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Fuzzy Optim Decis Making DOI 10.1007/s10700-014-9186-0 The existence and uniqueness of fuzzy solutions for hyperbolic partial differential equations Hoang Viet Long · Nguyen Thi Kim Son · Nguyen Thi My Ha · Le Hoang Son © Springer Science+Business Media New York 2014 Abstract Fuzzy hyperbolic partial differential equation, one kind of uncertain differential equations, is a very important field of study not only in theory but also in application This paper provides a theoretical foundation of numerical solution methods for fuzzy hyperbolic equations by considering sufficient conditions to ensure the existence and uniqueness of fuzzy solution New weighted metrics are introduced to investigate the solvability for boundary valued problems of fuzzy hyperbolic equations and an extended result for more general classes of hyperbolic equations is initiated Moreover, the continuity of the Zadeh’s extension principle is used in some illustrative examples with some numerical simulations for α-cuts of fuzzy solutions Keywords Fuzzy partial differential equation · Fuzzy solution · Integral boundary condition · Local initial condition · Fixed point theorem Mathematics Subject Classification 34A07 · 34A99 · 35L15 H V Long (B) Department of Basic Sciences, University of Transport and Communications, Hanoi, Vietnam e-mail: hvlong@utc.edu.vn N T K Son Department of Mathematics, Hanoi University of Education, Hanoi, Vietnam e-mail: mt02 02@yahoo.com N T M Ha Department of Mathematics, Hai Phong University, Hai Phong, Vietnam e-mail: halinhspt@gmail.com L H Son Centre for High Performance of Computing, VNU University of Science, Vietnam National University, Hanoi, Vietnam e-mail: sonlh@vnu.edu.vn 123 H V Long et al Introduction Fuzziness is a basic type of subjective uncertainty initialed by Zadeh via membership function in 1965 The complexity of the world makes events we face uncertain in various forms Besides randomness, fuzziness is also an important uncertainty, which plays an essential role in the real world The theory of fuzzy sets, fuzzy valued functions and necessary calculus of fuzzy functions have been investigated in the monograph by Lakshmikantham and Mohapatra (2003) and the references cited therein The concept of a fuzzy derivative was first introduced by Chang and Zadeh (1972), later Dubois and Prade (1982) defined the fuzzy derivative by using Zadeh’s extension principle Seikkala (1987) defined the concept of fuzzy derivative which is the generalization of Hukuhara derivative There are many different approaches to define fuzzy derivatives and they become a very quickly developing area of fuzzy analysis Moreover, in view of the development of calculus for fuzzy functions, the investigation of fuzzy differential equations (DEs) and fuzzy partial differential equations (PDEs) have been initiated (Buckley and Feuring 1999; Seikkala 1987) Fuzzy DEs were suggested as a way of modeling uncertain and incomplete information systems, and studied by many researchers In recent years, there has been a significant development in fuzzy calculus techniques in fuzzy DEs and fuzzy differential inclusions, some recent contributions can be seen for example in the papers of Chalco-Cano and Roman-Flores (2008), Lupulescu and Abbas (2012), RodríguezLópez (2013) and Nieto et al (2011), etc However, there