Math Sci DOI 10.1007/s40096-016-0205-x
ORIGINAL RESEARCH
Quadrature iterative method for numerical solution of two-dimensional linear fuzzy Fredholm integral equations H. Nouriani
1 •
R. Ezzati1
Received: 15 July 2016 / Accepted: 16 October 2016 Ó The Author(s) 2017. This article is published with open access at Springerlink.com
Abstract In this paper, first, we propose an iterative method based on quadrature formula for solving two-dimensional linear fuzzy Fredholm integral equations (2DLFFIE). Then, we prove the error estimation of the method. In addition, we show the numerical stability analysis of the method with respect to the choice of the first iteration. Finally, supporting examples are also provided. Keywords Fuzzy-number-valued functions Numerical method Two-dimensional linear fuzzy Fredholm integral equations Quadrature iterative method
Introduction The study of fuzzy differential and integral equations begins with the research of Kaleva [13] and Seikkala [21]. As we know, showing the existence and uniqueness solution of these equations is very important. So, to this end, many authors applied the Banach’s fixed point theorem and the method of successive approximations. Recently, many researchers proposed numerical methods for solving fuzzy Fredholm integral equations (FFIEs). To see more details about solving FFIE, one can refer to [2, 4–6, 8–12, 15, 16, 18, 19, 22, 23]. Solving linear and nonlinear FFIEs based on iterative method is done by many authors but for the first time, solving 2DLFFIEs was studied by authors of [18] that the authors proved the existence and uniqueness solution of & R. Ezzati
[email protected] 1
Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran
these equations by using Banach’s fixed point theorem. In addition, in [19], authors presented a numerical method for solving two-dimensional nonlinear FFIEs by using an iterative method. Recently, authors of [7, 14, 17, 20] proposed some numerical approaches to solve 2DLFFIEs. Here, we propose a numerical method for solving (3.1). In addition, we present the estimation error and the numerical stability analysis. The rest of the paper is organized as follows: In ‘‘Preliminaries’’, we review some properties for fuzzy-number-valued functions, such as continuity, boundedness, fuzzy Henstock and fuzzy Riemann integrability and fuzzy quadrature rules . In ‘‘2D linear fuzzy Fredholm integral equations’’, we propose a new approach based on quadrature rule and iterative method to solve 2DLFFIEs. In ‘‘Convergence analysis’’, we investigate convergence analysis of the proposed method. Considering the numerical stability analysis of the proposed method is done in ‘‘Numerical stability analysis’’. Finally, some numerical examples are presented in ‘‘Numerical examples’’.
Preliminaries At first, we present some basic definitions and necessary results about fuzzy set theory. Definition 2.1 [1] A fuzzy number is a function u : R ! ½0; 1 with the following properties: 1. 2. 3.
u is normal, i.e. 9x0 2 R; uðx0 Þ ¼ 1. uðgx þ ð1 gÞyÞ minfuðxÞ; uðyÞg8x; y 2 R; 8g 2 ½0; 1 (u is called a convex fuzzy subset). u is upper semicontinuous on R, i.e., 8x0 2 R and 8 [ 0, 9 neighborhood Vðx0 Þ : uðxÞ uðx0 Þ þ ; 8x 2 Vðx0 Þ.
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4.
The set suppðuÞ is compact suppðuÞ :¼ fx 2 R; uðxÞ [ 0g).
in
R
(where
The set of all fuzzy numbers is denoted by RF . Definition 2.2 [1] For 0\r 1 and u 2 RF define ½ur : ¼ fx 2 R : uðxÞ rg and ½u0 :¼ fx 2 R : uðxÞ [ 0g: It is well known that for each r 2 ½0; 1, ½ur is a closed and bounded interval of R. For u~, v~ 2 RF and k 2 R, we define uniquely the sum u~ v~ and the product k u~ by ½~ u v~r ¼ ½~ ur þ ½~ vr ; ½k u~r ¼ k½~ ur ; 8r 2 ½0; 1; vr means the usual addition of two intervals where ½~ ur þ ½~ (as subsets of R). Also, k½~ ur means the usual product between a scalar and a subset of R. Notice 1 u~ ¼ u~ and it holds u~ v~ ¼ v~ u~, k u~ ¼ u~ k. If 0 r1 r2 1 then ðrÞ
ðrÞ
ðrÞ ðrÞ ½~ ur2 ½~ ur1 . Actually ½~ ur ¼ ½~ u ; u~þ , where u~ u~þ , ðrÞ
u~ðrÞ ,
u~þ 2 R,
8r 2 ½0; 1.
For
k[0
one
has
5.
For any c, g 2 R and any u~ 2 RF , we have c ðg u~Þ ¼ ðc gÞ u~.
~ 8~ If we denote ku~kF :¼ Dð~ u; 0Þ, u 2 RF , then k:kF has the properties of a usual norm on RF , i.e., ~ kk u~k ¼ jkj:ku~k ; ku~kF ¼ 0 iff u~ ¼ 0; F F u; v~Þ: ku~ v~kF ku~kF þkv~kF ; ku~kF kv~kF Dð~ Notice that ðRF ; ; Þ is not a linear space over R, and P consequently ðRF ; k:kF Þ is not a normed space. Here denotes the fuzzy summation. Definition 2.6 [13] A fuzzy valued function f : ½a; b ! RF is said to be continuous at x0 2 ½a; b, if for each [ 0 there exists d [ 0 such that Dðf ðxÞ; f ðx0 ÞÞ\, whenever x 2 ½a; b and j x x0 j\d. We say that f is fuzzy continuous on [a, b] if f is continuous at each x0 2 ½a; b, and denotes the space of all such functions by CF ½a; b. Definition 2.7 [3] Let f : ½a; b ! RF be a bounded mapping. Then the function x½a;b ðf ; :Þ : Rþ [ f0g ! Rþ
respectively.
x½a;b ðf ; dÞ ¼ supfDðf ðxÞ; f ðyÞÞ; x; y 2 ½a; b; j x yj dg;
[1] Define D : RF RF ! Rþ by ðrÞ ðrÞ ðrÞ ~ ~ ~ Dð~ u; v~Þ :¼ sup max u~ðrÞ v v ; u þ þ
is called the modulus of oscillation of f on [a, b]. If f 2 CF ½a; b (i.e. f : ½a; b ! RF is continuous on [a, b]), then x½a;b ðf ; dÞ is called uniform modulus of continuity of f.
