Second order linear differential equations with generalized trapezoidal intuitionistic Fuzzy boundary value

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1 Journal of Linear and Topological Algebra Vol. 04, No. 0, 05, 5-9 Second order linear differential equations with generalized trapezoidal intuitionistic Fuzzy boundary value S. P. Mondal a, T. K. Roy b a Department of Mathematics, National Institute of Technology, Agartala, Jirania , Tripura, India; b Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-703, West Bengal, India. Received 9 June 05; Revised August 05; Accepted 7 September 05. Abstract. In this paper the solution of a second order linear differential equations with intuitionistic fuzzy boundary value is described. It is discussed for two different cases: coefficient is positive crisp number and coefficient is negative crisp number. Here fuzzy numbers are taken as generalized trapezoidal intutionistic fuzzy numbers (GTrIFNs). Further a numerical example is illustrated. c 05 IAUCTB. All rights reserved. Keywords: Fuzzy set, fuzzy differential equation, generalized trapezoidal intutionistic fuzzy number. 00 AMS Subject Classification: Primary: 34A07, Secondary: 03E7.. Introduction. Fuzzy and intutionistic fuzzy set theory Zadeh [] and Dubois and Parade [] were the first who introduced the conception based on fuzzy number and fuzzy arithmetic. The generalizations of fuzzy sets theory [] is considered as Intuitionistic fuzzy set (IFS). Out of several higher-order fuzzy sets, IFS was first introduced by Atanassov [3] and have been found to be suitable to deal with unexplored areas.the fuzzy set considers only the degree of belongingness and non belongingness. Fuzzy set theory does not incorporate the degree of hesitation (i.e.,degree Corresponding author. address: sankar.res07@gmail.com (S. P. Mondal). Print ISSN: 5-00 c 05 IAUCTB. All rights reserved. Online ISSN:

2 6 S. P. Mondal et al. / J. Linear. Topological. Algebra. 04(0) (05) 5-9. of non-determinacy defined as, - sum of membership function and non-membership function).to handle such situations, Atanassov [4] explored the concept of fuzzy set theory by intuitionistic fuzzy set (IFS) theory.the degree of acceptance is only considered in Fuzzy Sets but IFS is characterized by a membership function and a non-membership function so that the sum of both values is less than one [4].. Fuzzy derivative and fuzzy differential equation The topic fuzzy differential equation (FDE) has been rapidly developing in recent years. The appliance of fuzzy differential equations is a inherent way to model dynamic systems under possibilistic uncertainty [5]. The concept of the fuzzy derivative was first introduced by Chang and Zadeh [6] and it was followed up by Dubois and Prade [7]. Other methods have been discussed by Puri and Ralescu [8] and Goetschel and Voxman [9]. The concept of differential equations in a fuzzy environment was first formulated by Kaleva [0]. In fuzzy differential equation all derivative is deliberated as either Hukuhara or generalized derivatives. The Hukuhara differentiability has a imperfection (see [], [9]). The solution turns fuzzier as time goes by. Bede exhibited that a large class of BVPs has no solution if the Hukuhara derivative is used []. To overcome this difficulty, the concept of a generalized derivative was developed [ [], [3]) and fuzzy differential equations were discussed using this concept (see [4], [5], [6], [7]). Khastan and Nieto found solutions for a large enough class of boundary value problems using the generalized derivative [8].Bede in [6] discussed the generalized differentiability for fuzzy valued functions. Pointedly the disadvantage of strongly generalized differentiability of a function in comparison H-differentiability is that, a fuzzy differential equation has no unique solution [0]. Recently, Stefanini and Bede by the concept of generalization of the Hukuhara difference for compact convex set [], introduced generalized Hukuhara differentiability [] for fuzzy valued function and they demonstrated that, this concept of differentiability have relationships with weakly generalized differentiability and strongly generalized differentiability. Recently Gasilov et. all. [5] solve the fuzzy initial value problem by a new technique where Barros et. all. [33] solve fuzzy differential equation via fuzzification of the derivative operator. Recently the research on fuzzy boundary value is grew attention among all. Armand and Gouyandeh [3] solve two point fuzzy boundary value problems using variational iteration method. Gasilov et. all. [4] solve linear differential equation with fuzzy boundary value. Lopez [7] find the existence of solutions to periodic boundary value problems for linear differential equation..3 Intutionistic fuzzy differential equation Intutionistic FDE is very rare. Melliani and Chadli [8] solve partial differential equation with intutionistic fuzzy number. Abbasbandy and Allahviranloo [9] discussed numerical Solution of fuzzy differential equations by runge-kutta method and in the intuitionistic fuzzy environment.. Lata and Kumar [30] worked on time-dependent intuitionistic fuzzy differential equation and its application to analyze the intutionistic fuzzy reliability of industrial system. First order homogeneous ordinary differential equation with initial value as triangular intuitionistic fuzzy number is described by Mondal and Roy [3]. System of differential equation with initial value as triangular intuitionistic fuzzy number and its application is solved by Mondal and Roy [3].

