Asymptotic error estimates of a linearized projection-difference method for a differential equation with a monotone operator

We study a projection-difference method of solving the Cauchy problem for an operatordifferential equation with a selfadjoint leading operator A(t) and a nonlinear monotone subordinate operator K(·) in a Hilbert space. This method leads to a solution of a system of linear algebraic equations at each time level. Error estimates are derived for approximate solutions as well as for fractional powers of the operator A(t). The method is applied to a model parabolic problem.     

Feynman and quasi-Feynman formulas for evolution equations

New methods for obtaining representations of solutions of the Cauchy problem for linear evolution equations, i.e., equations of the form u _t '(t, x) = Lu(t, x), where the operator L is linear and depends only on the spatial variable x and does not depend on time t, are proposed. A solution of the Cauchy problem, that is, the exponential of the operator tL, is found on the basis of constructions proposed by the author combined with Chernoff’s theorem on strongly continuous operator semigroups.     

Cauchy problems for fractional differential equations with Riemann–Liouville fractional derivatives

In this paper, we are concerned with Cauchy problems of fractional differential equations with Riemann–Liouville fractional derivatives in infinite-dimensional Banach spaces. We introduce the notion of fractional resolvent, obtain some its properties, and present a generation theorem for exponentially bounded fractional resolvents. Moreover, we prove that a homogeneous α-order Cauchy problem is well posed if and only if its coefficient operator is the generator of an α-order fractional resolvent, and we give sufficient conditions to guarantee the existence and uniqueness of weak solutions and strong solutions of an inhomogeneous α-order Cauchy problem.     

Integration of some differential-difference nonlinear equations using the spectral theory of normal block Jacobi matrices

The following method for integrating the Cauchy problem for a Toda lattice on the half-line is well known: to a solution u(t), t ∈, [0, ∞), of the problem, one assigns a self-adjoint semi-infinite Jacobi matrix J(t) whose spectral measure dπ(λ; t) undergoes simple evolution in time t. The solution of the Cauchy problem goes as follows. One writes out the spectral measure dπ(λ; 0) for the initial value u(0) of the solution and the corresponding Jacobi matrix J(0) and then computes the time evolution dπ(λ; t) of this measure. Using the solution of the inverse spectral problem, one reconstructs the Jacobi matrix J(t) from dπ(λ; t) and hence finds the desired solution u(t). In the present paper, this approach is generalized to the case in which the role of J(t) is played by a block Jacobi matrix generating a normal operator in the orthogonal sum of finite-dimensional spaces with spectral measure dπ(ζ; t) defined on the complex plane. Some recent results on the spectral theory of these normal operators permit one to use the integration method described above for a rather wide class of differential-difference nonlinear equations replacing the Toda lattice.     

Conditions for the existence of a classical solution of a cauchy type problem for the diffusion equation with a Riemann-Liouville partial derivative

We study a Cauchy type problem for a differential equation containing a fractional Riemann-Liouville partial derivative of order α, 0 < α < 2. Conditions under which the solution of the problem tends to zero as |x| → ∞ are obtained. We prove an existence theorem for a classical solution of the Cauchy type problem and show that the solution has a singularity as t → 0 of order 1 ? α if 0 < α ≤ 1 and of order 2 ? α if 1 < α < 2.     

Degenerate parabolic systems of vector order Kolmogorov-type equations

We define a new class of degenerate \(\overrightarrow {2b}\)-parabolic systems of Kolmogorov-type equations with coefficients depending only on time. We also construct a fundamental matrix of solutions to a Cauchy problem for systems of this class and study its main properties.     

A class of degenerate fractional evolution systems in banach spaces

By using the notion of strongly (B, p)-sectorial operator and fractional differential calculus, we analyze the unique solvability of the Cauchy and Showalter problems for a class of degenerate fractional evolution systems. The results are used for the analysis of partial differential equations of fractional order with respect to the time variable.     

On Schrödinger type evolution equations with non-Lipschitz coefficients

We prove results of well-posedness of the global Cauchy problem in Sobolev spaces for a class of evolution equations with real characteristics that contains an Euler– Bernoulli vibrating beam model. We consider non-Lipschitz coefficients with respect to the time variable t and study the sharp rate of their oscillations. This is coupled with some necessary decay conditions as the spatial variable x → ∞.     

Game problems for fractional-order linear systems

The paper is concerned with studying approach game problems for linear conflict-controlled processes with fractional derivatives of arbitrary order. Namely, the classical Riemann-Liouville fractional derivatives, Dzhrbashyan-Nersesyan or Caputo regularized derivatives, and Miller-Ross sequential derivatives are considered. Under fixed controls of the players, solutions are presented in the form of analogs of the Cauchy formula with the use of generalized matrix Mittag-Leffler functions. The investigation is based on the method of resolving functions, which allows one to obtain sufficient conditions for the termination of the approach problem in some guaranteed time period. The results are exemplified by model game problems with a simple matrix and separated motions of fractional order π and e.     

Functions of finite fractional variation and their applications to fractional impulsive equations

We introduce a notion of a function of finite fractional variation and characterize such functions together with their weak σ-additive fractional derivatives. Next, we use these functions to study differential equations of fractional order, containing a σ-additive term—we prove existence and uniqueness of a solution as well as derive a Cauchy formula for the solution. We apply these results to impulsive equations, i.e. equations containing the Dirac measures.     

