Fuzzy solutions for general hyperbolic partial differential equations with local initial conditions

1. Introduction Fuzzy set theory was first introduced by Zadeh [15]. The ambition of fuzzy set theory is to provide a formal setting for incomplete, inexact, vague and uncertain information. Today, after its conception, fuzzy set theory has become a fashionable theory used in many branches of real life such as dynamics, computer technology, biological phenomena and financial forecasting, etc. The concepts of fuzzy sets, fuzzy numbers, fuzzy metric spaces, fuzzy valued functions and the necessary calculus of fuzzy functions have been investigated in papers [3, 7-10]. The fuzzy derivative was first introduced by Chang and Zadeh in [5]. The study of differential equations was considerd in [12-14]. The recent results on fuzzy differential equations and inclusion was presented in the monograph of Lakshmikantham and Mohapatra [11]. Nowadays, many fields of science can be presented using mathematical models, especially partial differential equations. When databases that are transformed from real life into mathematical models are incomplete or vague, we often use fuzzy partial differential equations. Hence, more and more authors have studied solutions for fuzzy partial differential equations. In [4], Buckley and Feuring found the existence of B-F solutions and Seikkala solutions for fuzzy partial differential equations by using crisp solution and the extension principle. Some other efforts have been recently made to find the numerical solutions for fuzzy partial differential equations by Allahviranloo [1]. With regards to the fuzzy hyperbolic partial differential equations with local and nonlocal initial conditions, Arara et. al. [2] used the Banach fixed point theorem to investigate the existence and uniqueness of fuzzy solutions. However, their results depended on the form and the size of the domain Ja × Jb. Meanwhile, it is absolutely not necessary. In this paper, by using the ideas of a new metric in a complete metric space, we show that fuzzy solutions of more general hyperbolic partial differential equations exist without any condition on the domain.

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JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2013, Vol. 58, No. 7, pp. 27-38 This paper is available online at FUZZY SOLUTIONS FOR GENERAL HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS WITH LOCAL INITIAL CONDITIONS Nguyen Thi My Ha1, Nguyen Thi Kim Son2 and Ha Thi Thanh Tam3 1Faculty of Mathematics, Hai Phong University 2Faculty of Mathematics, Hanoi National University of Education 3Diem Dien High School, Thai Binh Abstract. In this paper, we study the existence and uniqueness of fuzzy solutions for general hyperbolic partial differential equations with local conditions making use of the Banach fixed point theorem. Some examples are presented to illustrate our results. Keywords:Hyperbolic differential equations, fuzzy solution, local conditions, fixed point. 