Error estimates for series solutions to a class of nonlinear integral equations of mixed type

Abstract

In this paper, we prove that the accelerated Adomian polynomials formula suggested by Adomian (Nonlinear Stochastic Systems: Theory and Applications to Physics, Kluwer, Dordrecht, 1989) and the accelerated formula suggested by El-Kalla (Int. J. Differ. Equs. Appl. 10(2):225-234, 2005; Appl. Math. E-Notes 7: [214][215][216][217][218][219][220][221] 2007) are identically the same. The Kalla-iterates exhibit the same faster convergence exhibited by Adomian's accelerated iterates with the additional advantage of absence of any derivative terms in the recursion, thereby allowing for ease of computation. Moreover, the formula of El-Kalla is used directly to prove the convergence of the series solution to a class of nonlinear two dimensional integral equations. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of the Adomian series solution.

Approximate Solution of Urysohn Integral Equations Using the Adomian Decomposition Method

“…In the recent past, a lot of researchers [ 9 – 16 ] have expressed their interest in the study of ADM for various scientific models. Adomian [ 12 ] asserted that the ADM provides an efficient and computationally worthy method for generating approximate series solution for a large class of differential as well as integral equations.…”

Approximate Series Solution of Nonlinear Singular Boundary Value Problems Arising in Physiology

“…In 2010, Duan [25, 26] reported several new efficient algorithms for rapid computer generation of the Adomian polynomials. Recently, El-Kalla [27] suggested another programmable formula for Adomian polynomials: $\begin{array}{c}{A}_n=f\left(x,{\psi }_n\right)-\underset{j=\mathrm{normal0}}{\overset{n-\mathrm{normal1}}{{\displaystyle false\sum }}}{A}_j\hspace{1em}\text{or}\hspace{0.17em}\hspace{0.17em}f\left(x,{\psi }_n\right)=\underset{j=\mathrm{normal0}}{\overset{n}{{\displaystyle false\sum }}}{A}_j,\end{array}$ where ψ n = ∑ j =0 n y j is partial sum of the series solution ∑ j =0 ∞ y j .…”

Piece-wise continuous solution to a class of nonlinear boundary value problems

“…In this case it is difficult to integrate so, recursive (19) and (20) will be used in stead of (12) and (15). Using recursive (19) and (20), we can improve the accuracy of the solution not only by adding more terms to the solution series but also by increasing N in the recursive relation.…”

“…Using recursive (19) and (20), we can improve the accuracy of the solution not only by adding more terms to the solution series but also by increasing N in the recursive relation. Table 5 shows the relative absolute error (RAE) for different values of x at N = 3 and N = 5 for the same partial sum S 5 .…”

“…Formula (6) has the advantage of absence of any derivative terms in the recursion, thereby allowing for ease of computations. Also, Formula (6) was used successfully in one dimensional problems [16,17] and in two dimensional problems [18,19] to study the convergence of ADM when applied to some classes of nonlinear equations. In this work, formula (6) is used in the convergence analysis and all the calculations of the numerical examples.…”