Solution of a fuzzy differential equation with interactivity via Laplace transform

1 Introduction

Differential equations appear in many disciplines, such as engineering, physics, demography, economics, etc [8]. For instance, the authors in [6] modelled a period of crisis in the stock market using the equation x ″ ( t ) + a 2 x ( t ) = 0 .(1)

Since these types of phenomena often involve uncertainties, it is reasonable to deal with these problems using fuzzy set theory. Thus, in the case where a²x (t) (with a ∈ ℝ , a ≠ 0 ) and x″ (t) are fuzzy numbers for each t in Eq. (1), we are faced with a fuzzy differential equation given by an addition of two fuzzy numbers whose sum is zero. In order to give a suitable mathematical treatment to certain classes of fuzzy differential equations, it is necessary to use an arithmetic for fuzzy numbers that is compatible with the phenomenon under consideration. Therefore, if a²x (t) and x″ (t) in Eq. (1) are fuzzy numbers, then we have an addition of two fuzzy numbers, such that the sum is zero. This sum only makes sense if one uses the arithmetic for linearly correlated fuzzy numbers [2, 4].

Fuzzy differential equations with initial value conditions given by fuzzy numbers are called fuzzy initial value problems (FIVPs). Several authors have studied different techniques for solving FIVPs using, in many cases, fuzzy derivatives and the Zadeh extension principle [5]. However, other techniques have also been used. In [11], the authors introduce the concept of a fuzzy Laplace transform using the notion of strongly generalized differential. This operator can be used to obtain solutions of first and second orders FIVPs [11, 12]. Conditions for the existence of the fuzzy Laplace transform are established in [12]. Recently, in [4], the authors introduced the concept of interactive differentiability to solve a first order FIVP.

In this paper, we solve the differential equation given in Eq. (1), with a ∈ ℝ , a ≠ 0 , with uncertain initial conditions given by fuzzy numbers. To this end, we use the arithmetic for linearly correlated fuzzy numbers and the properties of fuzzy Laplace transform established in Proposition 7. It is worth noting that this paper extends the conference paper [10].

The definitions of fuzzy number, the extension principle and arithmetic for fuzzy numbers were extracted from [5] and the references therein.

Definition 1. Let U be a topological space. A fuzzy subset A of U is characterized by a membership function μ _A  : U → [0, 1], where μ _A  (x) denotes the degree that the element x belongs to the fuzzy set A.

Note that a classical subset A of U, can be associate with a fuzzy subset whose membership function is given by the characteristic function χ _A  (x). For notation convenience, we may use the symbol A (x) instead of μ _A  (x).

Definition 2. The α-levels of the fuzzy subset A are defined by [ A ] α = { { x ∈ U ; A ( x ) ≥ α } if 0 < α ≤ 1 , { x ∈ U ; A ( x ) > 0 } ¯ if α = 0 ,(2) where X ¯ denotes the closure of the subset X of the U.

Definition 3. A fuzzy subset A of ℝ is a fuzzy number if satisfies the following properties:

i) all α-levels of A are closed and nonempty intervals of ℝ .

ii) the set {x ; A (x)>0 } is a bounded set of ℝ .

From Definition 3, the α-level of a fuzzy number A is represented by theirs interval endpoints [ A ] α = [ a - α , a + α ] , ∀ α ∈ [ 0 , 1 ] .

We use the symbol F ( X ) to denote the fuzzy subsets of X and ℝ F to denote the set of all fuzzy numbers. An example of fuzzy number is the triangular fuzzy number, whose α-levels are given by [ A ] α = [ ( m - a - 0 ) α + a - 0 , ( m - a + 0 ) α + a + 0 ] , for all α ∈ [0, 1], where [ A ] 0 = [ a - 0 , a + 0 ] and {m} = [A] ¹ (a real number). A triangular fuzzy number is denoted by the triple ( a - 0 ; m ; a + 0 ) .

