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

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...

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Bibliographic Details
Published in:Nonlinear engineering Vol. 12; no. 1; pp. 15 - 29
Main Authors: Mirzaei G., F., Rostamy, Davood
Format: Journal Article
Language:English
Published: Berlin De Gruyter 08-02-2023
Walter de Gruyter GmbH
Subjects:
Collocation
Collocation methods
differentiation matrix
Domains
Elliptic functions
Gauss quadrature formulas
hyperbolic Volterra integrodifferential equations
Interpolation
Legendre-collocation spectral method
Nonlinear systems
Regularization
Regularization methods
Spectral methods
Volterra integral equations
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Summary: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.
ISSN:2192-8029
2192-8010
2192-8029
DOI:10.1515/nleng-2022-0250