ðrÞ k~ u
¼ ðk
ðrÞ u~Þ ,
Definition 2.3
r2½0;1
¼ sup Hausdorff distance ð½~ ur ; ½~ vr Þ; r2½0;1
The following properties will be very useful in what follows. ðrÞ
ðrÞ v ; v~þ ; u~, v~ 2 RF . Clearly, D is a metric on where ½~ vr ¼ ½~ RF . Also ðRF ; DÞ is a complete metric space, with the following properties [1]:
~ v~ wÞ ~ ¼ Dð~ ~ 2 RF ; Dð~ u w; u; v~Þ; 8~ u; v~; w 0 0 0 Dðk u~; k v~Þ ¼ jk jDð~ u; v~Þ; 8~ u; v~ 2 RF ; 8k0 2 R; ~ e~Þ Dð~ ~ þ Dð~ ~ e~ 2 RF : Dð~ u v~; w u; wÞ v; e~Þ; 8~ u; v~; w; Definition 2.4 [1] Let f ; g : R ! RF be fuzzy number valued functions. The distance between f, g is defined by D ðf ; gÞ :¼ sup Dðf ðxÞ; gðxÞÞ: x2R
Lemma 2.5 1.
2. 3.
4.
[1]
If we denote 0~ :¼ vf0g , then 0~ 2 RF is the neutral element with respect to , i.e., u~ 0~ ¼ 0~ u~ ¼ u~, 8~ u 2 RF . ~ none of u~ 2 RF , u~ 6¼ 0~ has opposite With respect to 0, in RF . Let a, b 2 R : a:b 0, and any u~ 2 RF , we have ða þ bÞ u~ ¼ a u~ b u~. For general a, b 2 R, the above property is false. For any c 2 R and any u~, v~ 2 RF , we have c ð~ u v~Þ ¼ c u~ c v~.
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Theorem 2.8 [3] The following statements, concerning the modulus of oscillation, are true: 1. 2. 3. 4. 5. 6.
Dðf ðxÞ; f ðyÞÞ x½a;b ðf ; j x yjÞ; 8x; y 2 ½a; b, x½a;b ðf ; dÞ is a nondecreasing mapping in d, x½a;b ðf ; 0Þ ¼ 0, x½a;b ðf ; d1 þ d2 Þ x½a;b ðf ; d1 Þ þ x½a;b ðf ; d2 Þ, d2 0, x½a;b ðf ; ndÞ nx½a;b ðf ; dÞ, 8d 0, n 2 N; x½a;b ðf ; gdÞ ðg þ 1Þx½a;b ðf ; dÞ, 8d, g 0.
8d1 ,
Definition 2.9 [1] Let f : ½a; b ! RF . We say that f is fuzzy-Riemann integrable to I 2 RF if for any [ 0, there exists d [ 0 such that for any division P ¼ f½u; v; ng of [a, b] with the norms DðpÞ\d, we have ! X D ðv uÞ f ðnÞ; I \ ; p
where
P
I :¼ ðFRÞ
denotes the fuzzy summation. We choose to write Z
b
f ðxÞdx: a
We also call an f as above (FR)-integrable.
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Lemma 2.10 [1] If f ; g : ½a; b R ! RF are fuzzy continuous functions, then the function F : ½a; b ! Rþ defined by FðxÞ :¼ Dðf ðxÞ; gðxÞÞ is continuous on [a, b], and Z b Z b Z b D ðFRÞ f ðxÞdx; ðFRÞ gðxÞdx Dðf ðxÞ; gðxÞÞdx: a
a
a
Theorem 2.11 [3] Let f : ½a; b ! RF be a Henstock integrable, bounded mapping. Then, for any division a ¼ x0 \x1 \ \xn ¼ b and any points ni 2 ½xi1 ; xi we have ! Z b n X D ðFHÞ f ðtÞdt; ðxi xi1 Þ f ðni Þ a
i¼1
n X ðxi xi1 Þx½xi1 ;xi ðf ; xi xi1 Þ:
Theorem 2.16 [18] If f and g are Henstock integrable mapping on ½a; b ½c; d and if D(f(s, t), g(s, t)) is Lebesgue integrable, then Z dZ b Z dZ b f ðs; tÞdsdt; ðFHÞ gðs; tÞdsdt D ðFHÞ ðLÞ
Z
c d
Z
c
a b
c
a
Dðf ðs; tÞ; gðs; tÞÞdsdt:
ð2:1Þ
a
Definition 2.17 [18] A function f : ½a; b ½c; d ! RF is said to be L-Lipschitz, if qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2:2Þ Dðf ðx; yÞ; f ðs; tÞÞ L ðx sÞ2 þ ðy tÞ2 ; 8x; s 2 ½a; b, y; t 2 ½c; d.