Theorem 1. [9] Let [ a - α , a + α ] , 0 < α ≤ 1, be a given family of non-empty intervals. If

(i) [ a - α , a + α ] ⊃ [ a - β , a + β ] for 0 < α ≤ β

and

(ii) [ lim k → ∞ a - α k , lim k → ∞ a + α k ] = [ a - α , a + α ] , whenever (α _k ) is a non-decreasing sequence converging to α ∈]0, 1], then the family [ a - α , a + α ] , 0 < α ≤ 1, represents the α- level sets of a fuzzy number A ∈ ℝ F . Conversely, if [ a - α , a + α ] , 0 < α ≤ 1, are the α-level sets of a fuzzy number A ∈ ℝ F then the conditions (i) and (ii) hold true.

The Zadeh’s extension principle is a mathematical method to extend classical functions to deal with fuzzy sets as arguments inputs.

Definition 4. Let f : X × Y → Z be a classical function and let A ∈ F ( X ) and B ∈ F ( Y ) . The Zadeh extension f ˆ of f, applied to (A, B), is the fuzzy subset f ˆ ( A , B ) of Z, whose membership function is defined by f ˆ ( A , B ) ( z ) = { sup ( x , y ) ∈ f - 1 ( z ) min { A ( x ) , B ( y ) } , if f - 1 ( z ) ≠ ∅ , 0 , if f - 1 ( z ) = ∅ ,(3) where f^-1 (z) = {(x, y) : f (x, y) = z} .

We can apply the Zadeh extension principle to define the usual addition ⊕ of two fuzzy numbers A and B and the usual multiplication ⊙ of a scalar λ and a fuzzy number A as follows:

Proposition 1. Let A and B be two fuzzy numbers with [ A ] α = [ a - α , a + α ] and [ B ] α = [ b - α , b + α ] . For all α ∈ [0, 1] and λ ∈ ℝ , we have [ B ⊕ A ] α = [ B ] α + [ A ] α = [ b − α + a − α , b + α + a + α ] , [ λ ⊙ A ] α = λ [ A ] α = { [ λ a − α , λ a + α ] , i f λ ≥ 0 , [ λ a + α , λ a − α ] , i f λ < 0.

The notion of interactivity among fuzzy numbers arises from a given joint possibility distribution.

Definition 5. [2] Let A 1 , … , A n ∈ ℝ F and let J be a fuzzy subset of ℝ n . The fuzzy subset J is called a joint possibility distribution of A₁, …, A _n if A i ( x i ) = sup x j ∈ ℝ , j ≠ i J ( x 1 , … , x n ) ,(4) for all x i ∈ ℝ and for all i = 1, …, n.

Next, we present the concept of sup-J extension principle.

Definition 6. [2] Let A 1 , … , A n ∈ ℝ F and f : ℝ n → ℝ . Given a joint possibility distribution J of A₁, …, A _n , the sup-J extension of f at (A₁, …, A _n ) is the fuzzy subset f ˜ ( J ) : = f J ( A 1 , … , A n ) of ℝ whose membership function is given by f ˜ ( J ) ( z ) = f J ( A 1 , … , A n ) ( z ) = sup ( x 1 , … , x n ) ∈ f - 1 ( z ) J ( x 1 , … , x n )(5) for all z ∈ ℝ , where f^-1 (z) = {(x₁, …, x _n ) : f (x₁, …, x _n ) = z} .

We can use the sup-J extension principle to generate an arithmetic on interactive fuzzy numbers. Here we focus on the special type of interactivity called linearly correlation.

Definition 7. [2, 4] Two fuzzy numbers A and B are linearly correlated if there exists q , r ∈ ℝ , with q ≠ 0, such that [ B ] α = q [ A ] α + r for each α ∈ [0, 1]. In this case, it is written B = qA + r.

Definition 8. [2] If A and B are linearly correlated fuzzy numbers, then one can define the addition and subtraction of A and B as follows

i) [A+  _L B]  ^α  = { a + b ;  a ∈ [A]  ^α , b = qa + r },

ii) [A -  _L B]  ^α  ={ a - b ;  a ∈ [A]  ^α , b = qa + r }.

