On the convergence of multistep collocation methods for integral-algebraic equations of index 1

Abstract

The multistep collocation method is introduced to solve integral-algebraic equations of index 1. The existence and uniqueness of the multistep collocation solution are proved. The convergence of the perturbed multistep collocation method is also investigated, which extends and includes the analysis of the multistep collocation method without perturbed terms. Some numerical experiments are given to illustrate the theoretical results. Keywords Integral-algebraic equations • Index 1 • Multistep collocation method • Existence and uniqueness • Convergence Mathematics Subject Classification 65R20 1 Introduction Volterra integral equations (VIEs) have rich applications in physics, biology, chemistry, etc. Generally, VIEs are divided into two classes: first-kind VIEs and second-kind VIEs. For first-kind VIEs, the unknown function only appears inside the integral, but for second-kind Communicated by Antonio José Silva Neto.

‎An effective computational approach based on‎ ‎new multi-step collocation method for solving integral-algebraic equations‎

“…Noting that ρ(W ) < 1, and ρ( Ã A A) < 1, then ρ(A A A) < 1, and by this recurrence relation for E E E 2,n+1 and a bound for E E E 2,n+1 can be obtained by the same way as described in [11,12,31,32] and this completes the proof.…”

“…Throughout this section, for all numerical experiments, we set T = 1 within the interval I := [0, T ], and the initial values were derived from the known exact solutions. The comparison of the obtained numerical results in Tables 1, 2, 3 and 4 shows that new multi-step method is more accurate than the one-step and multi-step methods used in [18,31]. Also in Figs.…”

“…We have demonstrated that the novel multi-step collocation technique provides an effective and highly precise numerical approach for approximating solutions to IAEs. The proper management of global convergence was ensured with the proposed scheme, where we observed that the collocation parameter and the c 1 = 0.7, c 2 = 1 c 1 = 0.7, c 2 = 1 c 1 = 0.7, c 2 = 1 N One-step method [18] Multi-step method [31] New multi-step method part of the text about the convergence analysis of the method and edited the contents. All authors discussed the results and revised the draft.…”

“…It's worth noting that, according to [31] (Theorem 2) and [32] (Lemma 2.1), it is established that the matrix A 1,n+1,h , is invertible and it's inverse is uniformly bounded. Also, we have…”

Stability and Convergence of the Canonical Euler Splitting Method for Nonlinear Composite Stiff Functional Differential-Algebraic Equations

“…In recent years, there has been extensively studied on the numerical stability and convergence of delay differential-algebraic equations [4][5][6][7][8][9][10][11][12][13][14][15]. Further, we can refer to [16][17][18][19] for details on the numerical stability and convergence of integral differential-algebraic equations. However, most of the above studies are focused on theoretical and numerical analysis of linear or non-stiff problems, we can refer to [20][21][22] for the stability of the more general nonlinear stiff FDAEs and theirs numerical methods.…”