is still lack of qualitative and quantitative researches for fuzzy PDEs Fuzzy PDEs were first introduced by Buckley and Feuring (1999) And up to now, the available theoretical results for this kind of equations are included in some researches of Allahviranloo et al (2011), Arara et al (2005), Bertone et al (2013) and Chen et al (2009) Some other efforts were succeeded in modeling some real world processes by fuzzy PDEs For instance, Jafelice et al (2011) proposed a model for the reoccupation of ants in a region of attraction using evolutive diffusion-advection PDEs with fuzzy parameters In Wang et al (2011), developed a fuzzy state-feedback control design methodology by employing a combination of fuzzy hyperbolic PDEs theory and successfully applied to the control of a nonisothermal plug-flow reactor via the existing LMI optimization techniques For a more comprehensive study of fuzzy PDEs in soft computing and oil industry we cite the book of Nikravesh et al (2004) Generally, industrial processes are often complex, uncertain processes in nature Consequently, the analysis and synthesis issues of fuzzy PDEs are of both theoretical and practical importance In the paper Bertone et al (2013) the authors considered the existence and uniqueness of fuzzy solutions for simple fuzzy heat equations and wave equations with concrete formulation of the solutions Arara et al (2005) considered the local and nonlocal initial problem for some classes of hyperbolic equations However, almost all of previous results based on some complicated conditions on data and domain Generally, existence theorems need a condition which may restrict the domain to a small scale This paper deal with some boundary value problems for hyperbolic with an improvement in technique to ensure that the fuzzy solutions exist without any condition on data and the boundary of the domain New weighted metrics are used 123 The existence and uniqueness of fuzzy solutions and suitable weighted numbers are chosen in order to prove that the existence and uniqueness of fuzzy solutions only depend on the Lipschitz property of the right side of the equations Moreover, the problems with fuzzy integral boundary conditions will be introduced and the solvability of these problems will be investigated for more general fuzzy hyperbolic equations As we know that fuzzy boundary value problems with integral boundary conditions constitute a very interesting and important class of problems They include two, three, multi-points boundary value problems and local, nonlocal initial conditions problems as the special cases We can see this fact in many references such as (Agarwal et al 2005; Arara and Benchohra 2006) Therefore the results of the present paper can be considered as a contribution to the subject In many cases, it is difficult to find the exact solution of fuzzy DEs and fuzzy PDEs In these cases, some optimal algorithms to find numerical solution must be introduced (see in Nikravesh et al 2004, Chapter: Numerical solutions of fuzzy PDEs and its applications in computational mechanics) In another example, Dostál and Kratochvíl (2010) used the two dimensional fuzzy PDEs to build up of a model for judgmental forecasting in bank sector The simulation solutions of this equations supports the managers optimize their decision making to close the branch of the bank for reducing the