Well-posedness of fractional parabolic equations

Ashyralyev Boundary Value Problems 2013, 2013:31 http://www.boundaryvalueproblems.com/content/2013/1/31 RESEARCH Open Access Well-posedness of fractional parabolic equations Allaberen Ashyralyev* * Correspondence: Department of Mathematics, Fatih University, Istanbul, 34500, Turkey Abstract In the present paper, we consider the abstract Cauchy problem for the fractional differential equation 1 du(t) + Dt2 u(t) + A(t)u(t) = f (t), dt 0 < t < 1, u(0) = 0 () in an arbitrary Banach space E with the strongly positive operators A(t). The well-posedness of this problem in spaces of smooth functions is established. The coercive stability estimates for the solution of problems for 2mth order multidimensional fractional parabolic equations and one-dimensional fractional parabolic equations with nonlocal boundary conditions in a space variable are obtained. The stable difference scheme for the approximate solution of this problem is presented. The well-posedness of the difference scheme in difference analogues of spaces of smooth functions is established. In practice, the coercive stability estimates for the solution of difference schemes for the fractional parabolic equation with nonlocal boundary conditions in a space variable and the 2mth order multidimensional fractional parabolic equation are obtained. MSC: 65M12; 65N12 Keywords: fractional parabolic equation; Basset problem; well-posedness; coercive stability 1 Introduction It is known that differential equations involving derivatives of noninteger order have shown to be adequate models for various physical phenomena in areas like rheology, damping laws, diffusion processes, etc. Methods of solutions of problems for fractional differential equations have been studied extensively by many researchers (see, e.g., [–] and the references given therein). The role played by coercive stability inequalities (well-posedness) in the study of boundary value problems for parabolic partial differential equations is well known (see, e.g., [– ]). In the present paper, the initial value problem  du(t) + Dt u(t) + A(t)u(t) = f (t), dt  < t < , u() =  () for the fractional differential equation in an arbitrary Banach space E with the linear (unbounded) operators A(t) is considered. Here u(t) and f (t) are the unknown and the given © 2013 Ashyralyev; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Ashyralyev Boundary Value Problems 2013, 2013:31 http://www.boundaryvalueproblems.com/content/2013/1/31 Page 2 of 18 functions, respectively, defined on [, T] with values in E. The derivative u (t) is understood as the limit in the norm of E of the corresponding ratio of differences. A(t) is a given closed linear operator in E with the domain D(A(t)) = D, independent of t and dense in E. Finally, u() = .    is the standard Riemann-Liouville derivative of order  . This fractional Here Dt = D+ differential equation corresponds to the Basset problem []. It represents a classical problem in fluid dynamics where the unsteady motion of a particle accelerates in a viscous fluid due to the gravity of force. Recently, fractional Basset equations with independent in t operator coefficients A(t) = A have been studied extensively (see, e.g., [–] and the references given therein). In the present paper, the well-posedness of problem () with dependent in t operator coefficients A(t) in spaces of smooth functions is established. In practice, the coercive stability estimates for the solution of problems for mth order multidimensional fractional parabolic equations and one-dimensional fractional parabolic equations with nonlocal boundary conditions in a space variable are obtained. The stable difference scheme for the approximate solution of initial value problem () ⎧ ⎨τ – (uk – uk– ) + Ak uk + ⎩f = f (t ), k k √ π Ak = A(tk ), k m= (k–m+  ) um –um–  (k–m)! τ tk = kτ , = fk ,  ≤ k ≤ N, () Nτ = , u =  ∞  is presented. Here (k – m +  ) =  t k–m–  e–t dt. The paper is organized as follows. The well-posedness of problem () in spaces of smooth functions is established in Section . In Section  the coercive stability estimates for the solution of problems for mth order multidimensional fractional parabolic equations and one-dimensional fractional parabolic equations with nonlocal boundary conditions are obtained. The well-posedness of () in difference analogues of spaces of smooth functions is established and the coercive stability estimates for the solution of difference schemes for the fractional parabolic equation with nonlocal boundary conditions in a space variable and the mth order multidimensional fractional parabolic equation are obtained in Section . 2 The well-posedness of problem (2) A function u(t) is called a solution of problem () if the following conditions are satisfied: (i) u(t) is continuously differentiable on the segment [, ]. The derivatives at the endpoints of the segment are understood as the appropriate unilateral derivatives. (ii) The element u(t) belongs to D(A(t)) for all t ∈ [, ] and the function A(t)u(t) is continuous on the segment [, ]. (iii) u(t) satisfies the equation and the initial condition (). A solution of problem () defined in this manner will from now on be referred to as a solution of problem () in the space C(E) = C([, ], E) of all continuous functions ϕ(t) defined on [, ] with values in E equipped with the norm   ϕC(E) = max ϕ(t)E . ≤t≤ () Ashyralyev Boundary Value Problems 2013, 2013:31 http://www.boundaryvalueproblems.com/content/2013/1/31 Page 3 of 18 In this paper, positive constants, which can differ in time, are indicated with an M. On the other hand, M(α, β, . . .) is used to focus on the fact that the constant depends only on α, β, . . . . The well-posedness in C(E) of boundary value problem () means that the coercive inequality     u  + A(·)uC(E) ≤ Mf C(E) C(E) () is true for its solution u(t) ∈ C(E). Suppose that for each t ∈ [, ] the operator –A(t) generates an analytic semigroup exp{–sA(t)} (s ≥ ) with an exponentially decreasing norm, when s → +∞, i.e., the following estimates       exp –sA(t)  , sA(t) exp –sA(t) E→E ≤ Me–δs E→E (s > ) () hold for some M ∈ [, +∞), δ ∈ (, +∞). From this inequality it follows the operator A– (t) exists and is bounded, and hence A(t) is closed in C(E). Suppose that the operator A(t)A– (s) is Hölder continuous in t in the uniform operator topology for each fixed s, that is,    A(t) – A(τ ) A– (s) ≤ M|t – τ |ε , E→E  < ε ≤ ,  ≤ t, s, τ ≤ . () An operator-valued function v(t, s), defined and strongly continuous jointly in t, s for  ≤ s < t ≤ , is called a fundamental solution of () if () the operator v(t, (...truncated)