Approximation of solutions for nonlinear functional integral equations

Mishra, Lakshmi Narayan; Pathak, Vijai Kumar; Băleanu, Dumitru

doi:10.3934/math.2022964

Cited by 13 publications

(3 citation statements)

References 47 publications

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“…have been done, and interesting qualitative results have been obtained. For further information, please go through Mishra et al (2022), Pathak et al (2023) and Pathak et al (2022).…”

Section: Introductionmentioning

confidence: 99%

Precision and efficiency of an interpolation approach to weakly singular integral equations

Bhat,

Mishra,

Mishra

et al. 2024

HFF

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Purpose This study aims to discuss the numerical solutions of weakly singular Volterra and Fredholm integral equations, which are used to model the problems like heat conduction in engineering and the electrostatic potential theory, using the modified Lagrange polynomial interpolation technique combined with the biconjugate gradient stabilized method (BiCGSTAB). The framework for the existence of the unique solutions of the integral equations is provided in the context of the Banach contraction principle and Bielecki norm. Design/methodology/approach The authors have applied the modified Lagrange polynomial method to approximate the numerical solutions of the second kind of weakly singular Volterra and Fredholm integral equations. Findings Approaching the interpolation of the unknown function using the aforementioned method generates an algebraic system of equations that is solved by an appropriate classical technique. Furthermore, some theorems concerning the convergence of the method and error estimation are proved. Some numerical examples are provided which attest to the application, effectiveness and reliability of the method. Compared to the Fredholm integral equations of weakly singular type, the current technique works better for the Volterra integral equations of weakly singular type. Furthermore, illustrative examples and comparisons are provided to show the approach’s validity and practicality, which demonstrates that the present method works well in contrast to the referenced method. The computations were performed by MATLAB software. Research limitations/implications The convergence of these methods is dependent on the smoothness of the solution, it is challenging to find the solution and approximate it computationally in various applications modelled by integral equations of non-smooth kernels. Traditional analytical techniques, such as projection methods, do not work well in these cases since the produced linear system is unconditioned and hard to address. Also, proving the convergence and estimating error might be difficult. They are frequently also expensive to implement. Practical implications There is a great need for fast, user-friendly numerical techniques for these types of equations. In addition, polynomials are the most frequently used mathematical tools because of their ease of expression, quick computation on modern computers and simple to define. As a result, they made substantial contributions for many years to the theories and analysis like approximation and numerical, respectively. Social implications This work presents a useful method for handling weakly singular integral equations without involving any process of change of variables to eliminate the singularity of the solution. Originality/value To the best of the authors’ knowledge, the authors claim the originality and effectiveness of their work, highlighting its successful application in addressing weakly singular Volterra and Fredholm integral equations for the first time. Importantly, the approach acknowledges and preserves the possible singularity of the solution, a novel aspect yet to be explored by researchers in the field.

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“…have been done, and interesting qualitative results have been obtained. For further information, please go through Mishra et al (2022), Pathak et al (2023) and Pathak et al (2022).…”

Section: Introductionmentioning

confidence: 99%

Precision and efficiency of an interpolation approach to weakly singular integral equations

Bhat,

Mishra,

Mishra

et al. 2024

HFF

Self Cite

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“…Many mathematical formulations in natural science, i.e., the study of fluids, biology, and chemical kinetics, contain integro-differential equations [ 5 , 21 , 23 , 24 ]. It can be classified into two types, i.e., Fredholm and Volterra equations.…”

Section: Introductionmentioning

confidence: 99%

Solving singularly perturbed fredholm integro-differential equation using exact finite difference method

Badeye,

Woldaregay,

Dinka

2023

BMC Res Notes

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Objectives In this paper, a numerical scheme is designed for solving singularly perturbed Fredholm integro-differential equation. The scheme is constructed via the exact (non-standard) finite difference method to approximate the differential part and the composite Simpson’s 1/3 rule for the integral part of the equation. Result The stability and uniform convergence analysis are demonstrated using solution bound and the truncation error bound. For three model examples, the maximum absolute error and the rate of convergence for different values of the perturbation parameter and mesh size are tabulated. The computational result shows, the proposed method is second-order uniformly convergent which is in a right agreement with the theoretical result.

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“…Delves and Mohamed [7], Kumar [8] and [9]. In [10], the approximation of solutions for nonlinear functional integral equations was examined by the authors.…”

Section: Introductionmentioning

confidence: 99%

Numerical Treatments of Functional Fredholm Integral Equation in 2D with Discontinuous Kernels

Al-Bugami

2023

JAMP

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This work proposes a new definition of the functional Fredholm integral equation in 2D of the second kind with discontinuous kernels (FT-DFIE). Furthermore, the work is concerned to study this new equation numerically. The existence of a unique solution of the equation is proved. In addition, the approximate solutions are obtained by two powerful methods Toeplitz Matrix Method (TMM) and Product Nyström Methods (PNM). The given numerical examples showed the efficiency and accuracy of the introduced methods.

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Approximation of solutions for nonlinear functional integral equations

Cited by 13 publications

References 47 publications

Precision and efficiency of an interpolation approach to weakly singular integral equations

Precision and efficiency of an interpolation approach to weakly singular integral equations

Solving singularly perturbed fredholm integro-differential equation using exact finite difference method

Numerical Treatments of Functional Fredholm Integral Equation in 2D with Discontinuous Kernels

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