Proposition 2. [2] The addition and subtraction operations of linearly correlated fuzzy numbers A and B, for all α ∈ [0, 1], are given respectively by [ B + L A ] α = ( q + 1 ) [ A ] α + r , [ B - L A ] α = ( q - 1 ) [ A ] α + r .(6)

Remark 1. Let A ∈ ℝ F ∖ ℝ . Note that B +  _L A = 0 if and only if q = -1 and r = 0.

Proposition 3. [2, 4] Let A and B be linearly correlated fuzzy numbers, where [B]  ^α  = q [A]  ^α  + r with q , r ∈ ℝ , q ≠ 0, A = [ a - α , a + α ] , and B = [ b - α , b + α ] . Then [ B + L A ] α = { [ b − α + a − α , b + α + a + α ] i f q > 0 , [ b + α + a − α , b − α + a + α ] i f − 1 ≤ q < 0 , [ b − α + a + α , b + α + a − α ] i f q < − 1. [ B − L A ] α = { [ b − α − a − α , b + α − a + α ] i f q ≥ 1 , [ b + α − a + α , b − α − a − α ] i f 0 < q < 1 , [ b − α − a + α , b + α − a − α ] i f q < 0.

Definition 9. [7] Let A , B ∈ ℝ F , the function D ∞ : ℝ F × ℝ F ⟶ [ 0 , + ∞ [ defined by D ∞ ( A , B ) = sup 0 ≤ α ≤ 1 max { ∣ a - α - b - α ∣ , ∣ a + α - b + α ∣ }

is called Pompeiu-Hausdorff distance between the fuzzy numbers A and B.

Definition 10. [7] Let A , B ∈ ℝ F . If there exists C ∈ ℝ F such that A = B ⊕ C, then C is called the H-difference (or Hukuhara difference) of A and B, and it is denoted by A ⊖ B.

Definition 11. [1] Let f : ( a , b ) → ℝ F and t₀ ∈ (a, b). We say that f is strongly generalized differentiable at t₀ if there exists an element f ′ ( t 0 ) ∈ ℝ F , such that for all h > 0 sufficiently small, one of these conditions holds true

i ) lim h ↘ 0 f ( t 0 + h ) ⊖ f ( t 0 ) h = lim h ↘ 0 f ( t 0 ) ⊖ f ( t 0 - h ) h = f ′ ( t 0 ) ii ) lim h ↘ 0 f ( t 0 ) ⊖ f ( t 0 + h ) - h = lim h ↘ 0 f ( t 0 - h ) ⊖ f ( t 0 ) - h = f ′ ( t 0 )

iii ) lim h ↘ 0 f ( t 0 + h ) ⊖ f ( t 0 ) h = lim h ↘ 0 f ( t 0 - h ) ⊖ f ( t 0 ) - h = f ′ ( t 0 )

iv ) lim h ↘ 0 f ( t 0 ) ⊖ f ( t 0 + h ) - h = lim h ↘ 0 f ( t 0 ) ⊖ f ( t 0 - h ) h = f ′ ( t 0 )

where all limits are taken in the metric D_∞.

Definition 12. [3] Let f : [ a , ∞ [ → ℝ F and [ f ( t ) ] α = [ f - α ( t ) , f + α ( t ) ] , for all t ∈ [a, ∞ [ and α ∈ [0, 1]. If f - α ( t ) and f + α ( t ) are Riemann integrable on [a, ∞ [ for each α ∈ [0, 1], then the improper fuzzy Riemann integral is the fuzzy number given by [ ∫ a ∞ f ( t ) dt ] α = [ ∫ a ∞ f - α ( t ) dt , ∫ a ∞ f + α ( t ) dt ] .(7)

Note that the improper fuzzy Riemann integral of a function f : [ a , ∞ [ → ℝ F is indeed a fuzzy number since the right side of Eq. (7) satisfies the requirements of Theorem 1. The next proposition provides sufficient conditions for a fuzzy function be improper fuzzy Riemann integrable.

