Solving fuzzy two-point boundary value problem using fuzzy Laplace transform

Transcription

1 Solving fuzzy two-point oundary value prolem using fuzzy Laplace transform arxiv: v1 [math.gm] Mar 014 Latif Ahmad, Muhammad Farooq, Saif Ullah, Saleem Adullah Octoer 5, 016 Astract A natural way to model dynamic systems under uncertainty is to use fuzzy oundary value prolems (FBVPs) and related uncertain systems. In this paper we use fuzzy Laplace transform to find the solution of two-point oundary value under generalized Hukuhara differentiaility. We illustrate the method for the solution of the well known two-point oundary value prolem Schrödinger equation, and homogeneous oundary value prolem. Consequently, we investigate the solutions of FBVPs under as a new application of fuzzy Laplace transform. Keywords: Fuzzy derivative, fuzzy oundary value prolems, fuzzy Laplace transform, fuzzy generalized Hukuhara differentiaility. 1 Introduction The theory of fuzzy differential equations (FDEs) has attracted much attentioninrecent years ecause thistheory represents a naturalway to model dynamical systems under uncertainty, Jamshidi and Avazpour [1].The concept 1. Shaheed Benazir Bhutto University, Sheringal, Upper Dir, Khyer Pakhtunkhwa, Pakistan.. Institute of Pure & Applied Mathematics, University of Peshawar, 510, Khyer Pakhtunkhwa, Pakistan. ahmad49960@yahoo.com Institute of Pure & Applied Mathematics, University of Peshawar, 510, Khyer Pakhtunkhwa, Pakistan. mfarooq@upesh.edu.pk Institute of Pure & Applied Mathematics, University of Peshawar, 510, Khyer Pakhtunkhwa, Pakistan. saifullah.maths@gmail.com Department of Mathematics, Quaid-i-Azam University, Islamaad, Pakistan. saleemadullah81@yahoo.com 1

2 of fuzzy set was introduce y Zadeh in 1965[]. The derivative of fuzzy-valued function was introduced y Chang and Zadeh in 197 [3]. The integration of fuzzy valued function is presented in [9]. Kaleva and Seikala presented fuzzy differential equations (FDEs) in [4, 5]. Many authors discussed the applications of FDEs in [6, 7, 8]. Two-point oundary value prolem is investigated in [10]. In case of Hukuhara derivative the funding Green s function helps to find the solution of oundary value prolem of first order linear fuzzy differential equations with impulses [11]. Wintner-type and superlinear-type results for fuzzy initial value prolems (FIVPs) and fuzzy oundary value prolems (FBVPs) are presented in [1]. The solution of FBVPs must e a fuzzy-valued function under the Hukuhara derivative [13, 14, 15, 16, 17]. Also two-point oundary value prolem (BVP) is equivalent to fuzzy integral equation [18]. Recently in [19, 0, 1] the fuzzy Laplace transform is applied to find the analytical solution of FIVPs. According to [] the fuzzy solution is different from the crisp solution as presented in [13, 14, 15, 16, 3, 4]. In [] they solved the Schrödinger equation with fuzzy oundary conditions. Further in [19] it was discussed that under what conditions the fuzzy Laplace transform (FLT) can e applied to FIVPs. For two-point BVP some of the analytical methods are illustrated in [, 5, 6] while some of the numerical methods are presented in [1, 7]. But every method has its own advantages and disadvantages for the solution of such types of fuzzy differential equation (FDE). In this paper we are going to apply the FLT on two-point BVP []. Moreover we investigate the solution of second order Schrödinger equation and other homogeneous oundary value prolems []. After applying the FLT to BVP we replace one or more missing terms y any constant and then apply the oundary conditions which eliminates the constants. The crisp solution of fuzzy oundary value prolem (FBVP) always lies etween the upper and lower solutions. If the lower solution is not monotonically increasing and the upper solution is not monotonically decreasing then the solution of the FDE is not a valid level set. This paper is organized as follows: In section, we recall some asics definitions and theorems. FLT is defined in section 3 and in this section the FBVP is riefly reviewed. In section 4, constructing solution of FBVP y FLT is explained. To illustrate the method, several examples are given in section 5. Conclusion is given in section 6. Basic concepts In this section we will recall some asics definitions and theorems needed throughout the paper such as fuzzy numer, fuzzy-valued function and the

