LAMBERTS W FUNCTION APPROACH ONTHE STABILITY ANALYSIS OF ONE DIMENSIONAL WAVE EQUATION VIA SECOND ORDER NEUTRAL DELAY DIFFERENTIAL EQUATION PJAEE, 17 (7) (2020) LAMBERTS W FUNCTION APPROACH ONTHE STABILITY ANALYSIS OF ONE DIMENSIONAL WAVE EQUATION VIA SECOND ORDER NEUTRAL DELAY DIFFERENTIAL EQUATION D. Piriadarshani1*, K. Sasikala2, Beena JAMES3 1*Department of Mathematics,Hindustan Institute of Technology and Science, Chennai,India. 2Department of Mathematics, Hindustan Institute of Technology and Science, Hindustan College of Arts and Science, Chennai, India. 3Department of Mathematics, Hindustan Institute of Technology and Science, Chennai, India. 1*[email protected],[email protected],3beenaj@hindustanuniv. ac.in D. Piriadarshani, K. Sasikala, Beena JAMES.Lamberts W Function Approach on the Stability Analysis of One Dimensional Wave Equation Via Second Order Neutral Delay Differential Equation-- Palarch’s Journal of Archaeology of Egypt/Egyptology 17(7), 4781-4790. ISSN 1567-214x Keywords: Neutral Delay Differential Equation, Lambert W Function, Nyquist Plot, Newton Raphson Method, Secant Method. ABSTRACT: In this article we have analyzed the stability of second order neutral delay differential equation of anone dimensional wave equation by means of the right most roots of its characteristic equation. The stability is referred by maximal real part of the characteristic root which shows the better perceptive of the model performance. Since Neutral delay differential equation has infinite number of roots with an open structure which can be calculated by numerical methods. This article estimates theleast significant roots of second order neutral delay differential equations by an efficient iterative methodand on the basis of Lambert W function stabilization was determined, whichis illustrated by a wave equation. INTRODUCTION With the advancement of the vigorous control practices, dynamical systems with time delays have given much importance over the earlier periods. Here the time delay is produced by the controllers, actuators, human–machine interaction, etc.[1-4] ,which creates the systemto beunstable. Many engineering models based on the delay differential equations of retarded type lack a time lags for the derivative of highest order but for neutral type, time 4781 LAMBERTS W FUNCTION APPROACH ONTHE STABILITY ANALYSIS OF ONE DIMENSIONAL WAVE EQUATION VIA SECOND ORDER NEUTRAL DELAY DIFFERENTIAL EQUATION PJAEE, 17 (7) (2020) lags exist inthe highest order of the derivative. The stability analysis is effectively significant in the dynamics and controllability of NDDE, and it is investigate by the Lyapunov’s technique which includes the linear matrix inequality (LMI) method and position of zeros in the characteristic equations of quasi-polynomials. Both methods have its own advantage. The Lyapunov’s methods give conservative outcome which operate on locally or globally in the region of equilibrium. Whereas the other carries out locally in the region of the equilibrium solitarily, it is used in the stability analysis. The abscissa is the maximum value of the real part of the characteristic