Bernoulli operational matrix method for the numerical solution of nonlinear two-dimensional Volterra–Fredholm integral equations of Hammerstein type

Bazm, Sohrab; Hosseini, Alireza

doi:10.1007/s40314-020-1077-0

Cited by 9 publications

(6 citation statements)

References 30 publications

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“…We have also established the existence and uniqueness of the numerical solution. Our numerical tests confirmed the theoretical findings, showing that an elevated rate of convergence in comparison with the numerical results reported in [6,[30][31][32]. Our spectral collocation method is more flexible with better accuracy than the existing ones.…”

Section: Discussionsupporting

confidence: 86%

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On the rate of convergence of the Legendre spectral collocation method for multi-dimensional nonlinear Volterra–Fredholm integral equations

Elkot¹,

Zaky²,

Doha³

et al. 2021

Commun. Theor. Phys.

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While the approximate solutions of one-dimensional nonlinear Volterra–Fredholm integral equations with smooth kernels are now well understood, no systematic studies of the numerical solutions of their multi-dimensional counterparts exist. In this paper, we provide an efficient numerical approach for the multi-dimensional nonlinear Volterra–Fredholm integral equations based on the multi-variate Legendre-collocation approach. Spectral collocation methods for multi-dimensional nonlinear integral equations are known to cause major difficulties from a convergence analysis point of view. Consequently, rigorous error estimates are provided in the weighted Sobolev space showing the exponential decay of the numerical errors. The existence and uniqueness of the numerical solution are established. Numerical experiments are provided to support the theoretical convergence analysis. The results indicate that our spectral collocation method is more flexible with better accuracy than the existing ones.

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Section: Discussionsupporting

confidence: 86%

“…Example 2. Consider the following two-dimensional Volterra integral equation [32]: The exact solution is given as…”

mentioning

confidence: 99%

On the rate of convergence of the Legendre spectral collocation method for multi-dimensional nonlinear Volterra–Fredholm integral equations

Elkot¹,

Zaky²,

Doha³

et al. 2021

Commun. Theor. Phys.

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“…Furthermore, orthogonal functions and polynomial series have attracted significant attention because they have been instrumental in treating various dynamical system problems. The main feature of this technique is that it reduces these problems to the solution of a system of algebraic equations by using the method of operational matrices based on orthogonal polynomials [25], such as Chebyshev polynomials [26], Bernoulli polynomials [27], and Laguerre polynomials [28], which significantly simplifies the problems and allows them to be solved by any computational program.…”

Section: Introductionmentioning

confidence: 99%

Novel Approximate Solutions for Nonlinear Initial and Boundary Value Problems

Mahdi Salih,

A. AL-Jawary,

Mustafa Turkyilmazoglu

2023

IHJPAS

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This paper investigates an effective computational method (ECM) based on the standard polynomials used to solve some nonlinear initial and boundary value problems appeared in engineering and applied sciences. Moreover, the effective computational methods in this paper were improved by suitable orthogonal base functions, especially the Chebyshev, Bernoulli, and Laguerre polynomials, to obtain novel approximate solutions for some nonlinear problems. These base functions enable the nonlinear problem to be effectively converted into a nonlinear algebraic system of equations, which are then solved using Mathematica®12. The improved effective computational methods (I-ECMs) have been implemented to solve three applications involving nonlinear initial and boundary value problems: the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation, and a comparison between the proposed methods has been presented. Furthermore, the maximum error remainder () has been computed to prove the proposed methods' accuracy. The results convincingly prove that ECM and I-ECMs are effective and accurate in obtaining novel approximate solutions to the problems.

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“…e study of a few more famous problems is presented by using the BC scheme such those including time-fractional cable equation [55], fourth-order Sturm-Liouville problem [56], stochastic Itô-Volterra integral equations of Abel type [57], and nonlinear two-dimensional Volterra-Fredholm integral equations of Hammerstein type [58].…”

Section: Introductionmentioning

confidence: 99%

Application of a Novel Collocation Approach for Simulating a Class of Nonlinear Third-Order Lane–Emden Model

Adel

Sabir

Rezazadeh

et al. 2022

Mathematical Problems in Engineering

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The present study aims to design a mathematical system based on the Lane–Emden third-order pantograph differential model by using the general forms of the pantograph as well as the Lane–Emden models. The designed model is divided into two types along with the various singularity details at each point. The shape factors and the pantograph points are discussed for each type of the newly designed nonlinear third-order pantograph differential model. The Bernoulli collocation scheme is implemented to find the numerical results of the novel model. To show the reliability of the designed novel nonlinear model, four different variants have been solved. Moreover, the comparison of the obtained results with the exact solutions is presented to check the accuracy of the designed novel model.

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Bernoulli operational matrix method for the numerical solution of nonlinear two-dimensional Volterra–Fredholm integral equations of Hammerstein type

Cited by 9 publications

References 30 publications

On the rate of convergence of the Legendre spectral collocation method for multi-dimensional nonlinear Volterra–Fredholm integral equations

On the rate of convergence of the Legendre spectral collocation method for multi-dimensional nonlinear Volterra–Fredholm integral equations

Novel Approximate Solutions for Nonlinear Initial and Boundary Value Problems

Application of a Novel Collocation Approach for Simulating a Class of Nonlinear Third-Order Lane–Emden Model

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