3 derivative of the fuzzy-valued functions. Definition.1. A fuzzy numer is defined in [] as the mapping such that u : R [0,1], which satisfies the following four properties 1. u is upper semi-continuous.. u is fuzzy convex that is u(λx+(1 λ)y) min{u(x),u(y)},x,y R and λ [0,1]. 3. u is normal that is x 0 R, where u(x 0 ) = A = {x R : u(x) > 0} is compact, where A is closure of A. Definition.. A fuzzy numer in parametric form given in [3, 4, 5] is an order pair of the form u = (u(r),u(r)), where 0 r 1 satisfying the following conditions. 1. u(r) is a ounded left continuous increasing function in the interval [0,1].. u(r) is a ounded left continuous decreasing function in the interval [0,1]. 3. u(r) u(r). If u(r) = u(r) = r, then r is called crisp numer. Now we recall a triangular fuzzy from [, 19, 0] numer which must e in the form of u = (l,c,r), where l,c,r R and l c r, then u(α) = l + (c r)α and u(α) = r (r c)α are the end points of the α level set. Since each y R can e regarded as a fuzzy numer if { 1, if y = t, ỹ(t) = 0, if 1 t. For aritrary fuzzy numers u = (u(α),u(α)) and v = (v(α),v(α)) and an aritrary crisp numer j, we define addition and scalar multiplication as: 1. (u+v)(α) = (u(α)+v(α)).. (u+v)(α) = (u(α)+v(α)). 3. (ju)(α) = ju(α), (ju)(α) = ju(α) j (ju)(α) = ju(α)α,(ju)(α) = ju(α)α, j < 0. 3

4 Definition.3. (See Salahshour& Allahviranloo, and Allahviranloo& Barkhordari [19, 0]) Let us suppose that x, y E, if z E such that x = y +z. Then, z is called the H-difference of x and y and is given y x y. Remark.4. (see Salahshour & Allahviranloo [19]). Let X e a cartesian product of the universes, X 1, X 1,,X n, that is X = X 1 X X n and A 1,,A n e n fuzzy numers in X 1,,X n respectively. Then, f is a mapping from X to a universe Y, and y = f(x 1,x,,x n ), then the Zadeh extension principle allows us to define a fuzzy set B in Y as; where u B (y) = B = {(y,u B (y)) y = f(x 1,,x n ),(x 1,,x n ) X}, { sup (x1,,x n) f 1 (y)min{u A1 (x 1 ), u An (x n )}, if f 1 (y) 0, 0, otherwise, where f 1 is the inverse of f. The extension principle reduces in the case if n = 1 and is given as follows: B = {(y,u B (y) y = f(x), x X)}, where { sup u B (y) = x f (y){u 1 A (x)}, if f 1 (y) 0, 0, otherwise. By Zadeh extension principle the approximation of addition of E is defined y (u v)(x) = sup y R min(u(y),v(x y)), x R and scalar multiplication of a fuzzy numer is defined y { u( x ), k > 0, k (k u)(x) = 0 otherwise, where 0 E. The Housdorff distance etween the fuzzy numers [7, 13, 19, 0] defined y d : E E R + {0}, d(u,v) = sup max{ u(r) v(r), u(r) v(r) }, r [0,1] where u = (u(r),u(r)) and v = (v(r),v(r)) R. We know that if d is a metric in E, then it will satisfies the following properties, introduced y Puri and Ralescu [8]: 4