i¼1
By the above theorem, the following result holds: Corollary 2.12 [3] Let f : ½a; b ! RF be a Henstock integrable, bounded mapping. Then Z D ðFHÞ
ba aþb f ðbÞ f ðaÞ 4 f 6 2 a ba : 3ðb aÞx½a;b f ; 6 b
Theorem 2.18 [18] Let f : ½a; b ½c; d ! RF be Henstock integrable, bounded mappings. Then, for any divisions a ¼ x0 \x1 \ \xn ¼ b and c ¼ y0 \y1 \ \yn ¼ d and any points ni 2 ½xi1 ; xi and gj 2 ½yj1 ; yj , one has
f ðtÞdt;
D ðFHÞ
Z
d
ðFHÞ
Z
c
n X
b
f ðs; tÞdsdt; a
n X n X
! ðxi xi1 Þðyj yj1 Þ f ðni ; gj Þ
j¼1 i¼1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxi xi1 Þðyj yj1 Þx½xi1 ;xi ½yj1 ;yj f ; ðxi xi1 Þ2 þ ðyj yj1 Þ2 :
i¼1
Definition 2.13 [18] Suppose that f : ½a; b ½c; d ! Rf is a bounded mapping. The function x½a;b ½c;d ðf ; :Þ : Rþ [ f0g ! Rþ defined by x½a;b ½c;d ðf ; dÞ ¼ supfDðf ðx; yÞ; f ðs; tÞÞ; x; s 2 ½a; b; y; t 2 ½c; d; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx sÞ2 þ ðy tÞ2 dg;
is called modules of oscillation of f on ½a; b ½c; d. In addition, if f 2 CF ð½a; b ½c; dÞ, then x½a;b ½c;d ðf ; dÞ is called uniform modules of continuity of f. [18] The following properties hold: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dðf ðx; yÞ; f ðs; tÞÞ x½a;b ½c;d ðf ; ðx sÞ2 þ ðy tÞ2 Þ;
Corollary 2.19 [18] Let f : ½a; b ½c; d ! RF be a twodimensional Henstock integrable, bounded mapping. Then Z d Z b ðb aÞðd cÞ f ða; cÞ f ða; dÞ D ðFHÞ ðFHÞ f ðs; tÞdsdt; 36 c a cþd aþb aþb aþc 4 f a; 4f ; c 16 f ; 2 2 2 2 aþb cþd 4f f ðb; cÞ f ðb; dÞ ; d 4 f b; 2 2 ðb aÞðd cÞ : ðb aÞðd cÞx½a;b ½c;d f ; 36
ð2:3Þ
Theorem 2.14 1. 2. 3. 4. 5. 6.
8x; s 2 ½a; b; y; t 2 ½c; d; x½a;b ½c;d ðf ; dÞ is a nondecreasing mapping in d; x½a;b ½c;d ðf ; 0Þ ¼ 0; x½a;b ½c;d ðf ; d1 þ d2 Þ x½a;b ½c;d ðf ; d1 Þ þ x½a;b ½c;d ðf ; d2 Þ, 8d1 ; d2 0; x½a;b ½c;d ðf ; ndÞ nx½a;b ½c;d ðf ; dÞ, 8d 0, n 2 N; x½a;b ½c;d ðf ; kdÞ ðk þ 1Þx½a;b ½c;d ðf ; dÞ, 8k; d 0.
Corollary 2.15 [18] We have Z dZ b Z dZ ðFRÞ f ðs; t; rÞdsdt ¼ c
ðFRÞ
Z c
a d
Z a
c b
f ðs; t; rÞdsdt ¼
Z c
Z
Consider 2DLFFIE as follows Z d Fðs; tÞ ¼ f ðs; tÞ k ðFRÞ ðFRÞ c Z b kðs; t; x; yÞ Fðx; yÞdxdy;
ð3:1Þ
a
f ðs; t; rÞdsdt;
where k [ 0, k(s, t, x, y) is an arbitrary positive function on ½a; b ½c; d ½a; b ½c; d and f : ½a; b ½c; d ! RF . We assume that k is continuous and, therefore, it is uniformly continuous with respect to (s, t). So, there exists M [ 0 such that M ¼ max jkðs; t; x; yÞj. Here, we
f ðs; t; rÞdsdt:
present a quadrature iterative method for solving this equation.
b
a d
2D linear fuzzy Fredholm integral equations
b
s;x2½a;b;t;y2½c;d
a
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Theorem 3.1 [18] Let the function k(s, t, x, y) be continuous and positive for s; x 2 ½a; b, and t; y 2 ½c; d, and let f : ½a; b ½c; d ! RF be continuous on ½a; b ½c; d. Also, suppose f(x,t) that is not zero. If B ¼ kMðb aÞðd cÞ\1 then the fuzzy integral equation (3.1) has a unique solution F 2 X where X ¼ ff : ½a; b ½c; d ! RF ; f is continousg; is the space of two-dimensional fuzzy continuous functions with the metric D ðf ; gÞ ¼
Convergence analysis In this section, we obtain an error estimate between the exact solution and the approximate solution of 2DLFFIE (3.1). Theorem 4.1 Under the hypotheses of Theorem 3.1 and k [ 0, the iterative procedure (3.6) converges to the unique solution of (3.1), F , and its error estimate is as follows D ðF ; um Þ
