Important Notes for a Fuzzy Boundary Value Problem

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Abstract

In this paper is studied a fuzzy Sturm-Liouville problem with the eigenvalue parameter in the boundary condition. Important notes are given for the problem. Integral equations are found of the problem.

Abstract

In this paper is studied a fuzzy Sturm-Liouville problem with the eigenvalue parameter in the boundary condition. Important notes are given for the problem. Integral equations are found of the problem.

1 Introduction

Firstly, Zadeh introduced the concept of fuzzy numbers and fuzzy arithmetic [22]. The major application of fuzzy arithmetic is fuzzy differential equations. Fuzzy differential equations are suitable models to model dynamic systems in which there exist uncertainties or vaguness. Fuzzy differential equations can be examined by several approach, such as Hukuhara differentiability, generalized differentiability, the concept of differential inclusion etc [1], [3], [4], [5], [6, 7, 8], [9], [11], [13, 14], [15] [17], [19], [20].

In this paper is on a fuzzy Sturm-Liouville problem with the eigenvalue parameter in the boundary condition. Important notes are given for the problem. Integral equations are found of the problem.

2 Preliminaries

Definition 1. [18] A fuzzy number is a function u : 0,1satisfying the properties:u is normal, u is convex fuzzy set, u is upper semi-continuous on ℝ, clxεux>0is compact, where cl denotes the closure of a subset.

Let RF denote the space of fuzzy numbers.

Definition 2. [15] Let uɛF. The a-level set of u, denoted, uα,0<α1,is

uα=xεuxα.

If a = 0; the support of u is defined

u0=clxεux>0.

The notation, uα=u¯α,u¯αdenotes explicitly the a-level set of u. We refer to u¯and ū as the lower and upper branches of u,respectively.

The following remark shows when u¯α,u¯α is a valid a-level set.

Remark 1. [10,15] The sufficient and necessary conditions for u¯α,u¯αto define the parametric form of a fuzzy number as follows:

u¯αis bounded monotonic increasing (nondecreasing) left-continuous function on (0;1] and right-continuous for a = 0,

u¯αis bounded monotonic decreasing (nonincreasing) left-continuous function on (0;1] and right-continuous fora = 0,

u¯αu¯α,0α1.

Definition 3. [15] For u,vεF and λ ∈R, the sum u+v and the product λ u are defined by u+vα=uα+vα,λuα=λuαwhere means the usual addition of two intervals (subsets) ofand λ [u]α means the usual product between a scalar and a subset of R.

The metric structure is given by the Hausdorff distance

D:F×F+0,

by

Du,v=supα0,1 maxu¯αv¯α,u¯αv¯α.

Definition 4. [18] If A is a symmetric triangular numbers with supports a¯,a¯,the α–level sets of A is Aα=a¯+a¯a¯2α,a¯a¯a¯2α.

Definition 5. [16] u,v 2F, uα=u¯α,u¯α,vα=v¯α,v¯α,the product uv is defined by

uvα=uαvα,α0,1,

where

uαvα=u¯a,u¯αv¯a,v¯α=w¯a,w¯α,w¯a=minu¯av¯a,u¯av¯α,u¯αv¯a,u¯αv¯α,w¯a=maxu¯av¯a,u¯av¯α,u¯αv¯a,u¯αv¯α.

Definition 6. [15, 21] Let u,v 2F.If there exists w 2F such that u = v+w; then w is called the Hukuhara difference of fuzzy numbers u and v;and it is denoted by w = u⊖v:

Definition 7. [2,15] Let f : [a;b]RF and t0 [a,b] :We say that f is Hukuhara differential at t0, if there exists an element f' 0 (t0) 2F such that for all h > 0 sufficiently small, ft0+hft0,ft0ft0hand the limits (in the metric D)

limh0ft0+hft0h=limh0ft0ft0hh=ft0.

Definition 8. [12] If px=0,rx=1and Ly=pxy˝+qxyin the fuzzy differential equation pxy+qxy+λrxy=0,px,px,qx,rx,are continuous functions and positive, the fuzzydifferential equation

Ly+λy=0

is called a fuzzy Sturm-Liouville equation.

Definition 9. [12] yx,λ0α=y¯x,λ0,y¯x,λ00,we say that λ = λ0 is eigenvalue of (2.1) if the fuzzy differential equation (2.1) has the nontrivial solutions y¯x,λ00,y¯x,λ00.

3 Findings and Main Results

Consider the fuzzy boundary value problem

τu=u˝+qxu,
τu+λu=0,x1,1
u1+u1=0,
λβu1+γu1=0,

where q(x) is continuous function and positive on [-1,1] , λ > 0 and β; γ > 0.

