The paper deals with stiff systems of differential equations. To solve this sort of system numerically is a diffult task. In spite of the fact that we come across stiff systems quite often in the common practice, a very interesting and promissing numerical method of solving systems of ordinary differential equations (ODE) based on Taylor series has appeared. The question was how to harness the ...

Variable-step (VS) second derivative $k$-step $3$-stage Hermite--Birkhoff--Obrechkoff (HBO) methods of order $p=(k+3)$, denoted by HBO$(p)$ are constructed as a combination of linear $k$-step methods of order $(p-2)$ and a second derivative two-step diagonally implicit $3$-stage Hermite--Birkhoff method of order 5 (DIHB5) for solving stiff ordinary differential equations. The main reason for co...

We study accelerated mirror descent dynamics in continuous and discrete time. Combining the original continuous-time motivation of mirror descent with a recent ODE interpretation of Nesterov’s accelerated method, we propose a family of continuous-time descent dynamics for convex functions with Lipschitz gradients, such that the solution trajectories converge to the optimum at a O(1/t2) rate. We...

We give a necessary and sufficient condition for an algebraic ODE to have a rational type general solution. For a first order autonomous ODE F = 0, we give an exact degree bound for its rational solutions, based on the connection between rational solutions of F = 0 and rational parameterizations of the plane algebraic curve defined by F = 0. For a first order autonomous ODE, we further give a p...

This paper deals with the problem of observers design for implicit state affine systems for which the observer synthesis depends on the input excitation. Two approaches can be used to solve this problem; The first one consists in designing an observer which requires solving at each time an algebraic equation combined with an ODE system (implicit observer). The second approach proposed in this p...

The novelty of this paper is to propose a numerical method for solving ordinary differential equations the first order that include both linear and nonlinear terms (ODEs). constructed in two stages, which may be called predictor corrector stages. stage uses dependent variable’s first- second-order derivative given equation. In literature, most predictor–corrector schemes utilize first-order var...

We present some variants of Burgers-type equations for incompressible and isothermal planar flow of viscous non-Newtonian fluids based on the Cross, the Carreau and the power-law rheology models, and on a symmetry assumption on the flow. We numerically solve the associated traveling wave equations by using industrial data and in order to validate the models we prove existence and uniqueness of ...

We suggest an approach for description of integrable cases of the Abel equations using the procedure of increasing the order and equivalence transformations for the induced secondorder equations. A diversity of methods were developed to date for finding solutions of nonlinear ordinary differential equations (ODE). Everybody who encounters integration of a particular ODE uses, as a rule, the acc...