Fixed‐Point Iterative Algorithm for the Linear Fredholm‐Volterra Integro‐Differential Equation

Berenguer, M.; Gámez, D.; Linares, A. J. López

doi:10.1155/2012/370894

Cited by 10 publications

(3 citation statements)

References 13 publications

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“…For Fredholm-Volterra IDEs, various methods were utilized. [7]employed the projection method based on a Bernstein collocation approach; [8] used the Bernstein collocation method; [9] applied a fixed-point iterative algorithm; and [10] employed a collocation method based on Bernstein polynomials. In [11], a new numerical method was developed specifically for solving systems of Volterra IDEs.…”

Section: Introductionmentioning

confidence: 99%

Vieta-Lucas Polynomial Computational Tecnique for Volterra Integro-Differential Equations

Oyedepo,

Ayinde,

Didigwu

2024

Electronic Journal of Mathematical Analysis and Applications

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In this study, we introduce a computational technique for tackling Volterra Integro-Differential Equations (VIDEs) using shifted Vieta-Lucas polynomials as the foundational basis functions. The approach involves adopting an approximative solution strategy through the utilization of Vieta-Lucas polynomials. These polynomials are then integrated into the pertinent VIDEs. Subsequently, the resulting equation is subjected to collocation at evenly spaced intervals, generating a system of linear algebraic equations with unspecified Vieta-Lucas coefficients. To solve this system, we employ a matrix inversion method to deduce the unknown constants. Once these constants are determined, they are incorporated into the earlier assumed approximate solution, thus yielding the sought-after approximated solution. To validate the accuracy and efficiency of this technique, we conducted numerical experiments. The obtained results underscore the outstanding performance of our method in comparison to outcomes found in existing literature. The precision and effectiveness of the approach are further illustrated through the utilization of tables.

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Section: Introductionmentioning

confidence: 99%

Vieta-Lucas Polynomial Computational Tecnique for Volterra Integro-Differential Equations

Oyedepo,

Ayinde,

Didigwu

2024

Electronic Journal of Mathematical Analysis and Applications

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“…For Fredholm-Volterra IDEs, various methods were utilized. [10] employed the projection method based on a Bernstein collocation approach; [11] used the Bernstein collocation method; [12] applied a fixed-point iterative algorithm; [13] utilized the Chebyshev polynomial approach; and [14] employed a collocation method based on Bernstein polynomials. In [15], a new numerical method was developed specifically for solving systems of Volterra IDEs.…”

Section: Introductionmentioning

confidence: 99%

Legendre Computational Algorithm for Linear Integro-Differential Equations

OYEDEPO,

AYOADE,

AJİLEYE

et al. 2023

Cumhuriyet Science Journal

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This work presents a collocation computational algorithm for solving linear Integro-Differential Equations (IDEs) of the Fredholm and Volterra types. The proposed method utilizes shifted Legendre polynomials and breaks down the problem into a series of linear algebraic equations. The matrix inversion technique is then employed to solve these equations. To validate the effectiveness of the suggested approach, the authors examined three numerical examples. The results obtained from the proposed method were compared with those reported in the existing literature. The findings demonstrate that the proposed algorithm is not only accurate but also efficient in solving linear IDEs. In order to present the results, the study employs tables and figures. These graphical representations aid in displaying the numerical outcomes obtained from the algorithm. All calculations were performed using Maple 18 software.

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“…Getting accurate approximations using numerical techniques will be very helpful because many IDEs cannot be solved analytically. The following are just a few of the authors who have offered numerical approaches to solve IDEs: Rationalized Haar function approach is used by [2] to solve a system of linear IDEs, Adomian decomposition method is implemented in [3] to solve BVPs for fourthorder IDEs, utilizing a variational iteration approach, [4] presented the solution of fourth order IDEs, Applying the differential transform method to solve high-order nonlinear Volterra-Fredholm IDEs is implemented by [5], For the solving linear FVIDE, [6] applied a fixed-point iterative algorithm, For solving Fredholm-Volterra Integro-Differential Equations (FVIDEs), [7], [8], and [9] used Chebyshev polynomials as basis functions, while [10] employed the Chebyshev wavelet approximation analytical solution for high-order IDEs. [11] presented a novel numerical method using the Chebyshev third-kind polynomials.…”

mentioning

confidence: 99%

Collocation Computational Algorithm for Volterra-Fredholm Integro-Differential Equations

Oyedepo¹,

Ishola

Ayoade

et al. 2023

Electronic Journal of Mathematical Analysis and Applications

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In this study, we present a collocation computational technique for solving Volterra-Fredholm Integro-Differential Equations (VFIDEs) via fourth kind Chebyshev polynomials as basis functions. The method assumed an approximate solution by means of the fourth kind Chebyshev polynomials, which were then substituted into the Volterra-Fredholm Integro-Differential Equations (VFIDEs) under consideration. Thereafter, the resulting equation is collocated at equally spaced points, which results in a system of linear algebraic equations with the unknown Chebyshev coefficients. The system of equations is then solved using the matrix inversion approach to obtain the unknown constants. The unknown constants are then substituted into the assumed approximate solution to obtain the required approximate solution.To test for the accuracy and efficiency of the scheme, six numerical examples were solved, and the results obtained show the method performs excellently compared to the results in the literature. Also, tables are used to outline the methods accuracy and efficiency.

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Fixed‐Point Iterative Algorithm for the Linear Fredholm‐Volterra Integro‐Differential Equation

Cited by 10 publications

References 13 publications

Vieta-Lucas Polynomial Computational Tecnique for Volterra Integro-Differential Equations

Vieta-Lucas Polynomial Computational Tecnique for Volterra Integro-Differential Equations

Legendre Computational Algorithm for Linear Integro-Differential Equations

Collocation Computational Algorithm for Volterra-Fredholm Integro-Differential Equations

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