Proposition 4. [3] Let f : [ a , ∞ [ → ℝ F and [ f ( t ) ] α = [ f - α ( t ) , f + α ( t ) ] for all t ∈ [a, ∞ [ and α ∈ [0, 1]. If f - α ( t ) and f + α ( t ) are Riemann integrable on [a, b] for α ∈ [0, 1] and there are two positive numbers M - α and M + α such that ∫ a b ∣ f - α ( t ) ∣ dt ≤ M - α and ∫ a b ∣ f + α ( t ) ∣ dt ≤ M + α , for all b ≥ a, then f is improper fuzzy Riemann integrable on [a, ∞ [.

The fuzzy Laplace transform method introduced in [11] can be used to solve linear fuzzy initial value problems.

Definition 13. [11] Let f : [ 0 , ∞ [ → ℝ F be a continuous function with [ f ( t ) ] α = [ f - α ( t ) , f + α ( t ) ] for all t ∈ [0, ∞ [ and α ∈ [0, 1]. If for every s > 0 the function f (t) ⊙ e^-st is improper fuzzy Riemann integrable on [0, ∞ [, then L { f ( t ) } = ∫ 0 ∞ f ( t ) ⊙ e - st dt(8)

is called fuzzy Laplace transform of the fuzzy function f.

Under the conditions of the Definition 13, we can show that [ L { f ( t ) } ] α = [ L { f - α ( t ) } , L { f + α ( t ) } ](9) where L denotes the usual Laplace transform, that is, if g : [ 0 , ∞ [ → ℝ , then L { g ( t ) } = ∫ 0 ∞ g ( t ) e - st dt .

The next propositions establish some properties of the fuzzy Laplace transform operator.

Proposition 5. [11] Let f , g : [ 0 , ∞ [ → ℝ F be continuous functions such that their fuzzy Laplace transform exist and let c 1 , c 2 ∈ ℝ . We have that L { ( c 1 ⊙ f ( t ) ) ⊕ ( c 2 ⊙ g ( t ) ) } = ( c 1 ⊙ L { f ( t ) } ) ⊕ ( c 2 ⊙ L { g ( t ) } ) .

Proposition 6. [11] If f , f ′ : [ 0 , ∞ [ → ℝ F are continuous and their fuzzy Laplace transform exist, then

1) L { f ′ ( t ) } = ( s ⊙ L { f ( t ) } ) ⊖ f ( 0 ) if f is differentiable in the sense (i);

2) L { f ′ ( t ) } = ( ( - 1 ) ⊙ f ( 0 ) ) ⊖ ( - s ⊙ L { f ( t ) } ) if f is differentiable in the sense (ii).

Proposition 7. [12] If f , f ″ : [ 0 , ∞ [ → ℝ F are continuous and their fuzzy Laplace transform exist, then

1) L { f ″ ( t ) } = ( s 2 ⊙ L { f ( t ) } ) ⊖ ( s ⊙ f ( 0 ) ) ⊖ f ′ ( 0 ) if f and f′ are differentiable in the sense (i);

2) L { f ″ ( t ) } = ( ( - 1 ) ⊙ f ′ ( 0 ) ) ⊖ ( - s 2 ⊙ L { f ( t ) } ) ⊕ ( - s ⊙ f ( 0 ) ) if f is differentiable in the sense (i) and f′ is differentiable in the sense (ii);

3) L { f ″ ( t ) } = ( - s ⊙ f ( 0 ) ) ⊖ ( - s 2 ⊙ L { f ( t ) } ) ⊖ ( f ′ ( 0 ) ) if f is differentiable in the sense (ii) and f′ is differentiable in the sense (i);

4) L { f ″ ( t ) } = ( s 2 ⊙ L { f ( t ) } ) ⊖ ( s ⊙ f ( 0 ) ) ⊕ ( ( - 1 ) ⊙ f ′ ( 0 ) ) if f and f′ are differentiable in the sense (ii).

