Symmetric Spectral Collocation Method for a Kind of Nonlinear Volterra Integral Equation

Wu, Nada; Zheng, Wei Xing; Gao, Wenjuan

doi:10.3390/sym14061091

Symmetry

2022

DOI: 10.3390/sym14061091

|View full text |Cite

Symmetric Spectral Collocation Method for a Kind of Nonlinear Volterra Integral Equation

Nada Wu

Wei Xing Zheng

Wenjuan Gao

Abstract: In this paper, we develop an efficient spectral method for numerically solving the nonlinear Volterra integral equation with weak singularity and delays. Based on the symmetric collocation points, the spectral method is illustrated, and the convergence results are obtained. In the end, two numerical experiments are carried out to confirm the theoretical results.

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2022

2023

Publication Types

Select...

Article2

Relationship

Self Cite0

Independent2

Authors

Journals

Cited by 2 publications

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

The regularization of spectral methods for hyperbolic Volterra integrodifferential equations with fractional power elliptic operator

Rostamy

2023

Nonlinear Engineering

View full text Add to dashboard Cite

In this study, a numerical approach is presented to solve the linear and nonlinear hyperbolic Volterra integrodifferential equations (HVIDEs). The regularization of a Legendre-collocation spectral method is applied for solving HVIDE of the second kind, with the time and space variables on the basis of Legendre-Gauss-Lobatto and Legendre-Gauss (LG) interpolation points, respectively. Concerning bounded domains, the provided HVIDE relation is transformed into three corresponding relations. Hence, a Legendre-collocation spectral approach is applied for solving this equation, and finally, ill-posed linear and nonlinear systems of algebraic equations are obtained; therefore different regularization methods are used to solve them. For an unbounded domain, a suitable mapping to convert the problem on a bounded domain is used and then apply the same proposed method for the bounded domain. For the two cases, the numerical results confirm the exponential convergence rate. The findings of this study are unprecedented for the regularization of the spectral method for the hyperbolic integrodifferential equation. The result in this work seems to be the first successful for the regularization of spectral method for the hyperbolic integrodifferential equation.

show abstract

The regularization of spectral methods for hyperbolic Volterra integrodifferential equations with fractional power elliptic operator

Rostamy

2023

Nonlinear Engineering

View full text Add to dashboard Cite

show abstract

Error Estimates of a Symmetric Spectral Method for a Linear Volterra Integral Equation

Zheng

Chen

2022

Symmetry

View full text Add to dashboard Cite

A symmetric spectral method is applied to investigate the two-dimensional Volterra integral equation with weakly singular kernels and delays. In this work, the solution of the equation we considered is assumed to be sufficiently smooth so that the spectral method can be applied naturally. Employing three couples of variable transformations, we apply the two-dimensional Gauss quadrature rule to approximate the weakly singular integral with delays and obtain the spectral discretization. Then we derive the convergence results of the proposed approximation scheme. We show that the errors of solution decay exponentially in both the infinity norm and weighted square norm. In the end, we carry out numerical experiments to verify the theoretical results.

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Symmetric Spectral Collocation Method for a Kind of Nonlinear Volterra Integral Equation

Cited by 2 publications

References 25 publications

The regularization of spectral methods for hyperbolic Volterra integrodifferential equations with fractional power elliptic operator

The regularization of spectral methods for hyperbolic Volterra integrodifferential equations with fractional power elliptic operator

Error Estimates of a Symmetric Spectral Method for a Linear Volterra Integral Equation

Contact Info

Product

Resources

About