costs However, before conducting a numerical method for any fuzzy equations, the question arises naturally whether the problems modeled by fuzzy PDEs are wellposed or not Thus, our study provides a theoretical foundation of numerical solution methods for some classes of fuzzy PDEs and ensures the consistency, stability and convergence of optimal algorithms The remainder of the paper is organized as follows Section presents some necessary preliminaries of fuzzy analysis, that will be used throughout this paper In Sect 3, we concern with the existence and uniqueness of fuzzy solutions for wave equations in the following form ∂ u(x, y) = f (x, y, u(x, y)), (x, y) ∈ [0, a] × [0, b], ∂ x∂ y (1) with local conditions u(0, 0) = u , u(x, 0) = η1 (x), u(0, y) = η2 (y), (x, y) ∈ [0, a] × [0, b] (2) The existence of fuzzy solutions of this problem is proved in Theorem 3.1 without any condition in the domain by using a new weighted metric H1 in the solutions space In this section, we also consider the Eq (1) with the integral boundary conditions b u(x, 0) + k1 (x)u(x, y)dy = g1 (x), x ∈ [0, a], (3) k2 (y)u(x, y)d x = g2 (y), (4) a u(0, y) + y ∈ [0, b] 123 H V Long et al The results about the solvability of this problem are given in Theorem 3.2 with metric H and Theorem 3.3 by using new weighted metric H2 The general hyperbolic PDEs in the form ∂ u(x, y) ∂ ( p1 (x, y)u(x, y)) ∂( p2 (x, y)u(x, y)) + + + c(x, y)u(x, y) ∂ x∂ y ∂x ∂y = f (x, y, u(x, y)) , (5) for (x, y) ∈ [0, a] × [0, b], are considered in the Sect Some results of the existence and uniqueness of fuzzy solutions for these equations with local initial conditions and integral boundary conditions are also exhibited in Theorems 4.1 and 4.2 Some illustrated examples for our results are given in Sect with some numerical simulations for α-cuts of the solutions Finally, some conclusions and future works are discussed in Sect Preliminaries This section will recall some concepts of fuzzy metric space used throughout the paper For a more thorough treatise on fuzzy analysis, we refer to Lakshmikantham and Mohapatra (2003) Let E n be space of functions u : Rn → [0, 1], which are normal, fuzzy convex, semi-continuous and bounded-supported functions We denote CC(Rn ) by the set of all nonempty compact, convex subsets of Rn The α-cuts of u are [u]α = {x ∈ Rn : u(x) ≥ α} for < α ≤ Obviously, [u]α is in CC(Rn ) And CC(Rn ) is complete metric space with Hausdorff metric defined by Hd (A, B) = max sup inf {||b − a||}, sup inf {||b − a||} , b∈B a∈A a∈A b∈B A, B ∈ CC(Rn ) where || · || is usual Euclidean norm in Rn Furthermore, (i) Hd (t A, t B) = |t| · Hd (A, B) (ii) Hd (A + A , B + B ) ≤ Hd (A, B) + Hd (A , B ); (iii) Hd (A + C, B + C) = Hd (A, B) where A, B, C, A , B ∈ CC(Rn ) and t ∈ R Let (E n , d∞ ) be a complete metric space with supremum metric d∞ defined by d∞ (u, u) = sup Hd [u]α , [u]α , u, u ∈ E n 0 0, there exists δ( , α) > such that Hd [ f (t, s, u)]α , [ f (t0 , s0 , u )]α < whenever max{|t − t0 | , |s − s0 |} < δ( , α) and Hd ([u]α , [u ]α ) < δ( , α), (t, s, u) ∈ J × En y y The integral of f : J = [x1 , y1 ] × [x2 , y2 ] → E n , denoted by x11 x22 f (t, s) dsdt, is defined by ⎡ ⎤α y1 y2 ⎣ y1 y2 f (t, s) dsdt ⎦ = x1 x2 = ⎧ ⎨ ⎩ f α (t, s) dsdt x1 x2 y1 y2 v (t, s) dsdt|v : J → Rn x1 x2 ⎫ ⎬ is a measurable selection for f α , for α ∈ (0, 1] ⎭ y y A function f : J → E n is integrable