Abstract:
In recent years, initial boundary value direct and inverse problems with fractional derivatives have become widespread for mathematical modeling in various fields of science. They are used in classical and quantum physics, field theory, solid mechanics, fluid and gas mechanics, general chemistry, nonlinear biology, stochastic analysis, nonlinear control theory, and image processing. The paper considers a one-dimensional mathematical model of anomalous diffusion, in which the order of the fractional time derivative is to be determined. The problem belongs to the class of inverse problems. The integral of the solution of the problem at the final moment of time with a non-negative weighting coefficient is given as a condition for redefinition. A discrete analogue of the problem posed is constructed by the finite-difference method; for the approximate calculation of a definite integral (overdetermination condition), the quadrature formula of trapezoids is used. For the numerical implementation of the obtained system of nonlinear equations, the iterative secant method is used.
Keywords:fractional time derivative, fractional diffusion equation, anomalous diffusion, inverse problem, finite difference method, identification of the order of fractional time derivative, iterative method.
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