Procedure for Exact Solutions of Sti Ordinary Dierential Equations Systems

Abstract

In this work, we present a technique for the analytical solution of systems of stiff ordinary differential equations (SODEs) using the power series method (PSM). Three SODEs systems are solved to show that PSM can find analytical solutions of SODEs systems in convergent series form. Additionally, we propose a post-treatment of the power series solutions with the Laplace-Padé (LP) resummation method as a powerful technique to find exact solutions. The proposed method gives a simple procedure based on a few straightforward steps.

Approximate Solutions for Flow with a Stretching Boundary due to Partial Slip

“…Nevertheless, sometimes, this type of solutions lacks large domains of convergence. Therefore, Laplace-Padé resummation method [ 54 ] is used in literature to enlarge the domain of convergence of solutions or inclusive to find exact solutions.…”

“…Let u ( t ) be an analytical function with the Maclaurin's expansion Then the Padé approximant to u ( t ) of order [ L , M ] which we denote by [ L / M ] u ( t ) is defined by [ 54 – 57 ] where we considered q 0 = 1, and the numerator and denominator have no common factors.…”

“…To guarantee the validity of the approximate solution ( 35 ) for large values of this variable, the series solution is transformed using the Padé approximation and Laplace transform (see Section 3 ). As first step, Laplace transform [ 54 ] is applied to ( 35 ): and then, s is substituted by 1/ x in the equation to obtain …”

Algebraic-based nonstandard time-stepping schemes

“…Step 3 -Integral expression of the numerical scheme Replacing in (1) the timederivative operator by (14) gives an integral expression of the proposed E-A-NSFD scheme:…”

“…[3,4,5,6,7,8,9,10]), including the solver routines 'ode45' and 'ode23s' [11,12,13]. Recent alternative methods propose for instance to use specific series, like power series [14] or Haar wavelets [15,16] to describe the solution of ODEs; the purpose is to substitute the standard finite difference by the numerical properties of the series. In this paper, we propose to extend the finite difference technique by an estimation of the derivatives using the algebraic estimation approach.…”