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Self-similarity of Burgers' equation with some stochastic advection is studied. In self-similar variables a stationary solution is constructed which establishes the existence of a stochastically self-similar solution for the stochastic…
The paper deals with a problem of asymptotic step-like solutions to the Burgers' equation with variable coefficients and a small parameter. By means of the non-linear WKB method, the algorithm of constructing these asymptotic solutions is…
This paper introduces a new framework for synthesizing time-varying control barrier functions (TV-CBFs) for general Signal Temporal Logic (STL) specifications using spatiotemporal tubes (STT). We first formulate the STT synthesis as a…
We consider an initial value problem for a quadratically nonlinear inviscid Burgers-Hilbert equation that models the motion of vorticity discontinuities. We use a modified energy method to prove the existence of small, smooth solutions over…
In this paper we propose and analyze explicit space-time discrete numerical approximations for additive space-time white noise driven stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities such as the…
The main result of this article establishes strong convergence rates on the whole probability space for explicit space-time discrete numerical approximations for a class of stochastic evolution equations with possibly non-globally monotone…
This paper solves the Sylvester equation in the form of AX+XB=C in a distributed way, and proposes three distributed continuous-time algorithms for three cases. We start with the basic algorithm for solving a least squares solution of the…
We consider an initial value problem for a quadratically nonlinear inviscid Burgers-Hilbert equation that models the motion of vorticity discontinuities. We use a normal form transformation, which is implemented by means of a near-identity…
The one dimensional Burgers equation in the inviscid limit with white noise initial condition is revisited. The one- and two-point distributions of the Burgers field as well as the related distributions of shocks are obtained in closed…
Spectral methods are renowned for their high accuracy and efficiency in solving partial differential equations. The Fourier pseudo-spectral method is limited to periodic domains and suffers from Gibbs oscillations in non-periodic problems.…
In this paper, we propose a numerical method for approximating the solution of a Cauchy singular integral equation defined on a closed, smooth contour in the complex plane. The coefficients and the right-hand side of the equation are…
By applying the Craig-Wayne-Bourgain (CWB) method, we establish the existence of periodic response solutions to multi-dimensional nonlinear Schr\"{o}dinger equations (NLS) with unbounded perturbation.
Admissible point transformations between Burgers equations with linear damping and time-dependent coefficients are described and used in order to exhaustively classify Lie symmetries of these equations. Optimal systems of one- and…
The Homotopy Perturbation Method (HPM) is used to solve the Burgers-Huxley non-linear differential equations. Three case study problems of Burgers-Huxley are solved using the HPM and the exact solutions are obtained. The rapid convergence…
Finding robot poses and trajectories represents a foundational aspect of robot motion planning. Despite decades of research, efficiently and robustly addressing these challenges is still difficult. Existing approaches are often plagued by…
Exactly solvable variable parametric Burgers type equations in one-dimension are introduced, and two different approaches for solving the corresponding initial value problems are given. The first one is using the relationship between the…
This article focuses on the space-time isogeometric method for a linear time dependent fourth order problem. Using an auxiliary variable, first the problem is split into a system of two second order differential equations and then the…
In the paper, a newly developed three-point fourth-order compact operator is utilized to construct an efficient compact finite difference scheme for the Benjamin-Bona-Mahony-Burgers' (BBMB) equation. Detailed derivation is carried out based…
In this study, a thorough investigation was conducted into the Homotopy Perturbation Method (HPM) and its application to solve the Burger and Blasius equations. The HPM is a mathematical technique that combines aspects of homotopy and…
We study the solution theory of the whole-space static (elliptic) Hamilton-Jacobi-Bellman (HJB) equation in spectral Barron spaces. We prove that under the assumption that the coefficients involved are spectral Barron functions and the…