Application of Fixed Point Theorem to Solvability for Non-Linear Fractional Hadamard Functional Integral Equations

Pathak, Vijai Kumar; Mishra, Lakshmi Narayan

doi:10.3390/math10142400

Cited by 17 publications

(6 citation statements)

References 33 publications

(4 reference statements)

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“…Ramadan et al (2020) improved the method by combining the homotopy perturbation approach with Taylor’s original power series formula to find the exact solution of integral equation. The results related to uniqueness (and existence) were mentioned in Agarwal and O’Regan (2000), Mishra and Sen (2016), Pathak and Mishra (2022), Pathak and Mishra (2023) and Wang et al (2014), resolvent methods (Becker, 2016; Farid et al , 2022; Paul et al , 2023; Paul et al , 2023a, 2023b), etc. In He (2000), the author provided a novel iteration method to solve autonomous ordinary differential systems.…”

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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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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“…It is worthwhile mentioning that Erdélyi-Kober integrals are applied to describe processes with continuously distributed scaling, medium with noninteger mass dimension, modeling of viscoelastic materials, and advection and dispersion of solutes in porous media. Quadratic integral equations have many potential applications in describing a great number of events, and problems of the real world, that can benefit both the understanding of profound complexities in the application and the field of fractional calculus itself [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40].…”

Section: Introductionmentioning

confidence: 99%

“…The practical applications of nonlinear integral equations have recently received a lot of attention from studies that incorporate the same in distinct areas of knowledge that include mathematical modeling of real‐world problems in various branches of science, like chemistry, physics, electrical networks, control of the dynamic system, optics, biological science, signal processing, and acoustic scattering [1–7]. Precisely, the recent development on these equations is focused on their solutions by using the measure of noncompactness technique [8–22].…”

Section: Introductionmentioning

confidence: 99%

On the solvability of a class of nonlinear functional integral equations involving Erdélyi–Kober fractional operator

Pathak

Mishra

2023

Math Methods in App Sciences

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In this paper, utilizing the technique of generalized Darbo's fixed‐point theorem associated with measure of noncompactness in Banach space, we analyze the existence of solution for a class of nonlinear functional integral equations involving Erdélyi–Kober fractional operator. The existing result was obtained to strengthen the ones mentioned previously in the literature. An example for a class of nonlinear functional integral equations is also presented to validate our main result. Finally, we propose the numerical method formed by the modified homotopy perturbation approach to resolving the problem with acceptable accuracy.

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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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Application of Fixed Point Theorem to Solvability for Non-Linear Fractional Hadamard Functional Integral Equations

Cited by 17 publications

References 33 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

On the solvability of a class of nonlinear functional integral equations involving Erdélyi–Kober fractional operator

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

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