Qualitative and Numerical Analysis to a Time-Fractional Stefan Convection-Diffusive Model Using Riemann-Liouville and Caputo Operators
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Caputo derivative, Convection- Diffusion equation, Riemann-Liouville derivative, self-similar solution, Stefan problem, Sumudu Transform Time-fractional derivativesAbstract
The importance of the time-fractional Stefan problem(SP) comes from its wide physical applications. In this study, we assume an advective-diffusive flux to derive the model of the fractional SP for linear advection and diffusion forces depending on a realistic ice-melting problem. The rescaling technique is significant in estimating the self-similar solutions for the Stefan model. Also, we consider the interface function to satisfy the time fractional SP including boundary and Stefan conditions. The fractional derivative method, particularly Riemann-Liouville and Caputo derivatives, is used to find approximated solutions. The SP and other related phase transition problems typically have a constitutive relation between quantity, enthalpy, and temperature that requires thorough derivation based on physical grounds. Since physical understanding requires mathematical analysis, we provide analytical formulas for weak solutions to the SP because the classical analytical approaches break down in the case of the changing interface. On the other hand, the time-fractional diffusion-convection equation is considered to estimate an approximated numerical solution by applying the Sumudu decomposition method (SDM). The proposed method depends on applying the Sumudu transform of the Caputo fractional derivative operator and then using the fractional integral of Riemann-Liouville. These processes are useful in handling the nonlinear term with ease. It is discovered that the Sumudu approach is precise and quick. The MATLAB software carried out all the computations and graphics. To show that the suggested technique is valid and applicable, illustrative examples are provided.
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