Related papers: Abstract non-scalar Volterra difference equations
This paper, first, we consider the Volterra integral equation for the remainder term in the asymptotic formula for the associated Euler totient function. Secondly, we solve the Volterra integral equation and we split the error term in the…
New problem is studied that is to find nonlinear differential equations with special solutions expressed via the Weierstrass function. Method is discussed to construct nonlinear ordinary differential equations with exact solutions. Main…
The set of common numerical and analytical problems is introduced in the form of the generalized multidimensional discrete Poisson equation. It is shown that its solutions with square-summable discrete derivatives are unique up to a…
In this work we prove that a family of explicit numerical finite-difference methods is convergent when applied to a nonlinear Volterra equation with a power-type nonlinearity. In that case the kernel is not of Lipschitz type, therefore the…
We extend the classical Bernstein technique to the setting of integro-differential operators. As a consequence, we provide first and one-sided second derivative estimates for solutions to fractional equations, including some convex fully…
In this PhD thesis we introduce a generalized fractional calculus of variations. We consider variational problems containing generalized fractional integrals and derivatives, and study them using standard (indirect) and direct methods. In…
This paper is in concern with Cauchy problems involving the fractional derivatives with respect to another function. Results of existence, uniqueness, and Taylor series among others are established in appropriate functional spaces. We prove…
The application of the approximation-operational approach to solving linear differential equations of fractional order with variable coefficients is considered. It is shown that the method can also be applied to solving differential…
We present a systematic study of higher-order Airy-type differential equations providing the explicit form of the solutions, deriving their power series expansions and a probabilistic interpretation. Under suitable convergence hypotheses,…
In this paper we obtain an It\^o differential representation for a class of singular stochastic Volterra integral equations. As an application, we investigate the rate of convergence in the small time central limit theorem for the solution.
In this paper are examined general classes of linear and non-linear analytical systems of partial differential equations. Indeed the integrability conditions are found and if they are satisfied, the solutions are given as functional series…
In this paper we study some cases of time-fractional nonlinear dispersive equations (NDEs) involving Caputo derivatives, by means of the invariant subspace method. This method allows to find exact solutions to nonlinear time-fractional…
In this paper, we analyze and discuss the well-posedness of a new variant of the so-called sweeping process, introduced by J.J. Moreau in the early 70's \cite{More71} with motivation in plasticity theory. In this variant, the normal cone to…
We discuss alternative iteration methods for differential equations. We provide a convergence proof for exactly solvable examples and show more convenient formulas for nontrivial problems.
In the paper stochastic Volterra equations with noise terms driven by series of independent scalar Wiener processes are considered. In our study we use the resolvent approach to the equations under consideration. We give sufficient…
We introduce a nabla, a delta, and a symmetric fractional calculus on arbitrary nonempty closed subsets of the real numbers. These fractional calculi provide a study of differentiation and integration of noninteger order on discrete,…
In this paper we explain how to use the Fast Fourier Transform (FFT) to solve partial differential equations (PDEs). We start by defining appropriate discrete domains in coordinate and frequency domains. Then describe the main limitation of…
We prove the existence of weak solutions for distribution-dependent stochastic Volterra equations under linear growth and continuity conditions on the coefficients and mild regularity assumptions on the kernels, including singular kernels.…
Fractional action-like variational problems have recently gained importance in studying dynamics of nonconservative systems. In this note we address multi-dimensional fractional action-like problems of the calculus of variations.
In this paper, we consider a nonlocal boundary-value problem for a mixed-type equation involving the bi-ordinal Hilfer fractional derivative in a rectangular domain. The main target of this work is to analyze the uniqueness and the…