if x11 x22 f (t, s) dsdt is in E n For any f : J → E n , fuzzy partial derivative of f with respect to x at point ∂ f (x0 , y0 ) (x0 , y0 ) ∈ J is a fuzzy set ∈ E n which is defined by ∂x ∂ f (x0 , y0 ) f (x0 + h, y0 ) − f (x0 , y0 ) = lim h→0 ∂x h Here the limitation is taken in the metric space (E n , d∞ ) and u − v is the Hukuhara difference of u and v in E n The fuzzy partial derivative of f with respect to y and higher order of partial derivatives of f are defined similarly 123 H V Long et al The fuzzy solutions of the hyperbolic PDEs Denote Ja = [0, a], Jb = [0, b], with a, b > In this part of the paper, we consider the hyperbolic PDE (1), where f : Ja × Jb × E n → E n is a given function, which satisfies following hypothesis Hypothesis (H) There exists K > such that Hd [ f (s, t, u)]α , [ f (s, t, u)]α ≤ K Hd [u]α , [u]α for all (s, t) ∈ Ja × Jb and all u, u ∈ E n In Arara et al (2005) studied the existence of fuzzy solutions of the equation (1) with local initial conditions (2), where η1 ∈ C(Ja , E n ), η2 ∈ C(Jb , E n ) are given functions and u ∈ E n Definition 3.1 (Arara et al 2005) A function u is called a solution of the prob2 = lem (1)–(2) if it is a function in the space C(Ja × Jb , E n ) satisfying ∂ ∂u(x,y) x∂ y f (x, y, u(x, y)) on Ja × Jb and u(0, 0) = u , u(x, 0) = η1 (x), u(0, y) = η2 (y) for all (x, y) ∈ Ja × Jb Proposition 3.1 (Arara et al 2005) Assume that the hypothesis (H) holds If K ab < 1, then the problem (1)–(2) has a unique fuzzy solution in the space C(Ja × Jb , E n ) Remark 3.1 This result based on the condition K ab < That condition is strict for the domain Ja × Jb to satisfy if the Lipschitz constant K is big enough On the other hand, it depends on the large scale of the domain To relax this restriction, we use a weighted metric H1 in the space C(Ja × Jb , E n ) H1 ( f, g) = sup (s,t)∈J d∞ ( f (s, t) , g (s, t)) e−λ(s+t) , where λ is a suitable positive number It is not difficult to check that (C(Ja × Jb , E n ), H1 ) is also a complete metric space Definition 3.2 A function u ∈ C(Ja × Jb , E n ) is called a solution of the problem (1), (2) if it satisfies y x u(x, y) = q1 (x, y) + f (s, t, u(s, t)) dsdt, 0 where q1 (x, y) = η1 (x) + η2 (y) − u , for (x, y) ∈ Ja × Jb Theorem 3.1 Assume that the condition (H) holds Then the problem (1)–(2) has a unique solution in C(Ja × Jb , E n ) 123 The existence and uniqueness of fuzzy solutions Proof From Definition 3.2, we realize that fuzzy solution of problem (1)–(2) (if it exists) is a fixed point of the operator N : C(Ja × Jb , E n ) → C(Ja × Jb , E n ) defined as follows y x N (u(x, y)) = q1 (x, y) + f (s, t, u(s, t)) dsdt (6) We will show that N is a contraction operator Indeed, for u, u ∈ C(Ja × Jb , E n ) and α ∈ (0, 1] then from the properties of supremum metric and (6), we have d∞ (N (u(x, y)), N (u(x, y))) ⎛ x y ≤ d∞ ⎝ x ⎞ y x f (s, t, u(s, t))dsdt ⎠ f (s, t, u(s, t))dsdt, 0 y ≤K 0 d∞ (u(s, t), u(s, t)) dsdt 0 It implies that ⎛ y x e−λ(x+y) d∞ ⎝ f (s, t, u(s, t)) dsdt ⎠ f (s, t, u(s, t)) dsdt, 0 x y ≤ K e−λ(x+y) ⎞ y x 0 d∞ (u(s, t), u(s, t))e−λ(s+t) eλ(s+t) dsdt 0 y x ≤ K H1 (u, u)e −λ(x+y) eλ(s+t) dsdt ≤ 0 K H1 (u, u) λ2 for all (x, y) ∈ Ja × Jb That shows e−λ(x+y) d∞ (N (u(x, y)), N (u(x, y))) ≤ K H1 (u, u) , (x, y) ∈ Ja × Jb λ2 Therefore H1 (N (u), N (u)) ≤ K H1 (u, u), for all u, u ∈ C(Ja × Jb , E n ) λ2 √ By choosing λ = 2K we have