We study the positivity preserving property for the Cauchy problem for the linear fourth order heat equation. Although the complete positivity preserving property fails, we show that it holds eventually on compact sets.

Abstract In this paper a fourth-order numerical scheme is developed and implemented for the solution of non-homogeneous heat equation ut = uxx + q(x, t) with integral boundary conditions. The results obtained show that the numerical method based on the proposed technique is fourth-order accurate as well as L-acceptable. Also the efficiency and the accuracy of the new scheme is in good agreement...

The heat transfer for a non-Newtonian power-law fluid over a moving surface is investigated by applying a uniform suction/injection velocity profile. The flow is influenced by internal heat generation/absorption. The energy equation is solved at constant surface temperature condition. The Merk-Chao series is applied to obtain a set of ODEs instead of a complicated PDE. The converted ordinary diffe...

Fourth-order Schrödinger equations have been introduced by Karpman and Shagalov to take into account the role of small fourth-order dispersion terms in the propagation of intense laser beams in a bulk medium with Kerr nonlinearity. In this paper we investigate the cubic defocusing fourth order Schrödinger equation i∂tu +∆ 2 u + |u|u = 0 in arbitrary space dimension R for arbitrary initial data....

and Applied Analysis 3 Motivated by 14, 15 , here we study the existence of AL-solutions for 1.1 . The approach is completely different from the one used in 15 , in which an iteration process, jointly with a comparison with the linear equation y 4 q t y 2 0, is employed. Our tools are based on a topological method, certain integral inequalities, and some auxiliary functions. In particular, for ...

In this presentation, we describe an implicit Closest Point Method [3] which allows large, stable time steps for high-order PDEs while retaining the principal benefits of the original method. Example computations (including the in-surface heat equation, reaction-diffusion on surfaces, Laplace–Beltrami eigenmodes, and fourth-order interface motion) on a variety of surfaces demonstrate the effect...

They are of first order, second order, third order 27 and fourth order, respectively. The third-order Burg28 ers equation, similar to the second-order ones, appear 29 in many physical and engineering fields [8–10], such 30 as the plasma physics and fluid mechanics. In addi31 tion, these equations play a key role in nonlinear the32 ory and mathematical physics, in particular in the inte33 grable...