Estimates on fractional power dissipative equations in function spaces

We study the fractional power dissipative equations, whose fundamental semigroup is given by e^{−t(−Δ)α} with α>0. By using an argument of duality and interpolation, we extend space-time estimates of the fractional power dissipative equations in Lebesgue spaces to the Hardy spaces and the modulation spaces. These results are substantial extensions of some known results. As applications, we study both local and global well-posedness of the Cauchy problem for the nonlinear fractional power dissipative equation u_t+(−Δ)^αu=|u|^mu for initial data in the modulation spaces.     

Équations faiblement hyperboliques du deuxième ordre et classes de fonctions ultradifférentiables

We consider a second order weakly hyperbolic equation and we study in which classes the corresponding Cauchy problem is well posed. We consider operators with coefficients depending only on the t variable and belonging to a class X between C^∞ and the real analytic class. We find then a class, strictly related to X, where the Cauchy problem is well posed. Finally, we prove by some counterexamples that these results are almost optimal.     

Nonexistence of global solutions of nonlinear evolution equations

We consider general nonlinear evolution equations of arbitrary order. For these equations, we find conditions under which the Cauchy problem has no solutions global in t > 0. We also estimate the time beyond which the solution of the considered Cauchy problem necessarily does not exist.     

Fractional Abstract Cauchy Problems

This paper is concerned with fractional abstract Cauchy problems with order \({\alpha\in(1,2)}\). The notion of fractional solution operator is introduced, its some properties are obtained. A generation theorem for exponentially bounded fractional solution operators is given. It is proved that the homogeneous fractional Cauchy problem (FACP ₀) is well-posed if and only if its coefficient operator A generates an α-order fractional solution operator. Sufficient conditions are given to guarantee the existence and uniqueness of mild solutions and strong solutions of the inhomogeneous fractional Cauchy problem (FACP _f ).     

On uniqueness of fractional powers of multi-valued linear operators and the incomplete Cauchy problem

We obtain uniqueness of additive families {A _t } _t>0 of fractional powers of a multi-valued sectorial linear operator A ?? A ₁ in a Banach space, satisfying a certain kind of continuity with respect to the exponent and a spectral property, from uniqueness of the solutions of the second-order incomplete Cauchy problem associated with A. We show the close relationship between the multiplicativity and the uniqueness of fractional powers.     

The fundamental solution of the space-time fractional advection-dispersion equation

A space-time fractional advection-dispersion equation (ADE) is a generalization of the classical ADE in which the first-order time derivative is replaced with Caputo derivative of order α ∈ (0, 1], and the second-order space derivative is replaced with a Riesz-Feller derivative of order β ∈ (0, 2]. We derive the solution of its Cauchy problem in terms of the Green functions and the representations of the Green function by applying its Fourier-Laplace transforms. The Green function also can be interpreted as a spatial probability density function (pdf) evolving in time. We do the same on another kind of space-time fractional advection-dispersion equation whose space and time derivatives both replacing with Caputo derivatives.     

Global weak solutions of the Cauchy problem for semilinear pseudo-hyperbolic equations

The Cauchy problem for a class of semilinear pseudo-hyperbolic equations is considered. For the corresponding linear problems, we obtain L _p → L _q estimates. By using these estimates, we prove global solvability theorems. We also establish the behavior of solutions as t → + ∞.     

A Stability Estimate for a Solution to the Wave Equation with the Cauchy Data on a Timelike Cylindrical Surface

We consider the linear wave equation in a domain of the x, t-space bounded from above and below by some smooth surfaces and from the sides by a cylindrical surface with generator parallel to the t-axis. We study the Cauchy problem for this equation with data on a piece of the timelike cylindrical surface and establish a stability estimate for a solution to the problem.     

L p solvability of divergence type parabolic and elliptic systems with partially BMO coefficients

We prove the ${{\mathcal{H}}^{1}_{p,q}}$ solvability of second order systems in divergence form with leading coefficients A ^???? only measurable in (t, x ¹) and having small BMO (bounded mean oscillation) semi-norms in the other variables. In addition, we assume one of the following conditions is satisfied: (i) A ¹¹ is measurable in t and has a small BMO semi-norm in the other variables; (ii) A ¹¹ is measurable in x ¹ and has a small BMO semi-norm in the other variables. The corresponding results for the Cauchy problem and elliptic systems are also established. Some of our results are new even for scalar equations. Using the results for systems in the whole space, we obtain the solvability of systems on a half space and Lipschitz domain with either the Dirichlet boundary condition or the conormal derivative boundary condition.     

Error estimates for projection-difference methods for differential equations with differentiable operators

We study the projection-difference methods for approximate solving the Cauchy problem for operator-differential equations with a leading self-adjoint operator A(t) and a subordinate linear operator K(t), whose definition domain is independent of t. Operators A(t) and K(t) are assumed to be sufficiently smooth. We obtain estimates for the rate of convergence of approximate solutions to the exact solution as well as those for fractional degrees of an operator similar to A(0).