DIFFERENCE METHODS FOR FUZZY PARTIAL DIFFERENTIAL EQUATIONS

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1 COMPUTATIONAL METHODS IN APPLIED MATHEMATICS, Vol.2(2002), No.3, pp c Institute of Mathematics of the National Academy of Sciences of Belarus DIFFERENCE METHODS FOR FUZZY PARTIAL DIFFERENTIAL EQUATIONS TOFIGH ALLAHVIRANLOO Department of Mathematics, Science and Research Branch Islamic Azad University Tehran, Iran alahviranlo@yahoo.com Abstract In this paper numerical methods for solving fuzzy partial differential equations (FPDE) is considered. Fuzzy reachable set can be approximated by proposed methods with complete error analysis which is discussed in details. The methods are illustrated by solving some linear and nonlinear FPDE Mathematics Subject Classification: 34A12, 65L05. Keywords: fuzzy partial differential equations, difference methods. 1. Introduction Knowledge about dynamical systems modeled by differential equations is often incomplete or vague. For example, for parametric quantities, functional relationships, or initial conditions, the well-known methods of solving FPDE analytically or numerically can only be used for finding the selected system behavior, e.g., by fixing unknown parameters to some plausible values. However, in this way it is not possible to characterize the hole set of system behaviors compatible with our partial knowledge. It motivates us all such systems as Fuzzy Input-Fuzzy Output (FIFO) systems. Here, we are going to operationalize our approach, i.e., we are going to propose a method for computing approximate solution for a fuzzy partial differential equation using numerical methods. Since finding this set of solutions analytically does only work with trivial examples, a numerical approach seems to be the only way of solving such problems. The topics of fuzzy differential equations which attracted growing interest for some time, in particular, in relation to fuzzy control, have been rapidly developed in recent years. Also the topics of numerical methods for solving fuzzy differential equations have been rapidly growing in recent years. Introduction to FPDE is presented by J. Buckley and T. Feuring in [2]. The paper is organized as follows: In Section 2 some basic definitions and results on fuzzy numbers and definition of a fuzzy derivative, which have been discussed by S. Seikkala in [4]. In Section 3 we define the problem, which is a fuzzy partial differential equation where a numerical solution is the main interest of this work. The difference method for fuzzy partial differential equation is discussed in Section 4. The proposed algorithm is illustrated by solving some examples in Section 5 and conclusions are drown in Section 6.

2 234 T. Allahviranloo 2. Preliminaries We begin this Section with defining the notation we use in the paper. We place sign the tilde over a letter to denote a fuzzy subset of real numbers. We write Ã(x), a number in [0, 1], for the membership function of à evaluated at x. An α cut of Ã, written Ã[α], is defined as {x Ã(x) α}, for 0 < α 1. The triangular fuzzy number Ñ is defined by three numbers a 1 < a 2 < a 3, where the graph of Ñ(x), the membership function of the fuzzy number Ñ, is a triangle with the base on the interval [a 1, a 3 ] and vertex at x = a 2. We specify Ñ as (a 1/a 2 /a 3 ). We write: (1) Ñ > 0 if a 1 > 0; (2) Ñ 0 if a 1 0; (3) Ñ < 0 if a 3 < 0; and (4) Ñ 0 if a 3 0. Let E be the set of all upper semicontinuous normal convex fuzzy numbers with bounded α level sets. Since the α cuts of fuzzy numbers are always closed and bounded, the intervals we write are Ñ[α] = [N(α), N(α)], for all α. We denote by κ the set of all nonempty compact subset of R and by κ c the subsets of κ consisting of nonempty convex compact sets. Recall that ρ(x, A) = min x a a A is the distance of the point x R from A κ and that the Hausdorff separation ρ(a, B) of A, B κ is defined as ρ(a, B) = max ρ(a, B). a A Note that the notation is consistent, since ρ(a, B) = ρ({a}, B). Now, ρ is not a metric. In fact, ρ(a, B) = 0 if and only if A B. The Hausdorff metric d H on κ is defined by The metric d H is defined on E as d H (A, B) = max{ρ(a, B), ρ(b, A)}. d (ũ, ṽ) = sup{d H (ũ[α], ṽ[α]) : 0 r 1}, ũ, ṽ E. 3. A fuzzy partial differential equation Consider the FPDE ϕ(d x, D y )Ũ(x, y) = F (x, y, K), (3.1) subject to certain boundary conditions where the operator ϕ(d x, D y ) is the polynomial, with a constant coefficient in D x and D y, where D x (D y ) stands for the partial derivative with respect to x(y). The boundary conditions can be of the form Ũ(0, y) = C 1, Ũ(x, 0) = C 2, Ũ(M 1, y) = C 3,..., Ũ(0, y) = C 1, Ũ(0, y) = g 1(y; C 4 ), Ũ(x, 0) = f 1 (x; C 5 ),.... F (x, y, K) is the fuzzy function which has K = ( k 1,..., k n ) for k i which is a triangular fuzzy number in J i, 1 i n. Let I 1 = [0, M 1 ], I 2 = [0, M 2 ]. The fuzzy function Ũ maps I 1 I 2 into fuzzy numbers. Also, let C = ( c 1,..., c m ) with c i being triangular fuzzy numbers in the intervals L i, 1 i m. Let n m K[α] = ki [α], C[α] = c i [α]. i=1 Let Ũ(x, y)[α] = [U(x, y; α), U(x, y; α)]. We assume that the U(x, y; α) and U(x, y; α) have continuous partial derivatives so that ϕ(d x, D y )U(x, y; α) and ϕ(d x, D y )U(x, y; α) are continuous for all (x, y) I 1 I 2, and all α. Define Γ(x, y; α) = ϕ(d x, D y )Ũ(x, y)[α] = [ϕ(d x, D y )U(x, y; α), ϕ(d x, D y )U(x, y; α)], (3.2) i=1

