Numerical solving of several types of two‐dimensional integral equations and estimation of error bound

Berenguer, M.; Gámez, D.

doi:10.1002/mma.4840

Cited by 8 publications

(4 citation statements)

References 34 publications

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“…Example 5.2 Berenguer and Gamez (2018), Mirzaee and Piroozfar (2010) Consider the following linear 2D Fredholm integral equation…”

Section: Illustrative Examplesmentioning

confidence: 99%

“…A method based on the use of 2D triangular orthogonal functions (2D-TFs), for the numerical solution of (5.4), was introduced in Mirzaee and Piroozfar (2010). The numerical results obtained by the presented method together with the results of Berenguer and Gamez (2018) and Mirzaee and Piroozfar (2010) are displayed in Table 2. The graph of the absolute error e N , for N = 2, 3, is also plotted in Fig.…”

Section: Illustrative Examplesmentioning

confidence: 99%

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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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“…Example 5.2 Berenguer and Gamez (2018), Mirzaee and Piroozfar (2010) Consider the following linear 2D Fredholm integral equation…”

Section: Illustrative Examplesmentioning

confidence: 99%

Section: Illustrative Examplesmentioning

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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“…For the usual Schauder bases chosen, we could estimate the convergence rate of the sequence of projections taking into account the properties of such bases and the Mean-Value Theorem in a similar way to [8] and [9].…”

Section: Numerical Examplesmentioning

confidence: 99%

“…with τ (n) = (i, j) (see [7]). The usual Schauder basis {B n } n≥1 of C(Ω 2 ) (see [9]) is defined as:…”

Section: Numerical Examplesmentioning

confidence: 99%

Projected Iterations of Fixed-Point Type to Solve Nonlinear Partial Volterra Integro-Differential Equations

Berenguer

Gámez

2020

Bull. Malays. Math. Sci. Soc.

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In this paper, we propose a method to approximate the fixed point of an operator in a Banach space.Using biorthogonal systems, this method is applied to build an approximation of the solution of a class of nonlinear partial integro-differential equations. The theoretical findings are illustrated with several numerical examples, confirming the reliability, validity and precision of the proposed method.

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Numerical Solution for Nonlinear Problems

Rabbani¹

2022

Approximation Theory, Sequence Spaces and Applications

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Numerical solving of several types of two‐dimensional integral equations and estimation of error bound

Abstract: Using fixed‐point techniques and Faber‐Schauder systems in Banach spaces, we obtain an approximation of the solution of 2‐dimensional nonlinear Volterra, Fredholm, and mixed Volterra‐Fredholm integral equations.

Cited by 8 publications

References 34 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

Projected Iterations of Fixed-Point Type to Solve Nonlinear Partial Volterra Integro-Differential Equations

Numerical Solution for Nonlinear Problems

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