Related papers: Gevrey estimates of formal solutions for certain m…
Partial differential equations have a wide range of applications in modeling multiple physical, biological, or social phenomena. Therefore, we need to approximate the solutions of these equations in computationally feasible terms. Nowadays,…
This paper is a continuation a previous work of the authors where parametric Gevrey asymptotics for singularly perturbed nonlinear PDEs has been studied. Here, the partial differential operators are combined with particular Moebius…
In this paper, we mainly propose improvements of the logarithmic difference lemma for meromorphic functions in several complex variables, and then investigate meromorphic solutions of partial difference equations from the viewpoint of…
Our purpose in this paper is to prove, under some regularity conditions on the datas, the solvability in a Gevrey class of bound -1 on the interval [-1,1] of a class of nonlinear fractional functional differential equations.
In this article we present some integrability conditions for partial difference equations obtained using the formal symmetries approach. We apply them to find integrable partial difference equations contained in a class of equations…
Partial differential equations are fundamental tools in mathematics,sciences and engineering. This book is mainly an exposition of the various algebraic techniques of solving partial differential equations for exact solutions developed by…
The notion of moment differentiation is extended to the set of generalized multisums of formal power series via an appropriate integral representation and accurate estimates of the moment derivatives. The main result is applied to…
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial…
We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order $\alpha\in(0,1)$ in time. The framework relies on three…
The existence and analyticity of solutions to linear systems of moment differential equations with analytic coefficients is studied. The relation of solutions of such systems with respect to linear moment differential equations is…
We state and prove here semiclassical results about the construction of asymptotic solutions by the WKB method for pseudo-differential equations of real principal type. It is a Gevrey version; the smooth $C^\infty$ and the analytic ones may…
We describe partial differential operators for which we can construct generalised integral means satisfying Pizzetti-type formulas. Using these formulas we give a new characterisation of summability of formal power series solutions to some…
The objective of this paper is to investigate the existence and the forms of the pair of finite order entire and meromorphic solutions of some certain systems of Fermat-type partial differential-difference equations of several complex…
Results providing conditions on a family of integro-differential operators to determine a formal automorphism are established. Equivalently, the problem can be read in terms of existence and uniqueness of formal solutions of Cauchy problems…
The analytic and formal solutions of certain family of $q$-difference-differential equations under the action of a complex perturbation parameter is considered. The previous study of the last two authors provides information in the case…
This research introduces a new method for the transition from partial to ordinary differential equations that is based on the Kolmogorov superposition theorem. In this paper, we discuss the numerical implementation of the Kolmogorov theorem…
We study the microlocal regularity of the analytic/Gevrey vectors for the following class of second order partial differential equations \begin{align*} P(x,D) = \sum_{\ell,j=1}^{n} a_{\ell,j}(x) D_{\ell} D_{j} + \sum_{\ell=1}^{n} i…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
We provide global gradient estimates for solutions to a general type of nonlinear parabolic equations, possibly in a Riemannian geometry setting. Our result is new in comparison with the existing ones in the literature, in light of the…
The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new…