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Abstract
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Two-dimensional Volterra integral equations with weakly singular kernels arise in numerous applications, including heat conduction, potential theory, fluid mechanics, and biological systems, where they model processes with memory or spatial interactions. The presence of weak singularities in the kernels poses significant challenges for traditional numerical methods, often leading to reduced accuracy, slow convergence, and high computational cost. This creates a pressing need for efficient and reliable solution techniques that can handle singular behavior while maintaining precision. A semi-analytical approach combining Laplace expansion with Padé approximants offers a promising solution by transforming the problem into a form that allows for accurate series representation and rational approximation, thereby enhancing convergence and stability. The importance of this research lies in providing a computationally efficient and high-accuracy framework for solving complex two-dimensional Volterra integral equations, contributing both to theoretical developments and practical applications in engineering and applied sciences.
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