Two-dimensional Euler polynomials solutions of two-dimensional Volterra integral equations of fractional order

Wang, Yifei; Huang, Jin; Wen, Xiaoxia

doi:10.1016/j.apnum.2021.01.007

Cited by 8 publications

(3 citation statements)

References 40 publications

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“…If the uniqueness of the solution of the boundary value problem is violated, then the loading can be interpreted as a strong perturbation. It turns out here that the character of the load is a perturbation (weak or strong perturbation) [13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 98%

On the Correctness of Boundary Value Problems for the Two-Dimensional Loaded Parabolic Equation

Attaev¹,

Ramazanov²,

Omarov³

2022

Bul. Kar. Univ. - Math.

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The paper studies the problems of the correctness of setting boundary value problems for a loaded parabolic equation. The a feature of the problems is that the order of the derivative in the loaded term is less than or equal to the order of the differential part of the equation, and the load point moves according to a nonlinear law. At the same time, the distinctive characteristic is that the line, on which the loaded term is set, is at the zero point. On the basis of the study the authors proved the theorems about correctness of the studied boundary value problems.

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Section: Introductionmentioning

confidence: 98%

On the Correctness of Boundary Value Problems for the Two-Dimensional Loaded Parabolic Equation

Attaev¹,

Ramazanov²,

Omarov³

2022

Bul. Kar. Univ. - Math.

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“…In [21], the second 2D nonlinear VIE was solved by the Galerkin method with the help of the moving least squares method. The paper proposed a numerical method to solve 2D-VIEs with fractional order, weakly singular kernels based on 2D Euler polynomials combined with the Gauss-Jacobi quadrature formula in [22]. In [23], an efficient method was presented for solving the second kind of 3D-VFIEs based on 3D Bernstein polynomials.…”

Section: Introductionmentioning

confidence: 99%

A High-Order Approximate Solution for the Nonlinear 3D Volterra Integral Equations with Uniform Accuracy

Wang

Long

Cao

2022

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In this paper, we present a high-order approximate solution with uniform accuracy for nonlinear 3D Volterra integral equations. This numerical scheme is constructed based on the three-dimensional block cubic Lagrangian interpolation method. At the same time, we give the local truncation error analysis of the numerical scheme based on Taylor’s theorem. Through theoretical analysis, we reach the conclusion that the optimal convergence order of this high-order numerical scheme is 4. Finally, we verify the effectiveness and applicability of the method through four numerical examples.

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“…In [1][2][3][4][5][6] authors for solving such problems used a technique which consists in reducing a non-cylindrical domain to a cylindrical one. There are a number of works devoted to numerical methods for solving such problems [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

Solution of a Two-Dimensional Boundary Value Problem of Heat Conduction in a Degenerating Domain

Ramazanov

Gulmanov

2021

JMMCS

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In the paper we consider the boundary value problem of heat conduction outside the cone, i.e. in the domain degenerating into a point at the initial moment of time. In this case, the boundary condition contain a derivative with respect to the time variable. The peculiarity of the problem under consideration consists precisely in the presence of a moving boundary and the degeneration of the solution domain into a point at the initial moment of time. The well-known classical methods are generally not applicable to this type of problems. By the method of heat potentials, such boundary value problems of heat conduction are reduced to the solution of singular Volterra type integral equations of the second kind A singular Volterra type equation is understood as an equation whose kernel has the following property: the integral of the kernel of the equation does not tend to zero as the upper limit tends to the lower one. Such integral equations cannot be solved by the method of successive approximations, and in most cases the corresponding homogeneous integral equations have nonzero solutions. We prove a theorem on the solvability of the considered boundary value problem in weighted spaces of essentially bounded functions. The issues of solvability of the singular Volterra integral equation of the second kind, to which the original problem is reduced, are studied. We found a nonzero solution of this singular integral equation.

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Two-dimensional Euler polynomials solutions of two-dimensional Volterra integral equations of fractional order

Cited by 8 publications

References 40 publications

On the Correctness of Boundary Value Problems for the Two-Dimensional Loaded Parabolic Equation

On the Correctness of Boundary Value Problems for the Two-Dimensional Loaded Parabolic Equation

A High-Order Approximate Solution for the Nonlinear 3D Volterra Integral Equations with Uniform Accuracy

Solution of a Two-Dimensional Boundary Value Problem of Heat Conduction in a Degenerating Domain

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