Volume 9 | Issue 2 | July 2024

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 Abstract

The stability and convergence analysis of solutions to Volterra integral equations, which are essential for simulating dynamic systems with memory or hereditary features, are thoroughly examined in this research. The behavior of numerical techniques for approximating the solutions of first and second kind Volterra integral equations is studied. Stability analysis highlights the resilience of the solutions by ensuring that minor adjustments to the original data or parameters do not result in appreciable variations in the outcomes. Convergence analysis measures how near numerical approximations are to the precise answer when the discretization is improved in order to examine their accuracy. The performance of approaches like iterative procedures and quadrature-based schemes is examined under various circumstances. The findings highlight how crucial step size, kernel characteristics, and numerical stability standards are to producing accurate approximations. The theoretical results are illustrated with case studies and real-world examples, showing how effective the suggested techniques are in producing stable and convergent solutions for Volterra integral equations. The numerical treatment of integral equations in applied mathematics, physics, and engineering is advanced by this study. A family of equations known as Volterra integral equations includes an unknown function under an integral sign. A wide range of engineering, biological, and physical phenomena are modeled using them. We give stability and convergence analysis of the Volterra integral equation results in this paper. The Volterra integral equation is approximated using a numerical approach based on the trapezoidal rule. The method's stability and convergence are then examined using a number of methods, such as the Banach fixed-point theorem and the Lipschitz condition. Our findings demonstrate that the method is convergent and stable, and that by adding more grid points, the method's accuracy may be raised.

 Keywords

Volterra integral equations, Stability analysis, Convergence analysis, Numerical methods, Trapezoidal rule, Lipschitz condition, Banach fixed- point theorem, Integral equations, Mathematical modelling, Numerical analysis, Computational mathematics

 Authors and Affiliations

Dr. Hetram Suryavanshi
Assistant Professor, Department of Mathematics, Vishwavidyalaya Engineering College, Ambikapur,Chhattisgarh, India
Dr.Gopi Sao
Associate Professor & Head, Department of Mathematics, School of Engineering, Eklavya University, Damoh, Madhya Pradesh, India

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