First Order Linear Non Homogeneous Ordinary Differential Equation in Fuzzy Environment Sankar Prasad Mondal * Tapan Kumar Roy Department of Mathematics Bengal Engineering and Science UniversityShibpurHowrah-711103 West Bengal India Abstract *Corresponding author-mail: sankar.res07@gmail.com In this paper the solution procedure of a first order linear non homogeneous ordinary differential equation in fuzzy environment is described. It is discussed for three different cases. They are i) Ordinary Differential Equation with initial value as a fuzzy number ii) Ordinary Differential Equation with coefficient as a fuzzy number and iii) Ordinary Differential Equation with initial value and coefficient are fuzzy numbers. Here fuzzy numbers are taken as Generalized Triangular Fuzzy Numbers (GTFNs). An elementary application of population dynamics model is illustrated with numerical example. Keywords: Fuzzy Ordinary Differential Equation (FODE) Generalized Triangular fuzzy number (GTFN) strong solution. 1. Introduction: The idea of fuzzy number and fuzzy arithmetic were first introduced by Zadeh [11] and Dubois and Parade [5]. The term Fuzzy Differential Equation (FDE) was conceptualized in 1978 by Kandel and Byatt [1] and right after two years a larger version was published [2]. Kaleva [16] and Seikkala [17] are the first persons who formulated FDE. Kaleva showed the Cauchy problem of fuzzy sets in which the Peano theorem is valid. The Generalization of the Hukuhara derivative which is based on fuzzy derivative was defined by Seikkala and brought that the fuzzy initial value problem (FIVP) 0 which has a unique fuzzy solution when f satisfies the generalized Lipschitz condition which confirms a unique solution of the deterministic initial value problem. Fuzzy differential equation and initial value problem were extensively treated by other researchers (see [41819138910]). Recently FDE has also used in many models such as HIV model [7] decay model [6] predatorprey model [15] population models [12] civil engineering [14] modeling hydraulic [3] etc. In this paper we have considered 1 st order linear non homogeneous fuzzy ordinary differential equation and have described its solution procedure in section-3. In section-4 we have applied it in a bio-mathematical model. 2. Preliminary concept: Definition 2.1: Fuzzy Set: Let X be a universal set. The fuzzy set is defined by the set of tuples as : : 01. The membership function of a fuzzy set is a function with mapping : 01. So every element x in X has membership degree in 01 which is a real number. As closer the value of is to 1 so much x belongs to. implies relevance of in is greater than the relevance of in. If = 1 then we say exactly belongs to if = 0 we say does not belong to and if = a where 0 < a < 1. We say the membership value of in is a. When is always equal to 1 or 0 we get a crisp (classical) subset of X. Here the term crisp means not fuzzy. A crisp set is a classical set. A crisp number is a real number. 85
Definition 2.2:!-Level or!