λK2 = 21 Hence, N is a contraction operator and by Banach fixed point theorem, N has a unique fixed point, that is the solution of the problem (1)–(2) The proof is completed 123 H V Long et al We continue concerning with the existence of fuzzy solutions for hyperbolic PDEs (1) with integral boundary conditions (3) and (4), where k1 (·) ∈ C(Ja , R), k2 (·) ∈ C(Jb , R)g1 (·) ∈ C(Ja , E n ), g2 (·) ∈ C(Jb , E n ) are given functions Definition 3.3 A function u ∈ C(Ja × Jb , E n ) is called a fuzzy solution of the problem (1), (3) and (4) if u satisfies the following integral equation b a u(x, y) = q(x, y) − k1 (x)u(x, y)dy − k2 (y)u(x, y)d x a b y x − k1 (0) k2 (y)u(x, y)d xd y + f (s, t, u(s, t)) dsdt, 0 b g2 (t)dt, where q(x, y) = g1 (x) + g2 (y) − g1 (0) + k1 (0) for (x, y) ∈ Ja × Jb Theorem 3.2 Let k1 = sups∈Ja |k1 (s)|, k2 = supt∈Jb |k2 (t)| Assume that the hypothesis (H) is satisfied If (1 + k1 b)(1 + k2 a) < − K ab, then the problem (1), (3), (4) has a unique fuzzy solution in C(Ja × Jb , E n ) Proof Integrating both sides of the equation (1) on [0, x] × [0, y], we have b a u(x, y) = q(x, y) − k1 (x)u(x, y)dy − k2 (y)u(x, y)d x a b y x − k1 (0) k2 (y)u(x, y)d xd y + 0 f (s, t, u(s, t)) dsdt 0 b where q(x, y) = g1 (x) + g2 (y) − g1 (0) + k1 (0) g2 (t)dt, for all (x, y) ∈ Ja × Jb We will prove that fuzzy solution of problem (1), (3), (4) is a fixed point of operator N : C Ja × Jb , E n → C Ja × Jb , E n defined as follows b N (u(x, y)) = q(x, y) − a k1 (x)u(x, y)dy − k2 (y)u(x, y)d x a b − k1 (0) y x k2 (y)u(x, y)d xd y + 0 f (s, t, u(s, t)) dsdt 0 For u, u ∈ C(Ja × Jb , E n ) arbitrary and α ∈ (0, 1], one gets 123 The existence and uniqueness of fuzzy solutions Hd [N (u(x, y))]α , [N (u(x, y))]α ⎛⎡ b ⎤α ⎡ ≤ Hd ⎝⎣ ⎛⎡ + Hd ⎝⎣ ⎛⎡ k1 (x)u(x, y)dy ⎦ ⎠ k1 (x)u(x, y)dy ⎦ , ⎣ a ⎤α ⎡ a a b 0 0 ⎤α ⎞ f (s, t, u(s, t)) dsdt ⎦ ⎠ b a Hd [u(x, y)]α , [u(x, y)]α dy + k2 ≤ k1 k2 (y)u(x, y)d xd y ⎦ ⎠ y x f (s, t, u(s, t)) dsdt ⎦ , ⎣ + Hd ⎝⎣ ⎤α ⎞ a b ⎤α ⎡ y x ⎤α ⎡ k2 (y)u(x, y)d xd y ⎦ , ⎣k1 (0) + Hd ⎝⎣k1 (0) ⎛⎡ ⎤α ⎞ k2 (y)u(x, y)d x ⎦ ⎠ k2 (y)u(x, y)d x ⎦ , ⎣ ⎤α ⎞ b Hd [u(x, y)]α , [u(x, y)]α d x b a Hd [u(x, y)]α , [u(x, y)]α d xd y + |k1 (0)|k2 0 y x Hd [ f (s, t, u(s, t))]α , [ f (s, t, u(s, t))]α dsdt + 0 ≤ (k1 b + k2 a + k1 k2 ab)d∞ (u(x, y), u(x, y)) y x K Hd [u(s, t)]α , [u(s, t)]α dsdt + 0 b a ≤ (k1 b + k2 a + k1 k2 ab)H (u, u) + K d∞ (u(x, y), u(x, y))d xd y 0 ≤ (k1 b + k2 a + k1 k2 ab + K ab)H (u, u), for all(x, y) ∈ Ja × Jb Hence H (N (u), N (u)) = = sup (x,y)∈Ja ×Jb sup (x,y)∈Ja ×Jb d∞ (N (u (x, y)), N (u (x, y))) sup Hd [N (u(x, y))]α , [N (u(x, y))]α 0 Lemma 3.1 Problem (1), (7) and (8) is equivalent to following integral equation x t u(x, y) = G(x, y) + K (x) k2 (t)u(s, t)dsdt y s + K (y) k1 (s)u(s, t)dtds 0 x − K (x) y ⎡ x ⎢ ⎣ ⎡ t ⎥ f (s, t, u(s, t)) dsdt ⎦ dt ⎤ y s f (s, t, u(s, t)) dsdt ⎦ ds ⎣ − K (y) ⎤ 0 y x + f (s, t, u(s, t)) dsdt, 0 where G(x, y), K (x), K (y) are defined by (15) Proof From Eq (1), we have y x u(x, y) = u(x, 0) + u(0, y) − u(0, 0) + f (s, t, u(s, t)) dsdt 123 (9) H V Long et al ⎛ x ⎜ + d∞ ⎝ K (x) ⎢ ⎣ × x ⎛ t 0 y 0 ⎤ ⎞ ⎤ y s f (s, t, u(s, t)) dsdt ⎦ ds, K (y) 0 ⎤ y s ⎞ f (s, t, u(s, t)) dsdt ⎦ ds ⎠ x ⎛ ⎥ f (s, t, u(s, t)) dsdt ⎦ dt, K (x) ⎣ ⎣ × ⎡ ⎡ ⎤ t ⎥ ⎟ f (s, t, u(s, t)) dsdt ⎦ dt ⎠ + d∞ ⎝ K (y) y x ⎢ ⎣ ⎡ x ⎡ y + d∞ ⎝ f (s, t, u(s, t)) dsdt ⎠ f (s, t, u(s, t)) dsdt, 0 ⎞ y x (16) For simplicity we