Polynomial Formulation of Second Derivative Multistep Methods
نویسندگان
چکیده
Following the work of Enright [3] there has been interest in studying second derivative methods for solving stiff ordinary differential equations. Successful implementations of second derivative methods have been reported by Enright [3], Sacks-Davis [9], [10] and Addison[l]. Wallace and Gupta [13] have suggested a polynomial formulation of the usual first-derivative multistep methods. Recently Skeel [11] has shown the equivalence of several formulations of multistep methods. The work of Wallace and Gupta [13] was extended to second derivative methods by Gupta [8]. The present work includes results obtained regarding the stability and truncation error of second derivative methods using the polynomial formulation.
منابع مشابه
Grid-independent Construction of Multistep Methods
A new polynomial formulation of variable step size linear multistep methods is presented, where each k-step method is characterized by a fixed set of k− 1 or k parameters. This construction includes all methods of maximal order (p = k for stiff, and p = k+1 for nonstiff problems). Supporting time step adaptivity by construction, the new formulation is not based on extending classical fixed step...
متن کاملOn the $s^{th}$ derivative of a polynomial
For every $1leq s< n$, the $s^{th}$ derivative of a polynomial $P(z)$ of degree $n$ is a polynomial $P^{(s)}(z)$ whose degree is $(n-s)$. This paper presents a result which gives generalizations of some inequalities regarding the $s^{th}$ derivative of a polynomial having zeros outside a circle. Besides, our result gives interesting refinements of some well-known results.
متن کاملOn the polar derivative of a polynomial
For a polynomial p(z) of degree n, having all zeros in |z|< k, k< 1, Dewan et al [K. K. Dewan, N. Singh and A. Mir, Extension of some polynomial inequalities to the polar derivative, J. Math. Anal. Appl. 352 (2009) 807-815] obtained inequality between the polar derivative of p(z) and maximum modulus of p(z). In this paper we improve and extend the above inequality. Our result generalizes certai...
متن کاملInverse Linear Multistep Methods for the Numerical Solution of Initial Value Problems of Ordinary Differential Equations
The well-known explicit linear multistep methods for the numerical solution of ordinary differential equations advance the numerical solution from xn+k_x to xn+k ky comPut'ng some numerical approximation from back values and then evaluating the problem defining function to obtain an approximation of the derivative. In this paper similar methods are proposed that first compute an approximation t...
متن کاملOn the $s^{th}$ Derivative of a Polynomial-II
The paper presents an $L^{r}-$ analogue of an inequality regarding the $s^{th}$ derivative of a polynomial having zeros outside a circle of arbitrary radius but greater or equal to one. Our result provides improvements and generalizations of some well-known polynomial inequalities.
متن کامل