First Order Linear Non Homogeneous Ordinary Differential Equation in Fuzzy Environment

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1 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 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 [ ]). 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

2 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. 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 is defined by ϕ Where L(x) denotes an increasing function and and R(x) denotes a decreasing function and Definition 2.5: Generalized Fuzzy number (GFN): Generalized Fuzzy number as ;: where 09:11 and ( 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 ) : for ) is strictly decreasing function for ) 0 for 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 ;:its membership function is given by 86

3 > < = < ; 0 12 :?@A B 2 A 1 12 B : 2 : A D@? A C &2 3 or E2FEGHF:?@A B A B :: 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 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

4 Case 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 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\ \ - F # ].1Ia -d b\ m tk@ K \ and "? \ t - k.? \ t -M -M 3 - # ] a.d b\ \. F # ].1Ia -d b\ m t K@K \. Here n n# " ] a.d b\ \ - ] a -d b\ m t K@K \ n n# " ] a.d b\ \. ] a -d b\ m t K@K \ 88

5 and :? \ t -F.? \ t -M \ : 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 [\. t Z [ \ u^[\ So the solution of the FODE (3.1.1) is a strong solution if - log [\. 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 [\. 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. #^ _ " ` 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

6 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 +Ž Œ - M-?\ +Ž D@ Š +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. _ " ` #^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 "a t 3. "d t š 1 - "a t 3. "d t š m Š Bu #Zˆ Š D@ #^ˆ K@K \ FŠ Bu #Zˆ IŠ D@ #^ˆ K@K \ - - "a t " "a t> 3. "d t 1 3. "d M 1. t= - "a. t ; - "a t "a t 3. "d t š m Š Bu #Zˆ FŠ D@ #^ˆ IK@K \

7 - œ Š Bu + Œ Š +Ž Œ 5M 1-œ Š D@ +Ž Œ Š B u + Œ ž. Š B u + Œ - žÿ FŠ Bu Š B u + Œ Š D@ +Ž Œ + Œ IŠ D@ +Ž Œ K@K \? - \ Š +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. #^ _ " ` 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 +Ž - FM 3. #^[\ Case II: when e 9 0 I-? \ vm FŠBu + Š B u + IK@K \ I-? \ vmfšd@ +Ž Š +Ž IK@K \ Let L.EW where EW 3 ; is a positive GTFN. Then EW # Y - #Zˆ 3. _ " ` #^ˆ 91

8 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 : m I - "at ; - "at 3. "dt Š Bu #Z Š D@ #^ K@K \ - - "at and ". 1 - "at > 2 3. "dt M - "a 3. "dt. M 3. "d = - "at - "at ; - "at 3. "dt š "at > 2 3. "dt M - "a 3. "dt - M 3. "d = - "at - "at ; - "at 3. "dt š -? \ Š +Ž. m Š Bu #Z Š D@ #^ K@K \ 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) ;0.8 and L (ii) 8000 and L ; (iii) ;0.7 and L ;

9 Solution: (i) Here # " ". Therefore solution of the model " "m. K and " "m. K. Table-1: Value of ± ²! and ± } ²! for different! and t=3 " " " From above table-1 we see that for this particular value of t=3 " is an increasing function " is a decreasing function and So Solution of above model for particular value of t is a strong solution. (ii) Here L # " ". Therefore solution of the model is " and " -k8000. lm.³u. #K.³u. #.³u. # -k @.µµ#. 4@.µµ# lm. 4@.µµ#K. Table-2: Value of ± ²! and ± } ²! for different! and t=3 " " "

10 From above table-1 we see that for this particular value of t=3 " is an increasing function " is a decreasing function and So Solution of above model for particular value of t is a strong solution. (iii) Here L # " " and # " " Therefore solution of the model is " -k "..µu. ³# and " -k ".. 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 " " " From above table-1 we see that for this particular value of t=3 " is an increasing function " is a decreasing function and 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 Tokyo Japan [2] A. Kandel and W. J. Byatt Fuzzy processes Fuzzy Sets and Systems vol. 4 no. 2 pp [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

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