1. Introduction Fuzzy set theory was first introduced by Zadeh [15]. The ambition of fuzzy set theory is to provide a formal setting for incomplete, inexact, vague and uncertain information. Today, after its conception, fuzzy set theory has become a fashionable theory used in many branches of real life such as dynamics, computer technology, biological phenomena and financial forecasting, etc. The concepts of fuzzy sets, fuzzy numbers, fuzzy metric spaces, fuzzy valued functions and the necessary calculus of fuzzy functions have been investigated in papers [3, 7-10]. The fuzzy derivative was first introduced by Chang and Zadeh in [5]. The study of differential equations was considerd in [12-14]. The recent results on fuzzy differential equations and inclusion was presented in the monograph of Lakshmikantham and Mohapatra [11]. Nowadays, many fields of science can be presented using mathematical models, especially partial differential equations. When databases that are transformed from real life into mathematical models are incomplete or vague, we often use fuzzy partial differential equations. Hence, more and more authors have studied solutions for fuzzy partial differential equations. In [4], Buckley and Feuring found the existence of B-F Received January 15, 2013. Accepted May 24, 2013. Contact Nguyen Thi Kim Son, e-mail address: mt02_02@yahoo.com 27 Nguyen Thi My Ha, Nguyen Thi Kim Son and Ha Thi Thanh Tam solutions and Seikkala solutions for fuzzy partial differential equations by using crisp solution and the extension principle. Some other efforts have been recently made to find the numerical solutions for fuzzy partial differential equations by Allahviranloo [1]. With regards to the fuzzy hyperbolic partial differential equations with local and nonlocal initial conditions, Arara et. al. [2] used the Banach fixed point theorem to investigate the existence and uniqueness of fuzzy solutions. However, their results depended on the form and the size of the domain Ja × Jb. Meanwhile, it is absolutely not necessary. In this paper, by using the ideas of a new metric in a complete metric space, we show that fuzzy solutions of more general hyperbolic partial differential equations exist without any condition on the domain. 2. Preliminaries In this section, we give some basic notations, necessary concepts and results which will be used later. We denote the set consisting of all nonempty compact, convex subsets ofRn byKnC . Let A and B be two nonempty bounded subsets of KnC . Denote by ||.|| a norm in Rn. The distance between A and B is defined by the Hausdorff metric dH(A,B) = max { sup a∈A inf b∈B ||a− b|| , sup b∈B inf a∈A ||a− b|| } and (KnC , dH) is a complete space [11]. Let En be the space of functions u: Rn → [0, 1] satisfying: i) there exists a x0 ∈ Rn such that u(x0) = 1; ii) u is fuzzy convex, that is for x, z ∈ Rn and 0 < λ ≤ 1, u(λx+ (1− λ)z) ≥ min[u(x), u(z)]; iii) u is semi-continuous; iv) [u]0 = {x ∈ Rn : u(x) > 0} is a compact set in Rn. If u ∈ En, u is called a fuzzy set and the α-level of u is defined