In this paper, a Jacobi spectral Galerkin method is developed for nonlinear Volterra integral equations (VIEs) of the second kind. The spectral rate of convergence for the proposed method is established in the $L∞$-norm and the weighted L2-norm. Global superconvergence properties are discussed by iterated Galerkin methods. Numerical results are presented to demonstrate the effectiveness of the proposed method.

## References

References
1.
Darania
,
P.
,
,
E.
, and
Oskoi
,
A. V.
,
2006
, “
Linearization Method for Solving Nonlinear Integral Equations
,”
Math. Probl. Eng.
,
2006
, p.
73714
.
2.
Darania
,
P.
, and
,
E.
,
2007
, “
A Method for the Numerical Solution of the Integro-Differential Equations
,”
Appl. Math. Comput.
,
188
(
1
), pp.
657
668
.
3.
Darania
,
P.
, and
,
M.
,
2006
, “
On the RF-Pair Operations for the Exact Solution of Some Classes of Nonlinear Volterra Integral Equations
,”
Math. Probl. Eng.
,
2006
, p.
97020
.
4.
Tang
,
T.
,
McKee
,
S.
, and
Diogo
,
T.
,
1992
, “
Product Integration Method for an Integral Equation With Logarithmic Singular Kernel
,”
Appl. Numer. Math.
,
9
, pp.
259
266
.
5.
Diogo
,
A. T.
,
McKee
,
S.
, and
Tang
,
T.
,
1991
, “
A Hermite-Type Collocation Method for the Solution of an Integral Equation With a Certain Weakly Singular Kernel
,”
IMA J. Numer. Anal.
,
11
(
4
), pp.
595
605
.
6.
Wazwaz
,
A. M.
, and
El-Sayed
,
S. M.
,
2001
, “
A New Modification of the Adomian Decomposition Method for Linear and Nonlinear Operators
,”
Appl. Math. Comput.
,
122
(
3
), pp.
393
404
.
7.
Darania
,
P.
, and
Ivaz
,
K.
,
2008
, “
Numerical Solution of Nonlinear Volterra–Fredholm Integro-Differential Equations
,”
Comput. Math. Appl.
,
56
(
9
), pp.
2197
2209
.
8.
Brunner
,
H.
,
2004
,
Collocation Methods for Volterra Integral and Related Functional Equations
,
Cambridge University Press
,
Cambridge, UK
.
9.
Fujiwara
,
H.
,
2003
, “
High-Accurate Numerical Method for Integral Equations of the First Kind Under Multiple-Precision Arithmetic
,”
Theor. Appl. Mech. Jpn.
,
52
, pp.
193
203
.
10.
Tang
,
T.
,
Xu
,
X.
, and
Cheng
,
J.
,
2008
, “
On Spectral Methods for Volterra Integral Equation and the Convergence Analysis
,”
J. Comput. Math.
,
26
(
6
), pp.
825
837
.
11.
Xie
,
Z.
,
Li
,
X.
, and
Tang
,
T.
,
2012
, “
Convergence Analysis of Spectral Galerkin Methods for Volterra Type Integral Equations
,”
J. Sci. Comput.
,
53
(
2
), pp.
414
434
.
12.
Yang
,
Y.
,
2016
, “
Jacobi Spectral Galerkin Methods for Volterra Integral Equations With Weakly Singular Kernel
,”
Bull. Korean Math. Soc.
,
53
(
1
), pp.
247
262
.
13.
Yang
,
Y.
,
2015
, “
Jacobi Spectral Galerkin Methods for Fractional Integro-Differential Equations
,”
Calcolo
,
52
(
4
), pp.
519
542
.
14.
Chen
,
Y.
, and
Tang
,
T.
,
2010
, “
Convergence Analysis of the Jacobi Spectral-Collocation Methods for Volterra Integral Equation With a Weakly Singular Kernel
,”
Math. Comput.
,
79
(
269
), pp.
147
167
.
15.
Yang
,
Y.
,
Chen
,
Y.
, and
Huang
,
Y.
,
2014
, “
Convergence Analysis of the Jacobi Spectral-Collocation Method for Fractional Integro-Differential Equations
,”
Acta Math. Sci.
,
34
(
3
), pp.
673
690
.
16.
Yang
,
Y.
, and
Huang
,
Y.
,
2013
, “
Spectral-Collocation Methods for Fractional Pantograph Delay-Integrodifferential Equations
,”
,
2013
, p.
821327
.
17.
Zhang
,
S.
,
Lin
,
Y.
, and
Rao
,
M.
,
2000
, “
Numerical Solutions for Second-Kind Volterra Integral Equations by Galerkin Methods
,”
Appl. Math.
,
45
(
1
), pp.
19
39
.
18.
Yi
,
L.
,
2015
, “
An h-p Version of the Continuous Petrov–Galerkin Finite Element Method for Nonlinear Volterra Integro-Differential Equations
,”
J. Sci. Comput.
,
65
(
2
), pp.
715
734
.
19.
Yang
,
Y.
,
Chen
,
Y.
,
Huang
,
Y.
, and
Yang
,
W.
,
2015
, “
Convergence Analysis of Legendre-Collocation Methods for Nonlinear Volterra Type Integro Equations
,”
,
7
(
1
), pp.
74
88
.
20.
Douglas
,
J.
,
Dupont
,
T.
, and
Wahlbin
,
L.
,
1975
, “
The Stability in Lq of the L2-Projection Into Finite Element Function Spaces
,”
Numer. Math.
,
23
(
3
), pp.
193
198
.
21.
Canuto
,
C.
,
Hussaini
,
M. Y.
,
Quarteroni
,
A.
, and
Zang
,
T.
,
2006
,
Spectral Methods: Fundamentals in Single Domains
,
Springer-Verlag
,
Berlin
.