3 .4 Novelties S. P. Mondal et al. / J. Linear. Topological. Algebra. 04(0) (05) In spite of above mentioned developments, following lacunas are still exists in the formulation and solution of the fuzzy boundary value problem, which are summarized below: (i) Though there are some articles of fuzzy boundary value problem was solved but till now none has solve boundary value problem with intuitionistic fuzzy number. (ii) Here also second order differential equation is solved with intuitionistic fuzzy number. (iii) The intutionistic fuzzy number is also taken as generalized trapezoidal intuitionistic fuzzy number..5 Structure of the paper The paper is organized as follows. In Section, the basic concept on fuzzy number and fuzzy derivative are discussed. Also basic definition and properties of generalized trapezoidal intuitionistic fuzzy number are defined. In Section 3 we solve the intutionistic fuzzy boundary value problem. In Section 4 the proposed method is illustrated by an example. The conclusion and future research scope is drawn in the last section 5.. Preliminary concept Definition.:Fuzzy Set: A fuzzy set à is defined by à {(x, µ à (x)) : xϵa, µã(x)ϵ[0, ]}. In the pair (x, µã(x)) the first element belongs to the classical set A, the second element µã(x), belongs to the interval [0, ], called membership function. Definition.:α-cut of a fuzzy set: The α-level set (or, interval of confidence at level α or α-cut) of the fuzzy set à of X that have membership values in A greater than or equal to α i.e., à {x : µã(x) α, xϵx, αϵ[0, ]}. Definition.3: Fuzzy number:a fuzzy number is an extension of a regular number in the sense that it does not refer to one single value but rather to a connected set of possible values, where each possible value has its own weight between 0 and.this weight is called membership function. Thus a fuzzy number is a convex and normal fuzzy set. Definition.4:Intuitionistic Fuzzy set: Let a set X be fixed. An IFS Ãi in X is an object having the form Ãi {< x, µãi(x), ϑãi(x) > 0 : xϵx}, where the µãi(x) : X [0, ] and ϑãi(x) : X [0, ] define the degree of membership and degree of non-membership respectively, of the element xϵx to the set Ãi, which is a subset of X, for every element of xϵx, 0 < µãi(x) + ϑãi(x). Definition.5:Intuitionistic Fuzzy number: An IFN Ãi is defined as follows (i) an intuitionistic fuzzy subject of real line. (ii) normal. i.e., there is any x 0 ϵr such that µãi(x 0 ) (so ϑãi(x 0 ) 0) (iii) a convex set for the membership function µãi(x), i.e., µãi(λx + ( λ)x ) min(µãi(x ), µãi(x )) x, x ϵr, λϵ[0, ] (iv)a concave set for the non-membership function ϑãi(x), i.e., ϑãi(λx + ( λ)x ) min(ϑãi(x ), ϑãi(x )) x, x ϵr, λϵ[0, ]