3 Difference methods for fuzzy partial differential equations 235 for all (x, y) I 1 I 2, and all α. If for each fixed (x, y) I 1 I 2, Γ(x, y; α) defines the α cut of a fuzzy number, then we will say that Ũ(x, y) is differentiable. Sufficient conditions for Γ(x, y; α) to define α cuts of a fuzzy number are: 1. ϕ(d x, D y )U(x, y; α) is an increasing function of α for each (x, y) I 1 I 2 ; 2. ϕ(d x, D y )U(x, y; α) is decreasing function of α for each (x, y) I 1 I 2 ; and 3. ϕ(d x, D y )U(x, y; 1) ϕ(d x, D y )U(x, y; 1) for all (x, y) I 1 I 2. Consider the system of partial differential equations for all (x, y) I 1 I 2 and all α [0, 1], where ϕ(d x, D y )U(x, y; α) = F (x, y; α), (3.3) ϕ(d x, D y )U(x, y; α) = F (x, y; α), (3.4) F (x, y; α) = min{f (x, y, k) k K[α]}, (3.5) F (x, y; α) = max{f (x, y, k) k K[α]}. (3.6) We append to equations (3.3) and (3.4) any boundary conditions, for example, if they are Ũ(0, y) = C 1 and Ũ(M 1, y) = C 2, then we add to equation (3.3) and U(0, y; α) = C 1 (α), U(M 1, y; α) = C 2 (α) (3.7) U(0, y; α) = C 1 (α), U(M 1, y; α) = C 2 (α) (3.8) to equation (3.4) where C i [α] = [C i (α), C i (α)], i = 1, 2. Let U(x, y; α) and U(x, y; α) solves equations (3.3) and (3.4), plus the boundary equations, respectively. If Ũ(x, y)[α] = [U(x, y; α), U(x, y; α)], defines the α cut of a fuzzy number, for all (x, y) I 1 I 2, then Ũ(x, y) is the solution for (3.1), see [2]. Let be the reachable set that is, := {x : I1 I 2 R x is solution of (3.1)} C(I 1 I 2 ), the set, consists of more than one element, that is we have a bundle of trajectories, [3]. 4. Difference methods In this section we solve the three types of FPDE as numerically. 1. Finite difference Poisson equation Consider the fuzzy elliptic partial differential equation (Poisson) 2 Ũ x (x, y) + 2 Ũ 2 y (x, y) = F (x, y, K), (4.1) 2

