The Super Stability of Third Order Linear Ordinary Differential Homogeneous Equation with Boundary Condition

1. International Journal of Trend in Scientific Research and Development (IJTSRD) Volume 3 Issue 6, October 2019 2019 Available Online: www.ijtsrd.com e International Journal of Trend in Scientific Research and Development (IJTSRD) International Journal of Trend in Scientific Research and Development (IJTSRD) e-ISSN: 2456 – 6470 The Super Stability of Third Order Linear Ordinary Homogeneous Equation Homogeneous Equation with Boundary Condition Dawit Kechine Menbiko f Third Order Linear Ordinary f Third Order Linear Ordinary Differential ith Boundary Condition Department of Mathematics, College f Mathematics, College of Natural and Computational Science, Wachemo University, Ethiopia Computational Science, Wachemo University, Ethiopia ABSTRACT The stability problem is a fundamental issue in the design of any distributed systems like local area networks, multiprocessor systems, distribution computation and multidimensional queuing systems. In Mathematics stability theory addresses the stability solutions of differential, integral and other equations, and trajectories of dynamical systems under small perturbations of initial conditions. Differential equations describe many mathematical models of a great interest in Economics, Control theo Engineering, Biology, Physics and to many areas of interest. In this study the recent work of Jinghao Huang, Qusuay. Yongjin Li in establishing the super stability of differential equation of second order with boundary condition was stability of differential equation third order with boundary condition. KEYWORDS: supper stability, boundary conditions, How to cite this paper Menbiko "The Super Stability of Third Order Linear Ordinary Homogeneous Equation with Boundary Condition" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456 6470, Volume Issue-6, October 2019, pp.697 https://www.ijtsrd.com/papers/ijtsrd29 149.pdf Copyright © 2019 by author(s) and International Journal Scientific Research and Development Journal. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/b y/4.0) How to cite this paper: Dawit Kechine Menbiko "The Super Stability of Third Order Linear Ordinary Homogeneous Equation with Boundary Condition" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456- 6470, Volume-3 | 6, October 2019, pp.697-709, https://www.ijtsrd.com/papers/ijtsrd29 The stability problem is a fundamental issue in the design of any distributed systems like local area networks, multiprocessor systems, distribution computation and multidimensional queuing systems. In Mathematics tability solutions of differential, integral and other equations, and trajectories of dynamical systems under small perturbations of initial conditions. Differential equations describe many mathematical models of a great interest in Economics, Control theory, Engineering, Biology, Physics and to many areas of interest. Differential Differential IJTSRD29149 In this study the recent work of Jinghao Huang, Qusuay. H. Alqifiary, and in establishing the super stability of differential equation of second order with boundary condition was extended to establish the super stability of differential equation third order with boundary condition. URL: boundary conditions, Intial conditions Copyright © 2019 by author(s) and International Journal Scientific Research and Development Journal. This is an Open Access article distributed under the terms of the Commons Attribution License http://creativecommons.org/licenses/b of of Trend Trend in in (CC (CC BY BY 4.0) 4.0) INTRODUCTION In recent years, a great deal of work has been done on various aspects of differential equations of third order. Third order differential equations describe many mathematical models of great iterest biology and physics. Equation ( ) ( ) ( ) x a x x b x x c x x ′′′ ′′ ′ + + + = entry-flow phenomena, a problem of hydrodynamics which is of considerable