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A new an‎d fast memory free method for approximation of the Caputo fractional derivatives

رياضي كاربردي و علوم كامپيوتر

As we know, fractional operators have a better an‎d more accurate description of existing phenomena. Many existing phenomena that require memory are better described by means of fractional derivatives. The set of theory of fractional calculus, which is a generalization of classical calculus, is widely used in all kinds of modeling such as viscoelastic systems, diffusion processes, generated spring energies, processing, control, etc. The methods of approximating Caputo’s fractional derivatives are divided into local methods (such as using (b) splines) an‎d global methods (such as using the polynomial interpolator method). Each of the mentioned methods has advantages an‎d disadvantages. Global methods are usually used for simple problems (linear problems); Also, their implementation is simple an‎d they have exponential accuracy. In local methods, the domains can be completely arbitrary an‎d the implementation of this method is simple for linear an‎d non-linear problems. Despite the merits that were said, this method leads to the production of a matrix with large dimensions, for its calculation we need a lot of memory, which is due to the non-locality of fractional derivatives (an‎d integrals). In this thesis, a novel approach based on the memory-free method for approximating Caputo fractional derivatives is introduced. The proposed method has been examined through several numerical examples, an‎d the results have been reported an‎d analyzed for accuracy an‎d efficiency. This innovative method demonstrates significant potential in improving the field of numerical analysis by providing a more effective means to calculate fractional derivatives.

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