Dðf ðs; tÞ; gðs; tÞÞ;
sup s2½a;b;t2½c;d
and it can be obtained by the following successive approximations method F0 ðs; tÞ ¼ f ðs; tÞ; Fm ðs; tÞ ¼ f ðs; tÞ k ðFRÞ
Z
d
ðFRÞ
Z
c
Fm1 ðx; yÞdxdy;
b
kðs; t; x; yÞ a
8m 1:
ð3:2Þ
Moreover, the sequence of successive approximations, ðFm Þm 1 converges to the solution F . Furthermore, the following error bound holds Bmþ1 M1 ; 8m 1; D ðF ; Fm Þ 1B
ð3:3Þ
where M1 ¼ sups2½a;b;t2½c;d kFðs; tÞkF . Now, we propose a numerical method to solve (3.1). In this way, we assume 2DLFFIE (3.1) and uniform partitions of the interval ½a; b ½c; d Dx : a ¼ s0 \s1 \ \s2n1 \s2n ¼ b;
ð3:4Þ
with si ¼ a þ ih, where h ¼ ba 2n , i ¼ 0; ; 2n and Dy : c ¼ t0 \t1 \ \t2n1 \t2n ¼ d;
ð3:5Þ
with tj ¼ c þ jh0 , where h0 ¼ dc 2n , j ¼ 0; ; 2n. So, the following iterative procedure gives the approximate solution of (3.1) in point (s, t) as follows
u0 ðs; tÞ
Bmþ1 M1 1B 10B ðx½a;b ½c;d ðf ; hh0 Þ þ 9ð1 BÞ B þ k f kxst ðk; hh0 Þ Mð1 BÞ 1 þ k f kxxy ðk; hh0 ÞÞ: Mð1 BÞ
ð4:1Þ
where xst ðk; dÞ ¼ supfjkðs1 ; t1 ; x; yÞ kðs2 ; t2 ; x; yÞj : qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðs2 s1 Þ2 þ ðt2 t1 Þ2 dg;
ð4:2Þ
8d 0; 8s1 ; s2 2 ½a; b; t1 ; t2 2 ½c; d xxy ðk; dÞ ¼ supfjkðs; t; x1 ; y1 Þ kðs; t; x2 ; y2 Þj : qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx2 x1 Þ2 þ ðy2 y1 Þ2 dg;
ð4:3Þ
8d 0; x1 ; x2 2 ½a; b; y1 ; y2 2 ½c; d Proof Considering iterative procedure (3.6), for all ðs; tÞ 2 ½a; b ½c; d, we have DðF1 ðs; tÞ; u1 ðs; tÞÞ Z Z d ðFRÞ ¼ D f ðs; tÞ k ðFRÞ c
b
kðs; t; x; yÞ
a
khh0 F0 ðx; yÞdxdy; f ðs; tÞ 9 n X n X ðkðs; t; s2i2 ; t2j2 Þ u0 ðs2i2 ; t2j2 Þ j¼1 i¼1
kðs; t; s2i2 ; t2j Þ u0 ðs2i2 ; t2j Þ kðs; t; s2i ; t2j2 Þ
¼ f ðs; tÞ
um ðs; tÞ
¼ f ðs; tÞ
n X n khh0 X kðs; t; s2i2 ; t2j2 Þ um1 ðs2i2 ; t2j2 Þ 9 j¼1 i¼1
kðs; t; s2i2 ; t2j Þ um1 ðs2i2 ; t2j Þ kðs; t; s2i ; t2j2 Þ um1 ðs2i ; t2j2 Þ kðs; t; s2i ; t2j Þ um1 ðs2i ; t2j Þ 4 kðs; t; s2i2 ; t2j1 Þ um1 ðs2i2 ; t2j1 Þ kðs; t; s2i1 ; t2j2 Þ um1 ðs2i1 ; t2j2 Þ kðs; t; s2i ; t2j1 Þ um1 ðs2i ; t2j1 Þ kðs; t; s2i1 ; t2j Þ um1 ðs2i1 ; t2j Þ 16kðs; t; s2i1 ; t2j1 Þ um1 ðs2i1 ; t2j1 Þ :
123
ð3:6Þ
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u0 ðs2i ; t2j2 Þ kðs; t; s2i ; t2j Þ u0 ðs2i ; t2j ÞÞ 4ðkðs; t; s2i2 ; t2j1 Þ u0 ðs2i2 ; t2j1 Þ kðs; t; s2i1 ; t2j2 Þ u0 ðs2i1 ; t2j2 Þ kðs; t; s2i ; t2j1 Þ u0 ðs2i ; t2j1 Þ kðs; t; s2i1 ; t2j Þ u0 ðs2i1 ; t2j ÞÞ
16kðs; t; s2i1 ; t2j1 Þ u0 ðs2i1 ; t2j1 Þ Z t2j Z s2i n X n X ¼D k ðFRÞ ðFRÞ kðs; t; x; yÞ f ðx; yÞdxdy; t2j2
j¼1 i¼1
s2i2
Mx½s2i2 ;s2i ½t2j2 ;t2j ðf ; hh0 Þ þ xxy ðk; hh0 Þk f k:
n X n khh0 X kðs; t; s2i2 ; t2j2 Þ f ðs2i2 ; t2j2 Þ 9 j¼1 i¼1
kðs; t; s2i2 ; t2j Þ f ðs2i2 ; t2j Þ kðs; t; s2i ; t2j2 Þ f ðs2i ; t2j2 Þ kðs; t; s2i ; t2j Þ f ðs2i ; t2j Þ 4ðkðs; t; s2i2 ; t2j1 Þ f ðs2i2 ; t2j1 Þ kðs; t; s2i1 ; t2j2 Þ f ðs2i1 ; t2j2 Þ kðs; t; s2i ; t2j1 Þ f ðs2i ; t2j1 Þ kðs; t; s2i1 ; t2j Þ f ðs2i1 ; t2j ÞÞ
t2j2
D ðF1 ; u1 Þ
Z
khh 9
c b
kðs; t; x; yÞ F1 ðx; yÞdxdy;
a n X n 0X
ðkðs; t; s2i2 ; t2j2 Þ
j¼1 i¼1
u1 ðs2i2 ; t2j2 Þ kðs; t; s2i2 ; t2j Þ u1 ðs2i2 ; t2j Þ kðs; t; s2i ; t2j2 Þ u1 ðs2i ; t2j2 Þ kðs; t; s2i ; t2j Þ u1 ðs2i ; t2j ÞÞ 4ðkðs; t; s2i2 ; t2j1 Þ u1 ðs2i2 ; t2j1 Þ kðs; t; s2i1 ; t2j2 Þ u1 ðs2i1 ; t2j2 Þ