4 236 T. Allahviranloo where ϕ(d x, D y ) = Let R = {(x, y) (x, y) I x 2 y 2 1 I 2 } and S denote the boundary of R. We must solve the following problems: x (x, y; α) + 2 U (x, y; α) = F (x, y; α), 2 y2 U(x, y; α) = g(x, y; α), x (x, y; α) + 2 U (x, y; α) = F (x, y; α), 2 y2 U(x, y; α) = g(x, y; α), (4.2) (4.3) for (x, y) S. Now we solve problems (4.2) and (4.3) numerically so that the first step is to choose the integers n and m and define the step size h and l by h = M 1 /n and l = M 2 /m where x i = ih, i = 0, 1,..., n and y j = jl, j = 0, 1,..., m. Let U(x i, y j ) V i,j and U(x i, y j ) V i,j. We obtain the difference method by using the Taylor series in x and y for (4.2) and (4.3), see [1], to form the difference quotient x (x i, y 2 j ; α) = 1 h [V i+1,j(α) 2V 2 i,j (α) + V i 1,j (α)] h2 4 U 12 x (ξ i, y 4 j ; α), (4.4) x (x i, y 2 j ; α) = 1 h [V i+1,j(α) 2V 2 i,j (α) + V i 1,j (α)] h2 4 U 12 x 4 (ξ i, y j ; α), (4.5) ξ i, ξ i (x i 1, x i+1 ), provided that e 2 U (x, y) E, it means we only need to check if x 2 ( 2 U (x, y; α)) > 0 and ( 2 U (x, y; α)) < 0, since the 2 U and 2 U are continuous α x 2 α x 2 x 2 x 2 and 2 U (x, y; 1) = 2 U (x, y; 1), then following equations defines the α cuts of fuzzy x 2 x 2 numbers, and also 2 Ũ x (x, y)[α] = U 2 [ 2 x (x, y; α), 2 U (x, y; α)], α (0, 1], (4.6) 2 x2 Ṽ i,j [α] = [V i,j (α), V i,j (α)], α (0, 1], (4.7) and where 4 Ũ x (ξ i, y 4 j )[α] = [ 4 U x (ξ i, y 4 j ; α), 4 U x (ξ i, y 4 j ; α)], α (0, 1], (4.8) x (x, y; α) = min{u u 2 Ũ (x, y)[α]}, 2 x2 (4.9) x (x, y; α) = max{u u 2 Ũ (x, y)[α]}, 2 x2 (4.10) V i,j (α) = min{v v Ṽi,j[α]}, (4.11) V i,j (α) = max{v v Ṽi,j[α]. (4.12)

5 Difference methods for fuzzy partial differential equations 237 Also for 2 e U y 2 and 4 e U y 4 we have the following formulas: y (x i, y 2 j ; α) = 1 l [V i,j+1(α) 2V 2 i,j (α) + V i,j 1 (α)] l2 4 U 12 y (x i, η 4 j ; α), (4.13) y (x i, y 2 j ; α) = 1 l [V i,j+1(α) 2V 2 i,j (α) + V i,j 1 (α)] l2 4 U 12 y (x i, η j; α), 4 (4.14) η j, η j (y j 1, y j+1 ). Using formulas (4.4) and (4.5) in (4.2) and (4.13) and (4.14) in (4.3), respectively, allows us to express the Poisson equations at the point (x i, y j ) as [ (h ) ] V l i,j (α) (V i+1,j (α) + V i 1,j (α)) ( ) 2 h (4.15) (V l i,j+1(α) + V i,j 1(α)) = h 2 F (x i, y j ; α), V 0,j (α) = g(x 0, y j ; α), V i,0 (α) = g(x i, y 0 ; α), V i,m (α) = g(x i, y m ; α), [ (h ) ] V i,j (α) ( V i+1,j (α) + V i 1,j (α) ) l ( ) 2 h ( V i,j+1(α) + V ) i,j 1(α) = h 2 F (x i, y j ; α), l V 0,j (α) = g(x 0, y j ; α), V n,j (α) = g(x n, y j ; α), V i,0 (α) = g(x i, y 0 ; α), (4.16) for i = 1, 2,..., n 1 and j = 1, 2,..., m 1. The local truncation error for this equations is τ i,j = O(h 2 + l 2 ). 2. Backward difference heat equation Consider the fuzzy heat equation which is an example of the fuzzy parabolic equations. Let ϕ(d x, D y ) = 2 β2. y x 2 Ũ t (x, t) 2 Ũ β2 (x, t) = 0, x2 0 < x < l, t > 0, (4.17) where F (t, x, K) = 0 subject to the conditions and Now we have Ũ(0, t) = Ũ(l, t) = 0, t > 0, Ũ(x, 0) = f(x), 0 x l. t (x, t; α) 2 U β2 (x, t; α) = 0, x2 U(0, t; α) = U(l, t; α) = 0, t > 0, U(x, 0; α) = f(x; α), 0 x l, t (x, t; α) 2 U β2 (x, t; α) = 0, x2 U(0, t; α) = U(l, t; α) = 0, t > 0, U(x, 0; α) = f(x; α), 0 x l. (4.18) (4.19)

