Related papers: Spectral solver for Cauchy problems in polar coord…

In this paper we apply the formal Inverse Spectral Transform for integrable dispersionless PDEs arising from the commutation condition of pairs of one-parameter families of vector fields, recently developed by S. V. Manakov and one of the…

Efficient and accurate spectral solvers for nonlocal models in any spatial dimension are presented. The approach we pursue is based on the Fourier multipliers of nonlocal Laplace operators introduced in a previous work. It is demonstrated…

In his monograph Chebyshev and Fourier Spectral Methods, John Boyd claimed that, regarding Fourier spectral methods for solving differential equations, ``[t]he virtues of the Fast Fourier Transform will continue to improve as the relentless…

We study a parabolic Ventsell problem for a second order differential operator in divergence form and with interior and boundary drift terms on the snowflake domain. We prove that under standard conditions a related Cauchy problem possesses…

In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving integral fractional Laplacian in $\mathbb{R}^d$, which is built upon two essential components: (i) the Dunford-Taylor formulation of the fractional…

In this paper, new boundary differential equations for the two-dimensional exterior scattering problem have been derived. It has been shown that the Helmholtz equation can be reduced to an inhomogeneous Bessel's equation in a body-fitted…

We define a class of pseudo-differential operators in a completely new way, which is called the abstract operators and expounded systematically the theory of abstract operators. By combining abstract operators with the Laplace transform, we…

In this paper we discuss the topic of correct setting for the equation $(-\Delta )^s u=f$, with $0<s <1$. The definition of the fractional Laplacian on the whole space $\mathbb R^n$, $n=1,2,3$ is understood through the Fourier transform,…

The two-dimensional Helmholtz equation separates in elliptic coordinates based on two distinct foci, a limit case of which includes polar coordinate systems when the two foci coalesce. This equation is invariant under the Euclidean group of…

In this work, we investigate a unique solvability of a direct and inverse source problem for a time-fractional partial differential equation with the Caputo and Bessel operators. Using spectral expansion method, we give explicit forms of…

We study the Cauchy problem for the reduced Maxwell-Bloch equations with initial data for the electric field in weighted Sobolev spaces, assuming that all atoms initially reside in their ground state. Using the d-bar steepest descent…

In this paper we present a pseudospectral method in the disk. Unlike the methods known until now, the disk is not duplicated. Moreover, we solve the Laplace equation subjected to nonhomogeneous Dirichlet, Neumann and Robin boundary…

We present a fast and numerically accurate method for expanding digitized $L \times L$ images representing functions on $[-1,1]^2$ supported on the disk $\{x \in \mathbb{R}^2 : |x|<1\}$ in the harmonics (Dirichlet Laplacian eigenfunctions)…

The Degasperis-Procesi (DP) equation \begin{align} &u_t-u_{txx}+3\kappa u_x+4uu_x=3u_x u_{xx}+uu_{xxx}, \nonumber \end{align} serving as an asymptotic approximation for the unidirectional propagation of shallow water waves, is an integrable…

Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decay very slowly, subject to certain power laws. Their numerical solutions are under-explored. This paper aims at developing accurate…

In this paper, we examine the harmonic oscillator problem in non-commutative phase space (NCPS) by using the Dunkl derivative instead of the habitual one. After defining the Hamilton operator, we use the polar coordinates to derive the…

This paper deals with the Cauchy-Dirichlet problem for the fractional Cahn-Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper…

We study the Cauchy problem for one-dimensional dispersive equations posed on $\mathbb{R} $, under the hypotheses that the dispersive operator behaves, for high frequencies, as a Fourier multiplier by $ i |\xi|^\alpha \xi $ with $ 1 \le…

Let $H$ be a norm of ${\bf R}^N$ and $H_0$ the dual norm of $H$. Denote by $\Delta_H$ the Finsler-Laplace operator defined by $\Delta_Hu:=\mbox{div}\,(H(\nabla u)\nabla_\xi H(\nabla u))$. In this paper we prove that the Finsler-Laplace…

We construct spectral decomposition of 3D Fokker - Planck differential operator in this paper. We use the decomposition to obtain solution of Cauchy problem - and especially the fundamental solution. Then we use the decomposition to…