We consider in this section a fuzzy initial value problem (FIVP) of the form

{ x ″ + a 2 x = 0 x ( 0 ) = A ∈ ℝ F x ′ ( 0 ) = B ∈ ℝ F(10)

where x , x ″ : [ 0 , ∞ [ → ℝ F , [ A ] α = [ a - α , a + α ] and [ B ] α = [ b - α , b + α ] for all α ∈ [0, 1], and a is a real number, a ≠ 0. Note that, for every t ∈ [0, ∞ [, a²x (t) and x″ (t) are fuzzy numbers whose the sum is a crisp number. This sum makes sense if x″ (t) and a²x (t) are linearly correlated with q = -1 and r = 0. This last observation implies that x″ (t) = - a²x (t) for all t ∈ [0, ∞ [. Thus, the FIVP (10) can be rewritten as follows:

{ x ″ = - a 2 x x ( 0 ) = A ∈ ℝ F x ′ ( 0 ) = B ∈ ℝ F(11)

With the assumption that x″ (t) and a²x (t) are linearly correlated we have that the Problem (10) is equivalent to the Problem (11). We can obtain a fuzzy solution, x (t), for (11) by applying the fuzzy Laplace transform in both sides of the differential equation x″ = - a²x. A solution of (10), or equivalently, of (11) is a function x : [ 0 , ∞ [ → ℝ F such that x″ (t) +  _L a²x (t) =0 for all t ∈ [0, ∞ [. Thus, using the Proposition 7, we have the following alternatives:

Case 1. If x is differentiable in the sense (i) and x′ is differentiable in the sense (ii), then, from Proposition 7, we have ( ( − 1 ) ⊙ x ′ ( 0 ) ) ⊖ ( − s 2 ⊙ L { x ( t ) } ) ⊕ ( − s ⊙ x ( 0 ) ) = ( − a 2 ⊙ L { x ( t ) } ) .(12)

Eq. (12) produces the system { L { x - α ( t ) } = 1 s 2 + a 2 b - α + s s 2 + a 2 a + α , L { x + α ( t ) } = 1 s 2 + a 2 b + α + s s 2 + a 2 a - α .(13)

Solving (13), we obtain

{ x - α ( t ) = a - α cos ( at ) + b - α sin ( at ) , x + α ( t ) = a + α cos ( at ) + b + α sin ( at ) .(14)

However, the diameter of 0-level is d ( t ) = ( a + 0 - a - 0 ) cos ( at ) + ( b + 0 - b - 0 ) sin ( at ) ,

which is not positive for all t ∈ [0, ∞ [. For example, if A , B ∈ ℝ F ∖ ℝ and a > 0, then d (t) <0 for t ∈ ] 1 a ( 2 π n + 3 π 4 ) , ( 2 π n + π ) 1 a [ , n∈ { 0, 1, 2, … }. Therefore, the expression (14) is not a solution of problem (11). This indicates that the hypotheses that x is differentiable in the sense (i) and x′ is differentiable in the sense (ii) do not hold true in the domain [0, ∞ [.

Case 2. If x is differentiable in the sense (ii) and x′ is differentiable in the sense (i), then, from Proposition 7, we have ( − s ⊙ x ( 0 ) ) ⊖ ( − s 2 ⊙ L { x ( t ) } ) ⊖ ( x ′ ( 0 ) ) = ( − a 2 ⊙ L { x ( t ) } ) .(15)

Eq. (15) yields the following system { L { x - α ( t ) } = s s 2 + a 2 a - α + 1 s 2 + a 2 b - α , L { x + α ( t ) } = s s 2 + a 2 a + α + 1 s 2 + a 2 b - α ,(16) and, from the system (16) above, we obtain the following expressions: { x - α ( t ) = a - α cos ( at ) + b + α sin ( at ) , x + α ( t ) = a + α cos ( at ) + b - α sin ( at ) .(17)

Again, the diameter of 0-level of x (t) is given by d ( t ) = ( a + 0 - a - 0 ) cos ( at ) - ( b + 0 - b - 0 ) sin ( at ) , which may not be positive for all t ∈ [0, ∞ [. If A , B ∈ ℝ F ∖ ℝ and a > 0, then d (t) <0 for t ∈ ] 1 a ( 2 π n + π 4 ) , ( 2 π n + π ) 1 a [ , n∈ { 0, 1, 2, … }. Therefore, the expression (17) does not represent a solution of Problem (11).

