A priori estimates for solutions of boundary value problems for fractional-order equations

Abstract

We consider boundary value problems of the first and third kind for the diffusionwave equation. By using the method of energy inequalities, we find a priori estimates for the solutions of these boundary value problems.Fractional calculus is used for the description of a large class of physical and chemical processes that occur in media with fractal geometry as well as in the mathematical modeling of economic and social-biological phenomena [1, Chap. 5]. It was proved in [1] that fractional differentiation is a positive operator; this result permits one to obtain a priori estimates for solutions of a wide class of boundary value problems for equations with fractional derivatives. The paper [2] deals with the generalization of the differentiation and integration operations from integer to fractional, real, and complex orders and with applications of fractional integration and differentiation to integral and differential equations and in function theory. In the paper [3], an a priori estimate in terms of a fractional Riemann-Liouville integral of the solution was obtained for the solution of the first initial-boundary value problem for the fractional diffusion equation. The general fractional diffusion equation (0 < α ≤ 1) with regularized fractional derivative was considered in [4]. A more detailed bibliography on fractional partial differential equations, including the diffusion-wave equation, can be found, for example, in [5].In the present paper, we use the method of energy inequalities to obtain a priori estimates for solutions of boundary value problems for the diffusion-wave equation with Caputo fractional derivative [6].

Dirichlet boundary value problem for a third order parabolic-hyperbolic equation with degenerating type and order in the hyperbolicity domain

“…We note that the proven Lemma 3.1 is an analogue of Lemma 1 in work [32]. As | | < 2 , by (3.17) and (3.20) we arrive at the inequality…”

Stability and convergence of difference schemes for boundary value problems for the fractional-order diffusion equation

“…As , the condition for stability (15) passes to the well-known condition (see [19]) for the classical diffusion equation:…”

“…In [12][13][14], from the maximum principle, a priori estimates for solutions of difference problems for both one-dimensional and multidimensional fractional-order diffusion equation were obtained. In [15], by the method of energetic inequalities, a priori estimates of solutions of boundary value problems for the diffusion-wave equation were obtained. In [16], a priori estimates for boundary value problems for the fractional-order diffusion equation for differential and finite-difference problems were found.…”

On a singular time-fractional order wave equation with Bessel operator and Caputo derivative

“…Our work can be considered as a further elaboration and generalization of the results obtained in [3], where the author proved the uniqueness of solution of an initial boundary value problem for a fractional wave equation subject to some homogeneous initial and classical boundary conditions. In our case, we have proved the uniqueness as well as the existence of solution for a mixed problem with Bessel operator.…”

“…Thanks to Lemma 3.3 in [3] which allows us to estimate the first term on the LHS of (4.8) as follows…”