Applying the modified block-pulse functions to solve the three-dimensional Volterra–Fredholm integral equations

Mirzaee, Farshid; Hadadiyan, Elham

doi:10.1016/j.amc.2015.05.125

Cited by 19 publications

(10 citation statements)

References 21 publications

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“…The goal of this section is to show the efficiency and ability of the method based on MNM and MF, the following example has been considered and MATLAB Rb 2013 has been used to get the results. Example 1: Consider the following 3D-VIOE 18…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

“…in 17 used a method that is based on the 3D block pulse function to establish an approximate solution for the 3D nonlinear mixed Fredholm-Volterra integral equations. In 18 , Mirzaee and Hadadiyan applied the modified block pulse approximation to solve the 3D nonlinear mixed Fredholm-Volterra integral equations of the second kind. While in 19 , Mirzaee and Hadadiyan found the solution of the 3D nonlinear mixed Fredholm-Volterra integral equations based on the 3D triangular functions.…”

Section: Introductionmentioning

confidence: 99%

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Three-Dimensional Nonlinear Integral Operator with the Modelling of Majorant Function

Hameed

Al-Saedi

2021

Baghdad Sci.J

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In this paper, the process for finding an approximate solution of nonlinear three-dimensional (3D) Volterra type integral operator equation (N3D-VIOE) in R 3 is introduced. The modelling of the majorant function (MF) with the modified Newton method (MNM) is employed to convert N3D-VIOE to the linear 3D Volterra type integral operator equation (L3D-VIOE). The method of trapezoidal rule (TR) and collocation points are utilized to determine the approximate solution of L3D-VIOE by dealing with the linear form of the algebraic system. The existence of the approximate solution and its uniqueness are proved, and illustrative examples are provided to show the accuracy and efficiency of the model.

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Section: Numerical Results and Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Three-Dimensional Nonlinear Integral Operator with the Modelling of Majorant Function

Hameed

Al-Saedi

2021

Baghdad Sci.J

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“…Discussing the first m 3 ‐items of 3D‐BPFs and show them by m 3 ‐vectors

Φ (x, y, z) = {([], ϕ_{1, 1, 1} ((), x, y, z), \dots, ϕ_{1, 1, m} ((), x, y, z), \dots, ϕ_{1, m, m} ((), x, y, z), \dots, ϕ_{m, m, m} ((), x, y, z))}^{T},

which ( x , y , z ) ∈ [0, 1). From (2.3), it can be concluded that

Φ (x, y, z) Φ {((), x, y, z)}^{T} = diag (Φ (x, y, z)) .

It is cleared that [29]

\begin{array}{l} \int_{0}^{x} \int_{0}^{y} \int_{0}^{z} Φ ((), s, t, r) italicdrdtds \approx ((), \int_{0}^{x} Φ ((), s) italicds) \otimes ((), \int_{0}^{y} Φ ((), t) italicdt) \otimes ((), \int_{0}^{z} Φ ((), r) italicdr) \\ \approx ((), P_{1} Φ ((), x)) \otimes ((), P_{1} Φ ((), y)) \otimes ((), P_{1} Φ ((), z)) = ((), P_{1} \otimes P_{1} \otimes P_{1}) Φ ((), x, y, z) = P Φ ((), x, y, z) \end{array}

where ⊗ is the Krone...…”

Section: D‐bpfsmentioning

confidence: 99%

Applying the three‐dimensional block‐pulse functions to solve system of Volterra–Hammerstein integral equations

Xie

Gong

Shi

et al. 2020

Numerical Methods Partial

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In this paper, a numerical scheme is utilized to solve three-dimensional nonlinear system of Volterra-Hammerstein integrals equations, which is based on the three-dimensional block-pulse functions (3D-BPFs) and their operational matrices. Then the primary nonlinear system is transferred into a linear system of algebraic equations by applying the approximate expression and operational matrices, which can be easily solved through any numerical techniques. According to the convergence of 3D-BPFs, the new convergence analysis and error estimation theorem of the research system is detailed investigated. Lastly illustrative examples are included to demonstrate the validity and applicability of the technique.

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“…Mirzaee and Hoseini [18] used Fibonacci polynomials and collocation points for numerical solution of Volterra-Fredholm IEs. Mirzaee and Hadadiyan [19] , applied the modified block-pulse functions for solution of three-dimensional Volterra-Fredholm IEs. Mirzaee et al [20] , used three-dimensional block-pulse functions for numerical solution of three-dimensional nonlinear mixed Volterra-Fredholm IEs.…”

Section: Introductionmentioning

confidence: 99%

Efficient numerical technique for solution of delay Volterra-Fredholm integral equations using Haar wavelet

Amin

Shah

Asif

et al. 2020

Heliyon

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In this article, a computational Haar wavelet collocation technique is developed for the solution of linear delay integral equations. These equations include delay Fredholm, Volterra and Volterra-Fredholm integral equations. First we transform the derived estimates for these equations. After that, we transform these estimates to a system of algebraic equations. Finally, we solve the obtained algebraic system by Gauss elimination technique. Numerical examples are taken from literature for checking the validity and convergence of the proposed technique. The maximum absolute and root mean square errors are compared with the exact solution. The convergence rate using distinct numbers of collocation points is also calculated, which is approximately equal to 2. All algorithms for the developed method are implemented in MATLAB (R2009b) software.

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Applying the modified block-pulse functions to solve the three-dimensional Volterra–Fredholm integral equations

Cited by 19 publications

References 21 publications

Three-Dimensional Nonlinear Integral Operator with the Modelling of Majorant Function

Three-Dimensional Nonlinear Integral Operator with the Modelling of Majorant Function

Applying the three‐dimensional block‐pulse functions to solve system of Volterra–Hammerstein integral equations

Efficient numerical technique for solution of delay Volterra-Fredholm integral equations using Haar wavelet

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