4 8 S. P. Mondal et al. / J. Linear. Topological. Algebra. 04(0) (05) 5-9. Definition.6: Trapezoidal Intuitionistic Fuzzy number: A TrIFN Ãi is a subset of IFN in R with following membership function and non membership function as follows: µãi (x) 0 otherwise x a a a if a x < a if a x a 3 a 4 x a 4 a 3 if a 3 < x a 4 and a x if a a a x < a 0 if a ϑãi (x) x a 3 x a 3 if a a 3 < x a 4 a 3 4 otherwise where a a a 3 a 4, a a a 3 a 4 and TrIFN is denoted by à T rif N (a, a, a 3, a 4 ; a, a, a 3, a 4 ) Note.: Here µãi (x) increases with constant rate for xϵ[a, a ] and decreases with constant rate for xϵ[a 3, a 4 ] but ϑãi (x) decreases with constant rate for xϵ[a, a ] and increases with constant rate for xϵ[a 3, a 4 ] Definition.7: Generalized Intuitionistic Fuzzy Number: An IFN Ãi is defined as follows (i) an intuitionistic fuzzy subject of real line. (ii) normal. i.e., there is any x 0 ϵr such that µãi(x 0 ) ω(so ϑãi(x 0 ) σ) for 0 < ω + σ. (iii) a convex set for the membership function µãi(x), i.e., µãi(λx + ( λ)x ) min(µãi(x ), µãi(x )) x, x ϵr, λϵ[0, ω] (iv)a concave set for the non-membership function ϑãi(x), i.e., ϑãi(λx + ( λ)x ) min(ϑãi(x ), ϑãi(x )) x, x ϵr, λϵ[σ, ] (v) µãi and ϑãi is continuous mapping from R to the closed interval [0, ω] and [σ, ] respectively and x 0 ϵr, the relation 0 µãi + ϑãi holds. Definition.8: Generalized Trapezoidal Intuitionistic Fuzzy number: A GTrIFN Ãi is a subset of IFN in R with following membership function and non membership function as follows:

5 S. P. Mondal et al. / J. Linear. Topological. Algebra. 04(0) (05) Figure. Generalized trapezoidal intutionistic fuzzy number ω x a a a if a x < a ω if a µãi (x) x a 3 ω a 4 x a 3 a 3 if a 3 < x a 4 0 otherwise and σ a x if a a a x < a σ if a ϑãi (x) x a 3 σ x a 3 if a a 3 < x a 4 a otherwise where a a a 3 a 4, a a a 3 a 4 and GTrIFN is denoted by à GT rif N ((a, a, a 3, a 4 ; ω), (a, a, a 3, a 4 ; σ)) Definition.9:Non negative GTrIFN: A GTrIFN à i GT rif N ((a, a, a 3, a 4 ; ω), (a, a, a 3, a 4 ; σ)) iff a 0. Definition.0:Equality of two GTrIFN: A GTrIFN Ãi GT IF N ((a, a, a 3, a 4 ; ω ), (a, a, a 3, a 4 ; σ )) and B i GT IF N ((b, b, b 3, b 4 ; ω ), (b, b, b 4, b 4 ; σ )) are said to be equal iff a b,a b,a 3 b 3,a 4 b 4,a b,a 4 b 4,ω ω and σ σ. Definition.: α-cut set: α-cut set of a GTrIFN Ãi GT rif N ((a, a, a 3, a 4 ; ω), (a, a, a 3, a 4 ; σ)) is a crisp subset of R which is defined as follows A α {x : µãi (x) α} [A (α), A (α)] [a + α ω (a a ), a 4 α ω (a 4 a 3 )] Definition.: β-cut set: β-cut set of a GTrIFN Ãi GT IF N ((a, a, a 3, a 4 ; ω), (a, a, a 3, a 4 ; σ)) is a crisp subset of R which is defined as follows A β {x : ϑãi (x) β} [A (β), A (β)] [a β(a a ), a 3 + β(a 4 a 3)]