importance in many branches of engineering. There are different problems concerning third order differential equations which have drawn the attention of researchers throughout the world. [17] In mathematics stability theory addresses the st solutions of differential equations, Integral equations, including other equations and trajectories of dynamical systems under small perturbations. Following this, stability means that the trajectories do not change too much under small perturbations [7].The stability problem is a fundamental issue in the design of any distributed systems like local area networks, multiprocessor systems, mega computations and multidimensional queuing systems and others. In the field of economics, stability is achieved by avoiding or limiting fluctuations in production, employment and price. The stability problem in mathematics started by Poland mathematician Stan Ulam for functional equations around 1940; and the partial solution of Hyers to the Ulam’s problem [2] and [20]. In 1940, Ulam [21] posed a problem concerning the stability of functional equations: “Give conditions in order for a linear function near an approximately linear function to exist.” A year later, Hyers answered to the problem of Ulam for additive functions defined on Banach space: be real Banach spaces andɛ ? 1 2 : f X X → ( ) ( ) ( ) f x y f x f y + − − ≤ ( ) ( ) ( ) f x y f x f y + − − ≤ In recent years, a great deal of work has been done on various aspects of differential equations of third order. order differential equations describe many The stability problem in mathematics started by Poland mathematician Stan Ulam for functional equations around 1940; and the partial solution of Hyers to the Ulam’s in engineering, biology and physics. Equation of of the the form form ( ) f t In 1940, Ulam [21] posed a problem concerning the stability of functional equations: “Give conditions in order for a linear function near an approximately linear function to exist.” arise in the study of arise in the study of a problem of hydrodynamics considerable importance in many branches of A year later, Hyers answered to the problem of Ulam for additive functions defined on Banach space: let ?? and ?? ? 0. Then for every function There are different problems concerning third order differential equations which have drawn the attention of Satisfying In mathematics stability theory addresses the stability of solutions of differential equations, Integral equations, including other equations and trajectories of dynamical systems under small perturbations. Following this, stability means that the trajectories do not change too ε ∈ ∈ , , x y x y X X 1 → : A X X There exists a unique additive function with the property ( ) ( ) f x A x ε − ≤ Thereafter, Rassias [14] attempted to solve the stability problem of the Cauchy additive functional equations in more general setting. A generalization of Ulam’s problem is recently proposed by replacing functional equations with differential equations with differential equations exists a unique additive function 1 2 − ≤ ∈ ∈ f x A x x x X X The stability problem 1 is a fundamental issue in the design of any distributed systems like local area networks, multiprocessor systems, mega computations and multidimensional queuing systems and others. In the field of economics, stability is ieved by avoiding or limiting fluctuations in Thereafter, Rassias [14] attempted to solve the stability problem of the Cauchy additive functional equations in more general setting. A generalization of Ulam’s problem is recently proposed by replacing functional equations ,... ′ ( , , f y y ϕ = ( ) n ) 0 y and @ IJTSRD | Unique Paper ID – IJTSRD29149 29149 | Volume – 3 | Issue – 6 | September 6 | September - October 2019 Page 697

2. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 0 ε > and a function y such that ( , , ,... f y y y There exists a solution such that ( ) ( ) ( ) y t y t k ε − ≤ And 0 ε → Those previous results were extended to the Hyers-Ulam stability of linear differential equations of first order and higher order with constant coefficients in [8, 18] and in [19] respectively. Rus investigated the Hyers-Ulam stability of differential and integral equations using the Granwall lemma and the technique of weakly Picard operators [15, 16]. Miura et al [13] proved the Hyers-Ulam stability of the first-order linear differential equations '( ) ( ) ( ) 0 y t g t y t + = , Where ( ) g t is a continuous function, while Jung (4) proved the Hyers-Ulam stability of differential equations of the form ( ) '( ) ( ) t y t y t ϕ = Motivation of this study comes from the work of Li [5] where he established the stability of linear differential equations of second order in the sense of the Hyers and Ulam. ' y y λ = Li and Shen [6] proved the stability of non-homogeneous linear differential equation of second order in the sense of the Hyers and Ulam '' ( ) ' ( ) ( ) 0 y p x y q x y r x + + + = While Gavaruta et al [1] proved the Hyers- Ulam stability of the equation '' ( ) ( ) 0 y x y x β + = with boundary and initial conditions. The recently, introduced notion of super stability [10,11,12,13] is utilized in numerous applications of the automatic control theory such as robust analysis, design of static output feedback, simultaneous stabilization, robust stabilization, and disturbance attenuation. Recently, Jinghao Huang, Qusuay H. Alqifiary, Yongjin Li[3] established the super stability of differential equations of second order with boundary conditions or with initial conditions as well as the super stability of differential equations of higher order in the form ( )( ) ( ) ( ) 0 y x x y x β + = with initial conditions, ( 1) ( ) '( ) ... ( ) 0 y a y a y a = = = = This study aimed to extend the super stability of second order to the third order linear ordinary differential homogeneous equations with boundary condition. Main result Super stability with boundary condition Definition: Assume that for any function y satisfies the differential inequalities ( , , ',..., f y y y [ , x a b ∈ conditions, then either y is a solution of the differential equation ( ) ( , , ',..., ) 0 f y y y ϕ = …… ……(2) ( ) y x ≤ kεfor any constant. Then, we say that (2) has super stability with boundary Preliminaries Lemma 1[10]. Let y c a b ∈ ( ( ) max 8 Proof Let M = max has the Hyers-Ulam stability if for a given ∈ n [ , ] c a b y ) ,if ′ ϕ ≤ ε ( ) n ) ϕ ≤ K ε ( ) n .....(1) 0y of the differential equation ] ε ≥ 0 for all and for some with boundary ε = lim ( ) k 0 0 n [ ] ∈ , where kis a , x a b or conditions. 2[ , ] = = , ( ) y a 0 ( ) y b ,then ) 2 b a − ≤ ''( ) . y x y x max { x } [ ] ∈ ( ) : y x ∈ , x a b Since = = M ( , ) a b ( ) y a ( ) y x 0 = ( ) y b , there exists such that 0 . By Taylor’s formula, we have 0 δ ''( ) 2! ''( ) 2! y = + − + − 2 ( ) y a ( ) '( )( ) ( ) , y x y x x a x a 0 0 0 0 η y = + − + − 2 ( ) y b ( ) '( )( ) ( ) y x y x b x b x , 0 0 ) M − 0 0 Where δ , η( ∈ , a b 2 ''( ) δ = , y Thus ( ) 2 x a 0 2 − M x ''( ) η = y ( ) 2 b 0 ( ) a b +     8 ∈ , x a In the case , we have 0 2  2 − 2 M − M M − ≥ = ( ) ( ) ( ) 2 2 2 x a b a b a 0 4 a b +   2 M b a −    ∈ , x b In the case , we have 0 2 2 − 8 M x M − ≥ = ( ) ( ) ( ) 2 2 2 b b a 0 4 n 8 8 M − ( ) ′′ ≥ = max max ( ) y x y x So, ( ) ( ) 2 2 − − n b a b a ( ) 2 b a − ≤ ( ) y x max ''( ) y x Therefore, max 8 @ IJTSRD | Unique Paper ID – IJTSRD29149 | Volume – 3 | Issue – 6 | September - October 2019 Page 698

3. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 In (2011) the three researchers Pasc Gavaruta, Soon-Mo, Jung and Yongjin Li, investigate the Hyers-Ulam stability of second order linear differential equation ''( ) ( ) ( ) 0 y x x y x β + = ………….. (3) with boundary conditions ( ) y a [ , , ( ) y c a b x β ∈ ∈ Definition: We say (3) has the Hyers-Ulam stability with boundary conditions ( ) ( ) 0 y a y b = = if there exists a positive constant K with the following property: For every [ 0, , y c a b ε > ∈ ''( ) ( ) ( ) y x x y x β ε + ≤ , And ( ) ( ) 0 y a y b = = , then [ , z c a b ∈ ''( ) ( ) ( ) 0 z x x z x β + = And ( ) 0 ( ) z a z b = = , such that ( ) Theorem 1[1]. Consider the differential equation ''( ) ( ) ( ) 0 y x x y x β + = --------------------- (4) With boundary conditions ( ) y a [ , , ( ) y c a b x β ∈ ∈ 8 max ( ) x b a − then the equation (4) above has the super stability with boundary condition ( ) ( ) 0 y a y b = = Proof: For every ε > 2 2 − 8 M − M M − ≥ = 2 2 2 ( ) b a ( ) ( ) x a b a 0 4   M − a b +    M − ( ), = = ( ) y b ] 0 ∈ x b On the case we have 0 ] [ 2 −∞ < < < +∞ 2 , , c a b a b where, 2 − 2 8 M x ≥ = 2 2 2 ( ) b a ( ) ( ) b b a 0 4 8 8 − 2 ) M − ≥ = ''( ) y x ( ) max y x So max 2 2 ( ) ( ) b a b a ] 2 , if b a − ( ≤ ( ) ''( ) max y x max y x Therefore 8 Thus there exists some b a − 2 ( ) ] ≤ − ( ) ( ) x y x β + ( ) x max y x β   ( ) ''( ) ( ) max y x max y x max 2  satisfying 8 b a − b a − 2 2 − < Kε ( ) ( )max ( ) z x y x ≤ ε + ( ) max x β ( ) y x 8 8 ( ) 2 b a − = k = = Let ( ) y b ] , 0 ( )      2 b a − ] [ −∞ < < < +∞ − β 2 8 1    , max ( ) x c a b a b Where 8 β < If 0( ) z x = x y x ( ) y b . ( ) z x ≤ 0 with the Boundary conditions Obviously, ''( ) y x ( ) y a ( ) y x Hence the differential equation has the super stability with boundary condition ( ) 0 ( ) y a y b = = In (2014) the researchers J. Huang, Q. H. Aliqifiary, Y. Li established the super stability of the linear differential equations. ''( ) ( ) '( ) ( ) ( ) y x p x y x q x y x + + With boundary conditions ( ) y a Where [ , , , , y c a b p c a b q ∈ ∈ Then the aim of this paper is to investigate the super stability of third-order linear differential homogeneous equations by extending the work of J. Huang, Q. H. Aliqifiary and Y. Li using the standard procedures of them. Lemma( 2) .Let [ , y c a b ∈ ( max ( ) max 48 is a solution of ( ) 2 ( ) ( ) β 0 = − − = 0 = Kε 0 + ( ) ( ) x y x β = ''( ) y x 0 [ ] ∈ 2 0, , y C a b , if + ( ) ( ) x y x β ≤ ε = = ''( ) y x and ( ) 0 ( ) y b y a Let M = { } [ ] ∈ = = max ( ) : y x ∈ , x a b , since ( ) 0 ( ) y b y a , ( , ) a b x there exists Such that 0 = = 0 0 ( ) y x =M . By Taylor formula, we have = ( ) y b 0 δ ''( ) 2! ''( ) 2! y = + − + − 2 ( ) y a ( ) '( )( ) ( ) , y x y x x a x a 0 0 0 0 ] [ ] [ ] ∈ −∞ < < < +∞ 2 1 0 , , c a b a b η y = + − + − 2 ( ) y b ( ) '( )( ) ( ) , y x y x b x b x 0 0 ) M − 0 0 ( δ η∈ , , a b Where Thus 2 ''( ) δ = , y ( ) 2 x a 0 ] 3 = = and ( ) 0 ( ) y b y a 2 M −  ∈ , then ( ) η = '' y 2 ( ) x b ) 3 b a − 0 ≤ '''( ) y x y x a b +    ( ) , x a On the case , we have 0 2  @ IJTSRD | Unique Paper ID – IJTSRD29149 | Volume – 3 | Issue – 6 | September - October 2019 Page 699

4. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 Proof { } [ ] = ∈ = = max ( ) : y x , M x a b since ( ) 0 ( ) y b y a Let there exists: ( ) ∈ = , ( ) y x x a b M Such that 0 0 By Taylor’s formula we have δ ''( 2! ) '''( ) 3! η y x y ( ) ( ) 2 3 = + − + − + − ( ) y a ( ) '( )( ) , 0 y x y x x a x a x a 0 0 0 0 0 ''( 2! ) '''( ) 3! y x y = + − + − + − 2 3 ( ) y b ( ) '( )( ) ( ) ( ) , 0 y x y x b x b x b x 0 0 0 0 0 δ ''( 2! ) '''( ) 3! y x y ( ) ( ) 2 3 ≤ + − + − + − ⇒ ( ) y a ( ) '( ) ( ) , 0 y x y x x a x a x a 0 0 0 0 0 η ''( 2! ) '''( ) 3! y x y ≤ + − + − + − ⇒ 2 3 ( ) y b ( ) '( ) ( ) ( ) ( ) , 0 y x y x b x b x b x And 0 0 0 0 0 ( ) δ η∈ , , a b Where 3! x 6 M − M − '''( ) δ = = y Thus 3 3 ( ) ( ) a x a 0 0 3! − 6 − M x M x '''( ) η = = y And 3 3 ( ) ( ) b b 0 0 + +      a b a b ∈ < ≤ , x a a x For the case that is ,we have 0 0 2 2 6 6 6 48 b M − M b M − M a ≥ = = − 3 3 3 3 ( ) b a ( ) ( ) x a +       a 0 − a 8 2 + +      a b a b ∈ ≤ < , x b x b And for the case ,that is we have 0 0 2 2 6 − 6 6 48 b M x M − M − M a ≥ = = ( ) ( ) 3 3 3 3 ( ) b −       b a b a 0 2 8 6 − 48 b a − M x M ≥ ⇒( ) ( ) 3 3 b 0 48 b a − 48 − M ≥ = max '''( ) max ( ) y x y x Thus 3 3 ( ) ( ) b a 48 − ≥ max '''( ) max ( ) y x y x from 3 ( ) b a − 3 ( ) b a ≤ ⇒ max ( ) y x max '''( ) y x 48 + + + = Theorem 2. Consider '''( ) ( ) ''( ) m x y x ( ) '( ) p x y x ( ) ( ) q x y x 0 y x ------------------ (5) @ IJTSRD | Unique Paper ID – IJTSRD29149 | Volume – 3 | Issue – 6 | September - October 2019 Page 700

5. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 with boundary conditions ( ) a =0= y b y [ , , , , ' , , , y c a b m c a b p c a b q c a b ∈ ∈ ∈ ∈ ( ) ] [ ] [ ] [ ] −∞ < < < +∞ 3 2 0 a b where ? ??? − ?′? + ? ????′− 2??′+ ???− 2??? − ? ???? < ?? If ??? ????? + ?????? -------(6) ( ) ( ) 0 y a y b = = Then (5) has the super stability with boundary conditions Proof [ ] ∈ 3 , y c a b Suppose that satisfies the inequality: ε > + + + ≤ ε 0 '''( ) ( ) ''( ) m x y x ( ) '( ) p x y x ( ) ( ) q x y x y x ------------ (7) for some Let ( ) U x = '''( ) + + ( ) ''( ) m x y x ( ) '( ) ( ) ( ) p x y x q x y x y x --------- (8) [ ] ∈ , x a b and define ( ) z x by For all x 1 2 −∫ ( ) τ τ p d = ( ) y x ( ) z x e --------- (9) a And by taking the first, second and third derivative of (9) That is x ∫ 1 2       1 2 − ( ) ′ ′ = − ( ) τ τ y x z zp e p d a x 1 2 −∫ ( ) τ τ p d       1 2 1 4 ( ) ′′ ′′ z p ′ ′ = − − + 2 y x z zp zp e a x 1 2 −∫ ( ) τ τ p d       1 2 1 2 1 2 1 2 1 4 1 4 1 2 1 8 ′ z p ′ z p ′ ′ ′′ ′ z p ′ = − − − + − − + + + − 2 2 3 '''( ) ''' '' '' ' y x z z p z p z p zp zpp zp zp e a And by substituting (9) and its first, second and third derivatives in (8) we get   1 2 1 2 1 2 1 2 1 4 1 4 1 2 1 8 = − − − + − − + + + − + 2 2 3 ( ) u x ''' '' '' ' ' z p ' ' ' z p '' ' ' z z p z p z p zp zpp z p pz zp x 1 2 −∫ ( ) τ τ p d         1 2 1 4 1 2 ] − − + + − + 2 '' ' ' ' e m z   z p zp zp p z   zp zq a   1 2 1 2 1 2 1 2 1 4 1 4 1 2 1 8 − − − + − − + + + − + 2 2 3 ''' '' '' ' ' z p ' ' ' z p '' ' ' z z p z p z p zp zpp z p zp zp = x 1 2 −∫ ( ) τ τ p d 1 2 1 4 1 2 ] − − + + − + 2 2 '' ' ' ' e mz z mp zmp zmp pz zp qz a   3 2 3 2 3 4 1 2 1 4 1 2 1 2 + − + − − + + + + − − + 2 ''' '' '' ' ' ' ' z p ' ' ' '' z mz z p z p z mp z p qz zp zpp zmp zp = x 1 2 −∫ ( ) τ τ p d 1 4 1 2 1 8 ] − − 2 2 3 e zmp zp zp a           3 2 3 2 3 4 = [ + − + − − + + 2 ''' '' ' ' z m p z   p mp p p z @ IJTSRD | Unique Paper ID – IJTSRD29149 | Volume – 3 | Issue – 6 | September - October 2019 Page 701

6. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 x 1 2 −∫ ( ) τ τ p d       1 2 1 4 1 8 ] + − + ' 2 − + − − 2 2 3 z ( '') ( ' 2 ) e q p p pp mp mp p p a 3 4 3 2 p − + 2 ' p p p 3 2 = = m m p Now choose and By doing this the coefficients of ''( ) x and '( ) z z x will vanish. To check whether the relation holds or not, for 3 4 3 2 p − + 2 ' p p p 3 2 = p 3 2 3 4 3 2 = − + ⇒ 2 2 ' p p p p 3 2 3 4 = − ⇒ 2 ' p p p       3 2 3 4 = − ⇒ ' 1 p p p       3 2 3 4 dp dx = − 1 p p 2 3 dp = ⇒ dx using partial fraction       3 4 − 1 p p 3 4 3 4 1 p 1 + = Since       3 4 − − 1 p 1 p p           3 4 3 4 1 p 2 3 + = dp dx --------(*) − 1 p By integrating both sides of (*) we have 3 4 2 3 − − = + ln ln 1 p p x c 1 2 3 p = x c + ⇒ ln 1 3 4 − 1 p 2 3 p x = ⇒ Ce 3 4 − 1 p 2 3       3 4 x = − 1 p Ce p @ IJTSRD | Unique Paper ID – IJTSRD29149 | Volume – 3 | Issue – 6 | September - October 2019 Page 702

7. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 2 3 2 3 3 4 x x = − Ce Cpe       2 3 2 3 2 3 3 4 3 4 x x x + = = + 1 p Cpe Ce p Ce 2 3 x Ce 3 2 = ⇒ = p m p 2 3 3 4 x + 1 Ce             2 3 x 3 2 Ce m = 2 3 3 4 x + 1 Ce x 1 2 −∫ ( ) τ τ p d       1 2 1 4 1 8 + + − + ' 2 − + − − = 2 2 3 ''' ( '') ( ' 2 ) ( ) u x z q p p pp mp mp p p ze a Then       1 2 1 4 1 8 + + − + ' 2 − + − − = 2 2 3 ''' ( '') ( ' 2 ) z q p p pp mp mp p p z Then from the inequality (7) we get x ∫ x ∫ 1 2 1 2 ( ) τ τ ( ) τ τ p d p d ≤ ε ( ) u x e e a a x 1 2 −∫ ( ) τ τ p d = = = ( ) y x ( ) z x e From the boundary condition ( ) 0 ( ) y b y a and a = = We have ( ) 0 ( ) z b z a 1 2 1 4 1 8 = + − + ' 2 − + − − β 2 2 3 ( ) x ( '') ( ' 2 ) q p p pp mp mp p p Define x ∫ x ∫ 1 2 1 2 ( ) τ τ ( ) τ τ p d p d + = ≤ ( ) z x β ε '''( ) ( ) u x e z x e a a Then Using lemma (2) ( ) 3 − b a ≤ max ( ) max '''( ) z x z x 48 ( ) 3 − b a ≤ + + β β     max '''( ) ( ) z x max max ( ) z x z x 48           x ∫ 1 2 ( ) ( ) 3 3 b a − b a − ( ) τ τ p d ≤ ε + β max max max ( ) e z x a 48 48 x ∫ 1 2 ( ) τ τ p d < ∞ on the interval [ ] max , a b e Since a K > ] 0 Hence, there exists a constant ( ) z x kε ≤ For all such that [ ∈ , x a b x 1 2 −∫ ( ) τ τ p d < ∞ on the interval [ ] K > ' 0 , a bwhich implies that there exists a constant such that max e More over a @ IJTSRD | Unique Paper ID – IJTSRD29149 | Volume – 3 | Issue – 6 | September - October 2019 Page 703

8. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470       x ∫ 1 2 = − ( ) τ τ ( ) y x ( )exp z x p d a              x ∫ 1 2 ≤ − ( ) τ τ ε ≤ ε max exp    ' p d k k a ≤ k ε ⇒ ( ) y x ' Then '''( ) + + + = ( ) ''( ) m x y x ( ) '( ) p x y x ( ) ( ) q x y x 0 y x has super stability with boundary conditions = = ( ) y a 0 ( ) y b Example Consider the differential equation below                         2 3 2 3 x x 3 2 e e + + + = ''' '' ' 0 y y y y ----------- (10) 2 3 2 3 3 4 3 4 x x + + 1 1 e e = = With boundary conditions ( ) 0 ( ) y b y a             2 3 2 3 x x 3 2 e e [ ] [ ] [ ] [ ] y c a b ∈ ∈ ∈ ∈ 3 2 1 0 −∞ < < < +∞ , , , , , ,1 , , c a b c a b c a b a b Where 2 3 2 3 3 4 3 4 x x + + 1 1 e e               ′′ ′ ′                                                 2 3 2 3 2 3 2 3 2 3 2 3 x x x x x x 1 2 1 4 e e e e e e + − + − + max 1 3 If 2 3 2 3 2 3 2 3 2 3 2 3 3 4 3 4 3 4 3 4 3 4 3 4 x x x x x x + + + + + + 1 1 1 1 1 1 e e e e e e 2 2 3                                           2 3 2 3 2 3 2 3 x x x x 3 2 1 8 48 − e e e e − − < 2 --------- (11) ( ) 2 3 2 3 2 3 2 3 3 3 4 3 4 3 4 3 4 b a x x x x + + + + 1 1 1 1 e e e e = = ( ) y a 0 ( ) y b Then (10) has super stability with boundary conditions [ , y c a b ∈ ] 3 Suppose that satisfies the inequality                         2 3 2 3 x x 3 2 e e ′′′ ′′ ′ ε > + + + ≤ ε 0 y y y y ,for some --------- (12) 2 3 2 3 3 4 3 4 x x + + 1 1 e e                         2 3 2 3 x x 3 2 e e ′′′ ′′ ′ = + + + y y y y Let v(x) ------- (13) 2 3 2 3 3 4 3 4 x x + + 1 1 e e [ ] ∈ , x a b and define ( ) z x by For all @ IJTSRD | Unique Paper ID – IJTSRD29149 | Volume – 3 | Issue – 6 | September - October 2019 Page 704

9. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 2 3 x x ∫ 1 2 e − τ d 2 3 3 4 x a + 1 e = ( ) y x ( ) z x e ----- (14) 2 3 x             x ∫             1 2 e − τ d 2 3 x 2 3 3 4 x 1 2 e a + 1 e ′ ′ = − y z z e 2 3 3 4 x + 1 e               ′ 2 3 2 x x ∫                                     1 2 e − 2 3 2 3 2 3 x x x 2 3 3 4 x 1 2 1 4 e e e a + 1 e ′′ ′′ ′ = − − + y z z z e 2 3 2 3 2 3 3 4 3 4 3 4 x x x + + + 1 1 1 e e e ' 2                                                       2 3 2 3 2 3 2 3 x x x x 1 2 1 2 e e e e ′′′ ′′′ ′′ ′′ ′ ′ = − − − + − y z z z z z 2 3 2 3 2 3 2 3 3 4 3 4 3 4 3 4 x x x x + + + + 1 1 1 1 e e e e ′ ′′ ′                                                 2 3 2 3 2 3 2 3 x x x x 1 2 1 2 1 4 e e e e ′ − + + z z z 2 3 2 3 2 3 2 3 3 4 3 4 3 4 3 4 x x x x + + + + 1 1 1 1 e e e e 2 3 2 3 x x ∫                                     1 2 e − τ d 2 3 2 3 2 3 x x x 2 3 3 4 x 1 4 1 2 1 8 e e e a + 1 e ) ′ + − z z z e 2 3 2 3 2 3 3 4 3 4 3 4 x x x + + + 1 1 1 e e e By substituting (14) and its first, second and third derivatives in (13) we get                                           2 3 2 3 x x 3 2 3 2 e e ′′′ ′′ + − + z z 2 3 2 3 3 4 3 4 x x + + 1 1 e e          ′     ′ 2                                           2 3 2 3 2 3 2 3 2 3 x x x x x 3 2 3 2 3 4 e e e e e − − + z 2 3 2 3 2 3 2 3 2 3 3 4 3 4 3 4 3 4 3 4 x x x x x + + + + + 1 1 1 1 1 e e e e e        ′′ ′ ′           −                                                   2 3 2 3 2 3 2 3 2 3 2 3 x x x x x x 1 2 1 4 e e e e e e + + + − 1 3 2 3 2 3 2 3 2 3 2 3 2 3 3 4 3 4 3 4 3 4 3 4 3 4 x x x x x x + + + + + + 1 1 1 1 1 1 e e e e e e 2 3 2 2 3 x x ∫                                           1 2 e − τ d 2 3 2 3 2 3 2 3 2 3 x x x x 3 4 x 3 2 1 8 e e e e a + 1 e ) + − − z e  2 2 3 2 3 2 3 2 3 3 4 3 4 3 4 3 4 x x x x + + + + 1 1 1 1 e e e e @ IJTSRD | Unique Paper ID – IJTSRD29149 | Volume – 3 | Issue – 6 | September - October 2019 Page 705

10. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 Since ' z and z′′                             ′′ ′ ′                                                 2 3 2 3 2 3 2 3 2 3 2 3 x x x x x x 1 2 1 4 e e e e e e ′′′ = + + − + − ( ) v x 1 3 z 2 3 2 3 2 3 2 3 2 3 2 3 3 4 3 4 3 4 3 4 3 4 3 4 x x x x x x + + + + + + 1 1 1 1 1 1 e e e e e e 2 3 2 2 3 x x ∫                                           1 2 e − τ d 2 3 2 3 2 3 2 3 2 3 x x x x 3 4 x 3 2 1 8 e e e e a + 1 e ) + − − z e  2 2 3 2 3 2 3 2 3 3 4 3 4 3 4 3 4 x x x x + + + + 1 1 1 1 e e e e Then from inequality (12) we get                      ′′ ′ ′                                                 2 3 2 3 2 3 2 3 2 3 2 3 x x x x x x 1 2 1 4 e e e e e e ′′′+ + − + − 1 3 z 2 3 2 3 2 3 2 3 2 3 2 3 3 4 3 4 3 4 3 4 3 4 3 4 x x x x x x + + + + + + 1 1 1 1 1 1 e e e e e e 2 2 3                                           2 3 2 3 2 3 2 3 x x x x 3 2 1 8 e e e e ) + − − 2 z 2 3 2 3 2 3 2 3 3 4 3 4 3 4 3 4 x x x x + + + + 1 1 1 1 e e e e                     2 3 2 3 x x x ∫ x ∫ 1 2 1 2 e e τ ε     = τ ≤ ( ) exp v x exp d d 2 3 2 3 3 4 3 4 x x + + a a 1 1 e e = = From the boundary condition ( ) 0 ( ) y b y a and             2 3 x x ∫ 1 2 e = − τ = = ( ) y x ( )exp z x d we have ( ) 0 ( ) z b z a 2 3 3 4 x + a 1 e Define:               ′′ ′ ′                                                 2 3 2 3 2 3 2 3 2 3 2 3 x x x x x x 1 2 1 4 e e e e e e + − + − β = 1 3 2 3 2 3 2 3 2 3 2 3 2 3 3 4 3 4 3 4 3 4 3 4 3 4 x x x x x x + + + + + + 1 1 1 1 1 1 e e e e e e 2 2 3                                           2 3 2 3 2 3 2 3 x x x x 3 2 1 8 e e e e + − − 2 2 3 2 3 2 3 2 3 3 4 3 4 3 4 3 4 x x x x + + + + 1 1 1 1 e e e e @ IJTSRD | Unique Paper ID – IJTSRD29149 | Volume – 3 | Issue – 6 | September - October 2019 Page 706

11. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470                     2 3 2 3 x x x ∫ x ∫ 1 2 1 2 e e ′′′+ ( ) z x β = τ ≤ τ ε     ( ) exp v x exp z d d Then 2 3 2 3 3 4 3 4 x x + + 1 1 e e a a using lemma (2) ( ) 3 − b a ≤ max ( ) max '''( ) z x z x 48 ( ) 3 b a − ′′′ ≤ + ( ) z x β + β     max max max ( ) z z x 48                       2 3 ( ) ( ) x 3 3 − − x ∫ b a b a 1 2 e ≤ τ ε + β max exp     max max ( ) d z x 2 3 48 48 3 4 x + a 1 e                    < ∞    2 3 x x ∫ 1 2 e on the interval [ ] τ , a b max exp     d since 2 3 3 4 x + a 1 e k > ≤ kε 0 such that ( ) z x Hence there exists a constants             2 3 x x ∫ 1 2 e = − τ ( ) y x ( )exp z x d 2 3 3 4 x + a 1 e                           2 3 x x ∫ 1 2 e max exp ≤ − τ ε ≤ ' ε d k k 2 3 3 4 x + a 1 e ≤ k ε ⇒ ' ( ) y x                         2 3 2 3 x x 3 2 e e ′′′ ′′ ′ + + + = 0 y y y y Then the differential equation 2 3 2 3 3 4 3 4 x x + + 1 1 e e on closed bounded interval [ ] = = , a b Has the super stability with boundary condition ( ) 0 ( ) y b y a                      ′′ ′ ′                                                 2 3 2 3 2 3 2 3 2 3 2 3 x x x x x x 1 2 1 4 e e e e e e + − + − + max 1 3 2 3 2 3 2 3 2 3 2 3 2 3 3 4 3 4 3 4 3 4 3 4 3 4 x x x x x x + + + + + + 1 1 1 1 1 1 e e e e e e To show the @ IJTSRD | Unique Paper ID – IJTSRD29149 | Volume – 3 | Issue – 6 | September - October 2019 Page 707

12. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 2 2 3                                           2 3 2 3 2 3 2 3 x x x x 3 2 1 8 48 − e e e e } − − < − − − − − − 2 (**) ( ) 2 3 2 3 2 3 2 3 3 3 4 3 4 3 4 3 4 b a x x x x + + + + 1 1 1 1 e e e e To make our work simple we simplify the expression (**) Then the simplified form of (**) is               2 3 4                      2 3 2 3 2 3 2 3 5 6 15 8 35 32 33 128 x x x x + + − e e e e 48 − + < max 1 ---------- (***) ( ) 4 3    b a 2 3 3 2 x + 1 e Figure:1               2 3 4                      2 3 2 3 2 3 2 3 5 6 15 8 35 32 33 128 x x x x + + − e e e e 48 − + < max 1 The graph which shows the ( ) 4 3    b a 2 3 3 2 x + 1 e Conclusion In this study, the super stability of third order linear ordinary differential homogeneous equation in the form of ( ) ( ) ( ) ( ) ( ) y x m x y x p x y x ′′′ ′′ + + boundary condition was established. And the standard work of JinghaoHuang, QusuayH.Alqifiary, Yongjin Li on investigating the super stability of second order linear ordinary differential homogeneous boundary condition is extended to the super stability of third order linear ordinary differential homogeneous equation with Dirchilet boundary condition. In this thesis the super stability of third order ordinary differential homogeneous equation was established using the same procedure coming researcher can extend for super stability of higher order differential homogeneous equation. ′ + = ( ) ( ) q x y x 0 with equation with @ IJTSRD | Unique Paper ID – IJTSRD29149 | Volume – 3 | Issue – 6 | September - October 2019 Page 708

13. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 [11]Polyak, B. T. and Shcherbakov, P. S.: Robastnaya ustojchivost I upravlenie (Robust Stability and control), Moscow: Nauka; 12 (2002). Reference [1]Gavrut’a P., JungS. Li. Y. “Hyers-Ulam stability for second- order linear differential Equations with boundary conditions, differential equations Vol.2011, No.80, (2011),pp.1-7. “Electronic Journal of [12]Polyak, B. T. and Shcherbakov, P.S.: Super stable Linear Control Systems. I. Analysis, Avton. Telemekh, no. 8 (2000) 37–53 [2]UlamS. M.: A collection of the Mathematical problems. Interscience publishers, NewYork1960. [13]Polyak, B. T. and Shcherbakov, P. S.: Super table Linear Control Systems II. Design, Avton. Telemekh; no. 11 (2002) 56–75. [3]Huang. J, Alqifiary. Q. H, Li. Y: On the super stability of deferential equations with boundary conditions Elect. J. of Diff.Eq;vol.2014(2014) 1-8 [14]RassiasTh. M, on the stability of the linear mapping in Banach spaces, pro.Amer. math.Soc. 72(1978), 297- 300. [4]Jung. S. M; Hyers-Ulam stability of linear differential equation of first order, Appl. Math. lett. 17(2004) 1135-1140. [15]Rus. I. A; remarks on Ulam stability of the operational equations, fixed point theory 10(2009), 305-320. [5]Li. Y, “Hyers-Ulam stability of linear differential equations, “Thai Journal of Mathematics, Vol8, No2, 2010, pp.215-219. [16]Rus. I. A; Ulam stability of differential equations, stud.univ. BabesBolyai Math.54 (2009), 125- 134. [6]Li. Y and Shen. Y, Hyers-Ulam stability of non- homogeneous linear differential equations of second order. Int. J. of math and math Sciences, 2009(2009), Article ID 576852, 7 pages. [17]Seshadev Padhi, and Smita Pati; Theory of Third- Order Differential Equations; Springer india , 2014. [18]Takahasi. S. E, Miura. T, Miyajima. S on the Hyers- Ulam stability of Banach space –valued differential equation ' y y λ = , Bull. Korean math. Soc.39 (2002), 309-315. [7]LaSalle J P., Stability theory of ordinary differential equations [J]. J Differential Equations; 4(1) (1968) 57–65. [19]Takahasi. S. E, takagi. H, Miura. T, Miyajima. S The Hyers-Ulam stability constants of first Order linear differential operators, J.Math.Anal.Appl.296 (2004), 403-409. [8]Miura. T, Miajima. S, Takahasi. S. E; Hyers-Ulam stability of linear differential operator with constant coefficients, Math. Nachr.258 (2003), 90-96. [9]Miura. Characterization of Hyers-Ulam stability of first order linear differential operators, J. Math. Anal. Appl. 286(2003), 136-146. T, Miyajima. S, Takahasi. S. E, a [20]UlamS. M.: A collection of the Mathematical problems. Interscience publishers,NewYork1960 [21]Ulam S.M.: Problems in Modern Mathematics, Chapter VI, Scince Editors, Wiley, New York, 1960 [10]Polyak, B., Sznaier, M., Halpern, M. and Scherbakov, P.: Super stable Control Systems, Proc. 15th IFAC World Congress. Barcelona, Spain; (2002) 799–804. @ IJTSRD | Unique Paper ID – IJTSRD29149 | Volume – 3 | Issue – 6 | September - October 2019 Page 709