16kðs; t; s2i1 ; t2j1 Þ f ðs2i1 ; t2j1 Þ
:
Using Corollary 2.19, we have ðs2i s2i2 Þðt2j t2j2 Þ
j¼1 i¼1
x½s2i2 ;s2i ½t2j2 ;t2j
10B 10B x½a;b ½c;d ðf ; hh0 Þ þ xxy ðk; hh0 Þk f k: 9 9M
DðF2 ðs; tÞ; u2 ðs; tÞÞ ¼ Dðf ðs; tÞ; f ðs; tÞÞ Z d ðFRÞ þ D k ðFRÞ
kðs; t; s2i1 ; t2j Þ f ðs2i2 ; t2j Þ kðs; t; s2i ; t2j2 Þ f ðs2i ; t2j2 Þ kðs; t; s2i ; t2j Þ f ðs2i ; t2j Þ 4ðkðs; t; s2i2 ; t2j1 Þ f ðs2i2 ; t2j1 Þ kðs; t; s2i1 ; t2j2 Þ f ðs2i1 ; t2j2 Þ
n X n X
ð4:4Þ
Now, for m ¼ 2, it follows that
hh kðs; t; s2i2 ; t2j2 Þ f ðs2i2 ; t2j2 Þ 9
DðF1 ðs; tÞ; u1 ðs; tÞÞ k
x½s2i2 ;s2i ½t2j2 ;t2j ðkf ; hh0 Þ Mx½s2i2 ;s2i ½t2j2 ;t2j ðf ; hh0 Þ
Therefore,
s2i2
kðs; t; s2i ; t2j1 Þ f ðs2i ; t2j1 Þ kðs; t; s2i1 ; t2j Þ f ðs2i1 ; t2j ÞÞ
Taking supremum from above inequality, we conclude that, þ xxy ðk; hh0 Þk f k:
16kðs; t; s2i1 ; t2j1 Þ f ðs2i1 ; t2j1 Þ Z t2j Z s2i n X n X D kðs; t; x; yÞ f ðx; yÞdxdy; k j¼1 i¼1 0
D f ða; cÞ kðs; t; a; cÞ; f ðb; gÞ kðs; t; b; gÞ D f ða; cÞ kðs; t; a; cÞ; f ðb; gÞ kðs; t; a; cÞ þ D f ðb; gÞ kðs; t; a; cÞ; f ðb; gÞ kðs; t; b; gÞ jkðs; t; a; cÞjD f ða; cÞ; f ðb; gÞ þ jkðs; t; a; cÞ kðs; t; b; gÞjD f ðb; gÞ; 0~
hh0 kf ; ; 9
n X n 10 X k ðs2i s2i2 Þðt2j t2j2 Þ 9 j¼1 i¼1
x½s2i2 ;s2i ½t2j2 ;t2j ðkf ; hh0 Þ: Since ða; cÞ; ðb; gÞ 2 ½s2i2 ; s2i ½t2j2 ; t2j qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ða bÞ þ ðc gÞ2 hh0 , we have
with
kðs; t; s2i ; t2j1 Þ u1 ðs2i ; t2j1 Þ kðs; t; s2i1 ; t2j Þ u1 ðs2i1 ; t2j ÞÞ
16kðs; t; s2i1 ; t2j1 Þ u1 ðs2i1 ; t2j1 Þ n X n Z t2j Z s2i X k kðs; t; x; yÞ D j¼1 i¼1
t2j2
s2i2
F1 ðx; yÞdxdy; hh0 ðkðs; t; s2i2 ; t2j2 Þ F1 ðs2i2 ; t2j2 Þ 9
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kðs; t; s2i2 ; t2j Þ F1 ðs2i2 ; t2j Þ
So, we have the following result:
kðs; t; s2i ; t2j2 Þ F1 ðs2i ; t2j2 Þ
D ðF2 ; u2 Þ
kðs; t; s2i ; t2j Þ F1 ðs2i ; t2j ÞÞ 4ðkðs; t; s2i2 ; t2j1 Þ F1 ðs2i2 ; t2j1 Þ kðs; t; s2i1 ; t2j2 Þ F1 ðs2i1 ; t2j2 Þ kðs; t; s2i ; t2j1 Þ F1 ðs2i ; t2j1 Þ
By induction, for m 3, we obtain
kðs; t; s2i ; t2j Þ u1 ðs2i ; t2j ÞÞ 4ðkðs; t; s2i2 ; t2j1 Þ u1 ðs2i2 ; t2j1 Þ
10B x½a;b ½c;d ðFm1 ; hh0 Þ 9 10B xxy ðk; hh0 ÞkFm1 k þ 9M þ BD ðFm1 ; um1 Þ; 10B x½a;b ½c;d ðFm2 ; hh0 Þ D ðFm1 ; um1 Þ 9 10B xxy ðk; hh0 ÞkFm2 k þ 9M þ BD ðFm2 ; um2 Þ; 10B x½a;b ½c;d ðFm3 ; hh0 Þ D ðFm2 ; um2 Þ 9 10B xxy ðk; hh0 ÞkFm3 k þ 9M þ BD ðFm3 ; um3 Þ; .. .. .. . . . 10B D ðF1 ; u1 Þ x½a;b ½c;d ðF0 ; hh0 Þ 9 10B xxy ðk; hh0 ÞkF0 k þ 9M þ BD ðF0 ; u0 Þ:
kðs; t; s2i1 ; t2j2 Þ u1 ðs2i1 ; t2j2 Þ
Then
D ðFm ; um Þ
kðs; t; s2i1 ; t2j Þ F1 ðs2i1 ; t2j ÞÞ
16kðs; t; s2i1 ; t2j1 Þ F1 ðs2i1 ; t2j1 Þ 0 hh þD ððkðs; t; s2i2 ; t2j2 Þ F1 ðs2i2 ; t2j2 Þ 9 kðs; t; s2i2 ; t2j Þ F1 ðs2i2 ; t2j Þ kðs; t; s2i ; t2j2 Þ F1 ðs2i ; t2j2 Þ kðs; t; s2i ; t2j Þ F1 ðs2i ; t2j ÞÞ 4ðkðs; t; s2i2 ; t2j1 Þ F1 ðs2i2 ; t2j1 Þ kðs; t; s2i1 ; t2j2 Þ F1 ðs2i1 ; t2j2 Þ kðs; t; s2i ; t2j1 Þ F1 ðs2i ; t2j1 Þ kðs; t; s2i1 ; t2j Þ F1 ðs2i1 ; t2j ÞÞ 16kðs; t; s2i1 ; t2j1 Þ F1 ðs2i1 ; t2j1 ÞÞ; 0
hh ððkðs; t; s2i2 ; t2j2 Þ u1 ðs2i2 ; t2j2 Þ 9 kðs; t; s2i2 ; t2j Þ u1 ðs2i2 ; t2j Þ kðs; t; s2i ; t2j2 Þ u1 ðs2i ; t2j2 Þ
kðs; t; s2i ; t2j1 Þ u1 ðs2i ; t2j1 Þ kðs; t; s2i1 ; t2j Þ u1 ðs2i1 ; t2j ÞÞ
D ðFm ; um Þ
16kðs; t; s2i1 ; t2j1 Þ u1 ðs2i1 ; t2j1 Þ