Let u1 (x;λ) and u2 (x;λ) be linearly independent solutions of the classical differential equation τu+λu=0. Then, the general solution of the fuzzy differential equation (3.1) is

ux,λα=u¯αx,λ,u¯αx,λ,
u¯αx,λ=aαλu1x+λ+bαλu2x+λ,

Also,

u¯αx,λ=cαλu1x,λ+dαλu2x,λ.
φx,λα=φ¯αx,λ,φ¯αx,λ

be the solution of the equation (3.1) satisfying the conditions

u1=1,u´1=1

and

χx,λα=χ¯αx,λ,χ¯αx,λ

be the solution of the equation (3.1) satisfying the conditions

u1=γ,u1=λβ

where

φ¯αx,λ+c¯1αλu1x,λ+c¯2αλu2x,λ,φ¯αx,λ+c¯1αλu1x,λ+c¯2αλu2x,λ
χ¯αx,λ=c¯3αλu1x,λ+c¯4αλu2x,λχ¯αx,λ=c¯3αλu1x,λ+c¯4αλu2x,λ.

From here, yields

Wφ¯α,χ¯αx,λ=c¯1αλc¯4αλc¯2αλc¯3αλWu1,u2x,λ
Wφ¯α,χ¯αx,λ=c¯1αλc¯4αλc¯2αλc¯3αλWu1,u2x,λ

Also, since u1 (x;λ) and u2 (x;λ) are linearly independent solutions of the classical differential equation τu+λu=0,the solution of the equation is

ux,λ=aλu1x,λ+bλu2x,λ.

φ(x;λ) be the solution of the classical differential equation τu+lu = 0 satisfying the conditions u1= 1,u1=1.Using boundary conditions, we have

aλu11,λ+bλu21,λ=1,
aλu11,λ+bλu21,λ=1.

From this, a(λ) , b(λ) are obtained as

aλ=u21,λ+u21,λWu1,u21,λ,bλ=u11,λu11,λWu1,u21,λ

Then,

φx,λ=1Wu1,u21,λu21,λ+u21,λu1x,λu21,λ+u11,λu2x,λ.

Again,χ (x,λ) be the solution of the classical differential equation τu+lu = 0 satisfying the conditions u1=γ,u1=λβ.Similarly, χ(x;λ) is obtained as

χx,λ=1Wu1,u21,λλβu21,λ+γu2´1,λu1x,λλβu11,λ+γu1´1,λu2x,λ

Thus,

φx,λα=φ¯αx,λ,φ¯αx,λ=c1α,c2αφx,λ

is the solution of the equation (3.1) satisfying the conditions (3.4) and

χx,λα=χ¯αx,λ,χ¯αx,λ=c1α,c2αχx,λ

is the solution of the equation (3.1) satisfying the conditions (3.5), where c1α,c2α=1α.We take 1α=α,2α.From here,

W¯αx,λ=α2φx,λχx,λχx,λφx,λ
W¯αx,λ=2α2φx,λχx,λχx,λφx,λ

are obtained. Computing the value φx,λχx,λχx,λφx,λ,we have

Wu1,u2x,λWu1,u21,λWu1,u21,λu11,λ+u1´1,λλβu21,λ+γu2´1,λu21,λ+u2´1,λλβu11,λ+γu1´1,λ

Considering the 3.7), the value is c¯1αλc¯4αλc¯2αλc¯3αλ

α2Wu1,u21,λWu1,u21,λu11,λ+u1´1,λλβu21,λ+γu2´1,λu21,λ+u2´1,λλβu11,λ+γu1´1,λ

and the value c¯1αλc¯4αλc¯2αλc¯3αλis

2α2Wu1,u21,λWu1,u21,λu11,λ+u1´1,λλβu21,λ+γu2´1,λu21,λ+u2´1,λλβu11,λ+γu1´1,λ

Consequently,

Wφ¯α,χ¯αx,λ=α22α2Wφ¯α,χ¯αx,λ.

Theorem 1. The Wronskian functions Wφ¯α,χ¯α(x;λ) and W φ¯α,χ¯α(x;λ) are independent of variable x for x ∈ (1;1), where functions φα¯,χα¯,φ¯α,χ¯αare the solution of the fuzzy boundary value problem (3.1)-(3.3).

Proof. Derivating of equations W φ¯α,χ¯α(x;λ) and W φ¯α,χ¯α(x;λ) according to variable x and using the functions φx,λα,χx,λαare the solutions of the equation (3.1)

W´φ¯α,χ¯αx,λ=0andW´φ¯α,χ¯αx,λ=0.

are obtained. The proof is complete.