5 1. d(u+w,v+w) = d(u,v), u, v, w E.. (k u,k v) = k d(u,v), k R, and u, v E. 3. d(u v,w e) d(u,w)+d(v,e), u, v, w, e E. Definition.5. (see Song and Wu [9]). If f : R E E, then f is continuous at point (t 0,x 0 ) R E provided that for any fixed numer r [0,1] and any ǫ > 0, δ(ǫ,r) such that d([f(t,x)] r,[f(t 0,x 0 )] r ) < ǫ whenever t t 0 < δ(ǫ,r) and d([x] r,[x 0 ] r ) < δ(ǫ,r) t R, x E. Theorem.6. (see Wu [30]). Let f e a fuzzy-valued function on [a, ) given in the parametric form as (f(x,r),f(x,r)) for any constant numer r [0,1]. Here we assume that f(x,r) and f(x,r) are Riemann-Integral on [a,] for every a. Also we assume that M(r) and M(r) are two positive functions, such that f(x,r) dx M(r) and f(x,r) dx M(r) for every a, then f(x) a a is improper integral on [a, ). Thus an improper integral will always e a fuzzy numer. In short r f(x)dx = ( f(x, r) dx, f(x, r) dx). a a a It is will known that Hukuhare differentiaility for fuzzy function was introduced y Puri & Ralescu in Definition.7. (see Chalco-Cano and Román-Flores [31]). Let f : (a,) E where x 0 (a,). Then, we say that f is strongly generalized differentiale at x 0 (Beds and Gal differentiaility). If an element f (x 0 ) E such that 1. h > 0 sufficiently small f(x 0 +h) f(x 0 ), f(x 0 ) f(x 0 h), then the following limits hold (in the metric d) f(x lim 0 +h) f(x 0 ) f(x h 0 = lim 0 ) f(x 0 h) h h 0 = f (x h 0 ), Or. h > 0 sufficiently small, f(x 0 ) f(x 0 +h), f(x 0 h) f(x 0 ), then the following limits hold (in the metric d) f(x lim 0 ) f(x 0 +h) f(x h 0 = lim 0 h) f(x 0 ) h h 0 = f (x h 0 ), Or 3. h > 0 sufficiently small f(x 0 +h) f(x 0 ), f(x 0 h) f(x 0 ) and the following limits hold (in metric d) (x lim 0 +h) f(x 0 ) f(x h 0 = lim 0 h) f(x 0 ) h h 0 = f (x h 0 ), Or 5

6 4. h > 0 sufficiently small f(x 0 ) f(x 0 +h), f(x 0 ) f(x 0 h), then the following limits holds(in metric d) f(x lim 0 ) f(x 0 +h) f(x h 0 = lim 0 h) f(x 0 ) h h 0 = f (x h 0 ). The denominators h and h denote multiplication y 1 h and 1 h respectively. Theorem.8. (Ses Chalco-Cano and Román-Flores [31]). Let f : R E e a function denoted y f(t) = (f(t,r),f(t,r)) for each r [0,1]. Then 1. If f is (i)-differentiale, then f(t,r) and f(t,r) are differentiale functions and f (t) = (f (t,r),f (t,r)),. If f is (ii)-differentiale, then f(t,r) and f(t,r) are differentiale functions and f (t) = (f (t,r),f (t,r)). Lemma.9. (see Bede and Gal [3, 33]). Let x 0 R. Then, the FDE y = f(x,y), y(x 0 ) = y 0 R and f : R E E is supposed to e a continuous and equivalent to one of the following integral equations. or y(x) = y 0 + x y(0) = y 1 (x)+( 1) x 0 f(t,y(t))dt x [x 0,x 1 ], x x 0 f(t,y(t))dt x [x 0,x 1 ], on some interval (x 0,x 1 ) R depending on the strongly generalized differentiaility. Integral equivalency shows that if one solution satisfies the given equation, then the other will also satisfy. Remark.10. (see Bede and Gal [3, 33]). In the case of strongly generalized differentiaility to the FDE s y = f(x,y) we use two different integral equations. But in the case of differentiaility as the definition of H-derivative, we use only one integral. The second integral equation as in Lemma.10 will e in the form of y 1 (t) = y0 1 ( 1) x x 0 f(t,y(t))dt. The following theorem related to the existence of solution of FIVP under the generalized differentiaility. Theorem.11. Let us suppose that the following conditions are satisfied. 1. Let R 0 = [x 0,x 0 +s] B(y 0,q),s,q > 0,y E, where B(y 0,q) = {y E : B(y,y 0 ) q} which denotes a closed all in E and let f : R 0 E e continuous functions such that D(0,f(x,y)) M, (x,y) R 0 and 0 E. 6