10B 10B x½a;b ½c;d ðF1 ; hh0 Þ þ xxy ðk; hh0 ÞkF1 k 9 9M þ kMðb aÞðd cÞD ðF1 ; u1 Þ
10B 10B x½a;b ½c;d ðF1 ; hh0 Þ þ xxy ðk; hh0 ÞkF1 k 9 9M n X n khh0 X kðs; t; s2i2 ; t2j2 Þ þ 9 j¼1 i¼1
D F1 ðs2i2 ; t2j2 Þ; u1 ðs2i2 ; t2j2 Þ
þ kðs; t; s2i2 ; t2j ÞD F1 ðs2i2 ; t2j Þ; u1 ðs2i2 ; t2j Þ
þ kðs; t; s2i ; t2j2 ÞD F1 ðs2i ; t2j2 Þ; u1 ðs2i ; t2j2 Þ
þ kðs; t; s2i ; t2j ÞD F1 ðs2i ; t2j Þ; u1 ðs2i ; t2j Þ
þ 4kðs; t; s2i2 ; t2j1 ÞD F1 ðs2i2 ; t2j1 Þ; u1 ðs2i2 ; t2j1 Þ
þ 4kðs; t; s2i1 ; t2j2 ÞD F1 ðs2i1 ; t2j2 Þ; u1 ðs2i1 ; t2j2 Þ
þ 4kðs; t; s2i ; t2j1 ÞD F1 ðs2i ; t2j1 Þ; u1 ðs2i ; t2j1 Þ
þ 4kðs; t; s2i1 ; t2j ÞD F1 ðs2i1 ; t2j Þ; u1 ðs2i1 ; t2j Þ
þ 16kðs; t; s2i1 ; t2j1 ÞD F1 ðs2i1 ; t2j1 Þ; u1 ðs2i1 ; t2j1 Þ
123
10B ðx½a;b ½c;d ðFm1 ; hh0 Þ þ Bx½a;b ½c;d ðFm2 ; hh0 Þ 9 2 þ B x½a;b ½c;d ðFm3 ; hh0 Þ þ þ Bm1 x½a;b ½c;d ðf ; hh0 ÞÞ 10B xxy ðk; hh0 ÞðkFm1 k þ BkFm2 k þ þ Bm1 kF0 kÞ: þ 9M
ð4:5Þ On the other hand, we have: DðFm ðs1 ; t1 Þ; Fm ðs2 ; t2 ÞÞ ¼ Dðf ðs1 ; t1 Þ k ðFRÞ Z d Z b ðFRÞ kðs1 ; t1 ; x; yÞ c
a
Fm1 ðx; yÞdxdy;
Z d ðFRÞ f ðs2 ; t2 Þ k ðFRÞ c Z b kðs2 ; t2 ; x; yÞ Fm1 ðx; yÞdxdyÞ a
Dðf ðs1 ; t1 Þ; f ðs2 ; t2 ÞÞ Z dZ b þk jkðs1 ; t1 ; x; yÞ kðs2 ; t2 ; x; yÞj c
a
~ DðFm1 ðx; yÞ; 0Þdxdy;
Math Sci
therefore, we see that x½a;b ½c;d ðFm ; hh0 Þ x½a;b ½c;d ðf ; hh0 Þ þ
B xst ðk; hh0 ÞkFm1 k; M
ð4:6Þ for any ðs1 ; t1 Þ; ðs2 ; t2 Þ 2 ½a; b ½c; d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðs1 s2 Þ þ ðt1 t2 Þ2 hh0 .
with
By using above inequality and (4.5), we obtain D ðFm ; um Þ
10B x½a;b ½c;d ðf ; hh0 Þð1 þ B þ B2 þ þ Bm1 Þ 9 10B2 þ xst ðk; hh0 ÞðkFm2 k þ BkFm3 k þ B2 kFm4 k 9M þ þ Bm2 kF0 kÞ 10B þ xxy ðk; hh0 ÞðkFm1 k þ BkFm2 k þ B2 kFm3 k 9M þ þ Bm1 kF0 kÞ:
c
b
kðs; t; x; yÞ Fm1 ðx; yÞdxdy; a
f ðs; tÞ k ðFRÞ
Z
d
ðFRÞ c
Z
b
kðs; t; x; yÞ a
Fm2 ðx; yÞdxdyÞ Z dZ b k jkðs; t; x; yÞjDðFm1 ðx; yÞ; Fm2 ðx; yÞÞdxdy c
Therefore, we see that D ðFm ; um Þ
10B ðx½a;b ½c;d ðf ; hh0 Þ 9ð1 BÞ B þ k f kxst ðk; hh0 Þ Mð1 BÞ 1 þ k f kxxy ðk; hh0 ÞÞ: Mð1 BÞ h
Numerical stability analysis
Now, by using (3.2), we conclude that DðFm ðs; tÞ; Fm1 ðs; tÞÞ ¼ Dðf ðs; tÞ k Z Z d ðFRÞ ðFRÞ
~ DðFm ðs; tÞ; F0 ðs; tÞÞ þ DðF0 ðs; tÞ; 0Þ ~ DðFm ðs; tÞ; 0Þ 1 D ðF1 ; F0 Þ þ k f k 1B 1 k f k: 1B
a
kMðd cÞðb aÞD ðFm1 ; Fm2 Þ Bm1 D ðF1 ; F0 Þ:
To show the numerical stability analysis of the proposed method in previous section, we consider another starting approximation gðs; tÞ ¼ Y0 ðs; tÞ such that 9 [ 0 for which DðF0 ðs; tÞ; Y0 ðs; tÞÞ\; 8s; t 2 ½a; b ½c; d. The obtained sequence of successive approximations is Z d Ym ðs; tÞ ¼ gðs; tÞ k ðFRÞ ðFRÞ c Z b kðs; t; x; yÞ Ym1 ðx; yÞdxdy; a
Consequently,
and using the same iterative method, the terms of produced sequence are
DðFm ðs; tÞ; F0 ðs; tÞÞ DðFm ðs; tÞ; Fm1 ðs; tÞÞ þ þ DðF1 ðs; tÞ; F0 ðs; tÞÞ: Taking supremum for s; t 2 ½a; b ½c; d from the above inequality, we have
v0 ðs; tÞ ¼ Y0 ðs; tÞ ¼ gðs; tÞ; n X n khh0 X vm ðs; tÞ ¼ gðs; tÞ kðs; t; s2i2 ; t2j2 Þ 9 j¼1 i¼1 vm1 ðs2i2 ; t2j2 Þ
D ðFm ; F0 Þ ðB
m1
m2
þB þ þ B þ 1ÞD ðF1 ; F0 Þ 1 D ðF1 ; F0 Þ: 1B
It is obvious that DðF1 ðs; tÞ; F0 ðs; tÞÞ ¼ Dðf ðs; tÞ kðFRÞ
Z
t
ðFRÞ c
Z
b
kðs; t; x; yÞ a
F0 ðx; yÞdxdy; f ðs; tÞÞ Z dZ b ~ k jkðs; t; x; yÞjDðf ðx; yÞ; 0Þdxdy c
a
kðd cÞðb aÞM k f k ¼ Bk f k;
