Linear Differential Equations with Fuzzy Boundary Values
Nizami Gasilov, Şahin Emrah Amrahov, Afet Golayoğlu Fatullayev
aa r X i v : . [ c s . NA ] J u l Linear Differential Equationswith Fuzzy Boundary Values
Nizami Gasilov
Baskent University,Eskisehir yolu 20. km, Baglica,06810 Ankara, TurkeyEmail: [email protected]
S¸ ahin Emrah Amrahov
Ankara University,Computer Engineering Department,06100 Ankara, TurkeyEmail: [email protected]
Afet Golayo ˘glu Fatullayev
Baskent University,Eskisehir yolu 20. km, Baglica,06810 Ankara, TurkeyEmail: [email protected]
Abstract —In this study, we consider a linear differentialequation with fuzzy boundary values. We express the solutionof the problem in terms of a fuzzy set of crisp real functions.Each real function from the solution set satisfies differentialequation, and its boundary values belong to intervals, determinedby the corresponding fuzzy numbers. The least possibility amongpossibilities of boundary values in corresponding fuzzy sets isdefined as the possibility of the real function in the fuzzy solution.In order to find the fuzzy solution we propose a method basedon the properties of linear transformations. We show that, ifthe corresponding crisp problem has a unique solution then thefuzzy problem has unique solution too. We also prove that if theboundary values are triangular fuzzy numbers, then the value ofthe solution at any time is also a triangular fuzzy number.We find that the fuzzy solution determined by our methodis the same as the one that is obtained from solution of crispproblem by the application of the extension principle.We present two examples describing the proposed method.
Keywords : fuzzy boundary value problem, fuzzy set, lineartransformation. I. I
NTRODUCTION
Approaches to fuzzy boundary value problems can be of twotypes. The first approach assumes that, even if only the bound-ary values are fuzzy in the handling problem, the solution isfuzzy function, consequently, the derivative in the differentialequation can be considered as a derivative of fuzzy function.This derivative can be Hukuhara derivative, or a derivative ingeneralized sense. Bede [1] has demonstrated that a large classof boundary value problems have not a solution, if Hukuharaderivative is used. To overcome this difficulty, in [2] and [3]the concept of generalized derivative is developed and fuzzydifferential equations have been investigated using this concept(see also [4], [5], [6]). Recently, Khastan and Nieto [7] havefound solutions for a large enough class of boundary valueproblems with the generalized derivative. However as it is seenfrom the examples in mentioned article, these solutions aredifficult to interpret because four different problems, obtainedby using the generalized second derivatives, often does notreflect the nature of the problem.The second approach is based on generating the fuzzysolution from the crisp solution. In particular case, for thefuzzy initial value problem this approach can be of three ways. The first one uses the extension principle. In thisway, the initial value is taken as a real constant, and theresulting crisp problem is solved. Then the real constant in thesolution is replaced with the initial fuzzy value. In the finalsolution, arithmetic operations are considered to be operationson fuzzy numbers ([8], [9]). The second way, offered byH¨ullemerier [10], uses the concept of differential inclusion.In this way, by taking an alpha-cut of initial value, the givendifferential equation is converted to a differential inclusion andthe obtained solution is accepted as the alpha-cut of the fuzzysolution. Misukoshi et al [11] have proved that, under certainconditions, the two main ways of the approach are equivalentfor the initial value problem. The third way is offered byGasilov et al [12]. In this way the fuzzy problem is consideredto be a set of crisp problems.In this study, we investigate a differential equation withfuzzy boundary values. We interpret the problem as a set ofcrisp problems. For linear equations, we propose a methodbased on the properties of linear transformations. We showthat, if the solution of the corresponding crisp problem existsand is unique, the fuzzy problem also has unique solution.Moreover, we prove that if the boundary values are triangularfuzzy numbers, then the value of the solution is a triangularfuzzy number at each time. We explain the proposed methodon examples.II. F
UZZY B OUNDARY V ALUE P ROBLEM