Case 3. If x and x′ are differentiable in the sense (ii), then, from Proposition 7, we have ( s 2 ⊙ L { x ( t ) } ) ⊖ ( s ⊙ x ( 0 ) ) ⊕ ( ( − 1 ) ⊙ x ′ ( 0 ) ) = ( − a 2 ⊙ L { x ( t ) } ) .(18) and Eq. (18) produces the system { L { x - α ( t ) } = s 3 s 4 - a 4 a - α + s 2 s 4 - a 4 b + α - a 2 s s 4 - a 4 a + α - a 2 s 4 - a 4 b - α , L { x + α ( t ) } = s 3 s 4 - a 4 a + α + s 2 s 4 - a 4 b - α - a 2 s s 4 - a 4 a + α - a 2 s 4 - a 4 b - α .(19) Solving (19), we obtain { x - α ( t ) = 1 2 ( cosh ( at ) + cos ( at ) ) a - α + 1 2 a ( sinh ( at ) + sin ( at ) ) b + α - 1 2 ( cosh ( at ) - cos ( at ) ) a + α - 1 2 a ( sinh ( at ) - sin ( at ) ) b - α , x + α ( t ) = 1 2 ( cosh ( at ) + cos ( at ) ) a + α + 1 2 a ( sinh ( at ) + sin ( at ) ) b - α - 1 2 ( cosh ( at ) - cos ( at ) ) a - α - 1 2 a ( sinh ( at ) - sin ( at ) ) b + α .(20) But, the diameter of 0-level is d ( t ) = ( a + 0 - a - 0 ) cosh ( at ) - 1 a ( b + 0 - b - 0 ) sinh ( at ) , which may not be positive for all t ∈ [0, ∞ [. For example, if A , B ∈ ℝ F ∖ ℝ and 0 < a < 1, then d (t) <0 for all t > ( log ( 1 + a 1 - a ) / 2 a ) . Therefore, the expression (20) does not represent a solution to the problem (11).

Case 4. Let us suppose that x and x′ are differentiable in the sense (i). From Proposition 7, we have ( s 2 ⊙ L { x ( t ) } ) ⊖ ( s ⊙ x ( 0 ) ) ⊖ x ′ ( 0 ) = ( - a 2 ⊙ L { x ( t ) } ) ,(21) or { L { x - α ( t ) } = s 3 s 4 - a 4 a - α + s 2 s 4 - a 4 b - α - a 2 s s 4 - a 4 a + α - a 2 s 4 - a 4 b + α , L { x + α ( t ) } = s 3 s 4 - a 4 a + α + s 2 s 4 - a 4 b + α - a 2 s s 4 - a 4 a - α - a 2 s 4 - a 4 b - α .(22) By applying the inverse of L in both sides of (22), we obtain { x - α ( t ) = 1 2 ( cosh ( at ) + cos ( at ) ) a - α + 1 2 a ( sinh ( at ) + sin ( at ) ) b - α - 1 2 ( cosh ( at ) - cos ( at ) ) a + α - 1 2 a ( sinh ( at ) - sin ( at ) ) b + α , x + α ( t ) = 1 2 ( cosh ( at ) + cos ( at ) ) a + α + 1 2 a ( sinh ( at ) + sin ( at ) ) b + α - 1 2 ( cosh ( at ) - cos ( at ) ) a - α - 1 2 a ( sinh ( at ) - sin ( at ) ) b - α .(23) In contrast to the formulas (14), (17) and (20), the formula (23) yields a fuzzy function x : [ 0 , ∞ [ ⟶ ℝ F . Note that the conditions (i) and (ii) of Theorem 1 are indeed satisfied:

(i) by analysing the signs of the terms of system (23), one can verify that x - α ( t ) < x - β ( t ) and x + α ( t ) > x + β ( t ) for all t > 0 and β > α.

