TY - JOUR

T1 - The Jacobi Collocation Method for a Class of Nonlinear Volterra Integral Equations with Weakly Singular Kernel

AU - Allaei, Sonia Seyed

AU - Diogo, Teresa

AU - Rebelo, Magda

N1 - Sem PDF conforme despacho.
info:eu-repo/grantAgreement/FCT/5876/136077/PT#
info:eu-repo/grantAgreement/FCT/3599-PPCDT/101867/PT#
info:eu-repo/grantAgreement/FCT/5876/147204/PT#
Fundacao para a Ciencia e a Tecnologia (Pest-OE/MAT/UI0822/2014 ; PTDC/MAT/101867/2008);
FCT (UID/MAT/00297/2013);
Hong Kong Research Grants Council (RGC)- HKBU 200113
1369648.

PY - 2016/11/1

Y1 - 2016/11/1

N2 - A Jacobi spectral collocation method is proposed for the solution of a class of nonlinear Volterra integral equations with a kernel of the general form xβ(z-x)-αg(y(x)), where α∈ (0 , 1) , β> 0 and g(y) is a nonlinear function. Typically, the kernel will contain both an Abel-type and an end point singularity. The solution to these equations will in general have a nonsmooth behaviour which causes a drop in the global convergence orders of numerical methods with uniform meshes. In the considered approach a transformation of the independent variable is first introduced in order to obtain a new equation with a smoother solution. The Jacobi collocation method is then applied to the transformed equation and a complete convergence analysis of the method is carried out for the L∞ and the L2 norms. Some numerical examples are presented to illustrate the exponential decay of the errors in the spectral approximation.

AB - A Jacobi spectral collocation method is proposed for the solution of a class of nonlinear Volterra integral equations with a kernel of the general form xβ(z-x)-αg(y(x)), where α∈ (0 , 1) , β> 0 and g(y) is a nonlinear function. Typically, the kernel will contain both an Abel-type and an end point singularity. The solution to these equations will in general have a nonsmooth behaviour which causes a drop in the global convergence orders of numerical methods with uniform meshes. In the considered approach a transformation of the independent variable is first introduced in order to obtain a new equation with a smoother solution. The Jacobi collocation method is then applied to the transformed equation and a complete convergence analysis of the method is carried out for the L∞ and the L2 norms. Some numerical examples are presented to illustrate the exponential decay of the errors in the spectral approximation.

KW - Convergence analysis

KW - Jacobi spectral collocation method

KW - Nonlinear Volterra integral equation

KW - Weakly singular kernel

UR - http://www.scopus.com/inward/record.url?scp=84965043198&partnerID=8YFLogxK

U2 - 10.1007/s10915-016-0213-x

DO - 10.1007/s10915-016-0213-x

M3 - Article

AN - SCOPUS:84965043198

VL - 69

SP - 673

EP - 695

JO - Siam Journal On Scientific Computing

JF - Siam Journal On Scientific Computing

SN - 1064-8275

IS - 2

ER -