-cut of a fuzzy set: Let X be an universal set. Let be a fuzzy set. " -cut of the fuzzy set is a crisp set. It is denoted by #. It is defined as # $ &" ) Note: # is a crisp set with its characteristic function * + (x) defined as * + (x) = 1 &" = 0 otherwise. Definition 2.3: Convex fuzzy set: A fuzzy set is called convex fuzzy set if all # are convex sets i.e. for every element # and # and for every " 01-1. # 01. Otherwise the fuzzy set is called non convex fuzzy set. Definition 2.4: Fuzzy Number: /0 is called a fuzzy number where R denotes the set of whole real numbers if i. is normal i.e. 0 exists such that 1. ii.! 01 # is a closed interval. If is a fuzzy number then is a convex fuzzy set and if 1 then is non decreasing for 1 and non increasing for &. The membership function of a fuzzy number 2 2 2 3 2 4 is defined by 1 2 2 3 6ϕ 5 8 2 1 1 2 0 2 3 1 12 4 Where L(x) denotes an increasing function and 098 11 and R(x) denotes a decreasing function and 010 91. Definition 2.5: Generalized Fuzzy number (GFN): Generalized Fuzzy number as 2 2 2 3 2 4 ;: where 09:11 and 2 2 2 3 2 4 (2 9 2 9 2 3 92 4 are real numbers. The generalized fuzzy number is a fuzzy subset of real line R whose membership function satisfies the following conditions: 1) : R [0 1] 2) 0 for 1 2 3) is strictly increasing function for 2 1 12 4) : for 2 1 12 3 5) is strictly decreasing function for 2 3 1 1 2 4 6) 0 for 2 4 1 Fig-2.1: Membership function of a GFN Definition 2.6: Generalized triangular fuzzy number (GTFN) : A Generalized Fuzzy number is called a Generalized Triangular Fuzzy Number if it is defined by 2 2 2 3 ;:its membership function is given by 86
> < = < ; 0 12 :?@A B 2 A C @A 1 12 B : 2 : A D@? A D @A C 2 1 12 3 0 &2 3 or E2FEGHF:?@A B A C @A B :: A D@? A D @A C I0I Definition 2.7: Fuzzy ordinary differential equation (FODE): Consider a simple 1 st Order Linear non-homogeneous Ordinary Differential Equation (ODE) as follows: J? JK L- with initial condition M The above ODE is called FODE if any one of the following three cases holds: (i) Only M is a generalized fuzzy number (Type-I). (ii) Only k is a generalized fuzzy number (Type-II). (iii) Both k and M are generalized fuzzy numbers (Type-III). Definition 2.8: Strong and Weak solution of FODE: Consider the 1 st order linear non homogeneous fuzzy ordinary differential equation Here k or (and) be generalized fuzzy number(s). J? JK L- with. Let the solution of the above FODE be N and its "-cut be " " ". If "1 " " 0: where 09:11 then N is called strong solution otherwise N is called weak solution and in that case the "-cut of the solution is given by "min$ " ")max$ " "). 3. Solution Procedure of 1 st Order Linear Non Homogeneous FODE The solution procedure of 1 st order linear non homogeneous FODE of Type-I Type-II and Type-III are described. Here fuzzy numbers are taken as GTFNs. 3.1. Solution Procedure of 1 st Order Linear Non Homogeneous FODE of Type-I Consider the initial value problem J? JK V-.(3.1.1) with Fuzzy Initial Condition (FIC) N MW M M M 3 ;: Let N be a solution of FODE (3.1.1). Let " " " be the "-cut of N and MW # " " YM - #Z [\ M ] 3. #^[\ _ " ` 0: 09:11 ] where a b\ M.M and d b\ M 3.M Here we solve the given problem for L 0 and L 9 0 respecively. 87
Case 3.1.1. When e0 The FODE (3.1.1) becomes a system of linear ODE J? f K JK L h "- for G 12..(3.1.2) with initial condition " M - #Z [\ and ] "M 3. #^[\ ] The solution of (3.1.2) is ".? \ j -k? \ j -M - #Z [\ ] lmjk@k \..(3.1.3) and ".? \ j -k? \ j -M 3. #^[\ ] lmjk@k \..(3.1.4) Now n n# " Z [\ ] mjk@k \ 0 n n# ".^[\ ] mjk@k \ 90 and :.? \ j -k? \ j -M lm jk@k \ :. So the solution of FODE (3.1.1) is a generalized fuzzy number N. The "-cut of the solution is ".? \ j -Y? \ j -FM - #Z [\ ] I? \ j -M 3. #^[\ ] _mjk@k \. Case 3.1.2. when e 90 Let L.E where m is a positive real number. Then the FODE (3.1.1) becomes a system of ODE as follows pqb r s@t? pr C K#u? \ v pqbr s@t? pr C K#u? \ (3.1.5) with initial condition " M - #Z [\ ] and "M 3. #^[\ ]. The solution of (3.1.5) is "? \ t - k.? \ t -M -M 3 - # ] a.d b\ lm @tk@k \ - F # ].1Ia -d b\ m tk@ K \ and "? \ t - k.? \ t -M -M 3 - # ] a.d b\ lm @tk@k \. F # ].1Ia -d b\ m t K@K \. Here n n# " ] a.d b\ m @tk@k \ - ] a -d b\ m t K@K \ n n# " ] a.d b\ m @tk@k \. ] a -d b\ m t K@K \ 88