denote c1 = sup (x,y)∈Ja ×Ja |K (x)| |k2 (y)|, c2 = c3 = K sup |K (x)| , y∈Ja ⎛ x ⎜ e−λ max{x,y} d∞ ⎝ K (x) x ≤e x 0 t d∞ (u(s, t), u(s, t))e−λ max{s,t } eλ max{s,t } dsdt 0 t x eλ max{s,t } dsdt −λ max{x,y} 0 x x t −λ max{x,y} λt e dsdt = c1 H2 (u, u)e 0 c1 x ac1 ≤ H2 (u, u)e−λ max{x,y} eλx ≤ H2 (u, u) λ λ 123 ⎟ k2 (t)u(s, t)dsdt ⎠ d∞ (u(s, t), u(s, t))dsdt x = c1 H2 (u, u)e ⎞ −λ max{x,y} = c1 H2 (u, u)e t t c1 = c1 e t k2 (t)u(s, t)dsdt, K (x) −λ max{x,y} |K (y)| |k1 (x)| c4 = K sup |K (y)| x∈Ja Then we have sup (x,y)∈Ja ×Ja −λ max{x,y} teλt dt (17) The existence and uniqueness of fuzzy solutions Similarly we have ⎛ y y s e−λ max{x,y} d∞ ⎝ K (y) k1 (s)u(s, t)dtds ⎠ k1 (s)u(s, t)dtds, K (y) 0 ⎞ s 0 ac2 ≤ H2 (u, u) λ (18) We now will estimate the third term in the right side of (16) ⎛ e −λ max{x,y} t K (x) 0 ⎞ x t −λ max{x,y} d∞ (u(s, t), u(s, t))dsdtdt 0 x = c3 e ⎟ f (s, t, u(s, t)) dsdtdt ⎠ x ≤ c3 e t f (s, t, u(s, t)) dsdtdt, x x x x ⎜ d∞ ⎝ K (x) x t −λ max{x,y} d∞ (u(s, t), u(s, t))e−λ max{s,t} eλ max{s,t} dsdtdt 0 x = c3 H2 (u, u)e x t −λ max{x,y} eλ max{s,t} dsdtdt 0 One gets x x t eλ max{s,t} dsdtdt 0 x = x = x = ⎡ t ⎢ ⎣ ⎡ t ⎝ t ⎢ ⎣ ⎡ ⎛ ⎛ eλ max{s,t} ds + t x eλt ds + t ⎢ ⎣ teλt + ⎞ t ⎤ ⎥ eλ max{s,t} ds ⎠ dt ⎦ dt t ⎝ x ⎞ ⎤ ⎥ eλs ds ⎠ dt ⎦ dt ⎤ λx λt ⎥ dt ⎦ dt e − e λ λ 123 H V Long et al x λt 2 te − eλt + eλx t + dt λ λ λ λ = = x x2 2x eλx + + − eλx + λ2 2λ λ λ ≤ x x2 2x eλx + + λ 2λ λ Thus we have x e x t −λ max{x,y} eλ max{s,t} dsdtdt 0 x x2 2x eλx + ≤ e−λ max{x,y} + λ 2λ λ ⎧ x x2 2x λx ⎨e−λx + 2λ e + λ2 if x ≥ y λ2 = x x2 ⎩e−λy eλx + 2x if y > x + 2λ λ2 λ2 ≤ 3a a2 + λ2 2λ That leads to x x c3 H2 (u, u) e t −λ max{x,y} 3a a2 + λ 2λ eλ max{s,t} dsdtdt ≤ c3 0 H2 (u, u) (19) Similarly, we have following estimation for the fourth term in the right side of (16) y e −λ max{x,y} d∞ K (y) f (s, t, u(s, t)) dsdtds, y 0 y s K (y) f (s, t, u(s, t)) dsdtds 0 3a a2 + λ2 2λ ≤ c4 y s H2 (u, u) (20) Finally, we have ⎛ y x e−λ max{x,y} d∞ ⎝ ⎞ f (s, t, u(s, t)) dsdt ⎠ f (s, t, u(s, t)) dsdt, 123 y x 0 The existence and uniqueness of fuzzy solutions y x ≤ Ke −λ max{x,y} d∞ (u(s, t), u(s, t))dsdt 0 y x = Ke −λ max{x,y} d∞ (u(s, t), u(s, t))e−λ max{s,t} eλ max{s,t} dsdt 0 y x = K H2 (u, u)e −λ max{x,y} eλ max{s,t} dsdt 0 x ⎡ +) If x < y then y x ⎣ eλ max{s,t} dsdt = 0 x ⎡ ⎣ = ⎤ s s x eλt dt ⎦ ds = eλs dt + eλ max{s,t} dt ⎦ ds eλ max{s,t} dt + y s ⎤ y s seλs + λy λs e − e ds λ λ 1 = xeλx + eλy x − (eλx − 1) ≤ xeλx + eλy x λ λ λ λ λ It follows y x e −λ max{x,y} e dsdt = e −λy eλ max{s,t} dsdt λx λy xe + e x λ λ ≤ e−λy y x λ max{s,t} 2a 2x ≤ ≤ λ λ +) If y ≤ x, we also get y x e −λ max{x,y} e ≤ e−λx y x λ max{s,t} dsdt = e −λx eλ max{s,t} dsdt λy λx ye + e y λ λ 2a 2y ≤ ≤ λ λ It implies y x K H2 (u, u)e −λ max{x,y} eλ max{s,t} dsdt ≤ 0 2K a H2 (u, u) λ 123 H V Long et al or ⎛ y x e−λ max{x,y} d∞ ⎝ f (s, t, u(s, t)) dsdt ⎠ f (s, t, u(s, t)) dsdt, 0 ⎞ y x 0 2K a H2 (u, u) ≤ λ (21) By multiplying e−λ max{x,y} to both sides of (16), and from (17) to (21), we receive a(c1 + c2 + 2K ) + λ H2 (N (u), N (u)) ≤ 3a a2 (c3 + c4 ) H2 (u, u) + λ2 2λ We can choose λ > satisfying a(c1 + c2 + K ) + λ 3a a2 (c3 + c4 ) < + λ2 2λ This follows that N has unique fixed point The theorem is proved completely Remark 3.3 From Theorems 3.1 and 3.3, by using different weighted metrics in the space C(Ja ×Jb , E n ), we can receive the various results in the existence