Abstract:
Recently, to describe various mathematical models of physical processes, fractional differential calculus has been widely used. In this regard, much attention is paid to partial differential equations of fractional order, which are a generalization of partial differential equations of integer order. In this case, various settings are possible.
Loaded differential equations in the literature are called equations containing values of a solution or its derivatives on manifolds of lower dimension than the dimension of the definitional domain of the desired function. Currently, numerical methods for solving loaded partial differential equations of integer and fractional orders are widely used, since analytical solving methods for solving are impossible. A fairly effective method for solving this kind of problem is the finite difference method, or the grid method.
We studied the initial-boundary value problem in the rectangle $\bar{D}=\{(x,t): 0\leq x\leq l, 0\leq t\leq T\}$ for the loaded differential heat equation with composition fractional derivative of Riemann – Liouville and Caputo – Gerasimov and with boundary conditions of the first and third kind. We have gotten an a priori assessment in differential and difference interpretations. The obtained inequalities mean the uniqueness of the solution and the continuous dependence of the solution on the input data of the problem. A difference analogue of the composition fractional derivative of Riemann – Liouville and Caputo –Gerasimov order $(2-\beta)$ is obtained and a difference scheme is constructed that approximates the original problem with the order $O(\tau+h^{2-\beta})$. The convergence of the approximate solution to the exact one is proven at a rate equal to the order of approximation of the difference scheme.
Keywords:boundary value problem, a priori estimate, method of energy inequalities, Caputo – Gerasimov fractional derivative, Riemann – Liouville fractional derivative
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