by [u]α = {x ∈ Rn : u(x) ≥ α} for each 0 < α ≤ 1. Then from (i) to (iv), it follows that [u]α is in KnC . The fuzzy sets u ∈ E1 is called fuzzy numbers. The triangular fuzzy numbers are those fuzzy sets in E1 for which the sendograph is a triangle. A triangular fuzzy number u is defined by three numbers a1 < a2 < a3 such that [u]0 = [a1, a3], u1 = a2. We write u > 0 if a1 > 0, u ≥ 0 if a1 ≥ 0, u < 0 if a3 < 0 and u ≤ 0 if a3 ≤ 0. The α-level set of a fuzzy number is presented by an ordered pair of function [u1(α), u2(α)], 0 < α < 1 which satisfies the following requirements: 28 Fuzzy solutions for general hyperbolic partial differential equations with local initial conditions i) u1(α) is a bounded left continuous non-decreasing function of α, ii) u2(α) is a bounded left continuous non-increasing function of α, iii) u1(1) ≤ u2(1). If g : Rn × Rn → Rn is a function, then, according to Zadeh’s extension principle we can extend g to En × En → En by the function defined by g(u, u)(z) = sup z=g(x,z) min{u(x), u(z)}. If g is continuous then [g (u, u)]α = g ([u]α , [u]α) for all u, u ∈ Rn, 0 ≤ α ≤ 1. Especially, we will define addition and scalar multiplication of fuzzy sets in En levelsetwises, that is, for all u, u ∈ Rn, 0 ≤ α ≤ 1, k ∈ R, k ̸= 0 [u+ u]α = [u]α + [u]α and [ky]α = k [u]α , where (u+ u)(x) = sup x=x1+x2 min{u(x1), u(x2)} and ku(x) = u(x/k). Supremum metric is the most commonly used metric on En defined by the Hausdorff metric distance between the level sets of the fuzzy sets d∞(u, u) = sup 0<α≤1 Hd ([u] α , [u]α) for all u, u ∈ En. It is obvious that (En, d∞) is a complete metric space [11]. From the properties of Hausdorff metric, we have: i) d∞(cu, cv) = |c|.d∞(u, v), ii) d∞(u+ u′, v + v′) ≤ d∞(u, v) + d∞(u′, v′), iii) d∞(u+ w, v + w) = d∞(u, v) for all u, v, u′, v′, w ∈ En and c ∈ R. 29 Nguyen Thi My Ha, Nguyen Thi Kim Son and Ha Thi Thanh Tam Definition 2.1. Let Ja = [0, a], Jb = [0, b], Jab = [0, a] × [0, b]. A map f : Jab → En is called continuous at (t0, s0) ∈ Jab if the multi-valued map fα (t, s) = [f (t, s)]α is continuous at (t0, s0) with respect to the Hausdorff metric dH for all α ∈ [0, 1]. In this paper, we denote C(Jab, En) be a space of all continuous functions f : J → En with the supremum weighted metric H1 defined by H1(f, g) = sup (s,t)∈J d∞(f (t, s) , g (t, s))e−λ(t+s). Since (En, d∞) is a complete metric space, it can shown that (C(Jab, En), H1) is also a complete metric space (see [11]). Definition 2.2. A map f : Jab × En → En is called continuous at point (t0, s0, x0) ∈ Jab × En provided, for any fixed α ∈ [0, 1] and arbitrary ϵ > 0, there exists δ(ϵ, α) > 0 such that dH ([f (t, s, x)] α , [f (t0, s0, x0)] α) < ϵ whenevermax |t− t0| , |s− s0| < δ(ϵ, α) and dH ([x]α , [x0]α) < δ(ϵ, α) for all (t, s, x) ∈ Jab × En. Definition 2.3. A function f : Jab → En is called integrably bounded if there exists an integrable function h ∈ L1(J,Rn) such that || y || ≤ h (s, t) for all y ∈ f0(s, t). Definition 2.4. Let f : Jab → En. The integral of f over Jab, denoted by ∫ a 0 ∫ b 0 f (t, s) dsdt is defined by(∫ a 0 ∫ b 0 f (t, s) dsdt )α = ∫ a 0 ∫ b 0 fα (t, s) dsdt = { ∫ a 0 ∫ b 0 v (t, s) dsdt|v : Jab → Rnis a measurable selection for fα} for all α ∈ (0, 1] (see [3]). A function f : Jab → En is integrable on Jab if∫ a 0 ∫ b 0 f (t, s) dsdt is in En. Let f, g : Jab → En be integrable and λ ∈ R. The intergral has the elementary properties as follows i) ∫ a 0 ∫ b 0 [f(t, s) + g(t, s)]dsdt = ∫ a 0 ∫ b 0 f(t, s)dsdt+ ∫ a 0 ∫ b 0 g(t, s)dsdt, ii) ∫ a 0 ∫ b 0 λf(t, s)dsdt = λ ∫ a 0 ∫ b 0 f(t, s)dsdt, ii) d∞( ∫ a 0 ∫ b 0 f(t, s)dsdt, ∫ a 0 ∫ b 0 (t, s)dsdt) ≤ ∫ a 0 ∫ b 0 d∞(f(t, s), g(t, s))dsdt. 30 Fuzzy solutions for general hyperbolic partial differential equations with local initial conditions Definition 2.5. Let x, y ∈ En. If there exists z ∈ En such that x = y + z then we call z the Hukuhara-difference of x and y, denoted x− y (see [11]). The definition of the fuzzy partial derivative is one of the most important concepts for fuzzy partial differential equation. Definition 2.6. Let f : Jab → En. The fuzzy partial derivative of f with respect to x at the point (x0, y0) ∈ J is a fuzzy set ∂f (x0, y0) ∂x ∈ En which is defined by ∂f (x0, y0) ∂x = lim h→0 f (x0 + h, y0)− f (x0, y0) h . Here the limit is taken in the metric space (En, d∞) and u− v is the Hukuhara-difference of u and v in En. The fuzzy partial derivative of f with respect to y and the higher order of fuzzy partial derivative of f at the point (x0, y0) ∈ Jab are defined similarly (see [6]). 3. Main result The aim of this section is to consider the existence and uniqueness of the fuzzy solutions for the general hyperbolic partial differential equation ∂2u(x, y) ∂x∂y + ∂(p1(x, y)u(x, y)) ∂x + ∂(p2(x, y)u(x, y)) ∂y + c(x, y)u(x, y) = f(x, y, u(x, y)) (3.1) for (x, y) ∈ Jab. The local initial conditions are u(0, 0) = u0, u(x, 0) = η1(x), u(0, y) = η2(y), (x, y) ∈ Jab, (3.2) where pi ∈ C(Jab,R); i = 1, 2, c ∈ C(Jab,R), η1 ∈ C(Ja, En), η2 ∈ C(Jb, En) are given functions; u0 ∈ En and f : Jab × C(Jab, En) → En which satisfies the following hypothesis: Hypothesis H There existsK > 0 such that d∞(f(x, y, u(x, y)), f(x, y, u(x, y))) ≤ Kd∞(u(x, y), u(x, y)) holds for all u, u ∈ En, (x, y) ∈ Jab. Definition 3.1. A function u ∈ C(Jab, En) is called a solution of the problem (3.1), (3.2) if it satisfies u(x, y) =q1(x, y)− ∫ y 0 p1(x, s)u(x, s)ds− ∫ x 0 p2(t, y)u(t, y)dt − ∫ x 0 ∫ y 0 c(t, s)u(t, s)dsdt+ ∫ x 0 ∫ y 0 f (t, s, u(t, s)) dsdt, 31 Nguyen Thi My Ha, Nguyen Thi Kim Son and Ha Thi Thanh Tam where q1(x, y) = η1(x) + η2(y)− u0 + ∫ y 0 p1(0, s)η2(s)ds+ ∫ x 0 p2(t, 0)η1(t)dt for all (x, y) ∈ Jab. By using the new weighted metric H1 in the space C(Jab, En), we receive the following result about the existence and uniqueness of solutions of the problem. Theorem 3.1. Assume that hypothesis H is satisfied. Then the problem (3.1), (3.2) has a unique solution in C(Jab, En) . Proof. Let p1 = sup(t,s)∈Ja×Jb |p1(t, s)|, p2 = sup(t,s)∈Ja×Jb |p2(t, s)|, c = sup(t,s)∈Ja×Jb |c(t, s)|. From Definition 3.1 for a fuzzy solution, we