Below, we use the notation e u = ( u L ( r ) , u R ( r )) , (0 ≤ r ≤ , to indicate a fuzzy number in parametric form. Wedenote u = u L (0) and u = u R (0) to indicate the left and theright limits of e u , respectively. We represent a triangular fuzzynumber as e u = ( l, m, r ) , for which we have u = l and u = r .In this paper we consider a fuzzy boundary value problem(FBVP) with crisp linear differential equation but with fuzzyboundary values. For clarity we consider second order differ-ential equation: x ′′ + a ( t ) x ′ + a ( t ) x = f ( t ) x (0) = e Ax ( T ) = e B (1)We note that the coefficients of the differential equation arenot necessary constant.et us represent the boundary values as e A = a cr + e a and e B = b cr + e b , where a cr and b cr are vectors with possibility 1and denote the crisp parts (the vertices) of e A and e B ; e a and e b denote the uncertain parts with vertices at the origin.We split the problem (1) to following two problems:1) Associated crisp problem (which is non-homogeneous): x ′′ + a ( t ) x ′ + a ( t ) x = f ( t ) x (0) = a cr x ( T ) = b cr (2)2) Homogeneous problem with fuzzy boundary values: x ′′ + a ( t ) x ′ + a ( t ) x = 0 x (0) = e ax ( T ) = e b (3)It is easy to see that, a solution of the given problem(1) is of the form x ( t ) = x cr ( t ) + x un ( t ) (crisp solution +uncertainty). Here x cr ( t ) is a solution of the non-homogeneouscrisp problem (2); while x un ( t ) is a solution of the homo-geneous problem (3) with fuzzy boundary conditions. x cr ( t ) can be computed by means of analytical or numerical methods.Hence, (1) is reduced to solving a homogeneous equation withfuzzy boundary conditions (3). Therefore, we will investigatehow to solve this problem.We assume the solution x un of the problem (3) be a fuzzyset e X of real functions such as x ( t ) . Each function x ( t ) mustsatisfy the differential equation and must have boundary values a and b from the sets e a and e b , respectively. We define thepossibility (membership) of the function x ( t ) to be equal tothe least possibility of its boundary values.Mathematically, the fuzzy solution set can be defined asfollows: e X = { x ( t ) | x ′′ + a ( t ) x ′ + a ( t ) x = 0; x (0) = a ; x ( T ) = b ; a ∈ e a ; b ∈ e b } (4)with membership function µ e X ( x ( t )) = min (cid:8) µ e a ( a ) , µ e b ( b ) (cid:9) (5)The solution e X , defined above, can be interpreted as a fuzzybunch of functions.One can also interpret that we consider a FBVP as a set ofcrisp BVPs whose boundary values belong to the fuzzy sets e a and e b . A. A matrix representation of the solution in the crisp case
Here we consider crisp BVP for second order homogeneouslinear differential equation: x ′′ + a ( t ) x ′ + a ( t ) x = 0 x (0) = ax ( T ) = b (6)Let x ( t ) and x ( t ) be linear independent solutions of thedifferential equation. Then the general solution is x ( t ) = c x ( t ) + c x ( t ) . For c and c we have the following linearsystem (cid:26) c x (0) + c x (0) = ac x ( T ) + c x ( T ) = b (7)Below we obtain a matrix representation for the solution ofthe BVP. We rewrite the linear system (7) in matrix form: M c = u where M = (cid:20) x (0) x (0) x ( T ) x ( T ) (cid:21) ; c = (cid:20) c c (cid:21) ; u = (cid:20) ab (cid:21) .The solution of the linear system is c = M − u (8)We constitute a vector-function of linear independent solutions s ( t ) = [ x ( t ) x ( t )] . Then the general solution can berewritten in matrix form as x ( t ) = [ x ( t ) x ( t )] (cid:20) c c (cid:21) = s ( t ) c Using (8) we have x ( t ) = s ( t ) M − u , or, x ( t ) = w ( t ) u = w ( t ) a + w ( t ) b (9)where w ( t ) = s ( t ) M − (10) B. The solution method for FBVP
Now we show how to find e X ( t ) (the value of the solutionfor the problem (3) at a time t ).Let linear independent solutions of the crisp equation(3), x ( t ) and x ( t ) , be known. Then we can constitute thevector w (see, formula (10)). According (4) and (9) we have: e X = (cid:26) x ( t ) = w ( t ) u | u = (cid:20) ab (cid:21) ; a ∈ e a ; b ∈ e b (cid:27) (11)Consider a fixed time t . Put v = w ( t ) . Then from (11) wehave: e X ( t ) = n v u | u = [ a b ] T ; a ∈ e a ; b ∈ e b o (12)To determine how is the set e X ( t ) we consider the transfor-mation T ( u ) = v u (here v is a fixed vector). One can see that T : R → R is a linear transformation. Therefore, e X ( t ) is theimage of the set e B = n u = [ a b ] T | a ∈ e a ; b ∈ e b o = ( e a, e b ) under the linear transformation T ( u ) .We remember some properties of linear transformations[13]:1. A linear transformation maps the origin (zero vector) tothe origin (zero vector).2. Under a linear transformation the images of a pair ofsimilar figures are also similar.3. Under a linear transformation the images of nested figuresare also nested.In addition, we shall reference a property of fuzzy numbervectors.4. The fuzzy set e B = ( e a, e b ) forms a fuzzy region in the ab -coordinate plane, vertex of which is located at the originnd boundary of which is a rectangle. Furthermore, the α -cutsof the region are rectangles nested within one another.The facts 1-4 allow us to derive the following conclusion.The vector e B , components of which are the boundary values e a and e b , form a fuzzy rectangle in the ab -coordinate plane.The linear transformation T ( u ) maps this fuzzy rectangle to afuzzy interval in R . Therefore, the solution at any time formsa fuzzy number. C. Particular case when boundary values are triangular fuzzynumbers
In particular, if e a and e b triangular fuzzy numbers, the α -cuts of the region e B = ( e a, e b ) are nested rectangles, further-more, they are similar. According to the discussion above,their images are intervals that also are nested and similar,consequently, form a triangular fuzzy number e X ( t ) . Therefore, e X ( t ) can be represented in the form e X ( t ) = ( x ( t ) , , x ( t )) .Now we investigate how to calculate x ( t ) and x ( t ) .Let e a = ( a, , a ) , e b = ( b, , b ) and w ( t ) = ( w ( t ) , w ( t )) .Since x ( t ) is the maximum value among all products w ( t ) · u = aw ( t ) + bw ( t ) , we have: x ( t ) = max { aw ( t ) , aw ( t ) } + max (cid:8) bw ( t ) , bw ( t ) (cid:9) x ( t ) = min { aw ( t ) , aw ( t ) } + min (cid:8) bw ( t ) , bw ( t ) (cid:9) Note that an α -cut of e X ( t ) can be determined by similarity: X α ( t ) = (cid:2) x α ( t ) , x α ( t ) (cid:3) = (1 − α ) [ x ( t ) , x ( t )] = (1 − α ) X ( t ) Formulas for x ( t ) and x ( t ) allow us to represent the solutionin a new way: e X ( t ) = w ( t ) e a + w ( t ) e b where the operations assumed to be multiplication of realnumber with fuzzy one, and addition of fuzzy numbers. D. General case when boundary values are parametric fuzzynumbers
In the general case, when e a and e b are arbitrary fuzzynumbers, the solution can be obtained by using α -cuts. Let a α = (cid:2) a α , a α (cid:3) and b α = (cid:2) b α , b α (cid:3) . Then B α = (cid:2) a α , a α (cid:3) × (cid:2) b α , b α (cid:3) . By similar argumentation to the preceding case, forthe α -cut of the solution we obtain the following formulas: X α ( t ) = (cid:2) x α ( t ) , x α ( t ) (cid:3) where x α ( t ) = max (cid:8) a α w ( t ) , a α w ( t ) (cid:9) +max (cid:8) b α w ( t ) , b α w ( t ) (cid:9) x α ( t ) = min (cid:8) a α w ( t ) , a α w ( t ) (cid:9) + min (cid:8) b α w ( t ) , b α w ( t ) (cid:9) Based on the formulas above we can conclude, that therepresentation for solution e X ( t ) = w ( t ) e a + w ( t ) e b (13)is valid in general. Thus, the solution, defined by formula(4), becomes the same as the solution obtained from (9) byextension principle. Remark : The approach is valid also for the general case,when n th-order m -point boundary value problem is consid-ered. E. Solution algorithm
Based on the arguments above, we propose the followingalgorithm to solve the problem (1):1. Represent the boundary values as e A = a cr + e a and e B = b cr + e b .2. Find linear independent solutions x ( t ) and x ( t ) ofthe crisp differential equation x ′′ + a ( t ) x ′ + a ( t ) x = 0 .Constitute the vector-function s ( t ) = [ x ( t ) x ( t )] , thematrix M and calculate the vector w ( t ) = ( w ( t ) , w ( t )) byformula (10).3. Find the solution x cr ( t ) of the non-homogeneous crispproblem (2).4. The solution of the given problem (1) is e x ( t ) = x cr ( t ) + w ( t ) e a + w ( t ) e b (14)III. E XAMPLES
Example 1 . Solve the FBVP: x ′′ − x ′ + 2 x = 4 t − x (0) = (1 . , , x (1) = ( 2 , , (15) Solution : We represent the solution as e x ( t ) = x cr ( t )+ e x un ( t ) .1. We solve crisp non-homogeneous problem x ′′ − x ′ + 2 x = 4 t − x (0) = 2 x (1) = 3 and find the crisp solution x cr ( t ) = 2 t + e − e (cid:2) e t − e t ) + ( e t − e t ) (cid:3) (the thickline in Fig.1). Fig. 1. The fuzzy solution and its α = 0 . -cut for Example 1. Dashed andthick bars represent the values of the fuzzy solution and its α = 0 . -cut atdifferent times, respectively.
