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Abstract
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Fractional sub-diffusion equations have become an essential tool in modeling complex transport processes that cannot be described adequately by classical diffusion equations. Unlike normal diffusion, sub-diffusion accounts for memory effects and anomalous transport behavior, which appear in various physical, chemical, and biological systems such as porous media, viscoelastic materials, groundwater flow, and cell biology. Traditionally, these equations were modeled using singular kernel operators, such as the Riemann–Liouville or Caputo derivatives. However, such formulations often pose mathematical and numerical difficulties due to weak singularities, high computational costs, and reduced accuracy in long-time simulations.
To address these limitations, researchers have proposed fractional derivatives with non-singular kernels, such as the Caputo–Fabrizio and Atangana–Baleanu operators. These models are advantageous because they avoid singularities and provide more realistic descriptions of memory effects in anomalous diffusion processes. Nevertheless, solving these equations analytically remains extremely difficult, and most practical problems require efficient and accurate numerical schemes.
The necessity of this research arises from the demand for reliable computational methods that can effectively approximate solutions of fractional sub-diffusion equations with non-singular kernels. Conventional numerical methods often struggle with stability, accuracy, and efficiency, particularly when dealing with fractional operators. In this context, exponential B-spline methods present a powerful alternative. Due to their flexibility, local support, and smoothness properties, exponential B-splines are well-suited for handling fractional models and capturing steep gradients or rapidly decaying solutions.
The importance of this research lies in its contribution to advancing numerical methods for fractional differential equations. By employing an exponential B-spline approach,
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