relize that the fuzzy solution of the problem (3.1), (3.2) (if it exists) is a fixed point of the operator N : C(Jab, E n)→ C(Jab, En) defined as follows: N(u)(x, y) =q1(x, y)− ∫ y 0 p1(x, s)u(x, s)ds− ∫ x 0 p2(t, y)u(t, y)dt − ∫ x 0 ∫ y 0 c(t, s)u(t, s)dsdt+ ∫ x 0 ∫ y 0 f (t, s, u(t, s)) dsdt. We will show thatN is a contraction operator. Indeed, if u, u ∈ C(Jab, En) and α ∈ (0, 1] then N(u(x, y)) =q1(x, y)− ∫ y 0 p1(x, t)u(x, t)dt− ∫ x 0 p2(s, y)u(s, y)dx − ∫ x 0 ∫ y 0 c(t, s)u(t, s)dsdt+ ∫ x 0 ∫ y 0 f (t, s, u(t, s)) dsdt and N(u(x, y)) =q1(x, y)− ∫ y 0 p1(x, t)u(x, y)dt− ∫ x 0 p2(s, y)u(s, y)ds − ∫ x 0 ∫ y 0 c(t, s)u(t, s)dsdt+ ∫ x 0 ∫ y 0 f (t, s, u(t, s)) dsdt. From the properties of supremum metric, we have: d∞(N(u(x, y)),N(u(x, y))) ≤ d∞( ∫ y 0 p1(x, s)u(x, s)ds, ∫ y 0 p1(x, s)u(x, s)ds) + d∞( ∫ x 0 p2(t, y)u(t, y)dt, ∫ x 0 p2(t, y)u(t, y)dt) + d∞( ∫ x 0 ∫ y 0 c(t, s)u(t, s)dsdt, ∫ x 0 ∫ y 0 c(t, s)u(t, s)dsdt) + d∞( ∫ x 0 ∫ y 0 f(t, s, u(t, s))dtds, ∫ x 0 ∫ y 0 f(t, s, u(t, s))dsdt). 32 Fuzzy solutions for general hyperbolic partial differential equations with local initial conditions Moreover d∞( ∫ y 0 p1(x, s)u(x, s)ds, ∫ y 0 p1(x, s)u(x, s)ds) ≤ sup (t,s)∈Ja×Jb |p1(t, s)|d∞( ∫ y 0 u(x, s)ds, ∫ y 0 u(x, s)ds) ≤ p1 ∫ y 0 d∞(u(x, s), u(x, s))ds. Hence for each (x, y) ∈ Ja × Jb, one gets e−λ(x+y)d∞( ∫ y 0 p1(x, s)u(x, s)ds, ∫ y 0 p1(x, s)u(x, s)ds) ≤ p1e−λ(x+y) ∫ y 0 d∞(u(x, s), u(x, s))e−λ(x+s)eλ(x+s)ds ≤ p1H1(u, u)e−λ(x+y) ∫ y 0 eλ(x+s)ds ≤ p1 λ H1(u, u). Similarly, we obtain: e−λ(x+y)d∞( ∫ x 0 p2(t, y)u(t, y)dt, ∫ x 0 p2(t, y)u(t, y)dt) ≤ p2 λ H1(u, u). Nevertheless d∞( ∫ x 0 ∫ y 0 c(s, t)u(s, t)dsdt, ∫ x 0 ∫ y 0 c(s, t)u(s, t)dsdt) ≤ sup (t,s)∈Ja×Jb |c(t, s)| ∫ x 0 ∫ y 0 d∞(u(s, t), u(s, t))dsdt ≤ c ∫ x 0 ∫ y 0 d∞(u(s, t), u(s, t))dsdt. Hence e−λ(x+y)d∞( ∫ x 0 ∫ y 0 c(t, s)u(t, s)dtds, ∫ x 0 ∫ y 0 c(t, s)u(t, s)dsdt) ≤ ce−λ(x+y) ∫ x 0 ∫ y 0 d∞(u(s, t), u(s, t))e−λ(t+s)eλ(t+s)dsdt ≤ cH2(u, u)e−λ(x+y) ∫ x 0 ∫ y 0 eλ(t+s)dsdt ≤ c λ2 H1(u, u). 33 Nguyen Thi My Ha, Nguyen Thi Kim Son and Ha Thi Thanh Tam Moreover, one gets d∞( ∫ x 0 ∫ y 0 f(t, s, u(t, s))dtds, ∫ x 0 ∫ y 0 f(t, s, u(t, s))dsdt) ≤ K ∫ x 0 ∫ y 0 d∞(u(t, s), u(t, s))dsdt. It implies that e−λ(x+y)d∞( ∫ x 0 ∫ y 0 f(t, s, u(t, s))dxdy, ∫ x 0 ∫ y 0 f(t, s, u(t, s))dsdt) ≤ Ke−λ(x+y) ∫ x 0 ∫ y 0 d∞(u(x, y), u(x, y))e−λ(t+s)eλ(t+s)dsdt ≤ KH1(u, u)e−λ(x+y) ∫ x 0 ∫ y 0 eλ(t+s)dsdt ≤ K λ2 H1(u, u). That shows H1(N(u(x, y)), N(u(x, y))) ≤ [p1 + p2 λ + c+K λ2 ]H1(u, u). Since we can choose λ > 0 satisfying p1 + p2 λ + c+K λ2 < 1, we receive N which is a contraction operator and by the Bannach fixed point theorem, N has a unique fixed point, that is a solution of the problem (3.1) - (3.2). The proof is completed. 