W¯αλ=Wφ¯α,χ¯αx,λ=W¯αλ=Wφ¯α,χ¯αx,λ.

Theorem 2. The eigenvalues of the fuzzy boundary value problem (3.1)-(3.3) if and only if are consist of the zeros of functionsW¯αλandW¯αλ.

Proof. Let be λ = λ0 is the eigenvalue. We show that W¯αλ0=0and W¯αλ0=0.We assume that W¯αλ00orW¯αλ00.Let be W¯αλ00.Then, the functions φ¯αx,λ0and χ¯αx,λ0are linearly independent. So, the general solution of the equation (3.1)

ux,λ0α=u¯αx,λ0,u¯αx,λ0,
u¯αx,λ0=aαλ0φ¯αx,λ0+bαλ0χ¯αx,λ0,
u¯αx,λ0=cαλ0φ¯αx,λ0+dαλ0χ¯αx,λ0.

Using the boundary condition (3.2) and using the solution function φx,λ0α=φ¯αx,λ0,φ¯αx,λ0satisfies the boundary condition (3.2),

bαλ0χ¯α1,λ0+χ¯α1,λ0=0,
dαλ0χ¯α1,λ0+χ¯α1,λ0=0

are obtained. Again, using (3.4), (3.10), we have

bαλ0W¯αλ0=0,dαλ0W¯αλ0=0.

From this, since W¯αλ00,bαλ0=0is obtained. Similarly, using the boundary condition (3.3), we obtained aαλ0=0.Thus, u¯αx.λ0=0,λ0is not an eigenvalue. That is, we have a contradiction. Similarly, ifW¯αλ00,u¯αx,λ0=0is obtained. λ0 is not an eigenvalue.

Let λ0 be zero ofW¯αλandW¯αλ.Then,

φ¯αx,λ0=k1χ¯αx,λ0,φ¯αx,λ0=k2χ¯αx,λ0.

That is, the functions φ¯α,χ¯αand φ¯α,χ¯αare linearly dependent. Also, since χx,λαsatisfies the boundary condition (3.3), χ¯αx,λ0and χ¯αx,λ0satisfy the boundary condition (3.3). In addition, from (3.14) the functions φ¯αx;λ0) and φ¯αx,λ0satisfy the boundary condition (3.3). So, φx,λ0αsatisfies the boundary condition (3.3). Hence, [φ(x;λ0)]a is the solution of the boundary value problem (3.1)-(3.3) for λ = λ0. Thus, λ = λ0 is the eigenvalue. The proof is complete.

Lemma 1. Let λ = s2. The lower and the upper solutions φ¯αx,λ,φ¯αx,λsatisfy the following integral equations for k=0 and k=1:

φ¯αx,λk=Cossx+1k1sSinsx+1k+1s1xSinsxykqyφ¯αx,λkdy,
φ¯αx,λk=Cossx+1k1sSinsx+1k+1s1xSinsxykqyφ¯αx,λkdy.

Proof. Since

φx,λα=φ¯αx,λ,φ¯αx,λ

is the solution of the equation (3.1), the equation

φ¯αy,λ,φ¯αy,λ+qyφ¯αy,λ,φ¯αy,λ+λφ¯αy,λ,φ¯αy,λ=0

is provided. Using the Hukuhara differentiability and fuzzy arithmetic,

φ¯αy,λ,φ¯αy,λ+qyφ¯αy,λ,qyφ¯αy,λ+λφ¯αy,λ,λφ¯αy,λ=0

is obtained. From here, yields

φ¯αy,λ+qyφ¯αy,λ+λφ¯αy,λ=0,φ¯αy,λ+qyφ¯αy,λ+λφ¯αy,λ=0.

Substituing the identity qyφ¯αy,λ=λφ¯αy,λφ¯ay,λin the right side of (3.15)

1xSinsxyqyφ¯αy,λdy=s21xSinsxyφ¯αy,λdy1xSinsxyφ¯αx,λdy

On integrating by parts twice and using (3.4)

1xSinsxyφ¯αx,λdy=Sinsx+1sφ¯αx,λ+sCossx+1+s21xSinsxyφ¯αy,λdy

is obtained. Substituing this back into the previous equality yields

1xSinsxyqyφ¯αy,λdy=Sinsx+1sφ¯αx,λ+sCossx+1

From here, we have

φ¯αx,λ=Cossx+11sSinsx+1+1xSinsxyqyφ¯αx,λdy.

