On the Existence of Solutions of Nonlinear Fredholm Integral Equations from Kantorovich’s Technique

Ezquerro, J. A.; Hernández‐Verón, M. A.

doi:10.3390/a10030089

Cited by 4 publications

(2 citation statements)

References 16 publications

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“…Modern publications are mainly related to non-linear Urysohn equations of the second kind. For numerically solving such types of equations, different modifications of the Newton-Kantorovich quadrature method are mainly used [8][9][10]. For solving NUIEs of the second kind, decompositions by different polynomials are used, for example: Chebyshev's polynomials [11], Bernoulli's polynomials [12], and Bernstein's polynomials [13], which are combined with artificial neural networks.…”

Section: Introductionmentioning

confidence: 99%

Application of Wavelet Transform to Urysohn-Type Equations

Lukianenko,

Kozlova,

Belozub

2023

Mathematics

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This paper deals with convolution-type Urysohn equations of the first kind. Finding a solution for such equations is an ill-posed problem. For it to be solved, regularization algorithms and the continuous wavelet transform are used. Similar to the Fourier transform, the continuous wavelet transform is applied to convolution-type equations (based on the Fourier and wavelet transforms) and to Urysohn equations with unknown shift. The wavelet transform is preferable for the cases with approximated right-hand sides and for type 1 equations. We demonstrated that the application of the wavelet transform to Urysohn-type equations with unknown shift translates into a solution of a nonlinear equation with an oscillating kernel. Depending on the availability of a priori information, a combination of regularization and iterative algorithms with the use of close equations are effective for solving convolution-type equations based on the continuous wavelet transform and Urysohn equation.

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Section: Introductionmentioning

confidence: 99%

Application of Wavelet Transform to Urysohn-Type Equations

Lukianenko,

Kozlova,

Belozub

2023

Mathematics

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“…Thus, the discussion on solving nonlinear Fredholm can be found in many literature. In the past several years, many analytical and numerical methods have been introduced to solve the nonlinear Fredholm integral equations, among these are the extrapolation method [4], parameter continuation method [5], successive approximations method [6], Newton's method [7], q-homotopy analysis method [8], Legendre spectral collocation method [9], least squares support vector regression [10], and Hermite wavelets collocation method [11]. In [12], the authors present a new technique based on a combination of a Newton-Kantorovich and Haar wavelet for solving nonlinear Fredholm integral equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Nonlinear Fredholm Integral Equations Using Half-Sweep Newton-PKSOR Iteration

Ali¹,

Sulaiman²,

Saudi³

et al. 2022

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This paper is concerned with producing an efficient numerical method to solve nonlinear Fredholm integral equations using Half-Sweep Newton-PKSOR (HSNPKSOR) iteration. The computation of numerical methods in solving nonlinear equations usually requires immense amounts of computational complexity. By implementing a Half-Sweep approach, the complexity of the calculation is tried to be reduced to produce a more efficient method. For this purpose, the steps of the solution process are discussed beginning with the derivation of nonlinear Fredholm integral equations using a quadrature scheme to get the half-sweep approximation equation. Then, the generated approximation equation is used to develop a nonlinear system. Following that, the formulation of the HSNPKSOR iterative method is constructed to solve nonlinear Fredholm integral equations. To verify the performance of the proposed method, the experimental results were compared with the Full-Sweep Newton-KSOR (FSNKSOR), Half-Sweep Newton-KSOR (HSNKSOR), and Full-Sweep Newton-PKSOR (FSNPKSOR) using three parameters: number of iteration, iteration time, and maximum absolute error. Several examples are used in this study to illustrate the efficiency of the tested methods. Based on the numerical experiment, the results appear that the HSNPKSOR method is effective in solving nonlinear Fredholm integral equations mainly in terms of iteration time compared to rest tested methods.

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Triangular functions for numerical solution of the nonlinear Volterra integral equations

Kazemi

2021

J. Appl. Math. Comput.

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On the Existence of Solutions of Nonlinear Fredholm Integral Equations from Kantorovich’s Technique

Cited by 4 publications

References 16 publications

Application of Wavelet Transform to Urysohn-Type Equations

Application of Wavelet Transform to Urysohn-Type Equations

Numerical Solution of Nonlinear Fredholm Integral Equations Using Half-Sweep Newton-PKSOR Iteration

Triangular functions for numerical solution of the nonlinear Volterra integral equations

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