(ii) since A and B are fuzzy numbers, we have that x - α k ( t ) ⟶ x - α ( t ) and x + α k ( t ) ⟶ x + α ( t ) whenever (α _k ) is a non-decreasing sequence converging to α ∈]0, 1].

Hence, from Theorem 1, x : [ 0 , ∞ [ ⟶ ℝ F is a fuzzy function, that is, x (t) is a fuzzy number for every t ≥ 0. Similarly, one can show that x ′ : [ 0 , ∞ [ ⟶ ℝ F is well defined and x, x^′ are differentiable in the sense (i) (that is, x and x^′ are Hukuhara differentiable). Moreover, it is easy to see that x^″ (t) = − a²x(t) for all t ∈[0, ∞[ and, therefore, x^″ (t) + _L a²x(t) = 0 with q = −1 and r = 0. Hence, the function x, given by (23), is a solution of the problem (11).

Figure 1 illustrates the geometric behavior of (23) with initial conditions given by triangular fuzzy numbers whose α-levels are given by [A]  ^α  = [0.1α + 0.9, - 0.1α + 1.1], [B]  ^α  = [0.05α - 0.05, 0] and a = 0.15.

OPEN IN VIEWER

Fig. 1

The figure exhibits the fuzzy solution x (t) of the FIVP (23) in the xt-plane, for a = 0.15 and initial conditions x (0) = A = (0.9 ; 1 ;1.1) and x′ (0) = B = (-0.05 ; 0 ;0). The greatest and least membership values are represented respectively by the black and white colors. In addition, the black dashed-line illustrate the 0-level of x (t) for all t.

It is easy to see that if the initial conditions are real numbers, that is, a - α = a + α = a ¯ and b - α = b + α = b ¯ , for all α ∈ [0, 1], then Eq. (24) becomes the classical solution of harmonic oscillator for all the four cases: x ( t ) = a ¯ cos ( at ) + b ¯ sin ( at ) .(24) Moreover, this solution is preferred in the sense that, for each t > 0, it belongs to fuzzy solution (23) with membership degree equal to 1. Deterministic solution (24) is depicted in Figure 2, for a ¯ = 1 , b ¯ = 0 and a = 0.15.

OPEN IN VIEWER

Fig. 2

Deterministic solution x (t) given by (24), for a ¯ = 1 , b ¯ = 0 and a = 0.15

Finally, we depicted in Figure 3 the fuzzy solution in the xtα-space.

OPEN IN VIEWER

Fig. 3

The image exhibits the fuzzy solution x (t) of the FIVP (23) in the xtα-space, for a = 0.15 and initial conditions x (0) = A = (0.9 ; 1 ;1.1) and x′ (0) = B = (-0.05 ; 0 ;0). The greatest and least membership values are represented respectively by the black and white colors. In addition, the black dashed-line illustrate the 0-level of x (t) for all t.

It is common to use differential equations to describe the temporal evolution of certain phenomena in areas such as demography, stock pricing, and so on, even those with oscillatory behaviour. In this work, we studied differential equations of the form x″ + a²x = 0,  a > 0, which are useful to model small oscillations of the so-called simple harmonic motion. By assuming that these phenomena can contain uncertainties in the state variables, we consider x (t) as being a fuzzy number. Thus, x″ + a²x = 0 is, typically, a fuzzy differential equation. However, in fuzzy arithmetic, the equation x″ + a²x = 0 makes sense if, and only if, the operands of the sum are linearly correlated fuzzy numbers. That is, x″ + a²x = 0, is indeed, x″ +  _L a²x = 0. In this last fuzzy differential equation we used the notion of strongly generalized differentiability and the fuzzy Laplace transform method to obtain a fuzzy solution.