A computational method based on hybrid of block-pulse functions and Taylor series for solving two-dimensional nonlinear integral equations

Mirzaee, Farshid; Hoseini, Ali Akbar

doi:10.1016/j.aej.2013.10.002

Cited by 22 publications

(15 citation statements)

References 21 publications

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“…The matrix P in (1.2) is called the operational matrix of integration. The operational matrix of integration P has already been obtained for several types of two-variable basis functions such as bivariate shifted Legendre Nemati et al (2013), bivariate shifted Chebyshev Nemati and Sedaghat (2016), two-dimensional triangular Khajehnasiri (2016), Babolian et al (2010), two-dimensional Bernstein Shekarabi et al (2015), twodimensional block pulse Maleknejad and Mahdiani (2011), Najafalizadeh and Ezzati (2017), hybrid of block-pulse functions and Taylor series Mirzaee and Hoseini (2014).…”

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confidence: 99%

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

Bazm

Hosseini

2020

Comp. Appl. Math.

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Two-dimensional Volterra-Fredholm integral equations of Hammerstein type are studied. Using the Banach Fixed Point Theorem, the existence and uniqueness of a solution to these equations in the space L ∞ ([0, 1] × [0, 1]) is proved. Then, the operational matrices of integration and product for two-variable Bernoulli polynomials are derived and utilized to reduce the solution of the considered problem to the solution of a system of nonlinear algebraic equations that can be solved by Newton's method. The error analysis is given and some examples are provided to illustrate the efficiency and accuracy of the method. Keywords Two-dimensional integral equations • Volterra-Fredholm integral equations of Hammerstein type • Bernoulli polynomials • Operational matrix method • Collocation method Mathematics Subject Classification 65R20 • 45D05 1 Introduction Two-dimensional (2D) integral equations appear in the mathematical modeling of various physical phenomena and engineering problems. For example, it is shown in McKee et al. (2000) that a type of nonlinear telegraph equations is equivalent to 2D Volterra integral equations (VIEs). Also, in Graham (1980/1981), an application of 2D Fredholm integral Communicated by Hui Liang.

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confidence: 99%

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

Bazm

Hosseini

2020

Comp. Appl. Math.

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“…In order to make full use of the advantage of BPFs and bypass their disadvantage, hybrid functions of BPFs and orthogonal functions or polynomials have been adopted to solve differential or integral equations. For example, the operational matrix of hybrid functions of BPFs and Bernoulli polynomials was developed to numerically solve the distributed fractional differential equations, 37 the operational matrix of hybrid functions of BPFs and Legendre polynomials was derived to solve a telegraph equation of fractional order, 38 the operational matrices of hybrid of BPFs and Taylor series, 39 Bernstein polynomials 40 were also developed. The hybrid functions are more smooth than BPFs and can approximate a function more accurately if the same number of basis functions is used.…”

Section: Introductionmentioning

confidence: 99%

Solving fractional differential equation using block‐pulse functions and Bernstein polynomials

Zhang

Tang

Liu

et al. 2021

Math Methods in App Sciences

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The method based on block pulse functions (BPFs) has been proposed to solve different kinds of fractional differential equations (FDEs). However, high accuracy requires considerable BPFs because they are piecewise constant and not so smooth. As a result, it increases the dimension of operational matrix and computational burden. To overcome this deficiency, a novel numerical method is developed to solve fractional differential equations. The method is based upon hybrid of BPFs and Bernstein polynomials (HBBPs), which are piecewise smooth. The HBBPs operational matrix of fractional‐order integral is derived to reduce the FDEs to a system of algebraic equations. Then the numerical solution of the FDEs is obtained through solving the system of algebraic equations. The convergence analysis is conducted for the suggested scheme, and the upper bound of error of the solution is given. Finally, illustrative examples are presented to demonstrate the validity, applicability, and efficiency of the proposed technique in contrast with other approaches.

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“…The existence and uniqueness theorems of these problems have been studied in (Mishra and Sen, 2016; Mishra, Agarwal and Sen, 2016; Mishra, Srivastava and Sen, 2016; El-Sayed et al , 2011; Brunner and Kauthen, 1989). Although, approximate solutions of 1D-integral equations have been investigated by many researchers, for two-dimensional cases only a few methods have been discussed (Mirzaee and Hoseini, 2014; Han and Zhang, 1994; Mirzaee et al , 2015). The analysis of numerical methods for two-dimensional integral equations, especially in the non-linear case are developed more recently.…”

Section: Introductionmentioning

confidence: 99%

Solving two-dimensional non-linear quadratic integral equations of fractional order via operational matrix method

Mirzaee

Alipour

2019

MMMS

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Purpose The purpose of this paper is to develop a new method based on operational matrices of two-dimensional delta functions for solving two-dimensional nonlinear quadratic integral equations (2D-QIEs) of fractional order, numerically. Design/methodology/approach For this aim, two-dimensional delta functions are introduced, and their properties are expressed. Then, the fractional operational matrix of integration based on two-dimensional delta functions is calculated for the first time. Findings By applying the operational matrices, the main problem would be transformed into a nonlinear system of algebraic equations which can be solved by using Newton's iterative method. Also, a few results related to error estimate and convergence analysis of the proposed method are investigated. Originality/value Two numerical examples are presented to show the validity and applicability of the suggested approach. All of the numerical calculation is performed on a personal computer by running some codes written in MATLAB software.

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A computational method based on hybrid of block-pulse functions and Taylor series for solving two-dimensional nonlinear integral equations

Cited by 22 publications

References 21 publications

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

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

Solving fractional differential equation using block‐pulse functions and Bernstein polynomials

Solving two-dimensional non-linear quadratic integral equations of fractional order via operational matrix method

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