Projection Method for Bounded and Unbounded Solution of Hypersingular Integral Equations of the First Kind

Eshkuvatov, Zainidin K.; Narzullaev, Anvar

doi:10.5958/1945-919x.2019.00002.1

Cited by 4 publications

(5 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Due to Theorem 5 and referring to Eshkuvatov et al [19] the following error estimation can be obtained.…”

Section: Preliminariesmentioning

confidence: 99%

Semi Bounded Solution of Hypersingular Integral Equations of the First Kind on the Rectangle

Eshkuvatov¹,

Kommuji²,

Aloev³

et al. 2020

View full text Add to dashboard Cite

A hypersingular integral equations (HSIEs) of the first kind on the interval [−1, 1] with the assumption that kernel of the hypersingular integral is constant on the diagonal of the domain is considered. Truncated series of Chebyshev polynomials of the third and fourth kinds are used to find semi bounded (unbounded on the left and bounded on the right and vice versa) solutions of HSIEs of first kind. Exact calculations of singular and hypersingular integrals with respect to Chebyshev polynomials of third and forth kind with corresponding weights allows us to obtain high accurate approximate solution. Gauss-Chebyshev quadrature formula is extended for regular kernel integrals. Three examples are provided to verify the validity and accuracy of the proposed method. Numerical examples reveal that approximate solutions are exact if solution of HSIEs is of the polynomial forms with corresponding weights.

show abstract

“…Due to Theorem 5 and referring to Eshkuvatov et al [19] the following error estimation can be obtained.…”

Section: Preliminariesmentioning

confidence: 99%

Semi Bounded Solution of Hypersingular Integral Equations of the First Kind on the Rectangle

Eshkuvatov¹,

Kommuji²,

Aloev³

et al. 2020

View full text Add to dashboard Cite

show abstract

“…Example 4. We investigate the following hypersingular integral equation of the form [20] The authors in [20] solved this example under consideration that the solution is unbounded at =  x 1, while we solve it for the bounded mode at =  x 1. For N = 4 the solution is approximated as follows, which is the where the exact solution is…”

Section: After Solving System (52) We Getmentioning

confidence: 99%

“…2 . This problem was solved in [20]. Table 2 compares the numerical results of the proposed method with [20].…”

Section: After Solving System (52) We Getmentioning

confidence: 99%

“…He obtained the semi-analytical solution of the hypersingular integral equation in these cases. Later, the authors in [20] used the projection method together with Chebyshev polynomials with the assumption that the kernel of the hypersingular integral equation k(x, s), is constant on the diagonal of the domain. They applied truncated series of Chebyshev polynomials of the third and fourth kinds to find semi bounded solutions for (1) with constant K(x, s) on the diagonal [21].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On projection method for numerical solution of hypersingular integral equations of the first kind combined with quadrature methods

et al. 2023

View full text Add to dashboard Cite

The Fredholm integral equations of the ﬁrst kind with two regular and hypersingular kernels are considered on [−1, 1]. The hypersingular kernel is considered smooth enough with no additional conditions. A projection method based on second kind Chebyshev polynomials approximation, combined with quadrature integration method, is developed to obtain high accurate approximations. The proposed method reduces the underlying integral equation to a system of algebraic equations. Several illustrative examples are provided to show the eﬃciency of the method.

show abstract

“…In certain cases, there exist some analytical methods for solving HSIEs, but in general, there is no known analytical method. Thus, different numerical methods are introduced and developed for solving HSIEs [8,9,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].…”

Section: Introductionmentioning

confidence: 99%

A Novel Collocation Method for Numerical Solution of Hypersingular Integral Equation with Singular Right-Hand Function

Elahi

Mahmoudi

Shamloo

et al. 2023

Advances in Mathematical Physics

View full text Add to dashboard Cite

In this study, the Fredholm hypersingular integral equation of the first kind with a singular right-hand function on the interval − 1 , 1 is solved. The discontinuous solution on the domain − 1 , 1 is approximated by a piecewise polynomial, and a collocation method is introduced to evaluate the unknown coefficients. This method, which can be applied to both linear and nonlinear integral equations, is very simple and straightforward. The presented illustrations relate that the results are very accurate compared to the other methods in the literature.

show abstract

Projection Method for Bounded and Unbounded Solution of Hypersingular Integral Equations of the First Kind

Cited by 4 publications

References 0 publications

Semi Bounded Solution of Hypersingular Integral Equations of the First Kind on the Rectangle

Semi Bounded Solution of Hypersingular Integral Equations of the First Kind on the Rectangle

On projection method for numerical solution of hypersingular integral equations of the first kind combined with quadrature methods

A Novel Collocation Method for Numerical Solution of Hypersingular Integral Equation with Singular Right-Hand Function

Contact Info

Product

Resources

About