Similarly φ¯αx,λis found. Derivating in these equations according to x, the derivative equations are obtained.

Lemma 2. Let λ = s2. The lower and the upper solutions χ¯αx,λ,χ¯αx,λsatisfy the following integral equations for k=0 and k=1:

χ_αx,λk=βCossx+1k+λαsSinsx+1k1sx1Sinsxykqyχ_αx,λkdy
χ¯αx,λk=βCossx+1k+λαsSinsx+1k1sx1Sinsxykqyχ¯αx,λkdy.

Proof. Substituing the identity qyχ¯αy,λ=λχ¯αy,λχ¯ay,λin the right side of (3.17), integrating by parts twice and using (3.5) yields (3.17) for k=0. Similarly, the equation (3.18) is found for k=0. Derivating in these equations according to x, the derivative equations are obtained.

Acknowledgement 1

This paper was supported by Scientific Research Projects Coordination Unit of Giresun University (The project number is FEN-BAP-A-230218-52). Thanks to Scientific Research Projects Coordination Unit of Giresun University.

References

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    B. Bede A note on ”two-point boundary value problems associated with non-linear fuzzy differential equations” Fuzzy Sets and Systems 157 (2006) 986-989.

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    B. Bede Note on ”Numerical solutions of fuzzy differential equations by predictor-corrector method” Inf. Sci. 178 (2008) 1917-1922.

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    B. Bede S.G. Gal Almost periodic fuzzy-number-valued functions Fuzzy Sets and Systems 147 (2004) 385-403.

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    B. Bede S. G. Gal Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations Fuzzy Sets and Systems 151 (2005) 581-599.

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    B. Bede I.J. Rudas A.L. Bencsik First order linear fuzzy differential equations under generalized differentiability Inform. Sci. 177 (2007) 1648-1662.

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    J.J. Buckley T. Feuring Fuzzy differential equations Fuzzy Sets and Systems 110 (2000) 43-54.

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    J.J. Buckley T. Feuring Fuzzy initial value problem for Nth-order linear differential equations Fuzzy Sets and Systems 121 (2001) 247-255.

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    Y. Chalco-Cano H. Roman-Flores On new solutions of fuzzy differential equations Chaos Solutions and Fractals 38 (2008) 112-119.

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    D. Dubois H. Prade Operations on fuzzy numbers International Journal of Systems Sciences 9 (1978) 613-626.

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    N. Gasilov ¸ S.E. Amrahov A.G. Fatullayev A geometric approach to solve fuzzy linear systems of differential equations Appl. Math. & Inf. Sci. 5 (2011) 484-495.

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    H. Gültekin Çitil N. Altını¸sık On the eigenvalues and the eigenfunctions of the Sturm-Liouville fuzzy boundary value problem Journal of Mathematical and Computational Science 7(4) (2017) 786-805.

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    E. Hüllermeir An approach to modeling and simulation of uncertain dynamical systems International Journal of Uncertanity Fuzziness and Knowledge-Based Systems 5 (1997) 117-137.

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    A. Khastan F. Bahrami K. Ivaz New Results on Multiple Solutions for Nth-order Fuzzy Differential Equations under Generalized Differentiability Boundary Value Problems (2009) 1-13.

    • Crossref
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    A. Khastan J.J. Nieto A boundary value problem for second order fuzzy differential equations Nonlinear Analysis 72 (2010) 3583-3593.

    • Crossref
    • Export Citation
  • [16]

    V. Lakshmikantham R. N. Mohapatra Theory of Fuzzy Differential Equations and Inclusions Taylor&Francis London New York 2003.

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    V. Lakshmikantham J.J. Nieto Differential Equations in Metric Spaces: An introduction and application to fuzzy differential equations Dynamics of Continuous. Discrete and Impulsive Systems Series A: Mathematical Analysis 10 (2003) 991-1000.

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    H.-K. Liu Comparison results of two-point fuzzy boundary value problems International Journal of Computational and Mathematical Sciences 5(1) (2011) 1-7.

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    J.J. Nieto R. Rodriguez-Lopez Euler polygonal method for metric dynamical systems Inform. Sci. 177 (2007) 42564270.

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    J.J. Nieto A. Khastan K. Ivaz Numerical solution of fuzzy differential equations under generalized differentiability Nonlinear Analysis: Hybrid Syst. 3 (2009) 700-707.

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    M.L. Puri D.A. Ralescu Differentials for fuzzy functions Journal of Mathematical Analysis and Applications 91 (1983) 552-558.

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    L.A. Zadeh Fuzzy Sets Information and Control 8 (1965) 338-353.

