The exact solution of a class of Volterra integral equation with weakly singular kernel

Zhong, Chen; Jiang, Wei

doi:10.1016/j.amc.2011.02.059

Cited by 8 publications

(4 citation statements)

References 16 publications

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“…In this section, we present a solution approach using our new force function formula for the nonlinear WSVIE, leading to a unique solution. Let us consider the general form of the weakly singular Volterra integral equation in [10]…”

Section: Implementation Of the Djm And The Force Function Formulamentioning

confidence: 99%

A Force Function Formula for Solutions of Nonlinear Weakly Singular Volterra Integral Equations

Sarfo,

Obeng-Denteh,

Takyi

et al. 2024

Eur. J. Pure Appl. Math.

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In this paper, we examine the nonlinear Weakly Singular Volterra Integral Equation(WSVIE), $u(x)=f(x)+\int_{0}^{x}\frac{t^{\mu-1}}{x^\mu}[u(t)]^\beta dt$. AL-Jawary and Shehan used Daftardar-Jafari Method(DJM) and solved the above integral equation for the investigation parameter $\mu>1$ using specific force functions with $\mu$ and $\beta$ values and obtained unique solutions. We have discovered a force function $f(x)=x^{k_1}-{\frac{x^{\gamma k_1}}{\gamma k_1+\mu}}$, that allows the introduction of noise terms phenomena discovered by Wazwaz; that cancel out the terms of the power series in the successive solution terms $u_m$, $m=0,1,2,...,n$: we thus obtain a maximum finite power series terms for each solution term called truncation point and denoted by $x^{g(n)}$. Such that the integral solution can be written as $u(x)=u_0+\sum_{m=1}^{n}u_m$, where $n$ is finite. Simplifying the solution terms we get the unique solution $u(x)=x^{k_1}$, irrespective of $n-$value in the truncation point. We discovered a formula relation between the last solution term $u_n$ and the truncation point as $u_n=a_nx^{g(n)}$. Our results confirm the results of the two solution examples of AL-Jawary and Shehan for the investigation parameter $\mu>1$. We extend the parameter range to include $\mu>1$ and $0<\mu\leq 1$ for our solution. In addition, for any chosen rational parameter $k_1$, the solution $u(x)=x^{k_1}$ is extrapolated to be valid for all integer parameter values $\beta\geq 2$, and positive rational parameter value $\mu>0$ and for any finite value of $n\geq2$.

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Section: Implementation Of the Djm And The Force Function Formulamentioning

confidence: 99%

A Force Function Formula for Solutions of Nonlinear Weakly Singular Volterra Integral Equations

Sarfo,

Obeng-Denteh,

Takyi

et al. 2024

Eur. J. Pure Appl. Math.

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“…It is difficult to solve these equations analytically, hence numerical solutions are required. Singular integral equations have been approached by different methods including Collocation method [2][3][4], Reproducing kernel method [17], Galerkin method [5], Adomian decomposition method [1], Homotopy perturbation method [6], Radial Basis Functions [10,11], Newton product integration method [7], and many others.…”

Section: Introductionmentioning

confidence: 99%

RBFs for Integral Equations with a Weakly Singular Kernel

Biazar

Asadi²

2015

AJAM

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In this paper a numerical method, based on collocation method and radial basis functions (RBF) is proposed for solving integral equations with a weakly singular kernel. Integrals appeared in the procedure of the solution are approximated by adaptive Lobatto quadrature rule. Illustrative examples are included to demonstrate the validity and applicability of the presented technique. In addition, the results of applying the method are compared with those of Homotopy perturbation, and Adomian decomposition methods.

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“…It is well known that there are many numerical methods for solving second kind Volterra integral equations such as the Runge-Kutta method and the collocation method based on piecewise polynomials; see, for example, Brunner [1] and references therein. For more information of the progress on the study of the problem, we refer the readers to [2][3][4][5][6][7][8]. Recently, a few works touched the spectral approximation to Volterra integral equations.…”

Section: Introductionmentioning

confidence: 99%

A Jacobi-Collocation Method for Second Kind Volterra Integral Equations with a Smooth Kernel

Guo

Cai

Xin

2014

Abstract and Applied Analysis

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The purpose of this paper is to provide a Jacobi-collocation method for solving second kind Volterra integral equations with a smooth kernel. This method leads to a fully discrete integral operator. First, it is shown that the fully discrete integral operator is stable in bothL∞and weightedL2norms. Then, the proposed approach is proved to arrive at an optimal (the most possible) convergent order in both norms. One numerical example demonstrates the efficiency and accuracy of the proposed method.

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The exact solution of a class of Volterra integral equation with weakly singular kernel

Cited by 8 publications

References 16 publications

A Force Function Formula for Solutions of Nonlinear Weakly Singular Volterra Integral Equations

A Force Function Formula for Solutions of Nonlinear Weakly Singular Volterra Integral Equations

RBFs for Integral Equations with a Weakly Singular Kernel

A Jacobi-Collocation Method for Second Kind Volterra Integral Equations with a Smooth Kernel

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