We wstepie t. 1: The 36th Semester Partial Differential Equations was held at the Stefan Banach International Mathematical Center in Warsaw from September 17 to December 17, 1990
We wstepie t. 1: The 36th Semester Partial Differential Equations was held at the Stefan Banach International Mathematical Center in Warsaw from September 17 to December 17, 1990
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We wstepie: The 36th Semester Partial Differential Equations was held at the Stefan Banach International Mathematical Center in Warsaw from September 17 to December 17, 1990
We wstepie: The 36th Semester Partial Differential Equations was held at the Stefan Banach International Mathematical Center in Warsaw from September 17 to December 17, 1990
Spatial decay estimates for an evolution equation on the range of Ural'tseva's axially symmetric operator in Sobolev spaces the use of a priori information in the solution of ill-posed problems …
Spatial decay estimates for an evolution equation on the range of Ural'tseva's axially symmetric operator in Sobolev spaces the use of a priori information in the solution of ill-posed problems allocation maps with general cost functions an elementary theorem in plane geometry and its multidimensional extension minimum problems for volumes of convex bodies on the continuous dependence of the solution of a linear parabolic partial differential equation on the boundary data and the solution at an interior spatial point decomposability of rectangular and triangular probability distributions nonlinear infinite networks with nonsymmetric resistances about a singular parabolic equation arising in thin film dynamics and in the Ricci flow for complete IR2 alternating-direction iteration for the p-version of the finite element method an integrodifferential analog of semilinear parabolic PDE's on solutions of mean curvature type inequalities an application of the calculus of variations to the study of optimal foraging a limit model of a soft thin joint projective invariants of complete intersections m-hyperbolicity, evenness and normality an extended variational principle instability criteria for solutions of second order elliptic quasilinear differential equations maximum principles for difference operators a generic uniqueness result for the Stokes system and its control theoretical consequences on a Stefan problem in a concentrated capacity total total internal reflection the reflector problem for closed surfaces upper bounds for Eigenvalues of elliptic operators stability for abstract evolution equations new techniques in critical point theory detecting underground gas sources conservative operators on the regularization of the antenna synthesis problem the problem of packaging the first digit problem and scale-invariance change of variable in the SL-integral the minimum energy configuration of a mixed-material column.
Introduction Preliminaries Green's functions for systems with constant coefficients Green's function for systems linearized along shock profiles Estimates on Green's function Estimates on crossing of initial layer Estimates on truncation …
Introduction Preliminaries Green's functions for systems with constant coefficients Green's function for systems linearized along shock profiles Estimates on Green's function Estimates on crossing of initial layer Estimates on truncation error Energy type estimates Wave interaction Stability analysis Application to magnetohydrodynamics Bibliography
This paper discusses an optimal control problem for the viscous Burgers equation using the Hopf-Cole transformation. Using this transformation, an optimal control problem for the Burgers equation is transformed into …
This paper discusses an optimal control problem for the viscous Burgers equation using the Hopf-Cole transformation. Using this transformation, an optimal control problem for the Burgers equation is transformed into that for a linear heat equation. Although this transformation makes the boundary condition complicated, we use a state feedback to overcome the difficulty. Using the state feedback, an exact solution of the heat equation is obtained by Laplace transformation which enables us to obtain the optimal control input for the heat equation exactly. In fact, it is obtained by a solution of a well-known Fredholm integral equation. Furthermore, we discuss how to choose an achievable desired terminal state from the controllability point of view. In addition, we exhibit the effectiveness of the proposed approach through a numerical simulation. It verifies that when the solution of the heat equation at terminal time converges to its desired value, that of the corresponding viscous Burgers equation also converges to the desired one.
We study directed polymers subject to a quenched random potential in d transversal dimensions. This system is closely related to the Kardar-Parisi-Zhang equation of nonlinear stochastic growth. By a careful …
We study directed polymers subject to a quenched random potential in d transversal dimensions. This system is closely related to the Kardar-Parisi-Zhang equation of nonlinear stochastic growth. By a careful analysis of the perturbation theory we show that physical quantities develop singular behavior for d to 4. For example, the universal finite size amplitude of the free energy at the roughening transition is proportional to (4-d)^(1/2). This shows that the dimension d=4 plays a special role for this system and points towards d=4 as the upper critical dimension of the Kardar-Parisi-Zhang problem.