7 . Let g : [x 0,x 0 +s] [0,q] R such that g(x,0) 0 and 0 g(x,u) M, x [x 0,x 0 + s],0 u q, such that g(x,u) is increasing in u, and g is such that the FIVP u (x) = g(x,u(x)),u(x) 0 on [x 0,x 0 +s]. 3. We have D[f(x,y),f(x,z) g(x,d(y,z))], (x,y), (x, z) R 0 and D(y,z) q. 4. d > 0 such that for x [x 0,x 0 +d], the sequence yn 1 : [x 0,x 0 +d] E given y y0 1(x) = y 0, yn+1 1 (x) = y 0 ( 1) x x 0 f(t,yn 1 )dt defined for any n N. Then the FIVP y = f(x,y), y(x 0 ) = y 0 has two solutions that is (1)-differentiale and the other one is ()-differentiale for y. y 1 = [x 0,x 0 + r] B(y 0,q), where r = min{s, q, q M M 1,d} and the successive iterations y 0 (x) = y 0, y n+1 (x) = y 0 + x x 0 f(t,y n (t))dt and yn+1 1 = y 0, yn+1 1 (x) = y 0 ( 1) x x 0 f(t,yn 1 (t))dt converge to these two solutions respectively. Now according to theorem (.11), we restrict our attention to function which are (1) or ()-differentiale on their domain except on a finite numer of points as discussed in [33]. 3 Fuzzy Laplace Transform Suppose that f is a fuzzy-valued function and p is a real parameter, then according to [19, 0] FLT of the function f is defined as follows: Definition 3.1. The FLT of fuzzy-valued function is [19, 0] F(p) = L[f(t)] = 0 e pt f(t)dt, (3.1) τ F(p) = L[f(t)] = lim e pt f(t)dt, (3.) τ 0 τ τ F(p) = [lim e pt f(t)dt, lim e pt f(t)dt], (3.3) τ 0 τ 0 whenever the limits exist. Definition 3.. Classical Fuzzy Laplace Transform: Now consider the fuzzy-valued function in which the lower and upper FLT of the function are represented y F(p;r) = L[f(t;r)] = [l(f(t;r)),l(f(t;r))], (3.4) 7

8 where l[f(t;r)] = l[f(t;r)] = 0 0 τ e pt f(t;r)dt = lim e pt f(t;r)dt, (3.5) τ 0 τ e pt f(t;r)dt = lim e pt f(t;r)dt. (3.6) τ Fuzzy Boundary Value prolem The concept of fuzzy numers and fuzzy set was first introduced y Zadeh []. Detail information of fuzzy numers and fuzzy arithmetic can e found in [13, 14, 15]. In this section we review the fuzzy oundary valued prolem (FBVP) with crisp linear differential equation ut having fuzzy oundary values. For example we consider the second order fuzzy oundary prolem as [10, 11,, 1]. ψ (t)+c 1 (t)ψ (t)+c (t)ψ(t) = f(t), ψ(0) = Ã, ψ(l) = B. (3.7) 4 Constructing Solutions Via FBVP In this section we consider the following second order FBVP in general form under generalized H-differentiaility proposed in []. We define suject to two-point oundary conditions y (t) = f(t,y(t),y (t)), (4.1) Taking FLT of (4.1) y(0) = (y(0;r),y(0;r)), y(l) = (y(l;r),y(l;r)). L[y (t)] = L[f(t,y(t),y (t))], (4.) which can e written as p L[y(t)] py(0) y (0) = L[f(t,y(t),y (t))]. The classical form of FLT is given elow: p l[y(t;r)] py(0;r) y (0;r) = l[f(t,y(0;r),y (0;r))], (4.3) 8

