Spectral methods for weakly singular Volterra integral equations with smooth solutions

Chen, Yanping; Tang, Tao

doi:10.1016/j.cam.2009.08.057

Cited by 135 publications

(67 citation statements)

References 18 publications

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“…To this end, we introduce the result proposed in [9][10][11]: for any ∈ ( ), there exists a positive constant independent of ,…”

Section: Journal Of Function Spacesmentioning

confidence: 99%

“…There are many numerical attempts based on the spline approximation to overcome the difficulty caused by the singularity of the solution of (2) (see [1][2][3][4][5][6][7][8]). Recently, spectral methods using Jacobi polynomial basis have received considerable attention to approximating the solution of integral equations due to their high accuracy and easy implementation (see [9][10][11][12][13][14][15][16][17]). In particular, Chen and Tang in [11] proposed a Jacobi-collocation spectral method for second kind Volterra integral equations with weakly singular kernels.…”

Section: Introductionmentioning

confidence: 99%

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Convergence Analysis of Generalized Jacobi-Galerkin Methods for Second Kind Volterra Integral Equations with Weakly Singular Kernels

Cai

2017

Journal of Function Spaces

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We develop a generalized Jacobi-Galerkin method for second kind Volterra integral equations with weakly singular kernels. In this method, we first introduce some known singular nonpolynomial functions in the approximation space of the conventional JacobiGalerkin method. Secondly, we use the Gauss-Jacobi quadrature rules to approximate the integral term in the resulting equation so as to obtain high-order accuracy for the approximation. Then, we establish that the approximate equation has a unique solution and the approximate solution arrives at an optimal convergence order. One numerical example is presented to demonstrate the effectiveness of the proposed method.

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“…To this end, we introduce the result proposed in [9][10][11]: for any ∈ ( ), there exists a positive constant independent of ,…”

Section: Journal Of Function Spacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Convergence Analysis of Generalized Jacobi-Galerkin Methods for Second Kind Volterra Integral Equations with Weakly Singular Kernels

Cai

2017

Journal of Function Spaces

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“…(29) Absolute errors of u(y,t) and v(y,t) related to (27)- (29) are introduced in Table 3 using C-GR-C method with two choices of N in the interval [−3, 3]. Moreover, maximum absolute errors of u(y,t) and v(y,t) related to (22)- (24) are introduced in Table 4 using C-GL-C method with various choices of N. The approximate solutions u(x,t) and v(x,t) of problem (27) at N = 12 are displayed in Figs.…”

Section: U(yt)(v(yt)) and U(yt)( V(ymentioning

confidence: 99%

“…Moreover, maximum absolute errors of u(y,t) and v(y,t) related to (22)- (24) are introduced in Table 4 using C-GL-C method with various choices of N. The approximate solutions u(x,t) and v(x,t) of problem (27) at N = 12 are displayed in Figs. 9 and 10, respectively.…”

Section: U(yt)(v(yt)) and U(yt)( V(ymentioning

confidence: 99%

“…Spectral methods have emerged as a powerful techniques used in applied mathematics and scientific computing to numerically solve linear and nonlinear differential equations [23]- [26], integral equations [27]- [30], integro-differential equations [31]- [34], function approximation and variational problems [35]- [38]. The main idea of spectral methods is to put the solution of the problem as a sum of certain basis functions and then to choose the coefficients in the sum in order to minimize the difference between the exact solution and approximate one as well as possible.…”

Section: Introductionmentioning

confidence: 99%

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A Chebyshev-Gauss-Radau Scheme For Nonlinear Hyperbolic System Of First Order

Doha¹,

Bhrawy²,

Hafez³

et al. 2014

Appl. Math. Inf. Sci.

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A numerical approximation of the initial-boundary system of nonlinear hyperbolic equations based on spectral collocation method is presented in this article. A Chebyshev-Gauss-Radau collocation (C-GR-C) method in combination with the implicit RungeKutta scheme are employed to obtain highly accurate approximations to the mentioned problem. The collocation points are the Chebyshev interpolation nodes. This approach reduces this problem to solve system of nonlinear ordinary differential equations which are far easier to be solved. Indeed, by selecting a limited number of collocation nodes, we obtain an accurate results. The numerical examples demonstrate the accuracy, efficiency, and versatility of the method.

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Numerical scheme for solving singular fractional partial integro‐differential equation via orthonormal Bernoulli polynomials

Samadyar

Mirzaee

2019

Int J Numerical Modelling

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In this paper, an efficient matrix method based on 2D orthonormal Bernoulli polynomials are developed to obtain numerical solution of weakly singular fractional partial integro‐differential equations (FPIDEs). Operational matrix of integration, almost operational matrix of integration, and product operational matrix are employed to transform the solution of Volterra singular FPIDEs to the system of linear or nonlinear algebraic equations. The obtained algebraic system can be easily solved by using an appropriate numerical method. Also, some useful theorems concerning the convergence and error analysis associated to the mentioned approach are proven in the current paper. Finally, some numerical examples are included to illustrate the accuracy, efficiency, and applicability of the proposed approach.

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Spectral methods for weakly singular Volterra integral equations with smooth solutions

Cited by 135 publications

References 18 publications

Convergence Analysis of Generalized Jacobi-Galerkin Methods for Second Kind Volterra Integral Equations with Weakly Singular Kernels

Convergence Analysis of Generalized Jacobi-Galerkin Methods for Second Kind Volterra Integral Equations with Weakly Singular Kernels

A Chebyshev-Gauss-Radau Scheme For Nonlinear Hyperbolic System Of First Order

Numerical scheme for solving singular fractional partial integro‐differential equation via orthonormal Bernoulli polynomials

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