4. Examples Example 4.1. The hyperbolic equation has the form ∂2u(x, y) ∂x∂y = −C1, (x, y) ∈ [0, 1]× [0, 1] (4.1) with the local conditions u(x, 0) = u(0, y) = u(0, 0) = C2, (4.2) where C1, C2 are triangular fuzzy numbers in [0,M ],M > 0 with the following level sets [C1] α = [Cα11, C α 12], [C2] α = [Cα21, C α 22] 34 Fuzzy solutions for general hyperbolic partial differential equations with local initial conditions for α ∈ [0, 1] and (x, y) ∈ [0, 1]× [0, 1]. In this problem, if f(x, y, u(x, y)) = −C1 then condition (H) is satisfied with K = 2. Therefore, from Theorem 3.1 there exists a solution to this problem. Next, we find this fuzzy solution. Assume that solution u has level sets [u]α = [u1(x, y) α, u2(x, y) α] for α ∈ [0, 1] and (x, y) ∈ [0, 1]× [0, 1]. We also have [ ∂2u(x, y) ∂x∂y ]α = [ ∂2uα1 (x, y) ∂x∂y , ∂2uα1 (x, y) ∂x∂y ]. Applying the extension principle, the fuzzy number −C1 has level sets [−C1]α = [min {−Cα11,−Cα12} ,max {−Cα11,−Cα12}] = [−Cα12,−Cα11] for α ∈ [0, 1] and (x, y) ∈ [0, 1]× [0, 1]. Thus the equation (4.1) is equivalent to the system ∂2uα1 (x, y) ∂x∂y = −Cα12, ∂2uα2 (x, y) ∂x∂y = −Cα11. (4.3) The local conditions (4.2) is equivalent to the following system uα1 (x, 0) = u α 1 (0, y) = u α 1 (0, 0) = C α 21, (4.4) uα2 (x, 0) = u α 2 (0, y) = u α 2 (0, 0) = C α 22. (4.5) The solutions of system (4.3) with conditions (4.4), (4.5) are uα1 (x, y) = −Cα12xy + Cα21, uα2 (x, y) = −Cα11xy + Cα22. Hence, the solution of problem (4.1), (4.2) has level sets [u]α = [−Cα12xy + Cα21,−Cα11xy + Cα22] for α ∈ [0, 1] and (x, y) ∈ [0, 1]× [0, 1]. We can write u(x, y) = −C1xy + C2. Example 4.2. Consider the fuzzy hyperbolic equation ∂2u(x, y) ∂x∂y + ∂u(x, y) ∂x + ∂u(x, y) ∂y + u(x, y) = 4Cex+y, (x, y) ∈ [0, 2]× [0, 2], (4.6) with the local conditions u(x, 0) = Cex, u(0, y) = Cey, u(0, 0) = C, (4.7) where C is a fuzzy triangular number in [0,M ],M > 0. C has level sets [C]α = [Cα1 , C α 2 ] for α ∈ [0, 1]. We have f(x, y, u(x, y)) = 4Cex+y then f satisfies condition (H) withK = 1. Hence, the 35 Nguyen Thi My Ha, Nguyen Thi Kim Son and Ha Thi Thanh Tam condition of Theorem 3.1 holds. Therefore, there exists a fuzzy solution of this problem. Next, we will give a clear solution. Suppose that solution u has level sets [u]α = [u1(x, y) α, u2(x, y) α] for α ∈ [0, 1] and (x, y) ∈ [0, 2]× [0, 2]. Define φ(Dx, Dy)U(x, y) = ∂2u(x, y) ∂x∂y + ∂u(x, y) ∂x + ∂u(x, y) ∂y + u(x, y) then φ(Dx, Dy)U(x, y) also has level sets [φ(Dx, Dy)U(x, y)] α =[ ∂2uα1 (x, y) ∂x∂y + ∂uα1 (x, y) ∂x + ∂uα1 (x, y) ∂y + uα1 (x, y), ∂2uα2 (x, y) ∂x∂y + ∂uα2 (x, y) ∂x + ∂uα2 (x, y) ∂y + uα2 (x, y)] for all α ∈ [0, 1] and (x, y) ∈ [0, 2]× [0, 2]. By the extension principle, we have [4Cex+y]α = [min{4Cα1 ex+y, 4Cα2 ex+y},max{4Cα1 ex+y, 4Cα2 ex+y}] = [4Cα1 e x+y, 4Cα2 e x+y] Similarly [Cex]α = [Cα1 e x, Cα2 e x], [Cey]α = [Cα1 e y, Cα2 e y] for all α ∈ [0, 1] and (x, y) ∈ [0, 2]× [0, 2]. Hence, the equation (4.6) is equivalent to the system ∂2uα1 (x, y) ∂x∂y + ∂uα1 (x, y) ∂x + ∂uα1 (x, y) ∂y + uα1 (x, y) = 4C α 1 e x+y, (4.8) ∂2uα2 (x, y) ∂x∂y + ∂uα2 (x, y) ∂x + ∂uα2 (x, y) ∂y + uα2 (x, y) = 4C α 2 e x+y. (4.9) The local conditions (4.7) are equivalent to the following uα1 (x, 0) = 4C α 1 e x, uα1 (0, y) = 4C α 1 e x, uα1 (0, 0) = 4C α 1 , (4.10) uα2 (x, 0) = 4C α 2 e x, uα1 (0, y) = 4C α 2 e x, uα2 (0, 0) = 4C α 2 . (4.11) Solving the problems (4.8), (4.10) and (4.9), (4.11) we have the solutions: uα1 (x, y) = 4C α 1 e x+y, uα2 (x, y) = 4C α 2 e x+y for all α ∈ [0, 1] and (x, y) ∈ [0, 2]× [0, 2]. Thus, the solution of problem (4.6), (4.7) is a fuzzy function u, which has level sets [u]α = [4Cα1 e x+y, 4Cα2 e x+y] and we can write u = 4Cex+y. 36 Fuzzy solutions for general hyperbolic partial differential equations with local initial conditions Example 4.3. We study the following hyperbolic equation: ∂2u(x, y) ∂x∂y − ∂u(x, y) ∂x = 2C1 − 2C1(x+ y), (x, y) ∈ [0, 1 8 ]× [0, 1 8 ], (4.12) u(x, 0) = C1x 2 + C3, u(0, y) = C1y 2 + C2siny + C3, u(0, 0) = C3, (4.13) with (x, y) ∈ [0, 1 8 ] × [0, 1 8 ] and Ci being triangular fuzzy numbers having level sets [Ci] α = [Cαi1, C α i2] for i = 1, 3, α ∈ [0, 1] and (x, y) ∈ [0, 18 ]× [0, 18 ]. We have f(x, y, u) = C1y2+C2siny+C3. It follows that the condition in Theorem 3.1 is satisfied withK = 1. Therefore there exists a solution of this problem. Suppose that solution u has level sets [u]α = [u1(x, y)α, u2(x, y)α] for α ∈ [0, 1] and (x, y) ∈ [0, 1 8 ]× [0, 1 8 ]. Using the extension principle, we have [ ∂2u(x, y) ∂x∂y − ∂u(x, y) ∂x ]α = [ ∂2uα1 (x, y) ∂x∂y − ∂u α 2 (x, y) ∂x , ∂2uα2 (x, y) ∂x∂y − ∂u α 1 (x, y) ∂x ] and [2C1 − 2C1(x+ y)]α = [2Cα11 − 2Cα12(x+ y)], 2Cα12 − 2Cα11(x+ y)] [C1x 2 + C3] α = [Cα11x 2 + Cα31, C α 12x 2 + Cα32], [C1y 2 + C2siny + C3] α = [Cα11y 2 + Cα21siny + C α 31, C α 12y 2 + Cα22siny + C α 32]. Thus equation (4.12) is equivalent to the following system ∂2uα1 (x, y) ∂x∂y − ∂u α 2 (x, y) ∂x = 2C11 − 2C12(x+ y), ∂2uα2 (x, y) ∂x∂y − ∂u α 1 (x, y) ∂x = 2Cα12 − 2Cα11(x+ y). The initial conditions (4.13) are equivalent to the system uα1 (x, 0) = C α 11x 2 + Cα31, u α 1 (0, y) = C α 11y 2 + Cα21siny + C α 31, u α 1 (0, 0) = C α 31 and uα2 (x, 0) = C α 12x 2 + Cα32, u α 1 (0, y) = C α 12y 2 + Cα22siny + C α 32, u α 1 (0, 0) = C α 32. The solutions of the system are given by uα1 = C α 11(x+ y) 2 + Cα21siny + C α 31 and uα2 = C α 12(x+ y) 2 + Cα22siny + C α 32. Therefore, the problem (4.12), (4.13) has a solution u(x, y) = C1(x+ y)2+C2siny+C3 with level sets [u]α = [Cα11(x+ y) 2 + Cα21siny + C α 31, C α 12(x+ y) 2 + Cα22siny + C α 32]. 37 Nguyen Thi My Ha, Nguyen Thi Kim Son and Ha Thi Thanh Tam 5. Conclusion We have investigated the existence and uniqueness of the fuzzy solution for the general hyperbolic partial differential equation with local conditions.This result is illustrated by some examples. The next step in the direction proposed here is to study the fuzzy solution for the general hyperbolic partial differential equation with nonlocal conditions and integral boundary conditions. REFERENCES [1] Allahviranloo T., 2002. Difference methods for fuzzy partial differential equations. Computational Methods in Applied Mathematics, 2(3), pp. 233-242. [2] Arara A., Benchohra M., Ntouyas S.K. and Ouahab A., 2005. Fuzzy Solutions for Hyperbolic Partial Differential Equations. International Journal of Applied Mathematical Sciences, 2(2), pp.
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