and :? \ t -F.? \ t -M Im @tk@k \ : Here three cases arise. Case1: When w xy z xy i.e. x{ y x x } x ~ ; is a symmetric GTFN n n# " a ] -d b\ m t K@K \ 0 n ". a n# ] -d b\ m t K@K \ 90 and : : So the solution of the FODE (3.1.1) is a strong solution. Case2: When w xy 9 z xy i.e. x{ y x x } x ~ ; is a non symmetric GTFN Here n n# "9 0 and : : but n n# "0 implies - log ^[\ @Z [\. t Z [ \ u^[\ So the solution of the FODE (3.1.1) is a strong solution if - log ^[\ @Z [\. t Z [ \ u^[\ Case3: When w xy z xy i.e. x{ y x x } x ~ ; is a non symmetric GTFN Here n n# "90 and : : but n "9 0 implies n# - log ^[\ @Z [\. t Z [ \ u^[\ So the solution of the FODE (3.1.1) is a strong solution if - t log Z [\ @^[\. Z [ \ u^[\ 3.2. Solution Procedure of 1 st Order Linear Non Homogeneous FODE of Type-II Consider the initial value problem J? JK L -.(3.2.1) with IC M. Here L 3 ;. Let N be the solution of FODE (3.2.1) Let " " " be the "-cut of the solution and the "-cut of L be L # L "L "Y - #Z ˆ 3. #^ _ " ` 0 0911 ˆ where a j. and d j 3.. Here we solve the given problem for L 0 and L 9 0 respecively. Case 3.2.1: when e 0 The FODE (3.2.1) becomes a system of linear ODE 89
J? f K JK L h " h - for G 12..(3.2.2) with IC M. The solution of (3.2.1) ".? \ Š B u + and Œ - M-?\ vm ŠBu + Š B u + Œ K@K \ Œ ".? \ Š D @ +Ž Œ - M-?\ +Ž D@ Š D @ +Ž Œ K@K \. Œ vmš Case 3.2.2: when 9 0 Let L.EW where EW 3 ; is a positive GTFN. So EW # E "E "Y - #Zˆ 3. _ " ` 00911 #^ˆ where a t. and d t 3. Then the FODE (3.2.1) becomes a system of ODE as follows pqbr s@t pr C #? C K#u? \ v pqc r s@t pr B #? B K#u? \ (3.2.3) with IC M Thus the solution is " 1 > 3. "d t 1 M 1. 2= - "a. t ; - "a t - 1 > 3. "d t 1 M 1-2= - "a. t ; - "a t. - 1 - "a t 3. "d t š 1 - "a t 3. "d t š m Š Bu #Zˆ Š D@ #^ˆ K@K \ m @ FŠ Bu #Zˆ IŠ D@ #^ˆ K@K \ - - "a t ". 1 2 - "a t> 3. "d t 1 3. "d M 1. t= - "a. t ; - "a t. 90 1 - "a t 3. "d t š m Š Bu #Zˆ FŠ D@ #^ˆ IK@K \
- œ Š Bu + Œ Š D @ +Ž Œ 5M 1-œ Š D@ +Ž Œ Š B u + Œ ž. Š B u + Œ - žÿ m @ FŠ Bu Š B u + Œ Š D@ +Ž Œ + Œ IŠ D@ +Ž Œ K@K \? - \ Š D @ +Ž Œ 3.3. Solution Procedure of 1 st Order Linear Non Homogeneous FODE of Type-III Consider the initial value problem J? JK V -.(3.3.1) With fuzzy IC N MW M M M 3 ;: where L 3 ; Let N be the solution of FODE (3.3.1). Let " " " be the "-cut of the solution. Also L # Y - #Z ˆ 3. #^ _ " ` 00911 ˆ where a j. and d j 3. and MW # " " YM - #Z [\ M ] 3. #^[\ _ " ` 0:09: 11 ] where a b\ M.M and d b\ M 3.M Let min: Here we solve the given problem for L 0 and L 9 0 respecively. Case I: when e 0 The FODE (3.1.1) becomes a system of linear ODE J? f K JK L h h "- for G 12..(3.3.2) with initial condition " M - #Z [\ and ] "M 3. #^[\ ] Therefore the solution is of (3.3.1) ".? \ - FM Š B u + - #Z[\ And ".? \ Š D @ +Ž - FM 3. #^[\ Case II: when e 9 0 I-? \ vm FŠBu + Š B u + IK@K \ I-? \ vmfšd@ +Ž Š D @ +Ž IK@K \ Let L.EW where EW 3 ; is a positive GTFN. Then EW # Y - #Zˆ 3. _ " ` 0 0911 #^ˆ 91