and uniqueness of fuzzy solutions of the problem without conditions on the boundary of the domain Fuzzy solutions of general hyperbolic PDEs We consider problem in general cases (5), where (x, y) ∈ Ja × Jb , f : Ja × Jb × E n → E n and c, pi ∈ C(Ja × Jb , R); i = 1, And the local initial conditions are in (2), in which η1 ∈ C(Ja , E n ), η2 ∈ C(Jb , E n ) are given functions and u ∈ E n Definition 4.1 A function u ∈ C(Ja × Jb , E n ) is called a fuzzy solution of the equations (5) with local condition (2) if it satisfies the equation y u(x, y) = q1 (x, y) − x p1 (x, t)u(x, t)dt − 0 y x y x c(s, t)u(s, t)dsdt + − 0 where q1 (x, y) = η1 (x) + η2 (y) − u + (x, y) ∈ Ja × Jb p2 (s, y)u(s, y)ds f (s, t, u(s, t)) dsdt, y 0 p1 (0, t)η2 (t)dt + x p2 (s, 0)η1 (s)ds, for Theorem 4.1 Let p1 = sup(s,t)∈Ja ×Jb | p1 (s, t)|, p2 = sup(s,t)∈Ja ×Jb | p2 (s, t)|, c = sup(s,t)∈Ja ×Jb |c(s, t)| If Hypothesis (H ) is satisfied and (bp1 +ap2 )+ab(c+ K ) < 1, then problem (5) and (2) has a unique solution in C(Ja × Jb , E n ) 123 The existence and uniqueness of fuzzy solutions Proof By doing the same previous arguments, we consider the operator N1 : C(Ja × Jb , E n ) → C(Ja × Jb , E n ) defined as follows y N1 (u(x, y)) = q1 (x, y) − x p1 (x, t)u(x, t)dt − 0 y x y x c(s, t)u(s, t)dsdt + − p2 (s, y)u(s, y)ds f (s, t, u(s, t)) dsdt 0 Let u, u ∈ C(Ja × Jb , E n ) and α ∈ (0, 1], (x, y) ∈ Ja × Jb Then Hd [N1 (u(x, y))]α , [N1 (u(x, y))]α ⎤α ⎡ ⎛⎡ y p1 (x, t)u(x, t)dt ⎦ , ⎣ ≤ Hd ⎝⎣ ⎤α ⎡ x + Hd ⎝⎣ ⎤α ⎞ x p2 (s, y)u(s, y)ds ⎦ ⎠ p2 (s, y)u(s, y)ds ⎦ , ⎣ ⎤α ⎡ y x + Hd ⎝⎣ ⎛⎡ p1 (x, t)u(x, t)dt ⎦ ⎠ ⎛⎡ ⎛⎡ ⎤α ⎞ y c(s, t)u(s, t)dsdt ⎦ ⎠ c(s, t)u(s, t)dsdt ⎦ , ⎣ 0 ⎤α ⎡ y x ⎤α ⎞ y x x f (s, t, u(s, t)) dsdt ⎦ , ⎣ + Hd ⎝⎣ 0 ⎤α ⎞ y f (s, t, u(s, t)) dsdt ⎦ ⎠ 0 b | p1 (x, t)|Hd [u(x, t)]α , [u(x, t)]α dt ≤ a | p2 (s, y)|Hd [u(s, y)]α , [u(s, y)]α ds + b a |c(s, t)|Hd [u(s, t)]α , [u(s, t)]α dsdt + 0 a b Hd [u(s, t)]α , [u(s, t)]α dsdt +K 0 ≤ (bp1 + ap2 + ab(c + K )) d∞ (u(x, y), u(x, y)) It implies H (N1 (u), N1 (u)) = sup (x,y)∈Ja ×Jb d∞ (N1 (u (x, y)), N1 (u ((x, y))) 123 H V Long et al = sup Hd [N1 (u(x, y))]α , [N1 (u)(x, y)]α sup (s,t)∈Ja ×Jb 0 And the local condition are u(0, 0) = u(x, 0) = 0; u(0, y) = cy (29) We have a = b = 18 , p1 (x, y) = p2 (x, y) = c(x, y) = We recognize that the right side of the equation is f (x, y, c) = x y + cy + y + x + c + 1, so the hypothesis (H) is satisfied with arbitrary positive number So all conditions of the Theorem 4.1 hold Therefore there exists a unique fuzzy solution of this problem Indeed, for any fixed c ∈ [0, M], the deterministic solution of (28)–(29) is u(x, y) = g(x, y, c) = cy + x y We fuzzify c, f and g by using the extension principle Fuzzy function F is computed from f and G is computed from g After that we will show that G is the fuzzy solution of (24)–(25) Indeed, let [G]α = [g1 (x, y, α), g2 (x, y, α)] = [c1 (α)y + x y, c2 (α)y + x y] and [F]α = [ f (x, y, α), f (x, y, α)] = [x y + c1 (α)y + y + x + c1 (α) + 1, x y + c2 (α)y + y + x + c2 (α) + 1] We consider differential operator ϕ(Dx , D y )u(x, y) = u x y + u x + u y + u 123 The existence and uniqueness of fuzzy solutions Fig Numerical simulation for fuzzy solution of (28)–(29) with Gaussian fuzzy number [C]α = [1 − 0.1 ln α1 , + 0.1 ln α1 ] We compute [ϕ(Dx , D y )g1 (x, y, α), ϕ(Dx , D y )g2 (x, y, α)] the result is [x y + c1 (α)y + y + x + c1 (α) + 