Analysis and Application of an Iterative Numerical Method for Volterra Integral Equations of the Second Kind

Nazhan Al-Din Ahmed Jasim Obeid

doi:10.32792/jeps.v16i1.749

المؤلفون

Nazhan Al-Din Ahmed Jasim Obeid Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran

DOI:

https://doi.org/10.32792/jeps.v16i1.749

الكلمات المفتاحية:

Volterra, Iteration, Numerical Method, Fixed Point, Singular Kernel, Approximation.

الملخص

Volterra integral equations of the second kind are important for studying systems that remember past events and change over time. They are often used in physics, biology, and engineering. These equations are usually hard to solve with exact methods because they are complicated, especially when the kernel function or source term includes nonlinearity or cannot be expressed in a simple formula. So, numerical methods are important tools for getting close answers that are very accurate. This study looks at a straightforward and useful method for solving a specific type of math problem called second kind Volterra integral equations. The method divides the time into equal parts and starts with an initial guess for the unknown function. The solution is gradually improved through several rounds until it is accurate enough. The method skips complicated math with matrices and instead uses the way the integral equation is set up to calculate values one by one in a forward way. To check if the method works, we used several test problems that have known answers. The results show that the numbers we calculated mostly match the exact answers, with very small mistakes and quick improvements. The method showed that it works well and is quick, needing only a few tries to get good results. These features make it really good for real-life situations where fast and dependable number solutions are required. In short, the suggested step-by-step numerical method is a trustworthy, precise, and simple way to solve second-kind Volterra integral equations. It is especially helpful when regular ways of analyzing data can't be used. This makes it a valuable tool for researchers and engineers in different scientific and technical areas.

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