تخمين هاي خطاي روشطيفيپتروف-گالركين براي معادلات انتشار-واكنشكسري دوطرفه

The increasing use of mathematical models based on fractional derivatives in diverse fields such as basic sciences, engineering, astronomy, medical sciences, an‎d other areas has highlighted the necessity an‎d importance of studying this class of equations more than ever. This thesis investigates the regularity an‎d numerical solution of the bidirectional fractional reaction–diffusion equation by employing the Petrov–Galerkin spectral method. In the first stage, the fundamental concepts of fractional calculus, including Riemann–Liouville an‎d Caputo derivatives, as well as the properties of special functions such as Gamma, Beta, an‎d Mittag–Leffler functions, are introduced. Functional spaces such as Hilbert, Banach, an‎d weighted Sobolev spaces are examined as the framework for analyzing the regularity of solutions. Moreover, classical orthogonal polynomials (Jacobi, Legendre, an‎d Chebyshev) an‎d Sturm–Liouville theory are presented as the central tools of spectral methods. Weighted residual methods an‎d numerical integration techniques, including Gauss–Jacobi quadrature, form the computational framework of this study. Subsequently, the bidirectional fractional reaction–diffusion equation is investigated, an‎d it is shown that the solution regularity in weighted Sobolev spaces is significantly improved compared to stan‎dard Sobolev spaces. By exploiting this improved regularity, the Petrov–Galerkin spectral method is implemented for the numerical solution of the equation, an‎d its discrete formulation together with the solution procedure is presented. Error analysis of the method, including projection error estimates an‎d stability considerations, is carried out, an‎d the optimal convergence rate of the method is established theoretically. Numerical results from simulations confirm the theoretical analysis an‎d validate the predicted convergence rate.

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