1, x y + c2 (α)y + y + x + c2 (α) + 1] which are the α-cuts of fuzzy number x y + C y + y + x + C + Hence G(x, y) is differentiable and ϕ(Dx , D y )G(x, y, C) = F(x, y, C) Since all partials of G and F with respect to c are positive and boundary conditions are all satisfied g1 (0, 0, α) = g2 (0, 0, α) = 0, g1 (x, 0, α) = g2 (x, 0, α) = 0, g1 (0, y, α) = c1 (α)y, g2 (0, y, α) = c2 (α)y Therefore, G(x, y, C) is the fuzzy solution of (28)–(29) In this example, we fuzzify crisp number c by Gaussian fuzzy numbers C with membership function C(t) = exp(−100(t − c)2 ) The α−cuts of C are [c1 (α), c2 (α)] = c − 0.1 ln 1 , c + 0.1 ln α α The continuity of extension principle states that fuzzy solution of (28)–(29) is [G(x, y, C)]α = c − 0.1 ln α y + x y, c + 0.1 ln α y + xy The simulation of some α-cuts of this fuzzy solution is shown in Fig 123 H V Long et al Example 5.3 In this example, we consider hyperbolic PDE u x y (x, y) − 2xu x (x, y) + 3y u y (x, y) − x yu(x, y) = c(−x y − 2x y + 3y x + 1), (30) where (x, y) ∈ Ja × Jb = [0, 16 ] × [0, 16 ], c a constant in J = [0, M], M > The boundary conditions are 1/6 xu(x, y)dy = u(x, 0) + cx , 72 (31) 1/6 y u(x, y)d x = u(0, y) + cy 72 (32) This example states a = b = 16 , p1 (x, y) = −2x, p2 (x, y) = 3y , c(x, y) = −x y, k1 (x) = x, k2 (y) = y and f (x, y, u) = c(−x y − 2x y + 3y x + 1) It implies Hd ([ f (s, t, u)]α , [ f (s, t, u)]α ) = and conditions in Theorem 4.2 hold for any positive number K , like K = 16 Therefore there exists a unique fuzzy solution of this problem By fuzzifying c we have fuzzy number C and fuzzy solution of (30)–(32) is [G(x, y, C)]α = [c1 (α)x y, c2 (α)x y] Now we consider Parabolic fuzzy number with membership function C(t) = 1− (b−a)2 t− a+b 2 if t ∈ [a, b] other Fig Numerical simulation for fuzzy solution of (30)–(32) with parabolic fuzzy number [C]α = √ √ [2 − 0.2 − α; − 0.2 − α] 123 The existence and uniqueness of fuzzy solutions Fuzzy solution of (30)–(32) is [G(x, y, C)]α = a + b b − a√ a + b b − a√ − α x y, − α xy − + 2 2 Fig shows some numerical simulations for level-sets of fuzzy solution of (30)– (32) with parabolic fizzy number Conclusions and future works The main goals of this article have been to investigate the fuzzy solutions of some class of hyperbolic PDEs with local initial conditions and integral boundary conditions We have achieved these goals by using Banach fixed point theorem and illustrated these results by some numerical examples Our results provide the theoretical foundation for many fuzzy PDEs optimal algorithms and support some decision making processes The next step in our future research is to investigate the existence and uniqueness of solution of some classes of fuzzy functional PDEs and extend the existence interval from a restricted domain to the whole domain by using some extension methods Acknowledgments The authors would like to thank Editor-in-Chiefs, Prof Shu-Cherng Fang; Associate Editor; anonymous reviewers; Prof Bui Cong Cuong (VAST), Dr Tran Dinh Ke (HNUE) for their comments and their valuable suggestions that improved the quality and clarity of the paper This work is supported by NAFOSTED, Vietnam under contract No.102.01-2012.14 References Allahviranloo, T., Abbasbandy, S., & Rouhparvar, H (2011) The exact solutions of fuzzy wave-like equations with variable 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