Let min: Then the FODE (3.3.1) becomes a system of ODE as follows pqb r s@t pr C? C K#u? \ v pqcr s@t pr B? B K#u? \ (3.3.3) with IC "M - #Z [\ ] and "M 3. #^[\ ]. Therefore the solution of (3.3.1) is " > 1< M - "a F 3. "dt I. M 3. "d 1 1 < 2 = F - "at I.. m < F - "at I F ; 0 - "at IF 3. "dt I FŠ Bu #Z IFŠ D@ #^ IK@K \ <š - 1 > M 2 - "a F 3. "dt I - M = F - "at 3. "d M 0 :. 1 1 - m I - "at ; - "at 3. "dt š @ Š Bu #Z Š D@ #^ K@K \ - - "at and ". 1 - "at > 2 3. "dt M - "a 3. "dt. M 3. "d 1 1.. = - "at - "at ; - "at 3. "dt š - 1 - "at > 2 3. "dt M - "a 3. "dt - M 3. "d 1 1. - = - "at - "at ; - "at 3. "dt š -? \ Š D @ +Ž. m Š Bu #Z Š D@ #^ K@K \ m @ ŠBu#Z ŠD@#^ K@K\ 4. Application: Population Dynamics Model Bacteria are being cultured for the production of medication. Without har-vesting the bacteria the rate of change of the population is proportional to its current population with a proportionality constant L per hour. Also the bacteria are being harvested at a rate of per hour. If there are initially bacteria in the culture solve the initial value problem: J JK L. 0 when (i) 780080008150;0.8 and L 0.2 1000 (ii) 8000 and L 0.170.20.24;0.6 1000 (iii) 790080008200;0.7 and L 0.160.20.23;0.8 1000. 92
Solution: (i) Here # 7800-250"8150.187.5". Therefore solution of the model "5000-2800-250"m. K and " 5000-3150.187.5"m. K. Table-1: Value of ± ²! and ± } ²! for different! and t=3 " " " 0 10101.9326 10739.6742 0.1 10147.4856 10705.5095 0.2 10193.0386 10671.3448 0.3 10238.5916 10637.1800 0.4 10284.1445 10603.0153 0.5 10329.6975 10568.8506 0.6 10375.2505 10534.6859 0.7 10420.8034 10500.5211 0.8 10466.3564 10466.3564 From above table-1 we see that for this particular value of t=3 " is an increasing function " is a decreasing function and 0.8 0.8 10466.3564. So Solution of above model for particular value of t is a strong solution. (ii) Here L # 0.17-0.050"0.26.0.066". Therefore solution of the model is " and " -k8000. lm.³u. #K.³u. #.³u. # -k8000.. 4@.µµ#. 4@.µµ# lm. 4@.µµ#K. Table-2: Value of ± ²! and ± } ²! for different! and t=3 " " " 0 9408.8519 12041.9940 0.1 9578.1917 11765.8600 0.2 9750.2390 11495.6109 0.3 9925.0361 11231.1346 0.4 10102.6256 10972.3224 0.5 10283.0510 10719.0687 0.6 10466.3564 10471.2714 93
From above table-1 we see that for this particular value of t=3 " is an increasing function " is a decreasing function and 0.89 0.8. So Solution of above model for particular value of t is a strong solution. (iii) Here L # 0.16-0.057"0.23.0.042" and # 7900-142.8"8200.285.7" Therefore solution of the model is " -k7900-142.8"..µu. ³# and " -k8200.285.7".. 3@.4 # lm.µu. ³#K.µu. ³# 3@.4 #K lm.. 3@.4 # Table3: Value of ± ²! and ± } ²! for different! and t=3 " " " 0 8916.5228 12027.9651 0.1 9124.4346 11797.2103 0.2 9336.5274 11570.1611 0.3 9552.8820 11346.7614 0.4 9773.5808 11126.9561 0.5 9998.7076 10910.6908 0.6 10228.3479 10697.9120 0.7 10462.5889 10488.5669 From above table-1 we see that for this particular value of t=3 " is an increasing function " is a decreasing function and 0.89 0.8. So Solution of above model for particular value of t is a strong solution. 5. Conclusion: In this paper we have solved a first order linear non homogeneous ordinary differential equation in fuzzy environment. Here fuzzy numbers are taken as GTFNs. We have also discussed three possible cases. For further work the same problem can be solved by Generalized L-R type Fuzzy Number. This process can be applied for any economical or bio-mathematical model and problems in engineering and physical sciences. References: [1] A. Kandel and W. J. Byatt Fuzzy differential equations in Proceedings of the International Conference on Cybernetics and Society pp. 1213 1216 Tokyo Japan 1978. [2] A. Kandel and W. J. Byatt Fuzzy processes Fuzzy Sets and Systems vol. 4 no. 2 pp. 117 152 1980. [3] A. Bencsik B. Bede J. Tar J. Fodor Fuzzy differential equations in modeling hydraulic differential servo cylinders in: Third Romanian_Hungarian Joint Symposium on Applied Computational Intelligence SACI Timisoara Romania 2006. 94