6 238 T. Allahviranloo By using the Taylor series in t t (x i, t j ; α) = 1 l [V i,j+1(α) V i,j (α)] l 2 t (x i, µ 2 j ; α), (4.20) t (x i, t j ; α) = 1 l [V i,j+1(α) V i,j (α)] l 2 t (x i, µ j; α), 2 (4.21) µ j, µ j (t j, t j+1 ), provided that U e (x, t) E. If we use equations (4.4) and (4.5) and (4.20) and (4.21) in t (4.18) and (4.19), we obtain this difference equations which constitute Crank-Nicolson Method V i,j+1 = (1 2β2 l h )V 2 i,j + β 2 l h (V 2 i+1,j + V i 1,j ), (4.22) V i,0 (α) = f(x i ; α), V 0,1 (α) = V m,1 (α) = 0, V i,j+1 = (1 2β2 l h )V 2 i,j + β 2 l h (V 2 i+1,j + V i 1,j ), V i,0 (α) = f(x i ; α), V 0,1 (α) = V m,1 (α) = 0. (4.23) The grid points are (x i, t j ), where x i = ih, h = l m for i = 1, 2,..., m 1 and t j = jl for j = 1, 2,.... Where f(x)[α] = [f(x; α), f(x; α)]. The local truncation error for this equations is τ i,j = O(l + h 2 ). 3. Finite difference wave equation Consider the wave equation, which an example of a hyperbolic fuzzy partial differential equation. The fuzzy wave equation is given by the differential equation 2 Ũ t (x, t) 2 Ũ 2 β2 (x, t) = 0, 0 < x < l, t > 0, x2 subject to the conditions Ũ(0, t) = Ũ(l, t) = 0, t > 0, Ũ(x, 0) = f(x), 0 x l, and Ũ (x, 0) = g(x), 0 x l, t where β is a constant. We have two problems t (x, y; α) 2 U 2 β2 (x, y; α) = F (x, y; α), x2 U(0, t; α) = U(l, t; α) = 0, t > 0, U(x, 0; α) = f(x; α), 0 x l, (x, 0; α) = g(x; α), t 0 x l, t (x, y; α) 2 U 2 β2 (x, y; α) = F (x, y; α), x2 U(0, t; α) = U(l, t; α) = 0, t > 0, U(x, 0; α) = f(x; α), 0 x l, (x, 0; α) = g(x; α), 0 x l. t (4.24) (4.25)