6 0 S. P. Mondal et al. / J. Linear. Topological. Algebra. 04(0) (05) 5-9. Definition.3: (α, β)-cut set: (α, β)-cut set of a GTrIFN Ãi GT rif N ((a, a, a 3, a 4 ; ω), (a, a, a 3, a 4 ; σ)) is a crisp subset of R which is defined as follows A α,β {[A (α), A (α)]; [A (β), A (β)]}, 0 < α + β ω, σ, αϵ[0, ω], βϵ[σ, ] Definition.4: Addition of two GTrIFN: Let two à i GT rif N ((a, a, a 3, a 4 ; ω ), (a, a, a 3, a 4 ; σ )) and Bi GT rif N ((b, b, b 3, b 4 ; ω ), (b, b, b 3, b 4 ; σ )) be GTrIFN, then the addition of two GTrIFN is given by à i GT rif N Bi GT rif N ((a + b, a + b, a 3 + b 3, a 4 + b 4 ; ω), (a + b, a + b, a 3 + b 3, a 4 + b 4 ; σ)) where ω min{ω, ω } and σ min{σ, σ }. Definition.5: Subtraction of two GTrIFN: Let two à i GT rif N ((a, a, a 3 m, a 4 ; ω ), (a, a, a 3, a 3 ; σ )) and B i GT rif N ((b, b, b 3, b 4 ; ω ), (b, b, b 3, b 4 ; σ )) be GTrIFN, then the subtraction of two GTrIFN is given by à i GT rif N B GT i rif N ((a b 4, a b 3, a 3 b, a 4 b ; ω), (a b 4, a b 3, a 3 b, a 4 b ; σ)) where ω min{ω, ω } and σ min{σ, σ }. Definition.6: Multiplication by a scalar: Let à i GT rif N ((a, a, a 3 ; ω), (a, a, a 3 ; σ)) and k is a scalar then kãi GT rif N is also a GTrIFN and is defined as{ kã i GT rif N ((ka, ka, ka 3 ; ω), (ka, ka, ka 3, ka 4 ; σ)) if k > 0 ((ka 3, ka, ka ; ω), (ka 4, ka 3, ka, ka ; σ)) if k < 0 where 0 < ω, σ. Definition.7: Multiplication of two GTIFN: Let two à i GT rif N ((a, a, a 3, a 4 ; ω ), (a, a, a 3, a 4 ; σ )) and Bi GT rif N ((b, b, b 3, b 4 ; ω ), (b, b, b 3, b 3 ; σ )) be GTrIFN, then the multiplication of two GTIFN is given by à i GT rif N Bi GT rif N ((a b, a b, a 3 b 3, a 4 b 4 ; ω), (a b, a b, a 3 b 3, a 4 b 4 ; σ)) where ω min{ω, ω } and σ min{σ, σ }. Definition.8: Division of two GTrIFN: Let two à i GT rif N ((a, a, a 3, a 4 ; ω ), (a, a, a 3, a 4 ; σ )) and Bi GT rif N ((b, b, b 3, b 4 ; ω ), (b, b, b 3, b 4 ; σ )) be GTrIFN, then the multiplication of two GTIFN is given by à i GT rif N / B GT i a rif N (( b 4, a b 3, a3 b, a4 b ; ω), ( a, a b b 4 3, a3 where ω min{ω, ω } and σ min{σ, σ }. b, a 4 Theorem.: [34] If ω and σ represent the maximum degree of membership and minimum degree of non-membership function respectively then they satisfy the conditions 0 ω, 0 σ and 0 < ω + σ. Definition.9: Generalized Hukuhara difference: [0] The generalized Hukuhara difference of two fuzzy number u, v R F is defines as follows b ; σ))