[4] B. Bede I.J. Rudas A.L. Bencsik First order linear fuzzy differential equations under generalized differentiability Information Sciences 177 (2007) 1648 1662. [5] Dubois D and H.Parade 1978 Operation on Fuzzy Number. International Journal of Fuzzy system 9:613-626 [6] G.L. Diniz J.F.R. Fernandes J.F.C.A. Meyer L.C. Barros A fuzzy Cauchy problem modeling the decay of the biochemical oxygen demand in water2001 IEEE. [7] Hassan Zarei Ali Vahidian Kamyad and Ali Akbar Heydari Fuzzy Modeling and Control of HIV Infection Computational and Mathematical Methods in Medicine Volume 2012 Article ID 893474 17 pages. [8] J. J. Buckley and T. Feuring Fuzzy differential equations Fuzzy Sets and Systems vol. 110 no. 1 pp.43 54 2000. [9] James J. Buckley Thomas Feuring Fuzzy initial value problem for Nth-order linear differential equations Fuzzy Sets and Systems 121 (2001) 247 255. [10] J.J. Buckley T. Feuring Y. Hayashi Linear System of first order ordinary differential equations: fuzzy initial condition soft computing6 (2002)415-421. [11] L. A. Zadeh Fuzzy sets Information and Control 8(1965) 338-353. [12] L.C. Barros R.C. Bassanezi P.A. Tonelli Fuzzy modelling in population dynamics Ecol. Model. 128 (2000) 27-33. [13] L.J. Jowers J.J. Buckley K.D. Reilly Simulating continuous fuzzy systems Information Sciences 177 (2007) 436 448. [14] M. Oberguggenberger S. Pittschmann Differential equations with fuzzy parameters Math. Modelling Syst. 5 (1999) 181-202. [15] Muhammad Zaini Ahmad Bernard De Baets A Predator-Prey Model with Fuzzy Initial Populations IFSA- EUSFLAT 2009. [16] O. Kaleva Fuzzy differential equations Fuzzy Sets and Systems 24 (1987) 301 317. [17] S. Seikkala On the fuzzy initial value problem Fuzzy Sets and Systems 24 (1987) 319 330. [18] W. Congxin S. Shiji Existence theorem to the Cauchy problem of fuzzy differential equations under compactness-type conditions Information Sciences 108 (1998) 123 134. [19] Z. Ding M. Ma A. Kandel Existence of the solutions of fuzzy differential equations with parameters Information Sciences 99 (1997) 205 217. 95
This academic article was published by The International Institute for Science Technology and Education (IISTE). The IISTE is a pioneer in the Open Access Publishing service based in the U.S. and Europe. The aim of the institute is Accelerating Global Knowledge Sharing. More information about the publisher can be found in the IISTE s homepage: http:// CALL FOR PAPERS The IISTE is currently hosting more than 30 peer-reviewed academic journals and collaborating with academic institutions around the world. There s no deadline for submission. Prospective authors of IISTE journals can find the submission instruction on the following page: http:///journals/ The IISTE editorial team promises to the review and publish all the qualified submissions in a fast manner. All the journals articles are available online to the readers all over the world without financial legal or technical barriers other than those inseparable from gaining access to the internet itself. Printed version of the journals is also available upon request of readers and authors. IISTE Knowledge Sharing Partners EBSCO Index Copernicus Ulrich's Periodicals Directory JournalTOCS PKP Open Archives Harvester Bielefeld Academic Search Engine Elektronische Zeitschriftenbibliothek EZB Open J-Gate OCLC WorldCat Universe Digtial Library NewJour Google Scholar