Related papers: The initial value problem for ordinary differentia…
In this paper, we introduce a novel semi-analytical method for solving a broad class of initial value problems involving differential, integro-differential, and delay equations, including those with fractional and variable-order…
The momentous objective of this work is to discuss some qualitative properties of solutions such as the estimate on the solutions, the continuous dependence of the solutions on initial conditions as well as the existence and uniqueness of…
The Cauchy problem for fractional derivatives linear systems of ordinary differential equations with constant coefficients is considered, where at first the analytic expressions are given through the matrix exponent of its corresponding…
In the article, in a rectangular domain, by the Fourier method, the initial boundary value problem for a high-order equation with two lines of degeneracy with a fractional derivative in the sense of Caputo is investigated for uniqueness and…
We develop an existence, regularity and potential theory for nonlinear integrodifferential equations involving measure data. The nonlocal elliptic operators considered are possibly degenerate and cover the case of the fractional…
In this manuscript we deal with existence/uniqueness and regularity issues of suitable weak solutions to nonlocal problems driven by fractional Laplace type operators. Different from previous researches, in our approach we consider gradient…
Using both fractional derivatives, defined in the Riemann-Liouville and Caputo senses, and classical derivatives of the integer order we examine different numerical approaches to ordinary differential equations. Generally we formulate some…
We prove multidimensional integration by parts formulas for generalized fractional derivatives and integrals. The new results allow us to obtain optimality conditions for multidimensional fractional variational problems with Lagrangians…
In this paper, we consider the indefinite fractional elliptic problem. A corresponding Liouville-type theorem for the indefinite fractional elliptic equations is established. Furthermore, we obtain a priori bound for solutions in a bounded…
Analytical and numerical techniques have been developed for solving fractional partial differential equations (FPDEs) and their systems with initial conditions. However, it is much more challenging to develop analytical or numerical…
An initial-boundary value problem for a time-fractional subdiffusion equation with an arbitrary order elliptic differential operator is considered. Uniqueness and existence of the classical solution of the posed problem are proved by the…
We consider the following class of fractional parametric problems \begin{equation*} \left\{ \begin{array}{ll} (-\Delta_{Dir})^{s} u= f(x, u)+t\varphi_{1}+h &\mbox{ in } \Omega\\ u=0 &\mbox{ on } \partial \Omega, \end{array} \right.…
Fractional calculus generalizes the derivative and antiderivative operations of differential and integral calculus from integer orders to the entire complex plane. Methods are presented for using this generalized calculus with Laplace…
Infinite order differential equations have come to play an increasingly significant role in theoretical physics. Field theories with infinitely many derivatives are ubiquitous in string field theory and have attracted interest recently also…
We study initial value problems having dynamics ruled by discontinuous ordinary differential equations with the property of possessing a unique solution. We identify a precise class of such systems that we call solvable intitial value…
Welcome to a beautiful subject in scientific computing: numerical solution of ordinary differential equations (ODEs) with initial conditions.
Further investigations of implicit solutions to non-linear partial differential equations are pursued. Of particular interest are the equations which are Lorentz invariant. The question of which differential equations of second order for a…
In this paper, we first deduce the explicit formulas for the projector of the $n$th level fractional derivative and for its Laplace transform. Then the fractional relaxation equation with the $n$th level fractional derivative is discussed.…
We study the inverse problem for the fractional Laplace equation with multiple nonlinear lower order terms. We show that the direct problem is well-posed and the inverse problem is uniquely solvable. More specifically, the unknown…
We introduce the concept of fractional derivative of Riemann-Liouville on time scales. Fundamental properties of the new operator are proved, as well as an existence and uniqueness result for a fractional initial value problem on an…