Numerical Stability Of Some Neutral Delay Differential Equations

In this paper, there are four chapters.Chapter 1: the delay differential equations are introduced in the history of researchand the main type of this equations.Chapter 2:we focus our attention on the stability of numerical methods for the linear pantograph equation.The linearθmethods and Runge-Kutta method are applied to the system,respectively.lt is proved that the numerical method preserve the delay pantograph equation stability of system if the linearθmethod satisfiesθ∈(1/2,1] and the Runge-Kutta method is stable.We consider the pantograph equation:Against the expansion of the equation with the delay of q, introduces a variable variable stepsize schemes.t_n = q^-1t_n-m,qh_n=h_n-m,(?)h_n=∞(2)Assume the existence and uniqueness of solutions to (1) has been done,and the system is asymptotically stable under some conditions.The linearθmethod to (1) give out the recurrence relation:y_n+1=y_n+h_n+1(aθy_n+1+a(1-θ)y_n + bθy_n-m+1 + b(1-θ)y_n-m(3) where y_n is an approximation to y(t_n),0 <θ< 1,and then give out the stability theorem,correspondly:Theorem 1 ([5]). Aθmethod, as applied to Eq. (1), is asymptotically stable if and only 1/2 <θ≤1.Furthermore show some researches on the stability of Runge-Kutta methods for the neutral pantograph equationsy'(t) =f(t,y(t),y(pt),y'(qt)),t>t₀ (4)where p, q∈(0,1),f : [t₀, +∞)×C^d + C^d×C^d→C^d is given such that the system(4) satisfies the existence and uniqueness of the solutions and y(t₀) = y₀∈C^d.Since the delay parameters p and q in system (4) may be different, we consider the Runge-Kutta methods in two cases:Case 1: 0≤q < p < 1There must exist l∈L⁺,such thatp^l+1≤q≤p'.Using the delay parameter p, we quote the variable stepsize as (2).Do the approximation of y'(qt) withδ₁, the s-stage Runge-Kutta method give out the recurrence relation:where t_n,i=t_n+c_ih_n+1,y_n,i=y_n+h_n+1(?)a_i,jK_n,j,η_n,i =δ₁y_n-m+1+(1-δ₁)y_n-mη'_n,i=δ₁k_n-lm+1,i+(1-δ₁)K_n-(l+1)m,iCase 2: 0 < p < q < 1Similar to the above analysis in case 1 ,the variable stepsize is defined by the delay parameter q. Do the approximation of y'(pt) withδ₂, get the same s-stage Runge-Kutta method with Eq.(5) whereη_n,i=δ₂(δ₂y_n-km+1+(1-δ₂)y_n-km) + (1 -δ₂)(δ₂yn-(k+1)m+1+(1-δ₂)yn-(k+1)m)η_n,i^'=K_n-m,iUsing the two recurrence relations respectively,we consider the linear neutral pantograph equationsy' (0 = Ly(t) + My(pt) + Ny' (qt) t>0 (6)There are stability theorems,respectively.Theorem 2 ([28]). Assume that the matrix I_sd - h₍n+1(A (?) L)is nonsingular, then the Runge-Kutta methods (A, b, c) applied to the neutral pantograph equation (6) with 0 < q≤p < 1 for any initial value, are asymptotically stable if|r_∞|<1Theorem 3 ([28]). Assume that the matrix I_sd - h_n+1(A (?) L)is nonsingular, then the Runge-Kutta methods (A, b, c) applied to the neutral pantograph equation (6) with 0 < p < q < 1 for any initial value, are asymptotically stable if|r_∞|<1Chapter 3: In this section,we consider the asymptotic stability of linear constant coefficient delay differential-algebraic equations and ofθmethods,Runge-Kutta meth-ods,linear multistep methods and Rosenbrock methods applied to these systems.the initial problem of linear delay differential-algebraic equations can be write aswhere A,B, C,D∈R^d×d, assume that the coefficient matrices can be transformed to triangular matrices simultaneously. When the diagonal elements of the corresponding matrices {a_i}, {b_i}, {c_i}, {d_i} meet some certain conditions,We have the proposition 3.1 to ensure the asymptotic stability of (7),where S_R is the stability region.For the general multistep methods < p, a > , we have the theorem as follow.Theorem 4 ([60]). If DDAEs system (10) satisfies the conditions of Proposition 3.1 and also the condition that-h b_i+d_iz/a_i+c_iz∈S_R,|z|≥1then if the multistep method satisfies that (?)β_jz^j is a Schur polynomial, the solution ofthe multistep method is asymptotically stable.Furthermore,it show that every linearθmethod withθ∈(1/2,1] for the DDAEs is asymptotically stable under some certain conditions.Using the s-stage Runge-Kutta method and giving out the theorem of numerical staby:Theorem 5 ([60]). For linear systems (10) which satisfy', the conditions of Proposition3.1 and also the condition-h b_i+d_iz^-m/a_i+c_iz^-m∈S_R,|z|≥1for a strictly stable Runge-Kutta method for which all the eigenvalues of its coefficient matrix Ahave positive real part, which has stability region S_R the numerical solution is asymptotically stable.Rosenbrock method is a special kind of semi-implicit Runge-Kutta method.Using this kind of method we can get the similar theorem with the Rung-Kutta method.Chapter 4:This section is concerned with the stability of numerical methods for linear neutral Volterra delay-integro-differential system.Asucientconditionsuchthatthesystem isasymptoticallystableisderived.Furthermore,it is proved that every linearθmethod withθ∈(1/2,1] and A-stable BDF method preserve the delay-independent stability of its exact solutions. where A, B, C,D,G∈R^d×,Ï„> 0,A is singular.Theorem 6 ([27]). Assume the matrix A is singular,then the linearθmethod is stable for the neutral delay-integro-differential system (8) ifθ∈(1/2,1] for the step size satisfying h=Ï„/m or h =Ï„/m-θ,m∈L⁺.where m is the smallest integer which is no less thenÏ„/hFor the BDF method,we first give out the definition of the A stability of BDF method, every A-stable BDF method is stable for the Volterra delay-integro-differential system (8) with some hypothetical conditions.