2. We consider fuzzy homogeneous problem x ′′ − x ′ + 2 x = 0 x (0) = ( − . , , x (1) = ( − , , ( t ) = e t and x ( t ) = e t are linear independent solutionsfor the differential equation. Then s ( t ) = (cid:2) e t e t (cid:3) , M = (cid:20) e e (cid:21) and w = s ( t ) M − = e − e (cid:2) e t − e t e t − e t (cid:3) .The formula (13) gives the solution of homogeneous problem: e x un ( t ) = e − e (( e t − e t ) ( − . , , e t − e t ) ( − , , (16)where the arithmetic operations are considered to be fuzzyoperations. We add this solution to the crisp solution and getthe fuzzy solution of the given FBVP (15): e x ( t ) = 2 t + e − e (( e t − e t ) (1 . , , e t − e t ) (0 , , (17)The fuzzy solution e x ( t ) forms a band in the tx -coordinateplane (Fig. 1). Since w ( t ) > and w ( t ) > for < t < T ,the upper border of the band, x ( t ) , becomes the solution ofthe crisp non-homogeneous problem with the upper boundaryvalues A = 3 and B = 4 , while the lower border x ( t ) corresponds to A = 1 . and B = 2 : x ( t ) = 2 t + 1 e − e (( e t − e t ) · e t − e t ) · x ( t ) = 2 t + 1 e − e (( e t − e t ) · . e t − e t ) · We can express the solution e x ( t ) also via α -cuts, whichare intervals x α ( t ) = h x α ( t ) , x α ( t ) i at any time t . Since theboundary values are triangular fuzzy numbers, e x un ( t ) also isa triangular fuzzy number, say e x un ( t ) = ( x un ( t ) , , x un ( t )) .Consequently, an α -cut of e x un ( t ) can be determined bysimilarity with coefficient (1 − α ) , i.e. x un, α ( t ) = (1 − α ) (cid:2) x un ( t ) , x un ( t ) (cid:3) Adding the crisp solution gives the α -cut of the solution e x ( t ) : h x α ( t ) , x α ( t ) i = 2 t +(1 − α ) · e − e (( e t − e t ) [1 . ,
3] +( e t − e t ) [0 , In Fig. 1 we show the fuzzy solution (dashed bars) and its α = 0 . -cut (thick bars) at different times. Example 2 . Solve the FBVP: x ′′ + 16 x = 47 − t x (0) = ( 2 , , . x (2) = (0 . , , . (18) Solution :Associated crisp non-homogeneous problem x ′′ + 16 x = 47 − t x (0) = 3 x (2) = 1 has the solution x cr ( t ) = 3 − . t (thick line in Fig. 2).To find the uncertain part of the fuzzy solution, e x un ( t ) , wesolve fuzzy homogeneous problem Fig. 2. The fuzzy solution and its α = 0 . -cut for Example 2. Dashed andthick bars represent the values of the fuzzy solution and its α = 0 . -cut atdifferent times, respectively. x ′′ + 16 x = 0 x (0) = ( − , , . x (2) = ( − . , , . x ( t ) = cos 4 t and x ( t ) = sin 4 t are linear independentsolutions for the differential equation. Then s ( t ) = [cos 4 t sin 4 t ] , M = (cid:20) (cid:21) and w = s ( t ) M − = [sin(8 − t ) sin 4 t ] .Using the formula (13) we obtain the solution of homo-geneous problem and adding the crisp solution we get thesolution of the given FBVP (18): e x ( t ) = 3 − . t + (sin(8 − t ) ( − , , . t ( − . , , . (19)Fuzzy solution generates a band in tx -plane (Fig. 2). UnlikeExample 1, the functions w ( t ) and w ( t ) takes both positiveand negative values in the interval < t < T . Because ofthat, in generation of upper and lower borders of the band, a and a , b and b take charge in alternately.IV. C ONCLUSION