H(g(t))] dt a ∫ b ∫ b = H(f (t))dt + H(g(t))dt a a (∫ b ) (∫ b ) f (t)dt + H g(t)dt =H ∫ ∫ b Thus, we obtain a ∫ b a a a b g(t)dt f (t)dt + [f (t) + g(t)]dt = a (ii) Similarly, for λ ∈ R, we also have ) ∫ b (∫ b ∫ b ∫ b λf (t)dt = H (λf (t)) dt = λH (f (t)) dt = λ H (f (t)) dt H a ∫ Hence, we obtain a ∫ b λf (t)dt = λ a a a b f (t)dt a (iii) Denote Dgr ([f (t)]α , [g(t)]α ) = max |f gr (t, α, αf ) − g gr (t, α, αg )|, αf ,αg α ∈ [0, 1] Since f and g are integrable fuzzy-valued function on [a, b], we directly imply that real-valued functions f gr (t, α, αf ), g gr (t, α, αg ) are integrable on [a, b], and then, f gr (t, α, αf ) − g gr (t, α, αg ) is also integrable function on [a, b] for all α, αf , αg ∈ [0, 1] Let α ∈ [0, 1] be fixed Then, we have Dgr ([f (t)]α , [g(t)]α ) is measurable on [a, b] From the formula (2.2), we have Dgr (f (t), g(t) = sup Dgr ([f (t)]α , [g(t)]α ), α α ∈ [0, 1] Furthermore, we have Dgr (f (t), g(t)) ≤ Dgr (f (t), ˆ0) + Dgr (g(t), ˆ0) ≤ h1 (t) + h2 (t), 24 where h1 (·), h2 (·) are integrable bounded functions for f and g, respectively Thus, we imply that Dgr (f (t), g(t)) is integrable The assertion (iii) holds (iv) From the definition of granular metric, we have ∫ b ∫ b gr D (f (t), g(t))dt = sup max |f gr (t, α, αf ) − g gr (t, α, αg )|dt α αf ,αg a a (∫ b ) ∫ b ∫ b ∫ b gr gr f (t, α, αf )dt − g gr (t, α, αg )dt f (t)dt, g(t)dt = sup max D a α a αf ,αg a a For all α, αf , αg ∈ [0, 1], since the fact that ∫ b ∫ b ∫ b gr gr f (t, α, αf )dt − g (t, α, αg )dt ≤ |f gr (t, α, αf ) − g gr (t, α, αg )| dt, a a a it follows that ∫ b ∫ b ∫ b gr gr |f gr (t, α, αf ) − g gr (t, α, αg )| dt, f (t, α, αf )dt − g (t, α, αg )dt ≤ max max αf ,αg α ,α g f a a a ∫ b ∫ b ∫ b gr gr |f gr (t, α, αf ) − g gr (t, α, αg )| dt, g (t, α, αg )dt ≤ max f (t, α, αf )dt − ⇒ sup max α ,α α ,α g g f f α a a a ∫ b ∫ b ∫ b gr gr |f gr (t, α, αf ) − g gr (t, α, αg )| dt g (t, α, αg )dt ≤ sup max f (t, α, αf )dt − ⇒ sup max α αf ,αg a α a αf ,αg a Therefore, we get the assertion (iv) (v) Based on Definition 2.4.2, we have ) ∫ b (∫ b f (t)dt = H(f (t))dt H a a ∫ c ∫ b = H(f (t))dt + H(f (t))dt a c (∫ c ) (∫ b ) f (t)dt + H f (t)dt =H c a (∫ c ) ∫ b =H f (t)dt + f (t)dt a c Hence, the assertion (v) holds Proposition 2.4.6 Assume that f and g are differentiable fuzzy − valued functions on an interval (a, b) Then, the following statements are fulfilled: i ii dgr u0 = 0, where u0 ∈ E dt dgr f (t) dgr g(t) dgr (αf (t) ± βg(t)) =α ±β for all t ∈ (a, b) and α, β ∈ R dt dt dt dgr [f (t)g(t)] dgr f (t) dgr g(t) iii = g(t) + f (t) for all t ∈ (a, b) dt dt dt 25 Example 2.4.2 Let u1 = + 2U and u2 = −4 + U be two interval fuzzy numbers with U = [0, 1] and function f (t) = u1 cos 3t + u2 t be defined on the interval [0, 10] It is easy to see that the horizontal membership function of f , given by H(f (t)) = (5 + 2α1 ) cos 3t + (−4 + α2 )t, is a differentiable function on [0, 10] Moreover, its derivative is ∂f gr (t, α1 , α2 ) = −3(5 + 2α1 ) sin 3t + (−4 + α2 ), ∂t where α1 , α2 ∈ [0, 1] Thus, we deduce that the function f is gr-differentiable on [0, 10] In addition, the gr-derivative of f can be given as follows: ) ( gr dgr f (t) ∂f (t, α1 , α2 ) =H dt ∂t [ ] = {(−15 − 6α1 ) sin 3t + α2 − 4}, max{(−15 − 6α1 ) sin 3t + α2 − 4} α1 ,α2 α1 ,α2 [ ] [ ] = min(−15 − 6α1 ), max(−15 − 6α1 ) sin 3t + min(α2 − 4), max(α2 − 4) α1 α1 α2 α2 = [−21, −15] sin 3t + [−4, −3] dgr f (t) Therefore, we obtain the gr-derivative = (−21 + 6U ) sin 3t + (−4 + U ), dt which graphical representation is shown in Figure 2.5 Figure 2.5: The gr-derivative dgr f (t) of fuzzy- valued function f (t) on [0, 10] dt 26 Chapter Applications to fuzzy differential equations In this chapter, fuzzy differential equations based on the horizontal membership functions are investigated Some interesting example are also given to illustrate the theoretical results 3.1 Theoretical results Consider the Cauchy problem for following fuzzy differential equation:  gr d x(t)    dt = f (t, x(t)) x(t0 ) = x(0)   t ∈ [t , t ] (3.1) f gr where x : [t0 , tf ] ⊆ R → E includes n ∈ N distinct fuzzy numbers u1 , u2 , , un , d dtx(t) means gr-derivative of x with respect to t, and x0 ∈ E is a fuzzy initial condition Based on Definition 2.1.2, fuzzy differential equation (3.1) can be rewritten as:  ( ) H dgr x(t) = H (f (t, x(t))) , t ∈ [t0 , tf ] dt (3.2) H (x(t0 )) = H (x(0)) Then, using Proposition 2.4.1, we have  gr u)   ∂x (t,α,α = f gr (t, xgr (t, α, αu ), α, αu ),  ∂t   xgr (t , α, α ) = xgr (α, α ), u u  α (αu1 , αu2 , ., αun ),     t ∈ [t0 , tf ] (3.3) Remark 3.1.1 As we know it, unlike g-differentiability, gH-differentiability, SGHdifferentiability and H-differentiability approaches, based on RDM fuzzy interval arithmetic and horizontal membership function approach, gr-differentiability enables us to solve always just one differential equation, i.e., granular differential equation, which is equivalent to the fuzzy differential equation It is clear that, the fuzzy differential equation (3.1) does not have any solution as long as its corresponding granular differential equation in the space of horizontal membership functions, i.e., differential equation (3.3), does not either Let xgr (t, α, α) be the solution of granular differential equation (3.3) Then, it is the full solution to fuzzy differential equation (3.1) However, if the span of the solution is meant to be obtained, then it is [x(t)]α = H−1 (xgr (t, α, αu )) 27 which is only the solution of the fuzzy differential equation in the space of vertical membership functions, and it is not as fruitful as the solution in the space of horizontal membership functions Remark 3.1.2 Existence and uniqueness of the solution to fuzzy differential equation (3.1) correspond to those of granular differential equation (3.3) 3.2 Numerical examples Example 3.2.1 Consider following triangular fuzzy-valued function ) ( 3 2t3 t t ; + t + 3; + for each t ∈ [−2; 2] f (t) = 3 For more clearly, we plot the graph of fuzzy-valued function f (see Figure 3.1) 3 Figure 3.1: Plot of the α-level sets of fuzzy function f (t) = ( t3 ; t3 + t + 3, 2t3 + 4) in [−2; 2] The blue and red curves show left and right end-points of the α-level sets of f , respectively and the black curve corresponds to the level α = According to Theorem 1.3.2, the derivatives of left and right end-points of the α-level sets of f can be given as follows  + df (t)   α = (2 − α)t2 + α, dt − (t) df  α  = t2 + α, dt df + (t) df − (t) and the plot of functions α and α have been shown in Figure 3.2 dt dt It can be seen that the function f (t) is neither gH-differentiable nor SGHdifferentiable in the interval [−2, 2] However, the function f (t) is gr-differentiable, and its granular derivative can be obtained stepwise as follows: [( ) ] t t3 gr + αf − t − , f (t; α; αf ) = + t + + (1 − α) 3 28 Figure 3.2: Representation of derivative of the α-level sets of the fuzzy function considered in Example 3.2.1 The blue and red curves show left and right endpoints of the α-level sets of the fuzzy function, respectively, and the black curve corresponds to the level α = for all α, αf ∈ [0, 1] , t ∈ [−2, 2] Then, ∂f gr (t, α, αf ) = t2 + + (1 − α)(t2 αf − 1), ∂t whose span is [ dgr f (t) dt ( ]α =H It follows that ( H−1 ∂f gr (t, α, αf ) ∂t ) = −1 ∂f gr (t, α, αf ) ∂t { [t2 + α, t2 + 1], ) |t| ≤ 1, [t2 + α, (2 − α)t2 + α)], |t| > Example 3.2.2 In this example, we will compare the proposed approach with other approaches using the following fuzzy differential equation  gr ( ) d x(t)  πt  = 0.1 sin( )x + u , t ∈ [0, 10],   dt (3.4) x(0) = (1, 2, 3),    u = (1, 1.2, 1.5) Next, by using the horizontal membership function approach, the solvability of the problem (3.4) is equivalent to the solvability of following granular differential equations:  gr ( ) ∂x (t,α,αx )   = 0.1 sin( πt )xgr (t, α, αx ) + ugr (α, αu ) , t ∈ [0, 10],  ∂t   xgr (0, α, α ) = − (1 − α) + 2(1 − α)α x u gr  u (α, α ) = 1.2 − 0.2(1 − α) + 0.5(1 − α)α  u u    αx (αx , αu ) Finally, by using numerical method, we can give explicit form of the solution of problem (3.4) with respect to different types of differentiability Its graphical representation is shown in Figure 3.3 29 Figure 3.3: The solution support closure to the FDE presented in Example 3.2.2 The cyan, red, brown, and green curves correspond to the approaches based on Hdifferentiability, gr-differentiability, gH-differentiability, and the second form of SGH-differentiability, respectively The black curve shows the solution in the level α = Example 3.2.3 (Ehrlich Ascites Tumor model) Experimental tumors have great importance in modeling, and Ehrlich ascites tumor (EAT) is one of the commonest tumors EAT is originally hyperdiploid, has high transplantable capability, no-regression, rapid proliferation, and also does not have tumor-specific transplantation antigen (TSTA) In this example, we describe the ETA model as a dynamical model with uncertainty:  gr ( )  d x(t) = rx(t) − r x(t) , t ∈ [0, 5], (3.5) dt K x(0) = (0.4, 0.6, 0.9), where r = 0.5 stands for the net reproduction rate of tumor, K = is the carrying capacity and Kr is social friction factor influenced by many uncertainty measurement Here, based on the concept of gr-differentiability and the approach introduced in the previous chapter, we have ∂xgr (t, α, αx ) = 0.5xgr (t, α, αx )(1 − xgr (t, α, αx )) ∂t (3.6) with the initial condition xgr (0, α, αx ) = 0.4 + 0.2α + 0.5(1 − α)αx , αx ∈ [0, 1] The analytic solution of the granular differential equation (3.6) subject to the initial data xgr (0, α, αx ) is xgr (t, α, αx ) = xgr (0, α, αx ) , xgr (0, α, αx ) + (1 − xgr (0, α, αx )e−0.5t whose span is given as follows ] [ 0.9 − 0.3α 0.4 + 0.2α α , [x(t)] = 0.4 + 0.2α + (0.6 − 0.2α)e−0.5t 0.9 − 0.3α + (0.1 + 0.3α)e−0.5t 30 Figure 3.4 shows the span of fuzzy solution of the fuzzy diferential equation (3.6) Figure 3.4: The α-level sets of the solution of the nonlinear fuzzy differential equation presented in Example 3.2.3 The blue and red curves show left and right end-points of the α-level sets of the fuzzy function, respectively, and the black curve corresponds to the level α = 31 Chapter Conclusions This mini graduation thesis is devoted to study the granular differentiability of fuzzy-valued functions and its applications Some obtained results of this thesis can be highlighted as follows: Recalling the concepts of fuzzy numbers on R and some characteristic properties of the space of fuzzy numbers such the arithmetic operationm, Zadeh’s Extension principle Especially, for more convenient for readers, we give some illustrating example about some type of fuzzy numbers such triangular fuzzy number, traperzoidal fuzzy numbers, Gaussian fuzzy number and etc Defining the concept of granular metric on the space of fuzzy numbers, that plays an important role to define further analysis properties on this space Defining the concepts of granular differentiability and granular integrability of fuzzy-valued functions Some new related properties of the gr-differentiability and gr-integrability of fuzzy-valued functions are proved in Chapter Fuzzy differential equations under granular computing are investigated in Chapter with some interesting numerical examples 32 REFERENCES [1] B Bede, Mathematics of Fuzzy Sets and Fuzzy Logic, Springer, 2013 [2] B Bede, L Stefanini, “Generalized differentiability of fuzzy-valued functions”, Fuzzy sets and systems, vol 230, pp 119-141, 2013 [3] N.P Dong, D.T Hoan, The solvability of Cauchy problem to fuzzy fractional evolution equations under Caputo generalized Hukuhara-differentiability, Journal of Science, HPU2, (2019), (REVISED) [4] M Mazandarani, N Pariz, A.V Kamyad, Granular Differentiability of FuzzyNumbers Valued Functions, IEEE Tran Fuzzy Syst., 26(1)(2018) 310-323 [5] H.V Long, N.P Dong, An extension of Krasnoselskii’s fixed point theorem and its application to nonlocal problems for implicit fractional differential systems with uncertainty J.Fixed point Theory and Appl., 2018 [6] N.T.K Son, N.P Dong, Asymptotic behavior of C -solutions of evolution equations with uncertainties, J Fixed Point Theory Appl., (2018) DOI: 10.1007/s11784-018-0633-3 33 ... concepts of granular differentiability and granular integrability of fuzzy- valued functions Some new related properties of the gr -differentiability and gr-integrability of fuzzy- valued functions. .. solve some classes of fuzzy differential equations and then simulate their solutions The object and scope of the study 4.1 The object of the study: fuzzy- valued functions and fuzzy differential... granular differentiability of fuzzy- valued functions and its applications Some obtained results of this thesis can be highlighted as follows: Recalling the concepts of fuzzy numbers on R and some