7 S. P. Mondal et al. / J. Linear. Topological. Algebra. 04(0) (05) 5-9. u gh v w is equivalent to { (i)u v w (ii)v u ( )w Consider [w] α [w (α), w (α)], then w (α) min{u (α) v (α), u (α) v (α)} and w (α) max{u (α) v (α), u (α) v (α)} Here the parametric representation of a fuzzy valued function f : [a, b] R F expressed by [f(t)] α [f (t, α), f (t, α)], t [a, b], α [0, ]. is Definition.0: Generalized Hukuhara derivative for first order: [0] The generalized Hukuhara derivative of a fuzzy valued function f : (a, b) R F at t 0 is defined as f f(t (t 0 )lim 0 +h) gh f(t 0 ) h 0 h If f (t 0 ) R F satisfying (.) exists, we say that f is generalized Hukuhara differentiable at t 0. Also we say that f(t) is (i)-gh differentiable at t 0 if [f (t 0 )] α [f (t 0, α), f (t 0, α)] and f(t) is (ii)-gh differentiable at t 0 if [f (t 0 )] α [f (t 0, α), f (t 0, α)] Definition.: Generalized Hukuhara derivative for second order: [3] The second order generalized Hukuhara derivative of a fuzzy valued function f : (a, b) R F at t 0 is defined as f (t 0 ) lim h 0 f (t 0 +h) gh f (t 0 ) h If f R F, we say that f (t 0 ) is generalized Hukuhara at t 0. Also we say that f (t 0 ) is (i)-gh differentiable at t 0 if { (f f (t 0, α) (t 0, α), f (t 0, α)) if f be (i)-gh differentiable on (a,b) (f (t 0, α), f (t 0, α)) if f be (ii)-gh differentiable on (a,b) for all α [0, ], and that f { (t 0 ) is (ii)-gh differentiable at t 0 if (f f (t 0, α) (t 0, α), f (t 0, α)) if f be (i)-gh differentiable on (a,b) (f (t 0, α), f (t 0, α)) if f be (ii)-gh differentiable on (a,b) for all α [0, ]. Definition.: Strong and week solution: If the solution of intutionistic fuzzy differential equation is of the form [x (t, α), x (t, α); x (t, β), x (t, β)], the solution is called strong solution when (i) dx(t,α) dα (ii) dx (t,β) dβ > 0, dx(t,α) dα < 0 α [0, ω], x (t, ω) x (t, ω) < 0, dx (t,β) dβ > 0 β [σ, ], x (t, σ) x (t, σ) Definition.3: (α, β)-cuts to Intutionistic fuzzy number: Let [A (α), A (α); A (β), A (β)] be the (α, β)-cuts of a trapezoidal intutionistic fuzzy number à and ω, σ be the gradation of membership and non membership function respectively then the intutionistic fuzzy number is given by

8 S. P. Mondal et al. / J. Linear. Topological. Algebra. 04(0) (05) 5-9. Ã ((A (α 0), A (α ω), A (α ω), A (α 0); ω); (A (β σ), A (β 0), A (β 0), A (β σ); σ)) 3. Second order intutionistic fuzzy boundary value problem Consider the differential equation d x(t) kx(t), with boundary condition x(0) ã and x(l) b. Where ã, b are generalized trapezoidal intutionistic fuzzy number. Let ã ((a, a, a 3, a 4 ; ω ), (a, a, a 3, a 4 ; σ )) and b ((b, b, b 3, b 4 ; ω ), (b, b, b 3, b 4 ; σ )) 3. Second order intutionistic fuzzy boundary value problem with coefficients is positive constant i.e., k > 0 Here two cases arise Case 3..: When x(t) is (i)-gh differentiable and dx(t) is (i)-gh then we have d x (t,α) kx (t, α) d x (t,α) kx (t, α) d x (t,β) kx (t, β) d x (t,β) kx (t, β) With boundary conditions x (0, α) a + αl ã ω, x (0, α) a 4 αr ã ω, x (0, β) a βl ã σ, x (0, β) a 3 + βr ã σ and x (L, α) b + αl b ω, x (L, α) b 4 αr b ω, x (L, β) b βl b σ, x (L, β) b 3 + βr b σ Where, lã a a,rã a 4 a 3, l ã a a, r ã a 4 a 3 and l b b b,r b b 4 b 3, l b b b, r b b 4 b 3 and ω min{ω, ω }, σ min{σ, σ }. Solution: The general solution of first equation is x (t, α) c e kt + c e kt Using initial condition we have c + c a + αl ã ω and c + c e kl b + αl b ω Solving we get c e kl {(b + αl b ω ) (a + αl ã ω )e kl }