7 Difference methods for fuzzy partial differential equations 239 Using equations (4.4) and (4.5) in (4.24) and (4.13) and (4.14) in (4.25), respectively, leads to the difference equations V i,j+1 = 2(1 λ 2 )V i,j + λ 2 (V i+1,j + V i 1,j ) V i,j 1, V 0,j = V m,j = 0, V i,0 = f(x i ; α), V i,1 = V i,0 + lg(x i ; α), V i,j+1 = 2(1 λ 2 )V i,j + λ 2 (V i+1,j + V i 1,j ) V i,j 1, V 0,j = V m,j = 0, V i,0 = f(x i ; α), V i,1 = V i,0 + lg(x i ; α), (4.26) (4.27) for each i = 1, 2,..., m 1, j = 1, 2,..., where λ = βl. The local truncation error for h this equations is τ i,j = O(l 2 + h 2 ). 5. Examples Example 5.1. Consider the fuzzy Poisson s equation 2 Ũ x (x, y) + 2 Ũ 2 y (x, y) = kxe y, 0 < x < 2, 0 < y < 1, (5.1) 2 where F (x, y, K) = kxe y, and k[α] = [k(α), k(α)] = [ α, α] with the boundary conditions Ũ(0, y) = 0, Ũ(2, y) = 2 ke y, 0 y 1, (5.2) Ũ(x, 0) = kx, Ũ(x, 1) = kex, 0 x 2. (5.3) The exact solution for x (x, y; α) + 2 U 2 y (x, y; α) = 2 k(α)xey, (5.4) x (x, y; α) + 2 U 2 y (x, y; α) = 2 k(α)xey, (5.5) are U(x, y; α) = k(α)xe y and U(x, y; α) = k(α)xe y. We will use equations (4.15) and (4.16) to approximate the exact solutions with n = 20 and m = 20. Fig. 1 approximate solutions at the point (0.5, 0.2) for each α (0, 1]. The Hausdorff distance between the solutions is

8 240 T. Allahviranloo exact approx Figure 1. h=0.025, l=0.01 Example 5.2. Consider the fuzzy parabolic equation Ũ t (x, t) 2 Ũ (x, t) = 0, x2 0 < x < l, t > 0 (5.6) with the boundary conditions Ũ(0, t) = Ũ(l, t) = 0, t > 0, and and Ũ(x, 0) = f(x) = k sin(πx), 0 x 1, k[α] = [k(α), k(α)] = [α 1, 1 α]. The exact solution for t (x, t; α) 2 U (x, t; α) = 0, x2 (5.7) t (x, t; α) 2 U (x, t; α) = 0, x2 (5.8) for 0 < x < l, t > 0 are U(x, y; α) = k(α)e π2t sin(πx) and U(x, y; α) = k(α)e π2t sin(πx). We use the equations (4.22) and (4.23) to approximate the exact solutions with n = 1 and m = 10. Fig. 2 shows the exact and the approximate solutions at the point (1, 0.01) for each α (0, 1]. The Hausdorff distance of solutions between the e 004.

9 Difference methods for fuzzy partial differential equations exact approx Figure 2. h=0.1, l=0.01 Example 5.3. Consider the fuzzy hyperbolic problem 2 Ũ t (x, t) Ũ (x, t) = 0, x2 0 < x < l, t > 0 with the boundary conditions Ũ(0, t) = Ũ(l, t) = 0, t > 0, Ũ(x, 0) = k sin(πx), 0 x 1, and the initial conditions Ũ (x, 0) = 0, 0 x 1, t where k[α] = [k(α), k(α)] = [ α, α]. The exact solutions are U(x, t; α) = k(α) sin(πx) cos(2πt), U(x, t; α) = k(α) sin πx) cos(2πt), for α (0, 1]. Using (4.26) and (4.27) with m = 10, T = 1, and N = 20, implies that h = 0.1, l = 0.05 and λ = 1. Fig. 3 shows the exact and the approximate solutions at the point (1, 1) for each α (0, 1]. The Hausdorff distance between the solutions is e 016.

10 242 T. Allahviranloo 1 exact approx Figure Conclusions We presented difference methods for solving FPDE. These numerical solutions are based on the seikkala derivative. If these solutions define α cuts of a fuzzy number, then the solutions of FPDE, would exist, which has been concluded from the numerical values. References [1] S. Abbasbandy and T. Allahviranloo, On the fuzzy initial value problem, CMAM, 2 (2002), No. 2, pp [2] J. J. Buckley and T. Feuring, Introduction to fuzzy partial differential equations, Fuzzy Sets and Systems, 105 (1999), pp [3] E. Hüllermeier, Int. J. Uncertainty, Fuzziness. ε Knowledge-Bases Systems, 35 (1990), pp [4] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems, 24 (1987), pp Received 18 Jul Revised 16 Sep. 2002