kðs; t; s2i2 ; t2j Þ vm1 ðs2i2 ; t2j Þ kðs; t; s2i ; t2j2 Þ vm1 ðs2i ; t2j2 Þ kðs; t; s2i ; t2j Þ vm1 ðs2i ; t2j Þ 4 kðs; t; s2i2 ; t2j1 Þ vm1 ðs2i2 ; t2j1 Þ kðs; t; s2i1 ; t2j2 Þ vm1 ðs2i1 ; t2j2 Þ kðs; t; s2i ; t2j1 Þ vm1 ðs2i ; t2j1 Þ kðs; t; s2i1 ; t2j Þ vm1 ðs2i1 ; t2j Þ 16kðs; t; s2i1 ; t2j1 Þ vm1 ðs2i1 ; t2j1 Þ :
and
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Definition 5.1 The proposed algorithm based on iterative method applied to solve 2DLFFIE (3.1) is said to be numerically stable with respect to the choice of the first iteration if there exist four independent constants k1 ; k2 ; k3 ; k4 [ 0 such that D ðum ;vm Þk1 þ k2 x½a;b ½c;d ðf ;hh0 Þ þ x½a;b ½c;d ðg;hh0 Þ þ k3 xst ðk;hh0 Þ þ k4 xxy ðk;hh0 Þ;
ð5:1Þ
0 dc where h ¼ ba 2n , h ¼ 2n ,
1 10B 10B2 ; k2 ¼ ; k3 ¼ 1B 9ð1 BÞ 9Mð1 BÞ2 10B ðk f k þ kgkÞ; k4 ¼ ðk f k þ kgkÞ: 9Mð1 BÞ2
k1 ¼
Theorem 5.2 Under the assumptions of Theorem 4.1, the presented method (3.6) is numerically stable with respect to the choice of the first iteration. Proof
At first, we obtain that
Dðum ðs; tÞ; vm ðs; tÞÞ Dðum ðs; tÞ; Fm ðs; tÞÞ þ DðFm ðs; tÞ; Ym ðs; tÞÞ þ DðYm ðs; tÞ; vm ðs; tÞÞ 10B x½a;b ½c;d ðf ; hh0 Þ 9ð1 BÞ B k f kxst ðk; hh0 Þ þ Mð1 BÞ 1 þ k f kxxy ðk; hh0 Þ Mð1 BÞ þ DðFm ðs; tÞ; Ym ðs; tÞÞ 10B x½a;b ½c;d ðg; hh0 Þ þ 9ð1 BÞ B þ kgkxst ðk; hh0 Þ Mð1 BÞ 1 kgkxxy ðk; hh0 Þ : þ Mð1 BÞ
We conclude that D ðFm ; Ym Þ þ k
c
Fm1 ðx; yÞdxdy; gðs; tÞ k ðFRÞ Ym1 ðx; yÞdxdy
Z
d
ðFRÞ
c
c
ðFRÞ
c
d
ðFRÞ c
þ kðFRÞ
b
D ðFm1 ; Ym1 Þ þ BD ðFm2 ; Ym2 Þ D ðFm2 ; Ym2 Þ þ BD ðFm3 ; Ym3 Þ .. .. . . D ðF1 ; Y1 Þ þ BD ðF0 ; Y0 Þ: So, D ðFm ; Ym Þ þ B þ BD ðFm2 ; Ym2 Þ 2 þ B þ B þ BD ðFm3 ; Ym3 Þ þ B þ B2 þ B3 þ BD ðFm4 ; Ym4 Þ
.. . þ B þ B2 þ B3 þ þ Bm D ðF0 ; Y0 Þ : 1 þ B þ B 2 þ B3 þ þ Bm 1B Therefore, D ðum ; vm Þ
kðs; t; x; yÞ
b
kðs; t; x; yÞ
b
kðs; t; x; yÞ Fm1 ðx; yÞdxdy; kðs; t; x; yÞ Ym1 ðx; yÞdxdy
d
ðFRÞ c
a
10B x½a;b ½c;d ðf ; hh0 Þ 9ð1 BÞ B þ k f kxst ðk; hh0 Þ Mð1 BÞ 1 0 k f kxxy ðk; hh Þ þ þ Mð1 BÞ 1B 10B þ x½a;b ½c;d ðg; hh0 Þ 9ð1 BÞ B þ kgkxst ðk; hh0 Þ Mð1 BÞ 1 þ kgkxxy ðk; hh0 Þ ; Mð1 BÞ
where
a
Z
MD ðFm1 ; Ym1 Þdxdy
D ðFm ; Ym Þ þ BD ðFm1 ; Ym1 Þ
b
a
Z
b
a
Dðf ðs; tÞ; gðs; tÞÞ Z Z d ðFRÞ þ kD ðFRÞ Z
Z
and thus
a
Z
d
¼ þ BD ðFm1 ; Ym1 Þ;
However, Z d Z DðFm ðs; tÞ; Ym ðs; tÞÞ ¼ D f ðs; tÞ k ðFRÞ ðFRÞ
Z
Z
b
jkðs; t; x; yÞj a
DðFm1 ðx; yÞ; Ym1 ðx; yÞÞdxdy:
1 10B ; k2 ¼ ; 1B 9ð1 BÞ 10B2 k3 ¼ ðk f k þ kgkÞ; 9Mð1 BÞ2 10B ðk f k þ kgkÞ: k4 ¼ 9Mð1 BÞ2 k1 ¼
h
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Numerical examples In this section, we apply the proposed method in ‘‘2D linear fuzzy Fredholm integral equations’’ for solving two examples. In addition, we compare the absolute errors of the obtained results with the results of the method [18]. Example 6.1
[14] Consider the linear integral equation
Fðs; tÞ ¼ f ðs; tÞ ðFRÞ
Z
1
ðFRÞ
Z
0
1
kðs; t; x; yÞ Fðx; yÞdxdy;
0
1 Table 3 The absolute errors on the level sets with h ¼ h0 ¼ 10 ,m¼5 for Example 6.1 by using the proposed method in ðs0 ; t0 Þ ¼ ð0:5; 0:5Þ r level er ¼ jF ðs0 ; t0 ; rÞ erþ ¼ F ðs0 ; t0 ; rÞ um ðs0 ; t0 ; rÞj um ðs0 ; t0 ; rÞj
1:42157 106
0.00
0
0.25
1:11060 107
1:32717 106
0.50
2:66545 10
7
1:19945 106
0.75
4:66454 10
7
1:00510 106
1.00
7:10787 107
7:10787 107
ð6:1Þ with kðs; t; x; yÞ ¼ s2 tx;