9 S. P. Mondal et al. / J. Linear. Topological. Algebra. 04(0) (05) and c e kl {(b + αl b ω ) (a + αl ã ω )e kl } Therefore the solution is x (t, α) e [{(b kl + αl b ω ) (a + αl ã ω )e kl }e kt {(b + αl b ω ) (a + αlã ω )e kl }e kt ] Similarly x (t, α) e [{(b kl 4 αr b ω ) (a 4 αr ã ω )e kl }e kt {(b 4 αr b ω ) (a 4 αrã ω )e kl }e kt ] x (t, β) e [{(b kl βl b σ ) (a βl ã σ )e kl }e kt {(b βl b σ ) (a βl ã σ )e kl }e kt ] x (t, β) e [{(b kl 3 + βl b σ ) (a 3 + βl ã σ )e kl }e kt {(b 3 + βl b σ ) (a 3 + βl ã σ )e kl }e kt ] Case 3..: When x(t) is (ii)-gh differentiable and dx(t) d x (t,α) kx (t, α) d x (t,α) kx (t, α) d x (t,β) kx (t, β) d x (t,β) kx (t, β) With same boundary conditions. Solution: The general solution is given by x (t, α) c e kt + c e kt + c 3 cos kt + c 4 sin kt x (t, α) c e kt + c e kt c 3 cos kt c 4 sin kt x (t, β) d e kt + d e kt + d 3 cos kt + d 4 sin kt x (t, β) d e kt + d e kt d 3 cos kt d 4 sin kt is (i)-gh then we have Where, c e kl [{ b +b 4 + α(l b r b) ω } { a +a 4 + α(l ã rã) ω }e kl ] c e b+b4 [{ kl e kl + α(l b r b) ω } { a+a4 + α(l ã rã) ω } ]

10 4 S. P. Mondal et al. / J. Linear. Topological. Algebra. 04(0) (05) 5-9. c 3 a a4 + α(l b+r b) ω c 4 sin b+b4 [{ kl + α(l b r b) ω d e b+b3 [{ kl e kl } { a+a4 + α(l ã rã) ω } cos kl] β(l b r b) σ } { a+a3 β(l ã r ) ã σ }e kl ] d e [{ b +b 3 kl β(l b r b) σ } { a +a 3 β(l ã r ) ã σ } ] d 3 a a 3 β(l ã +r ) ã σ d 4 sin [{ b b 3 kl β(l b r b) σ } { a a 3 β(l ã +r ) ã σ } cos kl] Case 3..3: When x(t) is (i)-gh differentiable and dx(t) d x (t,α) kx (t, α) d x (t,α) kx (t, α) d x (t,β) kx (t, β) d x (t,β) kx (t, β) With same boundary conditions. Solution: The result is same as Case 3... is (ii)-gh then we have Case 3..4: When x(t) is (ii)-gh differentiable and dx(t) is (ii)-gh then we have d x (t,α) kx (t, α) d x (t,α) kx (t, α) d x (t,β) kx (t, β) d x (t,β) kx (t, β) With same boundary conditions. Solution: The result is same as Case Second order intutionistic fuzzy boundary value problem with coefficients is negative constant i.e., k < 0, Consider k m, m > 0: In this problem four cases arise Case 3.. When x(t) is (i)-gh differentiable and dx(t) is (i)-gh then we have d x (t,α) mx (t, α)

