On a method of obtaining spectral relationships for integral operators of mixed problems of mechanics of continuous media

Александров, В. М.; Kovalenko, E. V.; Mkhitarian, S.M.

doi:10.1016/0021-8928(82)90067-3

Cited by 18 publications

(5 citation statements)

References 3 publications

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“…Therefore, Φ(z; 0) = u 0 (z). If p = 1, then N(Φ(z; p)) = 0, i.e., Φ(z; 1) is solution of the Equation (2). In this way, the change of parameter p from zero to one corresponds to the transition from a trivial task to the original task.…”

Section: The Homotopy Analysis Methodsmentioning

confidence: 99%

“…Fuzzy integral equations are one of the important branches of fuzzy analysis theory and they are applied as an adequate apparatus in mathematical modeling in biology, chemistry, physics, engineering, etc. (see, for example, [1][2][3][4]). One of the first applications of fuzzy integration was given by Wu and Ma [5], who investigated the fuzzy Fredholm integral equation of the second kind.…”

Section: Introductionmentioning

confidence: 99%

“…The paper presents an application of HAM for solving the following nonlinear Volterra-Fredholm fuzzy integral equations with two variables (2D-NVFFIE) u(s, t) =g(s, t) ⊕ (FR) The integral Equation (1) is called the Fredholm equation with respect to the position and Volterra with respect to the time. This equation is used in many problems of mathematical physics, theory of elasticity, contact problems and mixed problems of mechanics of continuous media (see [2,24]). In [25], the Adomian decomposition method is applied for solving this equation.…”

Section: Introductionmentioning

confidence: 99%

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Homotopy Analysis Method to Solve Two-Dimensional Nonlinear Volterra-Fredholm Fuzzy Integral Equations

Georgieva

Hristova

2020

Fractal Fract

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The main goal of the paper is to present an approximate method for solving of a two-dimensional nonlinear Volterra-Fredholm fuzzy integral equation (2D-NVFFIE). It is applied the homotopy analysis method (HAM). The studied equation is converted to a nonlinear system of Volterra-Fredholm integral equations in a crisp case. Approximate solutions of this system are obtained by the help with HAM and hence an approximation for the fuzzy solution of the nonlinear Volterra-Fredholm fuzzy integral equation is presented. The convergence of the proposed method is proved and the error estimate between the exact and the approximate solution is obtained. The validity and applicability of the proposed method is illustrated on a numerical example.

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Section: The Homotopy Analysis Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Homotopy Analysis Method to Solve Two-Dimensional Nonlinear Volterra-Fredholm Fuzzy Integral Equations

Georgieva

Hristova

2020

Fractal Fract

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“…This type of equation appears in many problems of mathematical physics, theory of elasticity, contact problems, and mixed problems of mechanics of continuous media see [11][12][13]. Several numerical methods for obtaining the approximate solution of (1) with continuous kernel are known [14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 98%

Error estimates for series solutions to a class of nonlinear integral equations of mixed type

El‐Kalla

2011

J. Appl. Math. Comput.

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In this paper, we prove that the accelerated Adomian polynomials formula suggested by Adomian (Nonlinear Stochastic Systems: Theory and Applications to Physics, Kluwer, Dordrecht, 1989) and the accelerated formula suggested by El-Kalla (Int. J. Differ. Equs. Appl. 10(2):225-234, 2005; Appl. Math. E-Notes 7: [214][215][216][217][218][219][220][221] 2007) are identically the same. The Kalla-iterates exhibit the same faster convergence exhibited by Adomian's accelerated iterates with the additional advantage of absence of any derivative terms in the recursion, thereby allowing for ease of computation. Moreover, the formula of El-Kalla is used directly to prove the convergence of the series solution to a class of nonlinear two dimensional integral equations. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of the Adomian series solution.

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“…Block pulse functions (BPFs) are easy to use and this simplicity allows one to use them to solve integral equations and differential equations [2]. The application of the integral equation that we solve is shown in [3][4][5].…”

Section: Introductionmentioning

confidence: 99%

A numerical method for solving Fredholm–Volterra integral equations in two-dimensional spaces using block pulse functions and an operational matrix

Babolian

Maleknejad

Mordad

et al. 2011

Journal of Computational and Applied Mathematics

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MSC: 45B05 45D05 45A05Keywords: Two-dimensional Fredholm-Volterra integral equations Block pulse functions Operational matrix a b s t r a c t In this paper, the block pulse functions (BPFs) and their operational matrix are used to solve two-dimensional Fredholm-Volterra integral equations (F-VIE). This method converts F-VIE to systems of linear equations whose solutions are the coefficients of block pulse expansions of the solutions of F-VIE.Finally some numerical examples are presented to show the efficiency and accuracy of the method.

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On a method of obtaining spectral relationships for integral operators of mixed problems of mechanics of continuous media

Cited by 18 publications

References 3 publications

Homotopy Analysis Method to Solve Two-Dimensional Nonlinear Volterra-Fredholm Fuzzy Integral Equations

Homotopy Analysis Method to Solve Two-Dimensional Nonlinear Volterra-Fredholm Fuzzy Integral Equations

Error estimates for series solutions to a class of nonlinear integral equations of mixed type

A numerical method for solving Fredholm–Volterra integral equations in two-dimensional spaces using block pulse functions and an operational matrix

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