roots of neutral delay system, it can be either a real root or a pair of complex conjugate characteristic root of a Quasi polynomial, which is also known as rightmost root. The delay system is asymptotically stable ifand only if the abscissa has negativereal part. Numerous methods are presented to check the sign of abscissa, which includes Pontryakin criterion, the Hassard criterion and the Nyquist criterion [4]. The stabilized or destabilized of the model is possible by regulating the delay. It is valuable to remind that; the stability can be unsatisfactory for few interval delays where the model isasymptotically stable [1]. Therefore when applying real model, both stability margin and asymptoticallystable should be considered.Stability Criteria of Networked Control System Using Lambert W Function [5] isanalyzed for second order DDE. In [6], an iterative method was planned for computing the rightmost root of an RDDE with proper initial guess, but this iterative sequence may not be convergent in all places. Recently, the author shows evidence when applied for Newton-Raphson or the Halley method with Lamberts W function to find the rightmost root of an NDDE. In this article we derive the stability test of second order Neutral delay differential equation ′′ ′′ ′ ′ 푛 푦 푡 − 퐶푦 푡 − 휏 + 푝1푦 푡 + 푝2푦 푡 − 휏 + 푞1푦 푡 + 푞2푦 푥 − 휏 , 푦 ∈ ℝ 푦 푡 = ∅ 푡 , −휏 ≤ 푡 ≤ 0. (1) And its characteristic equations can be written as 2 2 −푠휏 −푠휏 −푠휏 ∆ 푠 = 푠 − 퐶푠 푒 + 푝1푠 + 푝2푠푒 + 푞1 + 푞2푒 −푠휏 2 −푠휏 −푠휏 ∆ 푠 = 1 − 퐶푒 푠 + 푝1 + 푝2푒 푠 + 푞1 + 푞2푒 = 0 (2) Where the coefficients pj(z),qj(z)are polynomials in z andC is a constant. The solution is asymptoticallytable at C= 0,when 훥(휆)<max 푅푒푠: ∆ 푠 = 0, 푠 ∈ ℂ = 훼, (3) Δ(s) has the infinite roots at|C| = 1 and has positive real roots at |C|>1 having no negative real part,consequently becomes unstable. Hence |C| <1 is considered in this article, as Δ (s) has negative real roots. This paper enables to evaluate the rightmost root of RDDEs which analyses thestability of NDDEs of second order by an the iterative formula, which is depicted in wave equation. 4782 LAMBERTS W FUNCTION APPROACH ONTHE STABILITY ANALYSIS OF ONE DIMENSIONAL WAVE EQUATION VIA SECOND ORDER NEUTRAL DELAY DIFFERENTIAL EQUATION PJAEE, 17 (7) (2020) Calculation of the Rightmost Characteristic Root During this segment, we trace the rightmost root of Δ (s) by applying the Newton-Raphson method and Secant method along with Lambert W function. Second-order RDDEStabilization Criteria Second order delay differential equation of retarded type is depictedas ′′ ′ 푦 푡 +푝1푦 푡 +푞1푦 푡 − 푞2푦 푥 − 휏 = 0 (4) The resultant characteristic equation of equation (4) is 2 −푠휏 ∆ 푠 = 푠 + 푝1 푠 + 푞1 − 푞2푒 = 0 (5) Lambert function in general is multi-valued a complex function, which has an infinite number of branches, denoted as Wk(s), where the index k takes the values푘 = −∞, … , −2, −1,0,1, … , +∞. The fundamental branch of the Lambert function, which is denoted as W0(s) where index k=0. The fundamental branch takes real values on the interval [-1/e,+∞), while it is complex outside this interval. 