11 S. P. Mondal et al. / J. Linear. Topological. Algebra. 04(0) (05) d x (t,α) mx (t, α) d x (t,β) mx (t, β) d x (t,β) mx (t, β) With boundary conditions x (0, α) a + αl ã ω, x (0, α) a 4 αr ã ω, x (0, β) a βl ã σ, x (0, β) a 3 + βr ã σ and x (L, α) b + αl b ω, x (L, α) b 4 αr b ω, x (L, β) b βl b σ, x (L, β) b 3 + βr b σ Where, lã a a,rã a 4 a 3, l ã a a, r ã a 4 a 3 and l b b b,r b b 4 b 3, l b b b, r b b 4 b 3 and ω min{ω, ω }, σ min{σ, σ }. Solution: The general solution of this equation may be written as x (t, α) c e mt + c e mt + c 3 cos mt + c 4 sin mt x (t, α) c e mt c e mt + c 3 cos mt + c 4 sin mt x (t, β) d e mt + d e mt + d 3 cos mt + d 4 sin mt x (t, β) d e mt d e mt + d 3 cos mt + d 4 sin mt Where, c e ml e ml [{ b b + α(l b+r b) ω } { a a 4 + α(l ã+rã) ω }e ml ] c e ml e ml [{ b b 4 + α(l b+r b) ω } { a a 4 + α(l ã+rã) ω }e ml ] c 3 a+a4 + α(l b r b) ω c 4 sin b+b4 [{ ml + α(l b r b) ω d e ml e ml [{ b b3 } { a+a4 + α(l ã rã) ω } cos ml] β(l b+r b) σ } { a a3 β(l ã +r ) ã σ }e ml ] d e ml e ml [{ b b 3 β(l b+r b) σ } { a a 3 β(l ã +r ) ã σ }e ml ] d 3 a +a 3 β(l ã r ) ã σ d 4 sin ml [{ b +b 3 β(l b r b) σ } { a +a 3 β(l ã r ) ã σ } cos ml] Case 3..: When x(t) is (ii)-gh differentiable and dx(t) is (i)-gh then we have d x (t,α) mx (t, α) d x (t,α) mx (t, α)

12 6 S. P. Mondal et al. / J. Linear. Topological. Algebra. 04(0) (05) 5-9. d x (t,β) mx (t, β) d x (t,β) mx (t, β) With same boundary conditions. Solution: The solution is written as x (t, α) (a + αl ã ω ) cos mt + sin ml {(b + αl b ω ) (a + αl ã ω ) cos ml} sin mt x (t, α) (a 4 αr ã ω ) cos mt + sin ml {(b 4 αr b ω ) (a 4 αr ã ω ) cos ml} sin mt x (t, β) (a 3 + βr ã σ ) cos mt + sin ml {(b βl b σ ) (a 3 + βr ã σ ) cos ml} sin mt x (t, β) (a βl ã σ ) cos mt + sin ml {(b 3 + βr b σ ) (a βl ã σ ) cos ml} sin mt Case 3..3: When x(t) is (i)-gh differentiable and dx(t) d x (t,α) mx (t, α) d x (t,α) mx (t, α) d x (t,β) mx (t, β) d x (t,β) mx (t, β) With same boundary conditions. Solution: The result is same as Case 3... is (ii)-gh then we have Case 3..4: When x(t) is (ii)-gh differentiable and dx(t) we have is (ii)-gh differentiable then d x (t,α) mx (t, α) d x (t,α) mx (t, α) d x (t,β) mx (t, β) d x (t,β) mx (t, β) With boundary conditions Solution: The result is same as Case 3...

13 S. P. Mondal et al. / J. Linear. Topological. Algebra. 04(0) (05) Table. Solutions for t α x (t,α) x (t, α) β x (t, β) x (t, β) Figure. Graph of solutions for t 4. Numerical example: Consider the differential equation d x(t) 4x(t), with boundary conditions x(0) ((4.5, 5, 7, 7.5; 0.7), (4, 5, 7, 8; 0.)) and x() ((0, 0.5,.5, 3; 0.7), (9.5, 0.5,.5, 3.5; 0.)) Find the solution when x(t) and dx(t) is (i)-gh differentiable. Solution: The solution is given by x (t, α) e 4 e 4 [{(0 + 5α 7 ) ( α 7 )e 4 }e t {(0 + 5α 7 ) ( α 7 )e4 }e t ] x (t, α) e 4 e 4 [{(3 5α 7 ) (7.5 5α 7 )e 4 }e t {(3 5α 7 ) (7.5 5α 7 )e4 }e t ] x (t, β) e 4 e [{(0.5 5β) (5 5β)e 4 }e t {(0.5 5β) (5 5β)e 4 }e t ] 4 x (t, β) e 4 e [{(.5 + 5β) (7 + 5β)e 4 }e t {(.5 + 5β) (7 + 5β)e 4 }e t ] 4 Remarks: From the graph and table we conclude that x (t, α) is increasing and x (t, α) is decreasing whereas x (t, β) is decreasing and x (t, β) is increasing function for t. Thus in this case the solution is a strong solution.

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