9 p l[y(t;r)] py(0;r) y (0;r) = l[f(t,y(0;r),y (0;r))]. (4.4) Here we have to replace the unknown value y (0,r) y constant F 1 in lower case and y F in upper case. Then we can find these values y applying the given oundary conditions. In order to solve equations (4.3) and (4.4) we assume that A(p;r) and B(p; r) are the solutions of (4.3) and (4.4) respectively. Then the aove system ecomes l[y(t;r)] = A(p;r), (4.5) l[y(t;r)] = B(p;r). (4.6) Using inverse Laplace transform, we get the upper and lower solution for given prolem as: [y(t;r)] = l 1 [A(p;r)], (4.7) 5 Examples [y(t;r)] = l 1 [B(p;r)]. (4.8) In this section first we consider the Schrödinger equation [] with fuzzy oundary conditions under Hukuhara differentiaility. Example 5.1. The Schrödinger FBVP [] is as follows: ( h m )u (x)+v(x)u(x) = Eu(x), (5.1) where V(x) is potential and is defined as { 0, if x < 0, V(x) = l, if x > 0, suject to the following oundary conditions u(0) = (1+r,3 r), u(l) = (4+r,6 r). Now let a = h, = E. Then, (5.1) ecomes m au (x)+v(x)u(x) = u(x). (5.) In (5.1) for x < 0, we discuss (1,1) and (,)-differentiaility while in the case x > 0 we will discuss (1,) and (,1)-differentiaility. 9

10 5.1 Case-I: (1,1) and (,)-differentiaility For x < 0, (5.) ecomes au = u, au u = 0. (5.3) Now applying FLT on oth sides of equation (5.3), we get al[u (x)] L[u(x)] = 0, where L[u (x)] = p L[u(x)] pu(0) u (0). The classical FLT form of the aove equation is l[u (x,r)] = p l[u(x,r)] pu(0,r) u (0,r), l[u (x,r)] = p l[u(x,r)] pu(0,r) u (0,r). Solving the aove classical equations for lower and upper solutions, we have a{p l[u(x,r)] pu(0,r) u (0,r)} l[u(x,r)] = 0, or (ap )l[u(x,r)] = a{pu(0,r)+u (0,r)}. Applying the oundary conditions, we have (ap )l[u(x,r)] = a{p(1+r)+f 1 }, where F 1 = u (0,r). Simplifying and applying inverse Laplace we get u(x,r) = ( 1+r p )l 1 { }+F p 1 l 1 1 { }. p a a Using partial fraction u(x,r) = ( 1+r ){e a x +e a x }+ F 1 a {e a x e a x }. (5.4) 10

11 Now applying oundary conditions on (5.4) we get F 1 = Putting value of F 1 in (5.4) we get 1+r 4+r {e a l +e a l } 1 a {e a l e. a l } u(x,r) = ( 1+r ){e 1+r a x +e a x 4+r }+ {e a l +e a l } {e a l e {e a x e a x }. a l } Now solving the classical FLT form for u(x,r), we have a{p l[u(x,r)] pu(0,r) u (0,r)} l[u(x,r)] = 0, (ap )l[u(x,r)] = a{pu(0,r)+u (0,r)}. Using the oundary conditions, we have where (ap )l[u(x,r)] = a{p(3 r)+f }, F = u (0,r). Simplifying and applying inverse laplace we get Using partial fraction u(x,r) = ( 3 r p )l 1 { }+F p l 1 1 { }. ap a a u(x,r) = ( 3 r ){e a x +e a x }+ F a {e a x e a x }. (5.5) Now applying oundary conditions on (5.5) we have Putting value of F in (5.5) we get 3 r 6 r F = {e a l +e a l } 1 a {e a l e. a l } u(x,r) = ( 3 r ){e 3 r a x +e a x 6 r }+ {e a l +e a l } {e a l e {e a x e a x }. a l } 11