푝 2 Now, if훥(푠)= 0, 푞 = 1 ≜ 훽2then (푠 + 훽)2 = 푞 푒−푠휏 , 1 4 2 2 푠휏 (푠 + 훽) 푒 = 푞2 푠휏 (푠 + 훽)푒 2 = ± 푞2 푠휏 (푠 + 훽)휏푒 2 = ±휏 푞2 (푠 + 훽) 푠휏 훽휏 훽휏 휏푒 2 푒 2 = ±휏 푞 푒 2 2 2 By the definition of the Lambert W function 푊(퐻) 푒푊(퐻)= H, z ∈ C (6) 훽휏 푠 + 훽 휏 푞2푒 2 휏 = 푊 ± 2 2 Therefore, the roots of the characteristic equationisdemonstrated by means of Lambert W function 훽휏 2 휏 푞2푒 2 푠 = 푊 ± − 훽 휏 2 훽휏 2 휏 푞 푒 2 푠 = 푊 ± 2 − 훽,푘 = 0, ±1, ±2, ±3 … …. (7) 푘 휏 2 By the resultmax푘=0,±1,±2,..푅푒 푊푘 < 푅푒푊0(8) demonstrated in [10]. Thusthesolution y=0 of (2.1) is asymptotically stable if and only if the 훽휏 휏 푞 푒 2 휏Re 훽 rightmost root Re휆 <0, 푅푒푊 ( 2 )< (9) 0 0 2 2 4783 LAMBERTS W FUNCTION APPROACH ONTHE STABILITY ANALYSIS OF ONE DIMENSIONAL WAVE EQUATION VIA SECOND ORDER NEUTRAL DELAY DIFFERENTIAL EQUATION PJAEE, 17 (7) (2020) Hence using this formula, stability can be derived by applying Matlab,Maple and Mathematica software. Construction of NumericalAlgorithm for Neutral Delay Differential Equations An iterative algorithm is been projected in[12] for evaluating the rightmost root of RDDEs and its characteristic equation is given by 2 −푠휏 −푠휏 ∆ 푠 = 푠 + 푝1푠 + 푝2푠푒 + 푞1 + 푞2푒 (10) Where the coefficients푝1, 푝2.푞1푎푛푑푞2are polynomials in y. The most iterative method is reviewed as given below. To solve the NDDE we can choose any complex number푠0 as initial guess,with the푝푖’s and푞푖’s value are considered to be some fixed constant. 푝1푠+푞1 Let us describe퐹 푠 = 푝1푠 + 푞1 − 푊0 푝1푠 + 푞1 − ∆ 푠 푒 (11) 2 −푠휏 −푠휏 푝1푠+푞1 = 푝1푠 + 푞1 − 푊0 −푠 − 푝2푠푒 − 푞2푒 푒 Here푊0(푧)is the Lambert W function value on its principal branch, and numerically small values are considered as 푝1,푝2,푞1.and푞2.. Then we evaluate the least significant root of훥 푠 = 0 by applyingNewton-Raphson technique for unique roots and also convergequadratic ally. 퐹(푠푛 ) 푠푛+1=푠푛 − (푛 = 0,1,2, … . ) (12) 퐹′(푠푛 ) Secant method has linear convergence with the two initial guesses 퐹 푠푛 (푠푛 −푠푛 −1) 푠푛+1=푠푛 − (13) (퐹 푠푛 −퐹 푠푛 −1 ) The process of finding iterative procedure will be stopped when 푠푛 − 푠푛−1 < 휀, where ε is negligible. (14) VerificationUsing Graphical Test The right most roots of (8), Δ(s) = 0 is obtained from (12) or (13)and hasRes ≤ ResN. n The Nyquist plotΔ(iω + ResN )(1 + iω) of(14)shows the stability if itsurpasses the origin of the complex plane and unstable when it does not touch the origin when deviated by small μ > 0 forΔ(iω + ResN + μ)/(1 + iω) n. Similar results can be obtained by QPmR root finder method. The Wave Equation Model Let the performance of torsion with torque as a control input enforced in one node of a stretchy rod and other node is fastened by amass and its reaction to the torque is traced. The deterministic model oftorsional behavior of a flexible rod 2 휎 푢푡푡 푡, 푥 = 푢푥푥 푡, 푥 ,푢푥 푡, 0 = −푏 푡 , 푢푥 푡, 퐿 = −퐽푢푡푡 푡, 퐿 , 푢 0, 푥 = 푢0 푥 , 푢푡 0, 푥 = 푢1 푥 . Here 푢(푥, 푡)is the angular displacement of the rod from an unexcited point, L is the measurement lengthwise of a rod, 푦(푡) = 푢(푡, 퐿), 퐽is the moment of inertia of the weight, 휎is wave propagation speed’s inverse, and 푏(푡)is the control torque.푢0 푥 ,푢1 푥 depicted by the initial angular displacement and velocity, respectively.