1 Table 4 The absolute errors on the level sets with h ¼ h0 ¼ 20 ,m¼7 for Example 6.1 by using the proposed method in ðs0 ; t0 Þ ¼ ð0:5; 0:5Þ r level er ¼ jF ðs0 ; t0 ; rÞ erþ ¼ F ðs0 ; t0 ; rÞ um ðs0 ; t0 ; rÞj um ðs0 ; t0 ; rÞj
t f ðs; t; rÞ ¼ ðr þ rÞs sin ; 2 t f ðs; t; rÞ ¼ ð4 r 3 rÞs sin : 2 2
2:21362 108
The exact solution of this example is
0.00
0
t 16 1 F ðs; t; rÞ ¼ ðr þ rÞ s sin cos 1 s2 t ; 2 21 2 t 16 1 F ðs; t; rÞ ¼ ð4 r 3 rÞ s sin cos 1 s2 t : 2 21 2
0.25
1:72939 109
2:06662 108
0.50
4:15054 10
9
1:86774 108
0.75
7:26344 10
9
1:56510 108
1.00
1:10681 108
1:10681 108
2
By using the proposed method and the method [18] for 1 1 ; 20, m ¼ 5; 7 and r 2 f0:00; 0:25; 0:50; 0:75; h ¼ h0 ¼ 10 1:00g in ðs0 ; t0 Þ ¼ ð0:5; 0:5Þ, we present the absolute errors in Tables 1, 2, 3, and 4 1 Table 1 The absolute errors on the level sets with h ¼ h0 ¼ 10 ,m¼5 for Example 6.1 by using the method [18] in ðs0 ; t0 Þ ¼ ð0:5; 0:5Þ r level er ¼ jF ðs0 ; t0 ; rÞ erþ ¼ F ðs0 ; t0 ; rÞ um ðs0 ; t0 ; rÞj um ðs0 ; t0 ; rÞj
0
0.25
5:61213 106
3:92849 105
0.50
1:12243 10
5
3:36728 105
5
2:80606 105 2:24485 105
0.75
1:68364 10
1.00
2:24485 105
0
0.25
2:25737 105
2:69755 104
0.50
5:41768 10
5
4
0.75
9:48094 105
2:04292 104
1.00
1:44471 104
1:44471 104
2:43796 10
4:48970 105
0.00
4
0.00
2:88943 10
1 Table 5 The absolute errors on the level sets with h ¼ h0 ¼ 10 ,m¼5 for Example 6.2 by using the method [18] in ðs0 ; t0 Þ ¼ ð0:5; 0:5Þ, respectively r level er ¼ jF ðs0 ; t0 ; rÞ erþ ¼ F ðs0 ; t0 ; rÞ um ðs0 ; t0 ; rÞj um ðs0 ; t0 ; rÞj
1 Table 2 The absolute errors on the level sets with h ¼ h0 ¼ 20 ,m¼7 for Example 6.1 by using the method [18] in ðs0 ; t0 Þ ¼ ð0:5; 0:5Þ r level er ¼ jF ðs0 ; t0 ; rÞ erþ ¼ F ðs0 ; t0 ; rÞ um ðs0 ; t0 ; rÞj um ðs0 ; t0 ; rÞj
0.00
0
7:25207 105
0.25
5:66568 106
6:77049 105
0.50
1:35976 105
6:11894 105
0.75
2:37959 105
5:12744 105
1.00
3:62604 105
3:62604 105
1 Table 6 The absolute errors on the level sets with h ¼ h0 ¼ 10 ,m¼5 for Example 6.2 by using the proposed method in ðs0 ; t0 Þ ¼ ð0:5; 0:5Þ, respectively r level er ¼ jF ðs0 ; t0 ; rÞ erþ ¼ F ðs0 ; t0 ; rÞ r um ðs0 ; t0 Þ j um ðs0 ; t0 Þrþ j
0.00
9:20708 1013
0 1:15088 10
13
8:05633 1013
0.50
2:30177 10
13
6:90559 1013
0.75
3:45279 1013
5:75429 1013
1.00
13
4:60354 1013
0.25
4:60354 10
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Example 6.2 [14] Consider the following fuzzy Fredholm integral equation (6.1) with 1 2 2 ðs þ t 2Þ ; f ðs; t; rÞ ¼ r st þ 676 1 2 2 ðs þ t 2Þ ; f ðs; t; rÞ ¼ ð2 rÞ st þ 676 and kernel kðs; t; x; yÞ ¼
1 2 ðs þ t2 2Þðx2 þ y2 2Þ; 0 s; t; x; y 1: 169
The exact solution is F ðs; t; rÞ ¼ rst;
F ðs; t; rÞ ¼ ð2 rÞst: To compare the absolute errors for the proposed method and the method [18], see Tables 5, and 6
Conclusions To approximate the solution of 2DLFFIEs, we developed a quadrature iterative method. We prove the convergence analysis (Theorem 4.1) and the numerical stability analysis (Theorem 5.2) of the proposed method with respect to the choice of the first iteration. Obtained results show that the proposed method can be a suitable method for solving 2DLFFIEs. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://crea tivecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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