    • Crossref
    • Export Citation

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  • [1]

    B. Bede A note on ”two-point boundary value problems associated with non-linear fuzzy differential equations” Fuzzy Sets and Systems 157 (2006) 986-989.

    • Crossref
    • Export Citation
  • [2]

    B. Bede Note on ”Numerical solutions of fuzzy differential equations by predictor-corrector method” Inf. Sci. 178 (2008) 1917-1922.

    • Crossref
    • Export Citation
  • [3]

    B. Bede S.G. Gal Almost periodic fuzzy-number-valued functions Fuzzy Sets and Systems 147 (2004) 385-403.

    • Crossref
    • Export Citation
  • [4]

    B. Bede S. G. Gal Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations Fuzzy Sets and Systems 151 (2005) 581-599.

    • Crossref
    • Export Citation
  • [5]

    B. Bede I.J. Rudas A.L. Bencsik First order linear fuzzy differential equations under generalized differentiability Inform. Sci. 177 (2007) 1648-1662.

    • Crossref
    • Export Citation
  • [6]

    J.J. Buckley T. Feuring Fuzzy differential equations Fuzzy Sets and Systems 110 (2000) 43-54.

    • Crossref
    • Export Citation
  • [7]

    J.J. Buckley T. Feuring Fuzzy initial value problem for Nth-order linear differential equations Fuzzy Sets and Systems 121 (2001) 247-255.

    • Crossref
    • Export Citation
  • [8]

    Y. Chalco-Cano H. Roman-Flores On new solutions of fuzzy differential equations Chaos Solutions and Fractals 38 (2008) 112-119.

    • Crossref
    • Export Citation
  • [9]

    Y. Chalco-Cano H. Roman-Flores Comparation between some approaches to solve fuzzy differential equations Fuzzy Sets and Systems 160 (2009) 1517-1527.

    • Crossref
    • Export Citation
  • [10]

    D. Dubois H. Prade Operations on fuzzy numbers International Journal of Systems Sciences 9 (1978) 613-626.

    • Crossref
    • Export Citation
  • [11]

    N. Gasilov ¸ S.E. Amrahov A.G. Fatullayev A geometric approach to solve fuzzy linear systems of differential equations Appl. Math. & Inf. Sci. 5 (2011) 484-495.

  • [12]

    H. Gültekin Çitil N. Altını¸sık On the eigenvalues and the eigenfunctions of the Sturm-Liouville fuzzy boundary value problem Journal of Mathematical and Computational Science 7(4) (2017) 786-805.

  • [13]

    E. Hüllermeir An approach to modeling and simulation of uncertain dynamical systems International Journal of Uncertanity Fuzziness and Knowledge-Based Systems 5 (1997) 117-137.

    • Crossref
    • Export Citation
  • [14]

    A. Khastan F. Bahrami K. Ivaz New Results on Multiple Solutions for Nth-order Fuzzy Differential Equations under Generalized Differentiability Boundary Value Problems (2009) 1-13.

    • Crossref
    • Export Citation
  • [15]

    A. Khastan J.J. Nieto A boundary value problem for second order fuzzy differential equations Nonlinear Analysis 72 (2010) 3583-3593.

    • Crossref
    • Export Citation
  • [16]

    V. Lakshmikantham R. N. Mohapatra Theory of Fuzzy Differential Equations and Inclusions Taylor&Francis London New York 2003.

  • [17]

    V. Lakshmikantham J.J. Nieto Differential Equations in Metric Spaces: An introduction and application to fuzzy differential equations Dynamics of Continuous. Discrete and Impulsive Systems Series A: Mathematical Analysis 10 (2003) 991-1000.

  • [18]

    H.-K. Liu Comparison results of two-point fuzzy boundary value problems International Journal of Computational and Mathematical Sciences 5(1) (2011) 1-7.

  • [19]

    J.J. Nieto R. Rodriguez-Lopez Euler polygonal method for metric dynamical systems Inform. Sci. 177 (2007) 42564270.

  • [20]

    J.J. Nieto A. Khastan K. Ivaz Numerical solution of fuzzy differential equations under generalized differentiability Nonlinear Analysis: Hybrid Syst. 3 (2009) 700-707.

  • [21]

    M.L. Puri D.A. Ralescu Differentials for fuzzy functions Journal of Mathematical Analysis and Applications 91 (1983) 552-558.

    • Crossref
    • Export Citation
  • [22]

    L.A. Zadeh Fuzzy Sets Information and Control 8 (1965) 338-353.

    • Crossref
    • Export Citation
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