Absolute monotonicity of rational functions occurring in the numerical solution of initial value problems

1986 ◽  
Vol 49 (4) ◽  
pp. 413-424 ◽  
Author(s):  
J. A. van de Griend ◽  
J. F. B. M. Kraaijevanger
Keyword(s):  
Numerical Solution ◽  
Initial Value Problems ◽  
Rational Functions ◽  
Initial Value ◽  
Absolute Monotonicity

scholarly journals Rational functions with maximal radius of absolute monotonicity

2014 ◽  
Vol 17 (1) ◽  
pp. 159-205 ◽  
Author(s):  
Lajos Lóczi ◽  
David I. Ketcheson
Keyword(s):  
Initial Value Problems ◽  
Rational Functions ◽  
Strong Stability ◽  
Strong Stability Preserving ◽  
Algebraic Numbers ◽  
Initial Value ◽  
Stage Order ◽  
Absolute Monotonicity ◽  
Runge Kutta ◽  
Radius Of Absolute Monotonicity

AbstractWe study the radius of absolute monotonicity $R$ of rational functions with numerator and denominator of degree $s$ that approximate the exponential function to order $p$. Such functions arise in the application of implicit $s$-stage, order $p$ Runge–Kutta methods for initial value problems, and the radius of absolute monotonicity governs the numerical preservation of properties like positivity and maximum-norm contractivity. We construct a function with $p=2$ and $R>2s$, disproving a conjecture of van de Griend and Kraaijevanger. We determine the maximum attainable radius for functions in several one-parameter families of rational functions. Moreover, we prove earlier conjectured optimal radii in some families with two or three parameters via uniqueness arguments for systems of polynomial inequalities. Our results also prove the optimality of some strong stability preserving implicit and singly diagonally implicit Runge–Kutta methods. Whereas previous results in this area were primarily numerical, we give all constants as exact algebraic numbers.

scholarly journals A New Algorithm for Solving Terminal Value Problems of q-Difference Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Yong-Hong Fan ◽  
Lin-Lin Wang
Keyword(s):  
Exact Solution ◽  
Numerical Methods ◽  
Error Estimate ◽  
Numerical Solution ◽  
Numerical Method ◽  
Difference Equations ◽  
Initial Value Problems ◽  
Convergence Speed ◽  
Initial Value ◽  
Delta Derivatives

We propose a new algorithm for solving the terminal value problems on a q-difference equations. Through some transformations, the terminal value problems which contain the first- and second-order delta-derivatives have been changed into the corresponding initial value problems; then with the help of the methods developed by Liu and H. Jafari, the numerical solution has been obtained and the error estimate has also been considered for the terminal value problems. Some examples are given to illustrate the accuracy of the numerical methods we proposed. By comparing the exact solution with the numerical solution, we find that the convergence speed of this numerical method is very fast.

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