12 5. Case-II: (1) and ()-differentiaility, () and (1)- differentiaility For x > 0, (5.) ecomes au +(l )u = 0. (5.6) Applying FLT and inverse Laplace transform and then simplifying we get the following lower solution. u(x,r) = ( 1+r [ ) cos x l +cosh x ] l + H [ 1 a a a sin x l +sinh x ] l l a a (3 r) [ cos x l cosh x ] l H [ a a a sin x l sinh x l ], l a a or u(x,r) = 1+r (c 1)+ H 1 a l (c ) 3 r (c 3) H a l (c 4). The upper solution will e as follows: u(x,r) = ( 3 r [ ) cos x l +cosh x ] l + H a a a l (1+r) [ cos x l cosh x ] l H 1 a a a l u(x,r) = 3 r (c 1)+ H a l (c ) 1+r (c 3) H 1 a l (c 4) where c 1 = cos x l +cosh x l, a a c = sin x l a +sinh x l a, c 3 = cos x l a cosh x l a, [ sin x l +sinh x ] l a a [ sin x l sinh x ] l a a c 4 = sin x l sinh x l. a a H 1 = c [ 4+r r +1 c c 4 c 1+ 3 r ] c 3 + c [ 4 6 r 3 r c c 4 c 1+ 1+r ] c 3, and H = c [ 4 4+r r +1 c c 4 c 1+ 3 r ] c 3 + c [ 6 r 3 r c c 4 c 1+ 1+r ] c 3. 1

13 Example 5.. Consider the following fuzzy homogenous oundary value prolem x (t) 3x (t)+x(t) = 0, (5.7) suject to the following oundary conditions x(0) = (0.5r 0.5,1 r), x(1) = (r 1,1 r). Now applying fuzzy Laplace transform on oth sides of equation (5.7), we get We know that L[x (t)] = 3L[x (t)] L[x(t)]. (5.8) L[x (t)] = p L[x(t)] px(0) x (0). The classical FLT form of the aove equation is l[x (t,r)] = p l[x(t,r)] px(0,r) x (0,r), l[x (t,r)] = p l[x(t,r)] px(0,r) x (0,r). Now on putting in (5.8), we have p l[x(t,r)] px(0,r) x (0,r)} 3pl[x(t,r)]+3x(0,r)+l[x(t,r)] = 0, (5.9) p l[x(t,r)] px(0,r) x (0,r)} 3pl[x(t,r)]+3x(0,r)+l[x(t,r)] = 0. (5.10) Solving (5.9) for l[x(t,r)], we get (p 3p+)l[x(t,r)] = px(0,r)+x (0,r)+3[x(0,r)]. Applying oundary conditions, we have l[x(t,r)] = (0.5r 0.5)p p 3p+ 3(0.5r 0.5) p 3p+ + A p 3p+. Using partial fraction and then applying inverse Laplace, we get x(t,r) = (0.5r 0.5)[ e t +e t ] 3(0.5r 0.5)[ e t +e t ]+A[ e t +e t ]. Using oundary values, we get x(1,r) = r 1 = (0.5r 0.5)[ e+e ] 3(0.5r 0.5)[ e+e ]+A[ e+e ], 13

14 A = r 1+(0.5r 0.5)[ e+e ]. e e Finally on putting value of A we have x(t,r) = (0.5r 0.5)( e t +e t ) 3(0.5r 0.5)( e t +e t )+ r 1+(0.5r 0.5)( e+e ) ( e+e ) e e Now solving (5.10) for l[x(t,r)], we have (p 3p+)l[x(t,r)] = px(0,r)+x (0,r)}+3[x(0,r)]. Applying oundary condition we get l[x(t,r)] = (1 r)p p 3p+ 3(1 r) p 3p+ + A p 3p+. Using partial fraction and then applying inverse Laplace x(t,r) = (1 r)[ e t +e t ] 3(1 r)[ e t +e t ]+A[ e t +e t ]. (5.11) Using oundary values x(1,r) = 1 r = (1 r)[ e+e ] 3(1 r)[ e+e ]+A[ e+e ], Putting value of A in (5.11) we get A = 1 r +(1 r)[ e+e ]. e e x(t,r) = (1 r)[ e t +e t ] 3(1 r)[ e t +e t ]+ 1 r +(1 r)[ e+e ] [ e+e t ]. e e 6 Conclusion In this paper, we applied the fuzzy Laplace transform to solve FBVPs under generalized H-differentiaility, in particular, solving Schrödinger FBVP. We also used FLT to solve homogenous FBVP. This is another application of FLT. Thus FLT can also e used to solve FBVPs analytically. The method can e extended for an nth order FBVP. This work is in progress. 14

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