We present a mean-field theory for the dynamics of driven flow with exclusion in graphenelike structures, and numerically check its predictions. We treat first a specific combination of bond transmissivity …
We present a mean-field theory for the dynamics of driven flow with exclusion in graphenelike structures, and numerically check its predictions. We treat first a specific combination of bond transmissivity rates, where mean field predicts, and numerics to a large extent confirms, that the sublattice structure characteristic of honeycomb networks becomes irrelevant. Dynamics, in the various regions of the phase diagram set by open boundary injection and ejection rates, is then in general identical to that of one-dimensional systems, although some discrepancies remain between mean-field theory and numerical results, in similar ways for both geometries. However, at the critical point for which the characteristic exponent is z = 3/2 in one dimension, the mean-field value z = 2 is approached for very large systems with constant (finite) aspect ratio. We also treat a second combination of bond (and boundary) rates where, more typically, sublattice distinction persists. For the two rate combinations, in continuum or late-time limits, respectively, the coupled sets of mean-field dynamical equations become tractable with various techniques and give a two-band spectrum, gapless in the critical phase. While for the second rate combination quantitative discrepancies between mean-field theory and simulations increase for most properties and boundary rates investigated, theory still is qualitatively correct in general, and gives a fairly good quantitative account of features such as the late-time evolution of density profile differences from their steady-state values.
Among various real life models for controlled conservation laws, the dynamics of the open canal water flow is such that the nonlinear terms of the partial differential equations cannot be …
Among various real life models for controlled conservation laws, the dynamics of the open canal water flow is such that the nonlinear terms of the partial differential equations cannot be neglected. Starting from a rather complicated model deduced for a quite general case, there are considered some simpler models of the practice, relying on the model of the horizontal prismatic canal with simple cross-section (rectangular, trapezoidal). Basic theory (existence, uniqueness, continuous data dependence) is discussed for small deviations around a steady state. There is also discussed the problem of the invariant sets accounting for fluvialness - some kind of regular behavior of the flow reflected in the partial differential equations and their solutions: absence of cavitation, shock and/or rarefaction waves. Based on the Riemann invariants of the equations there is made the synthesis of the feedback structure for the controllers. These are nonlinear decentralized small gain controllers ensuring linearization and dissipativeness of the boundary conditions in their Riemann invariant forms.
From particular polynomials, we construct rational solutions to the Burgers' equation as a quotient of a polynomial of degree n -1 in x and n -1 -n 2 in t, …
From particular polynomials, we construct rational solutions to the Burgers' equation as a quotient of a polynomial of degree n -1 in x and n -1 -n 2 in t, by a polynomial of degree n in x and n 2 in t, |n| being the greater integer less or equal to n.We call these solutions, solutions of order n.We construct explicitly these solutions for orders 1 until 20.
This paper examines the properties of a regularization of Burgers equation in one and multiple dimensions using a filtered convective velocity, which we have dubbed as convectively filtered Burgers (CFB) …
This paper examines the properties of a regularization of Burgers equation in one and multiple dimensions using a filtered convective velocity, which we have dubbed as convectively filtered Burgers (CFB) equation. A physical motivation behind the filtering technique is presented. An existence and uniqueness theorem for multiple dimensions and a general class of filters is proven. Multiple invariants of motion are found for the CFB equation and are compared with those found in viscous and inviscid Burgers equation. Traveling wave solutions are found for a general class of filters and are shown to converge to weak solutions of inviscid Burgers equation with the correct wave speed. Accurate numerical simulations are conducted in 1D and 2D cases where the shock behavior, shock thickness, and kinetic energy decay are examined. Energy spectrum are also examined and are shown to be related to the smoothness of the solutions.
Dans cet exposé, on s’intéresse aux lois de conservation scalaires en dimension 1 d’espace, et aux propriétés de compacité associées au semi-groupe qu’elles engendrent.
Dans cet exposé, on s’intéresse aux lois de conservation scalaires en dimension 1 d’espace, et aux propriétés de compacité associées au semi-groupe qu’elles engendrent.
Abstract In this paper, we focus on the well‐posedness, blow‐up phenomena, and continuity of the data‐to‐solution map of the Cauchy problem for a two‐component higher order Camassa–Holm (CH) system. The …
Abstract In this paper, we focus on the well‐posedness, blow‐up phenomena, and continuity of the data‐to‐solution map of the Cauchy problem for a two‐component higher order Camassa–Holm (CH) system. The local well‐posedness is established in Besov spaces with , which improves the local well‐posedness result proved before in Tang and Liu [Z. Angew. Math. Phys. 66 (2015), 1559–1580], Ye and Yin [arXiv preprint arXiv:2109.00948 (2021)], Zhang and Li [Nonlinear Anal. Real World Appl. 35 (2017), 414–440], and Zhou [Math. Nachr. 291 (2018), no. 10, 1595–1619]. Next, we consider the continuity of the solution‐to‐data map, that is, the ill‐posedness is derived in Besov space with and . Then, the nonuniform continuous and Hölder continuous dependence on initial data for this system are also presented in Besov spaces with and . Finally, the precise blow‐up criteria for the strong solutions of the two‐component higher order CH system is determined in the lowest Sobolev space with , which improves the blow‐up criteria result established before in He and Yin [Discrete Contin. Dyn. Syst. 37 (2016), no. 3, 1509–1537] and Zhou [Math. Nachr. 291 (2018), no. 10, 1595–1619].
Abstract The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function …
Abstract The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design with pointwise bound constraints. This paper also introduces the latent variable proximal point (LVPP) algorithm, from which the proximal Galerkin method derives. When analyzing the classical obstacle problem, we discover that the underlying variational inequality can be replaced by a sequence of second-order partial differential equations (PDEs) that are readily discretized and solved with, e.g., the proximal Galerkin method. Throughout this work, we arrive at several contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation ; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field, density-based topology optimization. The complete proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. Open-source implementations of our methods accompany this work to facilitate reproduction and broader adoption.
In the paper, (n+1)-dimensional fractional M-Burgers equation with a force term in the sense of the Caputo fractional derivative is considered. We solve this equation using homotopy perturbation method (HPM) …
In the paper, (n+1)-dimensional fractional M-Burgers equation with a force term in the sense of the Caputo fractional derivative is considered. We solve this equation using homotopy perturbation method (HPM) and find its approximate analytical solution. We illustrate the method with some concrete examples. We also provide the graphical representation of the solutions. This paper extends some known results obtained by Sripacharasakullert et al. (2019).
Abstract In this paper, we consider an inverse backward problem for a nonlinear singularly perturbed parabolic equation of the Burgers’ type. We demonstrate how a method of asymptotic analysis of …
Abstract In this paper, we consider an inverse backward problem for a nonlinear singularly perturbed parabolic equation of the Burgers’ type. We demonstrate how a method of asymptotic analysis of the direct problem allows developing a rather simple algorithm for solving the inverse problem in comparison with minimization of the cost functional. Numerical experiments demonstrate the effectiveness of this approach.
Numerical solutions for Burgers’ equation based on the Galerkins’ method using cubic B‐splines as both weight and interpolation functions are set up. It is shown that this method is capable …
Numerical solutions for Burgers’ equation based on the Galerkins’ method using cubic B‐splines as both weight and interpolation functions are set up. It is shown that this method is capable of solving Burgers’ equation accurately for values of viscosity ranging from very small to large. Three standard problems are used to validate the proposed algorithm. A linear stability analysis shows that a numerical scheme based on a Cranck‐Nicolson approximation in time is unconditionally stable.
In the present work, a numerical scheme is constructed for approximation of time fractional Black-Scholes model governing European options.The present numerical scheme has the capability to overcome spurious oscillation in …
In the present work, a numerical scheme is constructed for approximation of time fractional Black-Scholes model governing European options.The present numerical scheme has the capability to overcome spurious oscillation in the case of volatility.In the present numerical method, the Laplace transform, radial kernels and quadrature rule are used.The time variable is eliminated by the use of Laplace transform which significantly reduced the computational cost as compared to the time-marching schemes.The spatial operator is discretized using radial kernels in the local setting which results in sparse differentiation matrices.By Laplace transform the solution is represented as integral along a smooth contour in the complex plane which is then evaluated by quadrature.The proposed numerical scheme is used to price several different European options.